Delta Complexes Code

 

Delta Complex

When we looked at the delta complex we got a chain of 'face maps' between each dimension and the next lower one. homology face maps

In homology we treat this as a chain of abelian groups (more detail on the page here).

The above representation defines faces, of any dimension, by their vertices. This is an efficient way to define them, the only disadvantage is that intermediate parts, such as edges, are not indexed. It is sometimes useful to be able to do this, for example, when generating homotopy groups such as the fundamental group.

To do this we represent the complex as a sequence of 'face maps' each one indexed into the next.

The ordering of the indexes is important. The indexes of any face are always ordered acording to their order in the next order face.

As an example consider the representation for a single tetrahedron:

The usual representation would be: (1,2,3,4)
which is very efficient but it does not allow us to refer to its edges and triangles.  

Here we have indexed:

  • The 4 vertices (in red)
  • The 6 edges (in green)
  • The 4 triangles (in blue)
indexed representation
This representation holds all these indexes so that we don't have to keep creating them and they can be used consistently.

Face Maps

So the tetrahedron indexes its 4 triangles, each triangle indexes its 3 edges and each edge indexes its 2 vertices.

Each face table therefore indexes into the next. To show this I have drawn some of the arrows (I could not draw all the arrows as that would have made the diagram too messy).

indexed table

Since edges, triangles, tetrahedrons, etc. are oriented we need to put the indexed in a certain order.

Edges are directed in the order of the vertex indices. That is low numbered index to high numbered index: edge numbering

For triangles we go in the order of the edges, except the middle edge is reversed.

On the diagram the edges are coloured as follows:

  • green arrows - not reversed
  • purple arrows - reversed

This gives the triangles a consistent winding

indexed triangle
However the whole tetrahedron is not yet consistent because some adjacent edges go in the same direction and some go in opposite directions. indexed tetrahedron
This gives the orientation of the faces as given by the right hand rule. That is: if the thumb of the right hand is outside the tetrahedion, pointing toward the face, then the fingers tend to curl round in the direction of the winding. right hand rule

The higher order faces may continue on in this manor, however it is difficult to visualize.

So to represent this structure we use the following:

Rep := Record(VERTS:VS,SIMP:List(List(Integer)))

where:

Example

Here is an example of a tetrahedron (2-sphere). indexed table
(1) -> DELTA := DeltaComplex(VertexSetAbstract)
 

   (1)  DeltaComplex(VertexSetAbstract)
                                                                   Type: Type
(2) -> b4 := sphereSolid(3)$SimplicialComplexFactory
 

   (2)  [1  2  3  4]
                             Type: FiniteSimplicialComplex(VertexSetAbstract)
(3) -> c4 := deltaComplex(b4)
 

   (3)
   edge:[[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]] triangle:
     [[1,2,4],[1,3,5],[2,3,6],[4,5,6]]
      tetrahedron:
     [[1,2,3,4]]
                                        Type: DeltaComplex(VertexSetAbstract)

 


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Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Mathematics++: Selected Topics Beyond the Basic Courses (Student Mathematical Library) Kantor, Ida.

Chapters:

  1. Measure
  2. High Dimensional Geometry
  3. Fourier Analysis
  4. Representations of Finite Groups
  5. Polynomials
  6. Topology

Chapter 6 - Topology. Contains a relatively gentle introduction to homology.

 

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