The simplicial complex defined by Wikipedia does not intersect itself and the winding number is thereby either 0 or 1.
Our complexes may intersect themselves, or at least their embedding in the Euclidean space will and the winding number may be any integer.
Nit Picking
I try here to bring some mathematical precision to the terminology used in this section.
I introduce “hedron” to refer to a particular form of simplicial complex.
Surely a better term is needed.
There may already be such a defined concept but I have not found it on the web.
In geometry it is normally natural to identify a construct by some set of ‘points’ in a space.
This does not work here.
I need to talk of distinct simplexes embedded in n-space even when the set of points within those simplexes are identical.
I even need more than the two simplexes of opposite orientation.
I will use the notation S(x) to denote the closed set of points associated with simplex x.
You could say that simplexes x and y are equivalent when S(x)=S(y) but I think I don’t need that notion.
The Wikipedia definition of simplicial complex would seem to say that two simplexes spanning the same set of points (in some Euclidean space) are identical.
This identification stems indirectly from the definition of ‘simplex’.
The discrimination I need may be natural to algebraic topology but not to the geometrical simplex concept.
I suspect that chain complexes may have to do with this but the definitions there seem circular.
The Abstract simplicial complex seems to be just what we need.
It is a complex outside any particular Euclidean space.
I think the axiom is better written:
∀X (X∊K ➞ ∀Y (Y⊂X ➞ Y∊K))
This axiom seems to ignore orientation issues.
The mechanisms I use in my code deal with orientation in an abstract simplicial complex.
(Google thinks this is interesting.)
This is nearer the root.