isSimplicial -- checks if a polyhedral object is simplicial

Synopsis

• Usage:

  b = isSimplicial X

• Inputs:
  • X, an instance of the type PolyhedralObject
• Outputs:
  • b, true if the PolyhedralObject is simplicial, false otherwise

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

See also

• 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