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isSimplicial -- checks if a polyhedral object is simplicial

Synopsis

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

See also

Ways to use isSimplicial :

For the programmer

The object isSimplicial is a method function.