# isSimplicial -- checks if a polyhedral object is simplicial

## Description

A Polyhedron of dimension $d$ is simplicial if it is compact and has $d+1$ vertices.

 i1 : P = convexHull matrix {{3,0,0,0,1},{0,3,0,0,1},{0,0,3,0,1}} o1 = {ambient dimension => 3 } dimension of lineality space => 0 dimension of polyhedron => 3 number of facets => 4 number of rays => 0 number of vertices => 4 o1 : Polyhedron i2 : isSimplicial P o2 = true i3 : P = hypercube 2 o3 = {ambient dimension => 2 } dimension of lineality space => 0 dimension of polyhedron => 2 number of facets => 4 number of rays => 0 number of vertices => 4 o3 : Polyhedron i4 : isSimplicial P o4 = false

A Fan of dimension $d$ is simplicial if it is pointed and has $d$ rays.

 i5 : C = posHull matrix {{1,0,0,1},{0,1,0,1},{0,0,1,1}} o5 = {ambient dimension => 3 } dimension of lineality space => 0 dimension of the cone => 3 number of facets => 3 number of rays => 3 o5 : Cone i6 : isSimplicial C o6 = true i7 : C = posHull matrix {{1,1,-1,-1},{1,-1,1,-1},{1,1,1,1}} o7 = {ambient dimension => 3 } dimension of lineality space => 0 dimension of the cone => 3 number of facets => 4 number of rays => 4 o7 : Cone i8 : isSimplicial C o8 = false

A Fan/PolyhedralComplex is simplicial if every Cone/Polyhedron of it is simplicial.

 i9 : F = normalFan hypercube 3 o9 = {ambient dimension => 3 } number of generating cones => 8 number of rays => 6 top dimension of the cones => 3 o9 : Fan i10 : isSimplicial F o10 = true i11 : PC = skeleton(2,polyhedralComplex crossPolytope 3) o11 = {ambient dimension => 3 } number of generating polyhedra => 8 top dimension of the polyhedra => 2 o11 : PolyhedralComplex i12 : isSimplicial PC o12 = true

• isCompact -- checks compactness of a Polyhedron
• isPointed -- checks if a Cone or Fan is pointed
• dim -- compute the Krull dimension
• vertices -- displays the vertices of a Polyhedron or a PolyhedralComplex
• rays -- displays all rays of a Cone, a Fan, or a Polyhedron

## Ways to use isSimplicial :

• "isSimplicial(PolyhedralObject)"

## For the programmer

The object isSimplicial is .