Generator functions

simpicial provides a set of functions for creating the common kinds of simplicial complex. These can be used as the basis for complexes built from other, simpler, complexes and glued together.

Simplices, skeletons, and voids

simplicial.k_simplex(k, id=None, attr=None, c=None)

Create a k-simplex. If no complex is provided, a new one is created. The top-level simplex can be named and given attributes; its faces will be “anonymous” and have names created for them.

Parameters

  • k (int) – the order of simplex to create

  • id (Optional[Any]) – (optional) the name of the simplex created

  • attr (Optional[Dict[str, Any]]) – (optional) attributes ot the top-level simplex

  • c (Optional[SimplicialComplex]) – (optional) the complex to create into

Return type

SimplicialComplex

Returns

the complex containing the new simplex

simplicial.k_skeleton(k, c=None)

Create the skeleton of a k-simplex, consisting of (k + 1) 0-simplices for the basis and k(k - 1)/2 1-simplices for the skeleton. Note that this doesn’t create the k-simplex itself, just a skeleton of points and lines: to get the simplex itself use k_simplex(). If no complex is provided, a new one is created.

Parameters

  • k (int) – the order of simplex to create

  • c (Optional[SimplicialComplex]) – (optional) the complex to create into

Return type

SimplicialComplex

Returns

a complex containing the skeleton

simplicial.k_void(k, c=None)

Create a (k + 1)-dimensional void or hole with a k-dimensional boundary – or in other words all the faces of a (k + 1)-simplex without filling in the (k + 1) simplex itself.

The numbering of holes is a bit confusing, but a cycle of edges (1-simplices) creates what is referred to as a 1-hole, even though it’s a 2-dimensional structure.

Parameters

  • k (int) – the order of simplex to create

  • c (Optional[SimplicialComplex]) – (optional) the complex to create into

Return type

SimplicialComplex

Returns

the complex containing the skeleton

Structures

simplicial.ring(n, c=None)

Create a closed ring of n lines (1-simplices), where n is strictly greater than 2.

Parameters

  • n (int) – the number of lines in the ring

  • c (Optional[SimplicialComplex]) – (optional) the complex to create into

Return type

SimplicialComplex

Returns

the complex containing the skeleton