Talk:Regular 4-polytope

2005 question

I'm not sure why we have discussion of infinite regular tessellations and semiregular polytopes made from Platonic solids on this page. These topics don't really fall under the heading of convex regular 4-polytope. This content should probably be moved elsewhere, although I am not sure where. Any suggestions? -- Fropuff 01:24, 18 September 2005 (UTC)

• Agreed. I expected the new content to be split into a new article. Polytopes don't have as much agreed upon terminology as polyhedrons - where there's a whole slew of geometric categorical articles: Semiregular polyhedra Platonic solid Kepler-Poinsot solid Archimedean solid Prism (geometry) Antiprism Bipyramid Tilings of regular polygons
• I could make a new article called Semiregular 4-polytopes which would include the 6 regular polytopes as a subset. However I don't have an ennumeration of all the semiregular polytopes. I only have the subset made from regular polyhedral cells. The larger set includes semiregular polyhedral cells.
• My priority now is to add pictures for all of these shapes, but I'm still working on getting some nice symmetric projective views.
• Okay, I suppose there's no harm in a new article Semiregular 4-polytopes that includes the regular and finite tessellations as well. I'll do it.
• Tom Ruen 18:44, 18 September 2005 (UTC)

You know, I got to thinking a couple of days ago, if you truncated a pentachoron wouldn't it result in a regular 10 celled polychoron? You should also get a regular shape from truncating a 24-cell.

Hmmm... Thinking in 4D by extrapolating 2-3D is mentally taxing!

• 2D: A truncated triangle can be a hexagon (if edges cut in thirds). (3->6 sides)
• 3D: A truncation of a tetrahedron can be an octahedron (if edges fully cut). (4->8 faces)
• 4D: Perhaps the semiregular polychoron Rectified 5-cell (Dispentachoron or rectified 5-cell) can be seen as a truncated pentachoron. It has 5 octahedra and 5 tetrahedral cells. I can't see it exactly, but seems sensible!
• Tom Ruen 19:11, 3 October 2005 (UTC)

Yeah I thought about that but that would really be more like the 4 dimensional equivelent of the truncated tetrahedron. Perhaps a better example is in order, say you intersected two 5-cells and then cut off all the points. Since the vertex figure of the 5-cell is a tetrehedron you'd end up with a polychoron with 10 tetrehedral cells. -RyanAH

Those are the bitruncated 5-cell (10-cell) and bitruncated 24-cell (48-cell). They are noble, but not regular, as they do not have regular polyhedra as cells. Double sharp (talk) 07:58, 10 April 2012 (UTC)

Symmetry groups

I'm glad for improvements. Any references to new symmetry group names? Existing header link not very helpful. Tom Ruen 20:08, 19 May 2006 (UTC)

The names refer to the finite Coxeter groups (as explained in the paragraph before the table). -- Fropuff 22:34, 19 May 2006 (UTC)

Got it, I added symbols to tables in regular polychora articles as well, and some of the uniform polychora linked at Uniform_polychoron. Tom Ruen 23:07, 19 May 2006 (UTC)

Cell-centered projections

The image for the projection of the 16-cell is wrong; at least, wrong with respect to the claim that these projections are cell-centered. The cell-centered projection of a 16-cell has a cubical envelope, not an octahedral envelope. The octahedral envelope only occurs in the vertex-first projection. Also, the pentakis dodecahedron is the envelope of 600-cell under a vertex-first projection, not a cell-centered projection!—Tetracube (talk) 15:56, 13 September 2008 (UTC)

OK, I've removed "cell-centered" from the heading and indicated what type of projection is used in each column. It seems a bit cluttered to do it this way; anyone has a better idea?—Tetracube (talk) 16:01, 13 September 2008 (UTC)

Pentakis icosidodecahedron

The solid projection of the 600-cell into 3-space is not a pentakis dodecahedron. A pentakis dodecahedron has three pentagonal pyramids sharing a vertex, but the 600-cell projection has another triangle where this vertex would be, and the pentagonal pyramids are rotated to the dual configuration. Note also that this shape is not Catalan.—Tetracube (talk) 20:11, 27 October 2008 (UTC)

Confirmed, thanks! Tom Ruen (talk) 00:02, 28 October 2008 (UTC)

Thanks. I accidentally changed it back, I'll self-revert. Professor M. Fiendish, Esq. 05:35, 4 September 2009 (UTC)

Schläfli–Hess polychoron: Point of view

Can we get other point of view which not in symmetry space or image of simple rotation? Because polytoped with same face arrangement are visually indistinguishable without representation of their cells.???‽‽‽!!!?‽!?‽!?‽!? 12:36, 20 April 2013 (UTC)

Agreed it is annoying that pairs of images are indistinguishable. I'm not sure perspective is the issue, since Stella is really drawing faces which is identical. You can download Stella (software) [1] and try yourself if you like. Tom Ruen (talk) 17:22, 20 April 2013 (UTC)

Stella cost money. I was try to download it, but saw this:"YOU SHOULD READ IT BEFORE PURCHASE THIS SOFTWARE" or something like that. - User:Trilelea

Sorry, I thought you could use it for an evaluation period, but I might be wrong. Tom Ruen (talk) 22:18, 29 April 2013 (UTC)

We have image of icosahedral 120-cell, I try to create small stellated 120-cell.???‽‽‽!!!?‽!?‽!?‽!? 16:45, 5 May 2013 (UTC)

Why not scale the cells like Stella allows you to? Double sharp (talk) 15:29, 14 April 2014 (UTC)

Regular 4-polytope

There is now an article at Regular 4-polytope. It contains all the material on this page and on Schläfli–Hess polychoron (regular star polychora). Can this article now become a redirect? — Cheers, Steelpillow (Talk) 12:57, 26 December 2014 (UTC)

I redirected two articles and talks. The convex article had a minimial edit history from a previous move, while the star article has an old history, but I don't know how you could merge an article history.

All done — Cheers, Steelpillow (Talk) 15:54, 26 December 2014 (UTC)

terminology: stellation

In the section "Names", I'd like to add a note to the effect that Conway's definition of stellation is much narrower than the common usage. Kepler's stellations of the dodecahedron fit it, but in many stellations – beginning with the stella octangula, which I guess Conway would call a great octahedron – the original edges appear, if at all, only as false edges.

How would you word such a note?

By the way, are Conway's three operators commutative? For example, is a great (grand thing) always the same as a grand (great thing) ? —Tamfang (talk) 22:46, 4 February 2023 (UTC)

Useful addition to article

It would be extremely useful if a definition of "Regular star (Schläfli–Hess) 4-polytopes" were included in that section of the article.

Currently, it just mentions in passing that such "polytopes" may intersect themselves.

It would be much better if a few sentences were devoted to an exact definition of such "polytopes".

I hope someone knowledgeable about this topic fills in this missing information.