Talk:Bitruncation

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Uniform Polytopes (inactive)

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Factual error?

The first paragraph says:

In solid geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves. (emphasis mine)

This last sentence is wrong: the original facets of the bitruncated n-polytope become their respective duals. Of course, in polyhedra, dual faces happen to be the same as the faces themselves, but this is not true in general. Now I realize that this paragraph is talking about solid geometry, i.e., 3D geometry; that's why I'm putting off correcting it. But I do think we should make the change, since this page is linked to from the polychora pages.—Tetracube 15:26, 1 September 2006 (UTC)[reply]

Thanks for the look! I believe what I wrote is correct for n-polytope faces, although not facets. Consider boh polyhedron table bitruncation=truncated dual Uniform_polyhedron#Convex_forms_and_fundamental_vertex_arrangements, and polychoron/honeycombs like Bitruncated_cubic_honeycomb - the original faces exist as smaller copies in both dimensions. Tom Ruen 19:03, 1 September 2006 (UTC)[reply]

Here's a quick test sequence of truncations from a cube to a birectified cube (octahedron). Image:Birectified_cube_sequence.png. Rectifying reduces edges to points. Birectifying reduces faces to points. (Trirectifying reduces cells to points, etc). Truncation is "half rectified". Bitruncation is between rectified and birecified. Tritruncated is between birectified and trirectified. Do you disagree with this? Tom Ruen 19:35, 1 September 2006 (UTC)[reply]

Ack. Not another instance of confusion between face and facet. Makes me want to avoid using face at all, for being overloaded with incompatible meanings. Anyways... for polyhedra, it happens to be true that the original faces become smaller versions of themselves—but actually, if you look at the way they're being truncated, they're actually becoming their duals, except that the dual of a polygon is itself, therefore it's equivalent to a smaller version of themselves. In higher dimensions, the dual of a facet is in general not the same as itself. But if you're talking about 2-faces, then I suppose the statement is correct, but I still think that it's more accurate to express it in terms of duals, because orientation-wise that's more accurate. (Look at the orientation of the squares in your diagram: they are actually dual to the faces of the original cube. Ditto with the triangular faces of the octahedron.)—Tetracube 17:31, 2 September 2006 (UTC)[reply]

You're right about about polygon faces becoming their own duals. I figured best to keep it simple and neglect that technical difference, although there is a re-orientation. I'm happy if you want to edit any of these operation articles for clarity and correctness. Much improvement possible I'm SURE. I added now mainly because cool to show self-duals bitruncated {p,q,p} form cell-uniform polytopes/honeycombs, and a first step into "hyperbolic honeycombs". Tom Ruen 23:11, 2 September 2006 (UTC)[reply]