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I need someone to add it because I would like to see what it looks like. LittleWhole ( talk) 00:46, 29 November 2016 (UTC)
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)
I mostly added the resource template because I'd like sources for this name. Tom Ruen 00:54, 29 July 2006 (UTC)
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)
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! [1]. JFSather ( talk) 15:20, 8 May 2022 (UTC)
References
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.
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||
|
I need someone to add it because I would like to see what it looks like. LittleWhole ( talk) 00:46, 29 November 2016 (UTC)
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)
I mostly added the resource template because I'd like sources for this name. Tom Ruen 00:54, 29 July 2006 (UTC)
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)
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! [1]. JFSather ( talk) 15:20, 8 May 2022 (UTC)
References
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.