# 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, 0, -1, 0}, {1, 0, 1, 0}, {2, -1, -1, 0}, {0, 0, 1, 0}, {1, 0, 0, 0}, {0,0,0,0}} 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 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); -- 0.242762 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, 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}} o7 : Triangulation i8 : elapsedTime Ts2 = generateTriangulations T; -- 0.864395 seconds elapsed i9 : #Ts2 == #Ts o9 = true

• 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

## 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.

## 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 .