Talk:Cross-polytope

Can someone add a hendecacross (11-orthoplex)?

I need someone to add it because I would like to see what it looks like. LittleWhole (talk) 00:46, 29 November 2016 (UTC)

Why stop at 11? All you need to do is draw a regular 22-gon, and connect all pairs of vertices, except opposite ones. Tom Ruen (talk) 04:30, 29 November 2016 (UTC)

Tessellation

Can someone tell me how many 16-cells (how many polychlora), how many facets (how many tetrahedra), how many ridges (how many triangular faces), and how many edges meet at any given vertex in a tessellation of regular 16-cells in Euclidean 4-space? (Untitled/unsigned question from Kevin Lamoreau 9 August 2005, belatedly moved into a properly titled section so it can be answered cleanly)

Probably 16-cell honeycomb would have been a better place to ask. But in case you're still monitoring this, nearly 10 years after having asked, let's close this out. You're asking about the k-faces incident to a single vertex of {3,3,4,3}; the dual question, with the same answers, asks about the (4-k)-faces incident to a single polychoron of {3,4,3,3}. I.e., "How many vertices, edges, faces, cells does a 24-cell have?" The answers are 24, 96, 96, 24, respectively. Joule36e5 (talk) 03:56, 23 June 2015 (UTC)

sources?

I mostly added the resource template because I'd like sources for this name. Tom Ruen 00:54, 29 July 2006 (UTC)

Names in higher dimensions

Where did all these -teron, -peton, etc. names come from? Who first called them that? I'm a little suspicious because triacontaditeron, for example, gets zero Google hits. —Keenan Pepper 04:37, 4 September 2006 (UTC)

See reply under Talk:Simplex. Search for polyteron' or triacontadi. No google hits for triacontadigon either, but it is a common 32-gon. Tom Ruen 04:55, 4 September 2006 (UTC)

Actually a 32-gon is a dotriacontagon or better yet a triacontakaidigon. —Keenan Pepper 05:22, 4 September 2006 (UTC)

Higher dimensions

Without warning, the author appears to have reverted to the secondary, parenthetical definition of a cross-polytope -- i.e. "(... [S]ome authors define a cross-polytope only as the boundary of this region.)". For the formula to be valid, k is restricted to values less than n. (Even if k=n is used, since 1/(-1)! = 0, the formula gives zero rather than one needed the number of n-dimensional components in an n-cross if the primary definition is used.) For consistency with the simplex and hypercube articles, the value 1 should be in table locations where k=n. Also for consistency with other dimensions and the hypercube article, shouldn't the graph for a one-dimensional cross-polytope include its single edge (Complete_graph_K2.svg)? I will add a similar question on the simplex article. 24.6.65.170 (talk) 16:46, 13 July 2009 (UTC)

The volume formula given is that of a cross polytope of edge length sqrt(2), i.e. circumradius 1. This should be clarified as most readers assume a polytope volume formula of edge length 1 if not otherwise stated. The general volume of a regular n-cross, edge length a, is a^n*2^(n/2)/n!. JFSather (talk) 15:20, 8 May 2022 (UTC)

References

Numbers of components for arbitrary n dimensions

After a row for each dimension n and a specific count for each column by dimension k of the component, the table ends on a last row with a single formula in k and n which isn't that easy to specialize to specific values of k. IOW, an intermediary row populating each column k with a corresponding simplified formula in n would be in style.

--83.76.122.54 (talk) 11:58, 31 March 2012 (UTC)