Recall what a simplicial complex is: First, an n-simplex delta_n is the convex hull of any n+1 points, in n-space, that lie in no hyperplane. The convex hull of any k+1 of these points is a k-simplex, and is called a k-face of delta_n. The faces of delta_n are all of its k-faces for all k = 0,1,...,n-1. A "simplical complex" is the result of gluing together a disjoint collection of simplices such that if two simplices are glued to each other, they are identified by an affine bijection from a k-face of one to a k-face of another, for some k. Thus in a simplicial complex, any two simplices that intersect do so in solely *one* common face. We shall assume that a simplicial complex K is *connected* -- i.e., for any two simplices s, t of K, there is a sequence of simplices s_0,...,s_L such that s = s_0, s_L = t, and that s_j and s_(j+1) are glued along a (now) common k-face for some k. We also assume a simplicial complex is built from only a *finite* collection of simplices. ---------------------------------------------------------------------------------------------------------- The dimension dim(K) of a simplical complex K is the highest dimension of any simplex of K. QUESTION 1: Is there a least dimension E(n) such that any simplical complex of dimension n can be *topologically* embedded in the Euclidean space of dimension f(n) ? I think this can be done by dipping the complex in goo so that it becomes a subset of a manifold (with boundary) of some dimension d >= n . . . and then use the result that all d-manifolds can be topologically embedded in 2d-dimensional Euclidean space (I think even lower for d-manifolds with nonempty boundary). If a topological embedding of a simplicial complex into Euclidean space is affine on each simplex (and hence must send each simplex to a simplex), we call the embedding *affine*. QUESTION 2: Can every simplicial complex K of dimension n be *affinely* embedded in some Euclidean space? (See Example below.) QUESTION 3: Among those simplicial complexes K (with dim(K) = n) that *do* embed affinely in some Euclidean space, is there a least dimension A(n) such that they can all be affinely embedded in the Euclidean space of dimension A(n) ? Example: Form a strip of three 2-simplices, using equilateral triangles to make an isosceles trapezoid. Identify its left edge with its right edge, abstractly, to form a topological Moebius band. Which can of course be embedded topologically in R^3. But surely this can't be embedded affinely in R^3. It probably can't be embedded affinely in any Euclidean space. (BUT, this is *not* a simplicial complex, since the two outer triangle intersect in both a 1-simplex and a separate 0-simplex.) --Dan