|WikiProject Mathematics||(Rated Start-class, Low-priority)|
Wow, that's eldritch. I wanna see it from the center! —Tamfang 00:26, 24 July 2006 (UTC)
icosahedra by 20 tetrahedra
The claim that "...icosahedron formed by 20 tetrahedron cells meeting at a point." is misleading. the 20 tetrahedrons, although very close, are not equalateral. the edge from the center of the icosahedron is smaller than the other edges.
- It is correct. 20 regular tetrahedra CAN meet at a point (and close) when they can be folded into the fourth dimension. Tom Ruen 22:43, 1 March 2007 (UTC)
- Think of the analogous situation in 3D with the icosahedron: there, you have 5 equilateral triangles that meet at a vertex. You cannot accomplish this in 2D, but if you fold the triangles into 3D, then it is possible for them to meet at a vertex and be equilateral at the same time. In fact, it is this folding that makes the icosahedron closed. Similarly, in 20 regular tetrahedra in 3D do not exactly meet at a point—but they can be made to do so by folding slightly into 4D. This causes a curvature in 4D which allows the polytope to be closed and bounded, since otherwise it would be a tiling, not a convex polytope. Hope this helps to clarify.—Tetracube (talk) 17:00, 28 May 2008 (UTC)
- "Me too!"... Saying "20 regular tetrahedra cannot meet at a point and form an icosahedron" is like saying "four triangles cannot meet at a point and form a square". The latter is true in a flat plane, but not true in three dimensions - nor on the surface of a sphere. Likewise, the former is true in 'flat' 3D space, but not true in flat 4D space - nor in curved (specifically, spherical) 3D space. mike40033 (talk) 06:46, 29 May 2008 (UTC)
How is it that it is possible to display the general drift of a polychoron by displaying it in only 2 dimensions? Because the 4th dimension is confusing like that. —Preceding unsigned comment added by Lighted Match (talk • contribs) 02:04, 28 May 2008 (UTC)
This new sentence has two features offensive to conservative grammarians. Due is an adjective, and adjectives apply to nouns, not verb clauses. Letting that go, since this use of due is not likely to go away, the name is not due to it but to its being, note the possessive. —Tamfang (talk) 01:14, 14 September 2008 (UTC)
- I don't have an opinion on grammar, but it shouldn't be implied that the 5-cell or 16-cell are called tetraplexes as well for having tetrahedral cells, since they are not called this. Tom Ruen (talk) 01:54, 14 September 2008 (UTC)
Nope, you could easily get a regular polytope with a larger number of facets/faces with a cross polytope of a sufficiently high dimension (a 10-dimensional cross polytope has 1024 facets). A 301-dimensional hypercube would have 602 facets, and a 600-dimensional simplex would have 601 facets. And you can just keep going.—Tetracube (talk) 23:53, 12 November 2008 (UTC)
New images for consideration
Please consider projections with a bit more color.
- Okay, what do the colors mean? —Tamfang (talk) 20:48, 2 February 2010 (UTC)
This seems wrong: In geometry, the 600-cell (or hexacosichoron). Should a 600-cell be called a hexahectachoron?
This seems wrong: In geometry, the 600-cell (or hexacosichoron)
Since icosi means 20, should hexacosi mean 6x20 = 120, then an hexacosichoron would be a 120-cell.