This is Triangulations.info, produced by Macaulay2, version 1.22

Triangulations -- generating and manipulating triangulations of point or vector configurations
**********************************************************************************************

Description
===========

Warning! This package is experimental, documentation is missing, and the
interface will be cleaned up and changed.  Use only if these issues don't
bother you!

Data of a triangulation
=======================

  * max(Triangulation) (missing documentation)
  * vectors(Triangulation) (missing documentation)

Creating triangulations
=======================

  * triangulation(Matrix,List) (missing documentation)
  * regularFineTriangulation(Matrix) (missing documentation)
  * generateTriangulations(Triangulation) (missing documentation)
  * allTriangulations(Matrix) -- use
    topcom to generate all triangulations of a point or vector configuration

Properties of triangulations
============================

  * isWellDefined(Triangulation) (missing documentation)
  * isRegularTriangulation(Triangulation), --
    determine if a given triangulation is a regular triangulation
  * regularTriangulationWeights(Triangulation) (missing documentation)
  * isStar(Triangulation) (missing documentation)
  * isFine(Triangulation) (missing documentation)
  * naiveIsTriangulation(Triangulation) (missing documentation)

Exploring the set of triangulations
===================================

  * bistellarFlip(Triangulation,List) (missing documentation)
  * flips(Triangulation) (missing documentation)
  * neighbors(Triangulation) (missing documentation)
  * affineCircuits(Triangulation) (missing documentation)
  * volumeVector(Triangulation) (missing documentation)
  * volumeVector(Triangulation) (missing documentation)
  * delaunayWeights(Matrix) (missing documentation)
  * delaunaySubdivision(Matrix) (missing documentation)

This package is designed to help compute and explore the set of all (or many)
triangulations of a point set or polytope.

We give a sample use of this package.

+-----------------------------------------------------------------------------+
|i1 : LP = {{-1, 0, -1, 1}, {-1, 0, 1, 0}, {-1, 0, 2, -1}, {-1, 1, -1, 0}, {1,|
|                                                                             |
|o1 = {{-1, 0, -1, 1}, {-1, 0, 1, 0}, {-1, 0, 2, -1}, {-1, 1, -1, 0}, {1, 0,  |
|     ------------------------------------------------------------------------|
|     -1, 0}, {1, 0, 1, 0}, {2, -1, -1, 0}, {0, 0, 1, 0}, {1, 0, 0, 0}, {0, 0,|
|     ------------------------------------------------------------------------|
|     0, 0}}                                                                  |
|                                                                             |
|o1 : List                                                                    |
|-----------------------------------------------------------------------------|
|0, -1, 0}, {1, 0, 1, 0}, {2, -1, -1, 0}, {0, 0, 1, 0}, {1, 0, 0, 0},         |
|-----------------------------------------------------------------------------|
|{0,0,0,0}}                                                                   |
+-----------------------------------------------------------------------------+
|i2 : A = transpose matrix LP                                                 |
|                                                                             |
|o2 = | -1 -1 -1 -1 1  1 2  0 1 0 |                                           |
|     | 0  0  0  1  0  0 -1 0 0 0 |                                           |
|     | -1 1  2  -1 -1 1 -1 1 0 0 |                                           |
|     | 1  0  -1 0  0  0 0  0 0 0 |                                           |
|                                                                             |
|              4        10                                                    |
|o2 : Matrix ZZ  <--- ZZ                                                      |
+-----------------------------------------------------------------------------+
|i3 : elapsedTime Ts = allTriangulations(A, Fine => true);                    |
| -- 0.290697 seconds elapsed                                                 |
+-----------------------------------------------------------------------------+
|i4 : select(Ts, T -> isStar T)                                               |
|                                                                             |
|o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,  |
|     ------------------------------------------------------------------------|
|     1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3,  |
|     ------------------------------------------------------------------------|
|     5, 7, 9}, {0, 3, 5, 8, 9}, {0, 4, 6, 8, 9}, {0, 5, 6, 7, 9}, {0, 5, 6,  |
|     ------------------------------------------------------------------------|
|     8, 9}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 6, 9}, {2, 3, 4, 8,  |
|     ------------------------------------------------------------------------|
|     9}, {2, 3, 5, 7, 9}, {2, 3, 5, 8, 9}, {2, 4, 6, 8, 9}, {2, 5, 6, 7, 9}, |
|     ------------------------------------------------------------------------|
|     {2, 5, 6, 8, 9}}}                                                       |
|                                                                             |
|o4 : List                                                                    |
+-----------------------------------------------------------------------------+
|i5 : #oo == 1                                                                |
|                                                                             |
|o5 = true                                                                    |
+-----------------------------------------------------------------------------+
|i6 : #Ts == 51                                                               |
|                                                                             |
|o6 = true                                                                    |
+-----------------------------------------------------------------------------+
|i7 : T = regularFineTriangulation A                                          |
|                                                                             |
|o7 = triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0, 1,|
|                                                                             |
|o7 : Triangulation                                                           |
|-----------------------------------------------------------------------------|
|6, 7, 9}, {0, 2, 3, 4, 6}, {0, 2, 3, 4, 9}, {0, 2, 4, 6, 9}, {0, 3, 4, 7, 8},|
|-----------------------------------------------------------------------------|
|{0, 3, 4, 7, 9}, {0, 3, 5, 7, 8}, {0, 4, 6, 7, 8}, {0, 4, 6, 7, 9}, {0, 5, 6,|
|-----------------------------------------------------------------------------|
|7, 8}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 7, 8}, {2, 3, 4, 7, 9},   |
|-----------------------------------------------------------------------------|
|{2, 3, 5, 7, 8}, {2, 4, 6, 7, 8}, {2, 4, 6, 7, 9}, {2, 5, 6, 7, 8}}          |
+-----------------------------------------------------------------------------+
|i8 : elapsedTime Ts2 = generateTriangulations T;                             |
| -- 0.44985 seconds elapsed                                                  |
+-----------------------------------------------------------------------------+
|i9 : #Ts2 == #Ts                                                             |
|                                                                             |
|o9 = true                                                                    |
+-----------------------------------------------------------------------------+

See also
========

  * Polyhedra -- for computations with convex polyhedra,
    cones, and fans
  * Topcom -- interface to selected functions from topcom
    package
  * ReflexivePolytopesDB -- simple access to
    Kreuzer-Skarke database of reflexive polytopes of dimensions 3 and 4

Author
======

  * Mike Stillman (see http://www.math.cornell.edu/~mike )
    <mike@math.cornell.edu>

Version
=======

This documentation describes version 0.1 of Triangulations.

Source code
===========

The source code from which this documentation is derived is in the file
Triangulations.m2 (see common/share/Macaulay2/Triangulations.m2 ).

Exports
=======

  * Types
      * Chirotope (missing documentation)
      * Triangulation (missing documentation)
  * Functions and commands
      * affineCircuits (missing documentation)
      * "allTriangulations" -- see allTriangulations(Matrix),
        -- use topcom to generate all
        triangulations of a point or vector configuration
      * bistellarFlip (missing documentation)
      * chirotope (missing documentation)
      * delaunaySubdivision (missing documentation)
      * delaunayWeights (missing documentation)
      * fineStarTriangulation (missing documentation)
      * flips (missing documentation)
      * generateTriangulations -- generate all
        triangulations with certain properties
      * gkzVector (missing documentation)
      * isFine (missing documentation)
      * isRegularTriangulation -- determine if a
        given triangulation is a regular triangulation
      * isStar (missing documentation)
      * naiveChirotope (missing documentation)
      * naiveIsTriangulation (missing documentation)
      * neighbors (missing documentation)
      * regularFineStarTriangulation (missing documentation)
      * regularFineTriangulation (missing documentation)
      * regularTriangulationWeights (missing documentation)
      * triangulation -- make a Triangulation object
      * vectors (missing documentation)
      * volumeVector (missing documentation)
  * Methods
      * allTriangulations(Matrix) -- use
        topcom to generate all triangulations of a point or vector
        configuration
      * "generateTriangulations(Matrix)" -- see generateTriangulations,
        -- generate all triangulations with certain
        properties
      * "isRegularTriangulation(Triangulation)" -- see
        isRegularTriangulation, -- determine if a given
        triangulation is a regular triangulation
  * Symbols
      * ConeIndex (missing documentation)

For the programmer
==================

The object Triangulations, is a package (Macaulay2Doc).

affineCircuits
**************

Ways to use affineCircuits :
============================

  * affineCircuits(Matrix,List) (missing documentation)
  * affineCircuits(Triangulation) (missing documentation)

For the programmer
==================

The object affineCircuits (missing documentation) is a method function (Macaulay2Doc).

allTriangulations(Matrix) -- use topcom to generate all triangulations of a point or vector configuration
*********************************************************************************************************

Synopsis
========

  * Function: allTriangulations,
  * Usage:
        allTriangulations A
        allTriangulations(A, Homogenize => true, Fine => true, RegularOnly => true)
  * Inputs:
      * A, a matrix (Macaulay2Doc),
  * Optional inputs (Macaulay2Doc),:
      * Homogenize => a Boolean value (Macaulay2Doc), default
        value true
      * ConnectedToRegular => a Boolean value (Macaulay2Doc),
        default value true
      * Fine => a Boolean value (Macaulay2Doc), default value
        false
      * RegularOnly => a Boolean value (Macaulay2Doc), default
        value true
  * Outputs:
      * a list (Macaulay2Doc),

Description
===========

This function constructs all triangulations of the point set corresponding to
$A$ (or triangulation of the cone over $A$, if Homogenize => false is given.
With no optional arguments, the default is to construct all regular
triangulations.

This function is a wrapper for the topcom function
topcomAllTriangulations (Topcom), and has
the same optional arguments as that function. This function returns a list of
Triangulation (missing documentation)'s.

A triangulation is a list of list of the indices of the maximal simplices in
the triangulation. (the index of the point corresponding to the $i$-th column
(starting at $0$) is simply $i$.

For example, the following point set is the smallest which has a non-regular
triangulation.

+-----------------------------------------------------------------------------+
|i1 : A = transpose matrix {{0,3},{0,1},{-1,-1},{1,-1},{-4,-2},{4,-2}}        |
|                                                                             |
|o1 = | 0 0 -1 1  -4 4  |                                                     |
|     | 3 1 -1 -1 -2 -2 |                                                     |
|                                                                             |
|              2        6                                                     |
|o1 : Matrix ZZ  <--- ZZ                                                      |
+-----------------------------------------------------------------------------+
|i2 : Ts = allTriangulations A                                                |
|                                                                             |
|o2 = {triangulation {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3},  |
|     ------------------------------------------------------------------------|
|     {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 2, 3}, {0, 2, 4}, {0, 3, 5},  |
|     ------------------------------------------------------------------------|
|     {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 1, 3}, {0, 1, 4}, {0, 3, 5},  |
|     ------------------------------------------------------------------------|
|     {1, 2, 3}, {1, 2, 4}, {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 1, 2},  |
|     ------------------------------------------------------------------------|
|     {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3}, {2, 3, 5}, {2, 4, 5}},      |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, 3}, {1, 2, 4}, {1, 3, 5},   |
|     ------------------------------------------------------------------------|
|     {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 1, 3}, {0, 1, 4}, {0, 3, 5},  |
|     ------------------------------------------------------------------------|
|     {1, 3, 4}, {3, 4, 5}}, triangulation {{0, 2, 3}, {0, 2, 4}, {0, 3, 5},  |
|     ------------------------------------------------------------------------|
|     {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4},  |
|     ------------------------------------------------------------------------|
|     {1, 2, 3}, {1, 3, 5}, {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 3, 4},  |
|     ------------------------------------------------------------------------|
|     {0, 3, 5}, {3, 4, 5}}, triangulation {{0, 2, 4}, {0, 2, 5}, {2, 4, 5}}, |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 4}, {0, 1, 5}, {1, 3, 4}, {1, 3, 5}, {3, 4, 5}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, 3}, {1, 2, 4}, {1, 3, 5},   |
|     ------------------------------------------------------------------------|
|     {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4},  |
|     ------------------------------------------------------------------------|
|     {1, 2, 5}, {2, 4, 5}}, triangulation {{0, 4, 5}}, triangulation {{0, 1, |
|     ------------------------------------------------------------------------|
|     4}, {0, 1, 5}, {1, 4, 5}}, triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2,  |
|     ------------------------------------------------------------------------|
|     4}, {1, 2, 5}, {2, 4, 5}}}                                              |
|                                                                             |
|o2 : List                                                                    |
+-----------------------------------------------------------------------------+
|i3 : #Ts == 16                                                               |
|                                                                             |
|o3 = true                                                                    |
+-----------------------------------------------------------------------------+
|i4 : netList Ts                                                              |
|                                                                             |
|     +-----------------------------------------------------------------------|
|o4 = |triangulation {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3}, {|
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 2, 3}, {0, 2, 4}, {0, 3, 5}, {2, 3, 4}, {3, 4, 5}}  |
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 1, 3}, {0, 1, 4}, {0, 3, 5}, {1, 2, 3}, {1, 2, 4}, {|
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3}, {|
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, 3}, {1, 2, 4}, {1, 3, 5}, {|
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 1, 3}, {0, 1, 4}, {0, 3, 5}, {1, 3, 4}, {3, 4, 5}}  |
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 2, 3}, {0, 2, 4}, {0, 3, 5}, {2, 3, 5}, {2, 4, 5}}  |
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4}, {1, 2, 3}, {1, 3, 5}, {|
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 3, 4}, {0, 3, 5}, {3, 4, 5}}                        |
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 2, 4}, {0, 2, 5}, {2, 4, 5}}                        |
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 1, 4}, {0, 1, 5}, {1, 3, 4}, {1, 3, 5}, {3, 4, 5}}  |
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, 3}, {1, 2, 4}, {1, 3, 5}, {|
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4}, {1, 2, 5}, {2, 4, 5}}  |
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 4, 5}}                                              |
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 1, 4}, {0, 1, 5}, {1, 4, 5}}                        |
|     +-----------------------------------------------------------------------|
|     |triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, 4}, {1, 2, 5}, {2, 4, 5}}  |
|     +-----------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|--------------------+                                                        |
|2, 3, 4}, {3, 4, 5}}|                                                        |
|--------------------+                                                        |
|                    |                                                        |
|--------------------+                                                        |
|2, 3, 4}, {3, 4, 5}}|                                                        |
|--------------------+                                                        |
|2, 3, 5}, {2, 4, 5}}|                                                        |
|--------------------+                                                        |
|2, 3, 4}, {3, 4, 5}}|                                                        |
|--------------------+                                                        |
|                    |                                                        |
|--------------------+                                                        |
|                    |                                                        |
|--------------------+                                                        |
|2, 3, 5}, {2, 4, 5}}|                                                        |
|--------------------+                                                        |
|                    |                                                        |
|--------------------+                                                        |
|                    |                                                        |
|--------------------+                                                        |
|                    |                                                        |
|--------------------+                                                        |
|2, 3, 5}, {2, 4, 5}}|                                                        |
|--------------------+                                                        |
|                    |                                                        |
|--------------------+                                                        |
|                    |                                                        |
|--------------------+                                                        |
|                    |                                                        |
|--------------------+                                                        |
|                    |                                                        |
|--------------------+                                                        |
+-----------------------------------------------------------------------------+
|i5 : tri = Ts#0                                                              |
|                                                                             |
|o5 = triangulation {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3},   |
|                                                                             |
|o5 : Triangulation                                                           |
|-----------------------------------------------------------------------------|
|{2, 3, 4}, {3, 4, 5}}                                                        |
+-----------------------------------------------------------------------------+
|i6 : isWellDefined tri                                                       |
|                                                                             |
|o6 = true                                                                    |
+-----------------------------------------------------------------------------+
|i7 : isRegularTriangulation tri                                              |
|                                                                             |
|o7 = true                                                                    |
+-----------------------------------------------------------------------------+
|i8 : Ts/isRegularTriangulation                                               |
|                                                                             |
|o8 = {true, true, true, true, true, true, true, true, true, true, true, true,|
|     ------------------------------------------------------------------------|
|     true, true, true, true}                                                 |
|                                                                             |
|o8 : List                                                                    |
+-----------------------------------------------------------------------------+
|i9 : regularTriangulationWeights tri                                         |
|                                                                             |
|      97  47  1                                                              |
|o9 = {--, --, -, 0, 0, 0}                                                    |
|       8   8  4                                                              |
|                                                                             |
|o9 : List                                                                    |
+-----------------------------------------------------------------------------+

The following code determines the support of each triangulation, and tallies
them. Thus for example, we see that there are 6 regular fine triangulations.

+----------------------------------------------------------+
|i10 : tally for tri in Ts list sort unique flatten max tri|
|                                                          |
|o10 = Tally{{0, 1, 2, 3, 4, 5} => 6}                      |
|            {0, 1, 2, 4, 5} => 2                          |
|            {0, 1, 3, 4, 5} => 2                          |
|            {0, 1, 4, 5} => 1                             |
|            {0, 2, 3, 4, 5} => 2                          |
|            {0, 2, 4, 5} => 1                             |
|            {0, 3, 4, 5} => 1                             |
|            {0, 4, 5} => 1                                |
|                                                          |
|o10 : Tally                                               |
+----------------------------------------------------------+

The method that topcom uses depends on the optional arguments Fine,
ConnectedToRegular and RegularOnly.

+---------------------------------------------+
|i11 : options allTriangulations              |
|                                             |
|o11 = OptionTable{ConnectedToRegular => true}|
|                  Fine => false              |
|                  Homogenize => true         |
|                  RegularOnly => true        |
|                                             |
|o11 : OptionTable                            |
+---------------------------------------------+

If the optional argument Fine is set to true, then only fine triangulations
(i.e. those that involve every column of $A$) will be generated.

+----------------------------------------------+
|i12 : Ts = allTriangulations(A, Fine => true);|
+----------------------------------------------+
|i13 : #Ts == 6                                |
|                                              |
|o13 = true                                    |
+----------------------------------------------+

If the optional argument RegularOnly is set to false, but ConnectedToRegular is
true, it will generally take less time, as the program doesn't need to check
each triangulation to see if it is regular.

+-----------------------------------------------------------------------------+
|i14 : T1s = allTriangulations(A, RegularOnly => true)                        |
|                                                                             |
|o14 = {triangulation {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3}, |
|      -----------------------------------------------------------------------|
|      {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 2, 3}, {0, 2, 4}, {0, 3, 5}, |
|      -----------------------------------------------------------------------|
|      {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 1, 3}, {0, 1, 4}, {0, 3, 5}, |
|      -----------------------------------------------------------------------|
|      {1, 2, 3}, {1, 2, 4}, {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 1, 2}, |
|      -----------------------------------------------------------------------|
|      {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3}, {2, 3, 5}, {2, 4, 5}},     |
|      -----------------------------------------------------------------------|
|      triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, 3}, {1, 2, 4}, {1, 3, 5},  |
|      -----------------------------------------------------------------------|
|      {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 1, 3}, {0, 1, 4}, {0, 3, 5}, |
|      -----------------------------------------------------------------------|
|      {1, 3, 4}, {3, 4, 5}}, triangulation {{0, 2, 3}, {0, 2, 4}, {0, 3, 5}, |
|      -----------------------------------------------------------------------|
|      {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4}, |
|      -----------------------------------------------------------------------|
|      {1, 2, 3}, {1, 3, 5}, {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 3, 4}, |
|      -----------------------------------------------------------------------|
|      {0, 3, 5}, {3, 4, 5}}, triangulation {{0, 2, 4}, {0, 2, 5}, {2, 4, 5}},|
|      -----------------------------------------------------------------------|
|      triangulation {{0, 1, 4}, {0, 1, 5}, {1, 3, 4}, {1, 3, 5}, {3, 4, 5}}, |
|      -----------------------------------------------------------------------|
|      triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, 3}, {1, 2, 4}, {1, 3, 5},  |
|      -----------------------------------------------------------------------|
|      {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4}, |
|      -----------------------------------------------------------------------|
|      {1, 2, 5}, {2, 4, 5}}, triangulation {{0, 4, 5}}, triangulation {{0, 1,|
|      -----------------------------------------------------------------------|
|      4}, {0, 1, 5}, {1, 4, 5}}, triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, |
|      -----------------------------------------------------------------------|
|      4}, {1, 2, 5}, {2, 4, 5}}}                                             |
|                                                                             |
|o14 : List                                                                   |
+-----------------------------------------------------------------------------+
|i15 : T2s = allTriangulations(A, RegularOnly => false)                       |
|                                                                             |
|o15 = {triangulation {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3}, |
|      -----------------------------------------------------------------------|
|      {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4}, |
|      -----------------------------------------------------------------------|
|      {1, 2, 3}, {1, 3, 5}, {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 2, 3}, |
|      -----------------------------------------------------------------------|
|      {0, 2, 4}, {0, 3, 5}, {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 1, 3}, |
|      -----------------------------------------------------------------------|
|      {0, 1, 4}, {0, 3, 5}, {1, 2, 3}, {1, 2, 4}, {2, 3, 4}, {3, 4, 5}},     |
|      -----------------------------------------------------------------------|
|      triangulation {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3},  |
|      -----------------------------------------------------------------------|
|      {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 2, 3}, {0, 2, 4}, {0, 3, 5}, |
|      -----------------------------------------------------------------------|
|      {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4}, |
|      -----------------------------------------------------------------------|
|      {1, 2, 3}, {1, 3, 5}, {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 1, 3}, |
|      -----------------------------------------------------------------------|
|      {0, 1, 4}, {0, 3, 5}, {1, 2, 3}, {1, 2, 4}, {2, 3, 5}, {2, 4, 5}},     |
|      -----------------------------------------------------------------------|
|      triangulation {{0, 3, 4}, {0, 3, 5}, {3, 4, 5}}, triangulation {{0, 1, |
|      -----------------------------------------------------------------------|
|      4}, {0, 1, 5}, {1, 2, 3}, {1, 2, 4}, {1, 3, 5}, {2, 3, 4}, {3, 4, 5}}, |
|      -----------------------------------------------------------------------|
|      triangulation {{0, 1, 3}, {0, 1, 4}, {0, 3, 5}, {1, 3, 4}, {3, 4, 5}}, |
|      -----------------------------------------------------------------------|
|      triangulation {{0, 1, 4}, {0, 1, 5}, {1, 3, 4}, {1, 3, 5}, {3, 4, 5}}, |
|      -----------------------------------------------------------------------|
|      triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, 3}, {1, 2, 4}, {1, 3, 5},  |
|      -----------------------------------------------------------------------|
|      {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 4, 5}}, triangulation {{0, 2,|
|      -----------------------------------------------------------------------|
|      4}, {0, 2, 5}, {2, 4, 5}}, triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, |
|      -----------------------------------------------------------------------|
|      4}, {1, 2, 5}, {2, 4, 5}}, triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, |
|      -----------------------------------------------------------------------|
|      4}, {1, 2, 5}, {2, 4, 5}}, triangulation {{0, 1, 4}, {0, 1, 5}, {1, 4, |
|      -----------------------------------------------------------------------|
|      5}}}                                                                   |
|                                                                             |
|o15 : List                                                                   |
+-----------------------------------------------------------------------------+
|i16 : #T1s                                                                   |
|                                                                             |
|o16 = 16                                                                     |
+-----------------------------------------------------------------------------+
|i17 : #T2s                                                                   |
|                                                                             |
|o17 = 18                                                                     |
+-----------------------------------------------------------------------------+

The following search would also yield all triangulations, even those not
connected via bistellar flips to regular triangulations.

+-----------------------------------------------------------------------------+
|i18 : T3s = allTriangulations(A, RegularOnly => false, ConnectedToRegular => |
|                                                                             |
|o18 = {triangulation {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3}, |
|      -----------------------------------------------------------------------|
|      {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, |
|      -----------------------------------------------------------------------|
|      {0, 3, 5}, {1, 2, 3}, {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 1, 2}, |
|      -----------------------------------------------------------------------|
|      {0, 1, 5}, {0, 2, 4}, {1, 2, 3}, {1, 3, 5}, {2, 3, 4}, {3, 4, 5}},     |
|      -----------------------------------------------------------------------|
|      triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4}, {1, 2, 3}, {1, 3, 5},  |
|      -----------------------------------------------------------------------|
|      {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4}, |
|      -----------------------------------------------------------------------|
|      {1, 2, 5}, {2, 4, 5}}, triangulation {{0, 1, 3}, {0, 1, 4}, {0, 3, 5}, |
|      -----------------------------------------------------------------------|
|      {1, 2, 3}, {1, 2, 4}, {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 1, 3}, |
|      -----------------------------------------------------------------------|
|      {0, 1, 4}, {0, 3, 5}, {1, 2, 3}, {1, 2, 4}, {2, 3, 5}, {2, 4, 5}},     |
|      -----------------------------------------------------------------------|
|      triangulation {{0, 1, 3}, {0, 1, 4}, {0, 3, 5}, {1, 3, 4}, {3, 4, 5}}, |
|      -----------------------------------------------------------------------|
|      triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, 3}, {1, 2, 4}, {1, 3, 5},  |
|      -----------------------------------------------------------------------|
|      {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, 3}, |
|      -----------------------------------------------------------------------|
|      {1, 2, 4}, {1, 3, 5}, {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 1, 4}, |
|      -----------------------------------------------------------------------|
|      {0, 1, 5}, {1, 2, 4}, {1, 2, 5}, {2, 4, 5}}, triangulation {{0, 1, 4}, |
|      -----------------------------------------------------------------------|
|      {0, 1, 5}, {1, 3, 4}, {1, 3, 5}, {3, 4, 5}}, triangulation {{0, 1, 4}, |
|      -----------------------------------------------------------------------|
|      {0, 1, 5}, {1, 4, 5}}, triangulation {{0, 2, 3}, {0, 2, 4}, {0, 3, 5}, |
|      -----------------------------------------------------------------------|
|      {2, 3, 4}, {3, 4, 5}}, triangulation {{0, 2, 3}, {0, 2, 4}, {0, 3, 5}, |
|      -----------------------------------------------------------------------|
|      {2, 3, 5}, {2, 4, 5}}, triangulation {{0, 2, 4}, {0, 2, 5}, {2, 4, 5}},|
|      -----------------------------------------------------------------------|
|      triangulation {{0, 3, 4}, {0, 3, 5}, {3, 4, 5}}, triangulation {{0, 4, |
|      -----------------------------------------------------------------------|
|      5}}}                                                                   |
|                                                                             |
|o18 : List                                                                   |
|-----------------------------------------------------------------------------|
|false)                                                                       |
+-----------------------------------------------------------------------------+
|i19 : #T3s                                                                   |
|                                                                             |
|o19 = 18                                                                     |
+-----------------------------------------------------------------------------+

Given the list of triangulations, we can query them using other topcom
functions. See also Triangulations, for other functionality.

+-----------------------------------------------------------------------------+
|i20 : netList Ts                                                             |
|                                                                             |
|      +----------------------------------------------------------------------|
|o20 = |triangulation {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3}, |
|      +----------------------------------------------------------------------|
|      |triangulation {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 5}, {1, 2, 3}, |
|      +----------------------------------------------------------------------|
|      |triangulation {{0, 1, 3}, {0, 1, 4}, {0, 3, 5}, {1, 2, 3}, {1, 2, 4}, |
|      +----------------------------------------------------------------------|
|      |triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, 3}, {1, 2, 4}, {1, 3, 5}, |
|      +----------------------------------------------------------------------|
|      |triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4}, {1, 2, 3}, {1, 3, 5}, |
|      +----------------------------------------------------------------------|
|      |triangulation {{0, 1, 4}, {0, 1, 5}, {1, 2, 3}, {1, 2, 4}, {1, 3, 5}, |
|      +----------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
|---------------------+                                                       |
|{2, 3, 4}, {3, 4, 5}}|                                                       |
|---------------------+                                                       |
|{2, 3, 5}, {2, 4, 5}}|                                                       |
|---------------------+                                                       |
|{2, 3, 4}, {3, 4, 5}}|                                                       |
|---------------------+                                                       |
|{2, 3, 4}, {3, 4, 5}}|                                                       |
|---------------------+                                                       |
|{2, 3, 5}, {2, 4, 5}}|                                                       |
|---------------------+                                                       |
|{2, 3, 5}, {2, 4, 5}}|                                                       |
|---------------------+                                                       |
+-----------------------------------------------------------------------------+
|i21 : for tri in Ts list isWellDefined tri                                   |
|                                                                             |
|o21 = {true, true, true, true, true, true}                                   |
|                                                                             |
|o21 : List                                                                   |
+-----------------------------------------------------------------------------+
|i22 : for tri in Ts list isRegularTriangulation tri                          |
|                                                                             |
|o22 = {true, true, true, true, true, true}                                   |
|                                                                             |
|o22 : List                                                                   |
+-----------------------------------------------------------------------------+
|i23 : for tri in Ts list regularTriangulationWeights tri                     |
|                                                                             |
|        97  47  1             87  41    1             157  15  1             |
|o23 = {{--, --, -, 0, 0, 0}, {--, --, - -, 0, 0, 0}, {---, --, -, 0, 0, 0},  |
|         8   8  4              8   8    4              20   4  2             |
|      -----------------------------------------------------------------------|
|       45  19  1             107  9    1             35  13    1             |
|      {--, --, -, 0, 0, 0}, {---, -, - -, 0, 0, 0}, {--, --, - -, 0, 0, 0}}  |
|        8   8  4              20  4    2              8   8    4             |
|                                                                             |
|o23 : List                                                                   |
+-----------------------------------------------------------------------------+

See also
========

  * topcomNumTriangulations(Matrix) (Topcom)
    -- the number of
    triangulations of a point or vector configuration
  * generateTriangulations -- generate all
    triangulations with certain properties

Ways to use this method:
========================

  * allTriangulations(Matrix) -- use
    topcom to generate all triangulations of a point or vector configuration

bistellarFlip
*************

Ways to use bistellarFlip :
===========================

  * bistellarFlip(List,List) (missing documentation)
  * bistellarFlip(Triangulation,List) (missing documentation)

For the programmer
==================

The object bistellarFlip (missing documentation) is a method function (Macaulay2Doc).

Chirotope
*********

Functions and methods returning an object of class Chirotope :
==============================================================

  * chirotope(Matrix) (missing documentation)
  * naiveChirotope(Matrix) (missing documentation)

Methods that use an object of class Chirotope :
===============================================

  * Chirotope == Chirotope (missing documentation)
  * toString(Chirotope) (missing documentation)

For the programmer
==================

The object Chirotope (missing documentation) is a type (Macaulay2Doc),
with ancestor classes HashTable (Macaulay2Doc),
< Thing (Macaulay2Doc).

chirotope
*********

Ways to use chirotope :
=======================

  * chirotope(Matrix) (missing documentation)
  * chirotope(String) (missing documentation)

For the programmer
==================

The object chirotope (missing documentation) is a method function with options (Macaulay2Doc).

ConeIndex
*********

Functions with optional argument named ConeIndex :
==================================================

  * fineStarTriangulation(...,ConeIndex=>...) (missing documentation)
  * regularFineStarTriangulation(...,ConeIndex=>...) (missing documentation)

For the programmer
==================

The object ConeIndex (missing documentation) is a symbol (Macaulay2Doc).

delaunaySubdivision
*******************

Ways to use delaunaySubdivision :
=================================

  * delaunaySubdivision(Matrix) (missing documentation)

For the programmer
==================

The object delaunaySubdivision (missing documentation) is a method function (Macaulay2Doc).

delaunayWeights
***************

Ways to use delaunayWeights :
=============================

  * delaunayWeights(Matrix) (missing documentation)

For the programmer
==================

The object delaunayWeights (missing documentation) is a method function (Macaulay2Doc).

fineStarTriangulation
*********************

Ways to use fineStarTriangulation :
===================================

  * fineStarTriangulation(Matrix,List) (missing documentation)

For the programmer
==================

The object fineStarTriangulation (missing documentation) is a method function with options (Macaulay2Doc).

flips
*****

Ways to use flips :
===================

  * flips(Triangulation) (missing documentation)

For the programmer
==================

The object flips (missing documentation) is a method function with options (Macaulay2Doc).

generateTriangulations -- generate all triangulations with certain properties
*****************************************************************************

Synopsis
========

  * Usage:
        Ts = generateTriangulations A
        Ts = generateTriangulations(A, tri)
        Ts = generateTriangulations T
  * Inputs:
      * A, a matrix (Macaulay2Doc), over the integers (or
        rationals?), whose columns are the points to use
      * tri, a list (Macaulay2Doc), A list of lists of indices
        representing a triangulation of the columns of $A$
      * T, an instance of the type Triangulation (missing documentation),
        packages both $A$ and tri in one object
  * Optional inputs (Macaulay2Doc),:
      * Limit => an integer (Macaulay2Doc), default value infinity,
        Stop after constructing this many triangulations
      * RegularOnly => a Boolean value (Macaulay2Doc), default
        value false, Only generate regular triangulations
      * Homogenize => a Boolean value (Macaulay2Doc), default
        value true, set to false in the case the columns form a vector
        configuration. The default case is that the columns of $A$ are
        considered a point configuration (not used in the variant
        generateTriangulations T)
  * Outputs:
      * Ts, a list (Macaulay2Doc), of lists of integers, each such
        list represents a triangulation

Description
===========

This function can be used to generate a set of triangulations of a point set or
vector configuration $A$ (the points are the columns of $A$).

It operates by starting with one triangulation (tri or T), if one is given, and
if not, it constructs a fine triangulation of the set of columns of $A$.

After this, it uses bistellar flips to generate neighbors, and continues, until
the limit is reached, or no new ones can be constructed.

Important note! This function generally starts with a fine triangulation (i.e.
one using all of the points in $A$), and only considers bistellar flips that
give fine triangulations.

+-----------------------------------------------------------------------------+
|i1 : A = vertices hypercube 3                                                |
|                                                                             |
|o1 = | -1 1  -1 1  -1 1  -1 1 |                                              |
|     | -1 -1 1  1  -1 -1 1  1 |                                              |
|     | -1 -1 -1 -1 1  1  1  1 |                                              |
|                                                                             |
|              3        8                                                     |
|o1 : Matrix QQ  <--- QQ                                                      |
+-----------------------------------------------------------------------------+
|i2 : T = topcomRegularFineTriangulation A                                    |
|                                                                             |
|o2 = {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 5, 6, 7}}                                                           |
|                                                                             |
|o2 : List                                                                    |
+-----------------------------------------------------------------------------+
|i3 : tri = regularFineTriangulation A                                        |
|                                                                             |
|o3 = triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},  |
|                                                                             |
|o3 : Triangulation                                                           |
|-----------------------------------------------------------------------------|
|{3, 4, 5, 6}, {3, 5, 6, 7}}                                                  |
+-----------------------------------------------------------------------------+
|i4 : Ts1 = generateTriangulations A -- list of Triangulation's.              |
|                                                                             |
|o4 = {triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {0, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 3, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},  |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 5}, {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6}, {3, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 6, 7}}}                                                          |
|                                                                             |
|o4 : List                                                                    |
+-----------------------------------------------------------------------------+
|i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets           |
|                                                                             |
|o5 = {{{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 3, 5, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, |
|     ------------------------------------------------------------------------|
|     {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {2, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 3, 5, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, |
|     ------------------------------------------------------------------------|
|     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 2, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, |
|     ------------------------------------------------------------------------|
|     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {1, 4, 6, 7}}}                                            |
|                                                                             |
|o5 : List                                                                    |
+-----------------------------------------------------------------------------+
|i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulatio|
|                                                                             |
|o6 = {triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {0, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 3, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},  |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 5}, {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6}, {3, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 6, 7}}}                                                          |
|                                                                             |
|o6 : List                                                                    |
|-----------------------------------------------------------------------------|
|ns                                                                           |
+-----------------------------------------------------------------------------+
|i7 : Ts4 = generateTriangulations tri -- list of Triangulations              |
|                                                                             |
|o7 = {triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {0, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 3, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},  |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 5}, {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, |
|     ------------------------------------------------------------------------|
|     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {1, 3, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {3, 4, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, |
|     ------------------------------------------------------------------------|
|     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, |
|     ------------------------------------------------------------------------|
|     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},  |
|     ------------------------------------------------------------------------|
|     triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, |
|     ------------------------------------------------------------------------|
|     {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6}, {3, 5, 6, 7}}, triangulation  |
|     ------------------------------------------------------------------------|
|     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, {1, 4, 5, 7},  |
|     ------------------------------------------------------------------------|
|     {1, 4, 6, 7}}}                                                          |
|                                                                             |
|o7 : List                                                                    |
+-----------------------------------------------------------------------------+
|i8 : all(Ts4, isFine)                                                        |
|                                                                             |
|o8 = true                                                                    |
+-----------------------------------------------------------------------------+
|i9 : all(Ts4, isStar)                                                        |
|                                                                             |
|o9 = false                                                                   |
+-----------------------------------------------------------------------------+
|i10 : all(Ts4, isRegularTriangulation)                                       |
|                                                                             |
|o10 = true                                                                   |
+-----------------------------------------------------------------------------+
|i11 : Ts4/isStar//tally                                                      |
|                                                                             |
|o11 = Tally{false => 66}                                                     |
|            true => 8                                                        |
|                                                                             |
|o11 : Tally                                                                  |
+-----------------------------------------------------------------------------+
|i12 : Ts4/gkzVector                                                          |
|                                                                             |
|           8  8  8  8  8  8       20  8     8     8  4          4  8     4   |
|o12 = {{8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -, 4, -,  |
|           3  3  3  3  3  3        3  3     3     3  3          3  3     3   |
|      -----------------------------------------------------------------------|
|      16     16    20     4        4     20       4  4  16  4  16  16        |
|      --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --, 4},   |
|       3      3     3     3        3      3       3  3   3  3   3   3        |
|      -----------------------------------------------------------------------|
|          8  16     4     16  4    8  8     20     8  4          4  8     20 |
|      {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8, --,|
|          3   3     3      3  3    3  3      3     3  3          3  3      3 |
|      -----------------------------------------------------------------------|
|         8  8       8  4     16     16  4    8  8  20     8        4    4    |
|      4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -}, {-, 4,|
|         3  3       3  3      3      3  3    3  3   3     3        3    3    |
|      -----------------------------------------------------------------------|
|         8     20  8  8    8     8  8  8  8     8    4  16  16        4  4   |
|      8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8, -, -,  |
|         3      3  3  3    3     3  3  3  3     3    3   3   3        3  3   |
|      -----------------------------------------------------------------------|
|      16       20  4     8  8     8    8        4  8  20  8       16     16  |
|      --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--, 4, --, |
|       3        3  3     3  3     3    3        3  3   3  3        3      3  |
|      -----------------------------------------------------------------------|
|      4     8  4       20     8  8     4  8          4  8     8     8  20    |
|      -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -, --},  |
|      3     3  3        3     3  3     3  3          3  3     3     3   3    |
|      -----------------------------------------------------------------------|
|          16  16  4  16  4  4       8        4     16  4  16          20  4  |
|      {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4, --, -, |
|           3   3  3   3  3  3       3        3      3  3   3           3  3  |
|      -----------------------------------------------------------------------|
|      4  20          16        8  16  4  4          20  4        4  20       |
|      -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, --, 4},  |
|      3   3           3        3   3  3  3           3  3        3   3       |
|      -----------------------------------------------------------------------|
|       16  16  4     4        8       4  4  16  8        16    16  4  16     |
|      {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, --, 4, |
|        3   3  3     3        3       3  3   3  3         3     3  3   3     |
|      -----------------------------------------------------------------------|
|      4        8    4  16     4     16  8       8  8     8  8     8  8       |
|      -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, -}, {4, |
|      3        3    3   3     3      3  3       3  3     3  3     3  3       |
|      -----------------------------------------------------------------------|
|      8  4     20  8     8    16  4  4        16  16  4       4  16  16  8   |
|      -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, --, -,  |
|      3  3      3  3     3     3  3  3         3   3  3       3   3   3  3   |
|      -----------------------------------------------------------------------|
|            4    4  20              20  4    4        20  20        4    4   |
|      8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, -}, {-,  |
|            3    3   3               3  3    3         3   3        3    3   |
|      -----------------------------------------------------------------------|
|            8  16  16  4          4     20  20     4             4  20  20   |
|      4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, --, --,  |
|            3   3   3  3          3      3   3     3             3   3   3   |
|      -----------------------------------------------------------------------|
|      4          8        16     4  4  16    8        4  8  8  20       8    |
|      -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, 4}, {-, 8,|
|      3          3         3     3  3   3    3        3  3  3   3       3    |
|      -----------------------------------------------------------------------|
|      8  8     4  20       20  4  4  20  4  20  20  4    4  20  20  4  20  4 |
|      -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, --, -, --, -,|
|      3  3     3   3        3  3  3   3  3   3   3  3    3   3   3  3   3  3 |
|      -----------------------------------------------------------------------|
|      4  20    20        4  4        20       8  4     4     16  16    8     |
|      -, --}, {--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, --, --}, {-, 8, |
|      3   3     3        3  3         3       3  3     3      3   3    3     |
|      -----------------------------------------------------------------------|
|         4     4  16  16    4  16     16     4  8       4        20     8  8 |
|      4, -, 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, 4, --, 8, -, -,|
|         3     3   3   3    3   3      3     3  3       3         3     3  3 |
|      -----------------------------------------------------------------------|
|      8    8  8  8     20        4    4        8     8  20  8    4     16  4 |
|      -}, {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, -}, {-, 8, --, -,|
|      3    3  3  3      3        3    3        3     3   3  3    3      3  3 |
|      -----------------------------------------------------------------------|
|         8  16          4  20     8     8  8    16  4  4     16        8     |
|      4, -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, 8, --, 4, 4, -},   |
|         3   3          3   3     3     3  3     3  3  3      3        3     |
|      -----------------------------------------------------------------------|
|       16     4  16  4  16     4       8  20  8  4        8    8  8     8    |
|      {--, 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, -}, {-, -, 8, -, 4,|
|        3     3   3  3   3     3       3   3  3  3        3    3  3     3    |
|      -----------------------------------------------------------------------|
|      20  4       4  16     4  16     4  16       20  8  8  4        8    20 |
|      --, -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, -, -, 4, 8, -}, {--,|
|       3  3       3   3     3   3     3   3        3  3  3  3        3     3 |
|      -----------------------------------------------------------------------|
|            4  8  8  8          8  8  8  4        20    16  16     4     4   |
|      4, 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -, 4, -,  |
|            3  3  3  3          3  3  3  3         3     3   3     3     3   |
|      -----------------------------------------------------------------------|
|      8          8  4     8  8     20       20     4  4     20       20  4   |
|      -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4}, {--, -,  |
|      3          3  3     3  3      3        3     3  3      3        3  3   |
|      -----------------------------------------------------------------------|
|                  4  20       4  20        20  4       8     8  20     4  8  |
|      4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --, 8, -, -, |
|                  3   3       3   3         3  3       3     3   3     3  3  |
|      -----------------------------------------------------------------------|
|              16  4  16  8        4    4     16  16     8  4          4  8   |
|      4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4}, {4, -, -,  |
|               3  3   3  3        3    3      3   3     3  3          3  3   |
|      -----------------------------------------------------------------------|
|         16  16     4    4     20        20     4    8  8  8        8  8  8  |
|      8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8, -, -, -},|
|          3   3     3    3      3         3     3    3  3  3        3  3  3  |
|      -----------------------------------------------------------------------|
|       8  20  8     8        4       16  8     4  16     4    4        8  16 |
|      {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8, 4, -, --,|
|       3   3  3     3        3        3  3     3   3     3    3        3   3 |
|      -----------------------------------------------------------------------|
|      4  16       16  4     16  4     16  4    4     16  4  16  4     16     |
|      -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -, 4, --}}   |
|      3   3        3  3      3  3      3  3    3      3  3   3  3      3     |
|                                                                             |
|o12 : List                                                                   |
+-----------------------------------------------------------------------------+
|i13 : volume convexHull A -- 8                                               |
|                                                                             |
|o13 = 8                                                                      |
|                                                                             |
|o13 : QQ                                                                     |
+-----------------------------------------------------------------------------+
|i14 : stars1 = select(Ts4, t -> (gkzVector t)#-1 == 8)                       |
|                                                                             |
|o14 = {triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},|
|      -----------------------------------------------------------------------|
|      {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},|
|      -----------------------------------------------------------------------|
|      {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation |
|      -----------------------------------------------------------------------|
|      {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, |
|      -----------------------------------------------------------------------|
|      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},|
|      -----------------------------------------------------------------------|
|      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7},|
|      -----------------------------------------------------------------------|
|      {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, |
|      -----------------------------------------------------------------------|
|      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, |
|      -----------------------------------------------------------------------|
|      {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7},|
|      -----------------------------------------------------------------------|
|      {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation |
|      -----------------------------------------------------------------------|
|      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, |
|      -----------------------------------------------------------------------|
|      {1, 4, 5, 7}}}                                                         |
|                                                                             |
|o14 : List                                                                   |
+-----------------------------------------------------------------------------+
|i15 : stars2 = select(Ts4, isStar)                                           |
|                                                                             |
|o15 = {triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},|
|      -----------------------------------------------------------------------|
|      {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},|
|      -----------------------------------------------------------------------|
|      {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation |
|      -----------------------------------------------------------------------|
|      {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, |
|      -----------------------------------------------------------------------|
|      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},|
|      -----------------------------------------------------------------------|
|      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7},|
|      -----------------------------------------------------------------------|
|      {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, |
|      -----------------------------------------------------------------------|
|      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, |
|      -----------------------------------------------------------------------|
|      {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7},|
|      -----------------------------------------------------------------------|
|      {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation |
|      -----------------------------------------------------------------------|
|      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, |
|      -----------------------------------------------------------------------|
|      {1, 4, 5, 7}}}                                                         |
|                                                                             |
|o15 : List                                                                   |
+-----------------------------------------------------------------------------+
|i16 : stars1 == stars2                                                       |
|                                                                             |
|o16 = true                                                                   |
+-----------------------------------------------------------------------------+

Caveat
======

This function is written in the top level Macaulay2 language, and so is much
slower than allTriangulations, which
calls the topcom code written in C++.  On the other hand, one can give this
function a limit for the number of triangulations to generate, so can be used
to generate triangulations in the case when the number is too large to write
down all of them.

See also
========

  * allTriangulations -- use topcom to
    generate all triangulations of a point or vector configuration

Ways to use generateTriangulations :
====================================

  * "generateTriangulations(Matrix)"
  * generateTriangulations(Matrix,List) (missing documentation)
  * generateTriangulations(Triangulation) (missing documentation)

For the programmer
==================

The object generateTriangulations, is a
method function with options (Macaulay2Doc).

gkzVector
*********

Ways to use volumeVector :
==========================

  * volumeVector(Matrix,List) (missing documentation)
  * volumeVector(Triangulation) (missing documentation)

For the programmer
==================

The object gkzVector (missing documentation) is a method function (Macaulay2Doc).

isFine
******

Ways to use isFine :
====================

  * isFine(Matrix,List) (missing documentation)
  * isFine(Triangulation) (missing documentation)

For the programmer
==================

The object isFine (missing documentation) is a method function (Macaulay2Doc).

isRegularTriangulation -- determine if a given triangulation is a regular triangulation
***************************************************************************************

Synopsis
========

  * Usage:
        isRegularTriangulation T
  * Inputs:
      * T, an instance of the type Triangulation (missing documentation), A
        triangulation of a point or vector configuration
  * Optional inputs (Macaulay2Doc),:
      * Homogenize (missing documentation) => a Boolean value (Macaulay2Doc),
        default value true, unused for this method
  * Outputs:
      * a Boolean value (Macaulay2Doc), whether the given
        triangulation is regular

Description
===========

A triangulation is called regular if it can be constructed in the following
way: place the point set in one higher dimension at various heights in the new
variable.  Compute the convex hull.  Collect the list of facets with downward
pointing normal (last coordinate of normal vector is negative).  If each of
these is a simplex, then these form a triangulation of the original point set.
A triangulation which arises this way is called regular.  See the book [deLoera
et al] for more details and many beautiful properties of such triangulations.

The following example is one of the simplest examples of a non-regular
triangulation.  Notice that tri is a triangulation of the polytope which is the
convex hull of the columns of $A$, which are the only points allowed in the
triangulation.

+-----------------------------------------------------------------------------+
|i1 : A = transpose matrix {{0,3},{0,1},{-1,-1},{1,-1},{-4,-2},{4,-2}}        |
|                                                                             |
|o1 = | 0 0 -1 1  -4 4  |                                                     |
|     | 3 1 -1 -1 -2 -2 |                                                     |
|                                                                             |
|              2        6                                                     |
|o1 : Matrix ZZ  <--- ZZ                                                      |
+-----------------------------------------------------------------------------+
|i2 : tri = {{0,1,2}, {1,3,5}, {2,3,4}, {0,1,5},                              |
|         {0,2,4}, {3,4,5}, {1,2,3}}                                          |
|                                                                             |
|o2 = {{0, 1, 2}, {1, 3, 5}, {2, 3, 4}, {0, 1, 5}, {0, 2, 4}, {3, 4, 5}, {1,  |
|     ------------------------------------------------------------------------|
|     2, 3}}                                                                  |
|                                                                             |
|o2 : List                                                                    |
+-----------------------------------------------------------------------------+
|i3 : T = triangulation(A, tri)                                               |
|                                                                             |
|o3 = triangulation {{0, 1, 2}, {0, 1, 5}, {0, 2, 4}, {1, 2, 3}, {1, 3, 5},   |
|                                                                             |
|o3 : Triangulation                                                           |
|-----------------------------------------------------------------------------|
|{2, 3, 4}, {3, 4, 5}}                                                        |
+-----------------------------------------------------------------------------+

We check that T is indeed a triangulation, and whether it is a regular
triangulation.

+-----------------------------+
|i4 : isWellDefined T         |
|                             |
|o4 = true                    |
+-----------------------------+
|i5 : isRegularTriangulation T|
|                             |
|o5 = false                   |
+-----------------------------+

Many of the functions in this package are wrappers for topcom functions.
Setting the global variable debugLevel to either 1,2, or 5 will give more
detail about what files are written to Topcom, and what the executable is.
Setting debugLevel to 0 means that the function will run silently.

Caveat
======

Does topcom check that the triangulation is actually well defined?  I'm not
sure...  This is why we call isWellDefined(Triangulation) (missing
documentation) first.

See also
========

  * regularTriangulationWeights(Triangulation) (missing documentation)
  * regularFineTriangulation (missing documentation)
  * isWellDefined(Triangulation) (missing documentation)

Ways to use isRegularTriangulation :
====================================

  * "isRegularTriangulation(Triangulation)"
  * isRegularTriangulation(Matrix,List) (missing documentation)

For the programmer
==================

The object isRegularTriangulation, is a
method function with options (Macaulay2Doc).

isStar
******

Ways to use isStar :
====================

  * isStar(Matrix,List) (missing documentation)
  * isStar(Triangulation) (missing documentation)

For the programmer
==================

The object isStar (missing documentation) is a method function (Macaulay2Doc).

naiveChirotope
**************

Ways to use naiveChirotope :
============================

  * naiveChirotope(Matrix) (missing documentation)

For the programmer
==================

The object naiveChirotope (missing documentation) is a method function with options (Macaulay2Doc).

naiveIsTriangulation
********************

Ways to use naiveIsTriangulation :
==================================

  * naiveIsTriangulation(Matrix,List) (missing documentation)
  * naiveIsTriangulation(Matrix,List,List) (missing documentation)
  * naiveIsTriangulation(Triangulation) (missing documentation)

For the programmer
==================

The object naiveIsTriangulation (missing documentation) is a method function (Macaulay2Doc).

neighbors
*********

Ways to use neighbors :
=======================

  * neighbors(Triangulation) (missing documentation)

For the programmer
==================

The object neighbors (missing documentation) is a method function (Macaulay2Doc).

regularFineStarTriangulation
****************************

Ways to use regularFineStarTriangulation :
==========================================

  * regularFineStarTriangulation(Matrix) (missing documentation)

For the programmer
==================

The object regularFineStarTriangulation (missing documentation) is a
method function with options (Macaulay2Doc).

regularFineTriangulation
************************

Ways to use regularFineTriangulation :
======================================

  * regularFineTriangulation(Matrix) (missing documentation)

For the programmer
==================

The object regularFineTriangulation (missing documentation) is a method function with options (Macaulay2Doc).

regularTriangulationWeights
***************************

Ways to use regularTriangulationWeights :
=========================================

  * regularTriangulationWeights(Matrix,List) (missing documentation)
  * regularTriangulationWeights(Triangulation) (missing documentation)

For the programmer
==================

The object regularTriangulationWeights (missing documentation) is a
method function with options (Macaulay2Doc).

Triangulation
*************

Functions and methods returning an object of class Triangulation :
==================================================================

  * regularFineTriangulation(Matrix) (missing documentation)
  * triangulation(Matrix,List) (missing documentation)

Methods that use an object of class Triangulation :
===================================================

  * affineCircuits(Triangulation) (missing documentation)
  * bistellarFlip(Triangulation,List) (missing documentation)
  * flips(Triangulation) (missing documentation)
  * generateTriangulations(Triangulation) (missing documentation)
  * isFine(Triangulation) (missing documentation)
  * "isRegularTriangulation(Triangulation)" -- see
    isRegularTriangulation, -- determine if a given
    triangulation is a regular triangulation
  * isStar(Triangulation) (missing documentation)
  * isWellDefined(Triangulation) (missing documentation)
  * matrix(Triangulation) (missing documentation)
  * max(Triangulation) (missing documentation)
  * naiveIsTriangulation(Triangulation) (missing documentation)
  * neighbors(Triangulation) (missing documentation)
  * net(Triangulation) (missing documentation)
  * regularTriangulationWeights(Triangulation) (missing documentation)
  * toExternalString(Triangulation) (missing documentation)
  * toString(Triangulation) (missing documentation)
  * Triangulation == Triangulation (missing documentation)
  * vectors(Triangulation) (missing documentation)
  * volumeVector(Triangulation) (missing documentation)

For the programmer
==================

The object Triangulation (missing documentation) is a type (Macaulay2Doc),
with ancestor classes HashTable (Macaulay2Doc),
< Thing (Macaulay2Doc).

triangulation -- make a Triangulation object
********************************************

Synopsis
========

  * Usage:
        triangulation(A, T)
  * Inputs:
      * A, a matrix (Macaulay2Doc),
      * T, a list (Macaulay2Doc), representing a triangulation of
        the columns of $A$ (each element in the list is a list of indices in
        the range $0, \ldots, n-1$, where $n$ is the number of columns of $A$)
  * Optional inputs (Macaulay2Doc),:
      * Homogenize (missing documentation) => a Boolean value (Macaulay2Doc),
        default value true, if true, add a row of ones
        to think of this as a vector configuration in one higher dimension.
  * Outputs:
      * an instance of the type Triangulation (missing documentation), A
        Triangulation (missing documentation) object.  Very little computation
        is performed.  The matrix and list representing a triangulation is
        packaged into an object to make clear that it is a triangulation

Description
===========

+-----------------------------------------------------------------------------+
|i1 : P = hypercube 3                                                         |
|                                                                             |
|o1 = P                                                                       |
|                                                                             |
|o1 : Polyhedron                                                              |
+-----------------------------------------------------------------------------+
|i2 : A = vertices P                                                          |
|                                                                             |
|o2 = | -1 1  -1 1  -1 1  -1 1 |                                              |
|     | -1 -1 1  1  -1 -1 1  1 |                                              |
|     | -1 -1 -1 -1 1  1  1  1 |                                              |
|                                                                             |
|              3        8                                                     |
|o2 : Matrix QQ  <--- QQ                                                      |
+-----------------------------------------------------------------------------+
|i3 : T = topcomRegularFineTriangulation A                                    |
|                                                                             |
|o3 = {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 5, 6, 7}}                                                           |
|                                                                             |
|o3 : List                                                                    |
+-----------------------------------------------------------------------------+
|i4 : tri = triangulation(A, T)                                               |
|                                                                             |
|o4 = triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},  |
|                                                                             |
|o4 : Triangulation                                                           |
|-----------------------------------------------------------------------------|
|{3, 4, 5, 6}, {3, 5, 6, 7}}                                                  |
+-----------------------------------------------------------------------------+
|i5 : matrix tri                                                              |
|                                                                             |
|o5 = | -1 1  -1 1  -1 1  -1 1 |                                              |
|     | -1 -1 1  1  -1 -1 1  1 |                                              |
|     | -1 -1 -1 -1 1  1  1  1 |                                              |
|     | 1  1  1  1  1  1  1  1 |                                              |
|                                                                             |
|              4        8                                                     |
|o5 : Matrix QQ  <--- QQ                                                      |
+-----------------------------------------------------------------------------+
|i6 : vectors tri                                                             |
|                                                                             |
|o6 = {{-1, -1, -1, 1}, {1, -1, -1, 1}, {-1, 1, -1, 1}, {1, 1, -1, 1}, {-1,   |
|     ------------------------------------------------------------------------|
|     -1, 1, 1}, {1, -1, 1, 1}, {-1, 1, 1, 1}, {1, 1, 1, 1}}                  |
|                                                                             |
|o6 : List                                                                    |
+-----------------------------------------------------------------------------+
|i7 : max tri                                                                 |
|                                                                             |
|o7 = {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},  |
|     ------------------------------------------------------------------------|
|     {3, 5, 6, 7}}                                                           |
|                                                                             |
|o7 : List                                                                    |
+-----------------------------------------------------------------------------+
|i8 : isWellDefined tri                                                       |
|                                                                             |
|o8 = true                                                                    |
+-----------------------------------------------------------------------------+
|i9 : netList affineCircuits tri                                              |
|                                                                             |
|     +------+------+                                                         |
|o9 = |{1, 2}|{0, 3}|                                                         |
|     +------+------+                                                         |
|     |{3, 4}|{2, 5}|                                                         |
|     +------+------+                                                         |
|     |{3, 4}|{1, 6}|                                                         |
|     +------+------+                                                         |
|     |{5, 6}|{4, 7}|                                                         |
|     +------+------+                                                         |
+-----------------------------------------------------------------------------+
|i10 : isFine tri                                                             |
|                                                                             |
|o10 = true                                                                   |
+-----------------------------------------------------------------------------+
|i11 : isStar tri                                                             |
|                                                                             |
|o11 = false                                                                  |
+-----------------------------------------------------------------------------+
|i12 : isRegularTriangulation tri                                             |
|                                                                             |
|o12 = true                                                                   |
+-----------------------------------------------------------------------------+

Caveat
======

See also
========

  * max(Triangulation) (missing documentation)
  * vectors(Triangulation) (missing documentation)
  * regularFineTriangulation (missing documentation)
  * isWellDefined(Triangulation) (missing documentation)

Ways to use triangulation :
===========================

  * triangulation(Matrix,List) (missing documentation)

For the programmer
==================

The object triangulation, is a method function with options (Macaulay2Doc).

vectors
*******

Ways to use vectors :
=====================

  * vectors(Triangulation) (missing documentation)

For the programmer
==================

The object vectors (missing documentation) is a method function (Macaulay2Doc).

volumeVector
************

Ways to use volumeVector :
==========================

  * volumeVector(Matrix,List) (missing documentation)
  * volumeVector(Triangulation) (missing documentation)

For the programmer
==================

The object volumeVector (missing documentation) is a method function (Macaulay2Doc).