# isSimplicial -- checks if a polyhedral object is simplicial

## Description

A Polyhedron of dimension $d$ is simplicial if it is compact and every facet is (isomorphic to) a simplex.

 i1 : P = convexHull matrix {{3,0,0,0,1},{0,3,0,0,1},{0,0,3,0,1}} o1 = P o1 : Polyhedron i2 : isSimplicial P o2 = true i3 : P = hypercube 2 o3 = P o3 : Polyhedron i4 : isSimplicial P o4 = true

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

 i5 : C = coneFromVData matrix {{1,0,0,1},{0,1,0,1},{0,0,1,1}} o5 = C o5 : Cone i6 : isSimplicial C o6 = true i7 : C = coneFromVData matrix {{1,1,-1,-1},{1,-1,1,-1},{1,1,1,1}} o7 = C 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 = F o9 : Fan i10 : isSimplicial F o10 = true i11 : PC = skeleton(2,polyhedralComplex crossPolytope 3) o11 = PC 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 -- calculates the extreme rays of the cone; invokes "rays" from 4ti2

## Ways to use isSimplicial :

• "isSimplicial(PolyhedralObject)"

## For the programmer

The object isSimplicial is .