# polyhedralComplex(Matrix,Matrix,Matrix,List) -- Constructing a polyhedral complex.

## Synopsis

• Function: polyhedralComplex
• Usage:
PC = polyhedralComplex(V,R,N,L)
• Inputs:
• V, , Matrix containing the vertices as columns
• R, , Matrix containing rays as columns (optional)
• N, , Matrix containing generators of the lineality space as columns (optional)
• L, a list, List contiaining lists with indices of the vertices of the maximal cells.
• Outputs:

## Description

Basic constructor for polyhedral complices that takes a matrix containing the vertices of the polyhedral complex and a list of lists with the indices of the vertices and rays in the maximal cells. Both the rays and the lineality space are optional arguments. If two matrices are provided, then the second matrix is considered to contain rays. To input a lineality space, one must provide three matrices.

This constructor does not check well-definedness, see isWellDefined.

 i1 : M = matrix {{0,1,2}} o1 = | 0 1 2 | 1 3 o1 : Matrix ZZ <--- ZZ i2 : L = {{0,1},{1,2}} o2 = {{0, 1}, {1, 2}} o2 : List i3 : PC = polyhedralComplex(M,L) o3 = PC o3 : PolyhedralComplex
 i4 : C = hypercube 2 o4 = C o4 : Polyhedron i5 : F = faces(1,C) o5 = {({0, 2}, {}), ({1, 3}, {}), ({0, 1}, {}), ({2, 3}, {})} o5 : List i6 : V = vertices C o6 = | -1 1 -1 1 | | -1 -1 1 1 | 2 4 o6 : Matrix QQ <--- QQ i7 : L = linealitySpace C o7 = 0 2 o7 : Matrix QQ <--- 0 i8 : PC = polyhedralComplex(V,L,F) o8 = PC o8 : PolyhedralComplex i9 : vertices PC o9 = | -1 1 -1 1 | | -1 -1 1 1 | 2 4 o9 : Matrix QQ <--- QQ i10 : maxPolyhedra PC o10 = {({0, 2}, {}), ({1, 3}, {}), ({0, 1}, {}), ({2, 3}, {})} o10 : List i11 : dim PC o11 = 1