Cubical Complexes Factory

As an alternative to simplicial complexes we can base our topology on squares rather than triangles.

Examples of Complexes

Solid Sphere - In this case 2 dimentional square:

(1) -> sp :=
         sphereSolid(2)$CubicalComplexFactory

   (1)
        (1..2,1..2)
            Type: FiniteCubicalComplex(Integer)
 

Sphere Surface - In this case 2 dimentional square sides:

(2) -> sps :=
      sphereSurface(2)$CubicalComplexFactory

   (2)
        -(1..1,1..2)
        (2..2,1..2)
        (1..2,1..1)
        -(1..2,2..2)
            Type: FiniteCubicalComplex(Integer)
 

Band - Cylinder without ends:

(3) -> bnd := band()$CubicalComplexFactory

   (3)
        (1..1,1..2,1..2)
        (2..2,1..2,1..2)
        (1..2,1..1,1..2)
        (1..2,2..2,1..2)
            Type: FiniteCubicalComplex(Integer)
 

Torus -

(4) ->tor := torusSurface()$CubicalComplexFactory

   (4)
        (1..1,1..2,1..1,1..2)
        (1..1,1..2,2..2,1..2)
        (1..1,1..2,1..2,1..1)
        (1..1,1..2,1..2,2..2)
        (2..2,1..2,1..1,1..2)
        (2..2,1..2,2..2,1..2)
        (2..2,1..2,1..2,1..1)
        (2..2,1..2,1..2,2..2)
        (1..2,1..1,1..1,1..2)
        (1..2,1..1,2..2,1..2)
        (1..2,1..1,1..2,1..1)
        (1..2,1..1,1..2,2..2)
        (1..2,2..2,1..1,1..2)
        (1..2,2..2,2..2,1..2)
        (1..2,2..2,1..2,1..1)
        (1..2,2..2,1..2,2..2)
      Type: FiniteCubicalComplex(Integer)

cubical torus

Möbius band: One boundary:

not yet implemented

schlegel

Projective Plane - No boundaries, every edge is connected to two faces:

(6) -> pp :=
       projectiveSpace(2)$CubicalComplexFactory

   (5)
        (1..2,1..1,1..1,1..2,1..1)
        (1..2,1..1,1..1,1..1,1..2)
        (1..1,1..2,1..2,1..1,1..1)
        (1..1,1..2,1..1,1..2,1..1)
        (1..1,1..1,1..2,1..1,1..2)
        (1..2,1..2,2..2,1..1,1..1)
        (1..2,2..2,1..2,1..1,1..1)
        (2..2,1..2,1..2,1..1,1..1)
        (1..2,1..2,1..1,1..1,2..2)
        (1..2,2..2,1..1,1..1,1..2)
        (2..2,1..2,1..1,1..1,1..2)
        (1..2,1..1,1..2,2..2,1..1)
        (1..2,1..1,2..2,1..2,1..1)
        (2..2,1..1,1..2,1..2,1..1)
        (1..1,1..2,1..1,1..2,2..2)
        (1..1,1..2,1..1,2..2,1..2)
        (1..1,2..2,1..1,1..2,1..2)
        (1..1,1..1,1..2,1..2,2..2)
        (1..1,1..1,1..2,2..2,1..2)
        (1..1,1..1,2..2,1..2,1..2)
             Type: FiniteCubicalComplex(Integer)

 

Further Information

For corresponding factory for simpectial complexes see page here.

For more general information about cubical complexes see page here .

My code for this page is on github here.


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see also:
  • I have put the code here.
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