Re: [math-fun] Embedding simplicial complexes in Euclidean space
<< Finally, turning a previous comment on its head, this result seems to give a very cheap proof of a weaker version of the manifold embedding theorem: just triangulate a d-manifold arbitrarily densely, then continuity ensures it can be embedded in (2d+1)-space! [Have I overlooked anything here?] Now then, how might the extra dimension be unloaded, I wonder ...
Only that with a little more work, it can be seen that d-manifolds embed in 2d-dimensional space. But I suspct there's a simplicial complex whose highest-dimensional simplex is d, but which doesn't embed in 2d-space. Quite possibly the "Hilbert heptahedron" consisting of every other face of an octahedron, plus three squares (topologically a projective plane) -- after each square is subdivided into two triangles to get a simplical complex -- does not embed (affinely on simplices) in 2*2 = 4-space. (But -- as I now realize -- it must embed AOS in 5-space since it has only 6 vertices and is a subcomplex of the 5-simplex.) --Dan
On 11/25/06, Daniel Asimov <dasimov@earthlink.net> wrote:
<< Finally, turning a previous comment on its head, this result seems to give a very cheap proof of a weaker version of the manifold embedding theorem: just triangulate a d-manifold arbitrarily densely, then continuity ensures it can be embedded in (2d+1)-space! [Have I overlooked anything here?] Now then, how might the extra dimension be unloaded, I wonder ...
Only that with a little more work, it can be seen that d-manifolds embed in 2d-dimensional space.
That's what I meant --- but for instance does any sufficiently dense triangulation of a given manifold also embed in 2d-space? Or does the extra dimension somehow shrivel away faster than the others, say as the square of the edge-length?
But I suspct there's a simplicial complex whose highest-dimensional simplex is d, but which doesn't embed in 2d-space.
Try the set of d-faces of a (2d+2)-simplex --- e.g. for d = 1, the 10 edges of a pentatope constitute a non-planar graph (Kuratowski). But don't ask me to prove it ... WFL
participants (2)
-
Daniel Asimov -
Fred lunnon