Talk:Simplicial complex

Merge simplicial complex & abstract simplicial complex?

It may eliminate some redundancy (and confusion) if simplicial complex and abstract simplicial complex are put on the same page -- especially since the difference between the two is fairly minor.

Trevorgoodchild 03:31, 1 February 2007 (UTC)[reply]

Disagreed. "Some" redundancy does not hurt. What kind of confusion you are talking about? `'mikka 23:05, 1 February 2007 (UTC)[reply]

I guess I don't know why we need simplicial complex:geometry AND simplicial complex:algebraic topology AND abstract simplicial complex as three separate things, each with its own independent definition. The differences among the three are extremely superficial. Trevorgoodchild 15:44, 2 February 2007 (UTC)[reply]

Not at all superficial. To a pure algebraic topologist it may appear so (I am guessing here), but to a student of partially ordered sets the difference between an abstract simplicial complex and any other kind is important; the questions and examples are different. To a geometer the difference between a truly geometrical realization and an abstract or topological complex is important, for instance, the geometrical realization of a simplex may be a rigid body like a convex polytope. It is nonetheless true that there are very close relationships and for some purposes the concepts are not significantly different. Zaslav 11:34, 20 March 2007 (UTC)[reply]

what is not a simplicial complex

I believe that it should be emphasized that the intersection of two faces must be a SINGLE simplex. e.g.,, two edges cannot form a cycle. This has caused me some confusion in the past. —The preceding unsigned comment was added by 169.233.53.82 (talk) 21:11, 15 April 2007 (UTC).[reply]

Relationship to a Manifold

A manifold has fixed dimension; obviously a simplicial complex does not. Is there a generalization of a simplicial complex that is more "manifold-like". For example, the set { a circular disk, a point, a line segment connected to the surface of an ellipsoid } is homeomorphic to the set { a triangle, a point, a line segment connected to the vertex of a tetrahedron }. What would you call that first set? —Ben FrantzDale 16:34, 6 May 2007 (UTC)[reply]

It looks like the answer is "a CW complex". Is that right? —Ben FrantzDale 18:57, 6 May 2007 (UTC)[reply]

Closure, star, and link

It seems that the given definition of a link is only valid for a vertex (or a union of vertex). The correct definition of the link of S should be the union of all faces of the closed star of S that don't intersect S, and it is no longer a kind of boundary. A picture of the link of a 2-dimensional simplex in a 3-dimensional complex should help clarify the situation. Groug (talk) 08:58, 4 October 2010 (UTC)[reply]

I see two different definitions of the star in the literature. One says that the star of sigma contains all simplices that contain sigma as a face (e.g. here or here), another definition says that it simply consists of all simplices containing sigma (e.g. Hatcher, p. 178). The definitions are not equivalent, it seems to be confusing. How to present it here, any idea? Franp9am (talk) 14:38, 16 January 2013 (UTC)[reply]

Sphere packings ?

I don't understand the paragraph about sphere packings. The sentence "Simplicial complexes can be seen to have the same geometric structure as the contact graph of a sphere packing" is very ambiguous and seems wrong to me: it is well known that $K_{6}$ can not be realized as the contact structure of a sphere packing in $\mathbb {R} ^{3}$ , so furthermore not every simplicial complex is a contact complex of a sphere packing. There is not a single reference (and googling did not provide much additional insight on this), so it seems to me that this part is both inaccurate and not mainstream enough to be featured on the wikipedia page. Ventricule (talk) 13:02, 26 September 2013 (UTC)[reply]

Star definition

All sources I see define the star of S as the union of all simplices of K which contain S. That's different from what our article says. Is this a convention that I've never heard of? Or is our article wrong? I'll change it soon if nobody objects, since I can't find any sources that agree with our article. (though I haven't looked extensively) Staecker (talk) 15:17, 13 March 2015 (UTC)[reply]

Nah, you broke it! :-) The confusion likely comes from the fact that all the references you're looking at consider an "S" that is a single simplex. In that case, yes, the star consists of all simplices that contain "S". But now think about what happens if "S" consists of, say, two distant vertices. Your definition, that "S" must be a subset of every simplex in the star, implies that the star is empty! Sure, I guess you could define it that way... but it is far more useful (and quite conventional) if the star of two vertices is the union of the stars of each vertex. Which is why the original definition was the set of all simplices in "K" that have any faces in "S" (rather than all the faces, as your definition implies).

Yes maybe this is the confusion. But the two definitions do not agree when S is a single simplex. If S is, say, an edge in a triangulation of a surface, then the star of S (according to every source I've ever seen except this article) is the union of the two faces which have S as an edge. According to "your" definition, the star of S would additionally contain all edges and faces which intersect with any of the two vertices of S. Do you have a specific source for your definition?

If not, I would suggest that the definition say something like: the star of a single simplex s is the set of simplices having an edge in s. If S is a set of simplices, then the star of S is the union of the stars of each simplex in S. Staecker (talk) 23:17, 4 July 2015 (UTC)[reply]

Empty set

"Note that the empty set is a face of every simplex." Really? Following the link, we see that in the definition of face, it is the convex hull of a non-empty subset of the simplex's vertices. Also in the preamble at the top of this article it says "... a simplicial complex is a set composed of points, ..." - the empty set not being an element of the set of simplices which define a simplicial complex. Davyker (talk) 19:33, 5 December 2017 (UTC)[reply]

The people doing combinatorics

The article calls them both "Combinatorialists" and a bit later "Combinatorists". Is there really a difference? I don't think so, but am unsure which, if any, are correct. --CommonTypoHunter (talk) 13:08, 24 July 2018 (UTC)[reply]

The space is not specified

The article currently doesn't specify the space in which these simplices live. The simplex article suffers from the same problem, but at least it mentions $\mathbb {R} ^{n}$ in some places, so you can guess that it's meant to be implied that the simplices live in $\mathbb {R} ^{n}$ . Here, no space at all is mentioned, and one has to look at one of the linked articles to guess that $\mathbb {R} ^{n}$ is appaerently meant to be implied. (I'm not editing it myself because I'm not sure whether this is indeed implied or whether simplicial complexes can also live in other spaces (e.g. $\mathbb {C} ^{n}$ ).) Joriki (talk) 13:13, 2 December 2023 (UTC)[reply]