Talk:Regular polyhedron

New article

I hope I haven't upset anyone's careful organization of articles regarding polyhedra by replacing the former redirect for this page with a brief descriptive article instead. It seemed like a good idea to me, but I know that Wikipedians have a highly variable sense of "rightness". If I've undone someone else's good work I apologize. andersonpd 01:15, 3 August 2006 (UTC)

I think you are absolutely right. Just moved some more stuff over. Way still to go! Steelpillow 22:14, 1 February 2007 (UTC)

Explanation of reversion

I have reverted the recent changes by User:Ppatel43026 because:

1. The meaning of face-transitive is that all faces are identical - there is no need to repeat this definition.
2. Information on Euler's formula belongs in the planar graph article, which has a section on Euler's formula.
3. The style is unencyclopedic (e.g. "Euler was a busy man"); there are numerous digressions; references are incomplete.

Gandalf61 09:21, 18 April 2007 (UTC)

Enumeration of regular polyhedra

User:4 has just added some "topological" examples to the list of regular polyhedra. This makes me wonder, what should be the scope of this article? If we address these hyperbolics then what about the elliptic, the infinite skew, the multiply-wrapped and so on? Or, should we stick with the nine elementary examples that are regular in any locally Euclidean space? I think that the History section could usefully summarise/link to articles on the more exotic types, but I do feel that there are benefits in the main article to the previous treatment of only discussing the nine in any detail. — Cheers, Steelpillow (Talk) 18:11, 30 August 2011 (UTC)

I'd lead to keeping the basis 5+4, and these so-called topological ones seem like trouble - mixing pentagons and pentagrams. I have thought to add the regular skew polyhedra to list of regular polytopes, but still sorting out the finite ones! SockPuppetForTomruen (talk) 02:16, 31 August 2011 (UTC)

The excavated dodecahedron is weird as a polyhedron, faces are hexagonal but connected in a different order. . So it does not have regular star polygon faces, but they are vertex-transitive. SockPuppetForTomruen (talk) 19:59, 31 August 2011 (UTC)

I expanded the section, showing faces, and added finite skew regular polyhedra. I wondered the regular tilings as infinite polyhedra deserve a section too, but added an intro statement at least. The whole thing needs a look, to see how the following sections are limited by each extended solution class. Maybe, if these cause too many conflicts in the wider sections, these should instead be added as subsections to Regular_polyhedron#Further_generalisations? SockPuppetForTomruen (talk) 00:49, 1 September 2011 (UTC)

Tom, not quite sure how your addition of yet more examples relates to your comment that "I'd lead to keeping the basis 5+4". It appears to me that you have just done the exact opposite, or do I misunderstand? BTW, these "topological" examples are all about regularity of abstract structure - they do not have geometric regularity. Problem with enumeration is, there are shedloads of the beasts, for example User:4 has not mentioned any of the elliptic examples such as the hemicube, hemidodecahedron, hemicosahedron, etc. Also the theory is quite extensive and esoteric - see for example McMullen & Schulte's book Abstract regular polytopes. So I am really in favour of returning to my original approach: cut back the main article to where it was and, in the Further generalisations section, stick to short remarks and reference out with links to the more specialised articles. Otherwise this article will just sprawl and sprawl, losing its elementary focus in the process. Any objections? — Cheers, Steelpillow (Talk) 10:56, 1 September 2011 (UTC)

Yes, I'm a contradiction, rethought and attempted to expand to see how it looked, maybe because I was excited about the skew forms. I tried moving the new sections inside of Further generalisations. Seems good to me, although I agree these sections could be cut down further. SockPuppetForTomruen (talk) 17:55, 1 September 2011 (UTC)

That's working for me. I have tidied up a bit and abandoned the description "topological polyhedron" in favour of "abstract polyhedron". The latter term has a precise mathematical meaning, whereas topologists talk about various incompatible types of object as "polyhedra". Once fleshed out, the "Abstract regular polytope" section can be either moved to its own article or merged into the abstract polytope article. — Cheers, Steelpillow (Talk) 20:42, 1 September 2011 (UTC)

Good, abstract is much better! I merged the infinite/finite skew polyhedra sections. Should we include the regular projective polyhedron: hemicube, hemi-octahedron, hemi-dodecahedron, hemi-icosahedron? Also there's the hosohedron {2,n}, and dihedron {n,2} which are degenerate polyhedra, except as spherical tilings. I guess these are all covered under abstract regular polytope, as a section of abstract polytope. SockPuppetForTomruen (talk) 21:55, 1 September 2011 (UTC)

The hosohedra and dihedra are best introduced after spherical polyhedra, as they can be realised in this form. I'm not sure whether to discuss the regular spherical polyhedra under History or Further Generalisations. — Cheers, Steelpillow (Talk) 21:42, 2 September 2011 (UTC)

I've placed them under Further Generalisations for now, because we have to include them somewhere, and there's not much stuff about spherical polyhedra in the History section. Yes, there are the Scottish balls, but those are doubtful - many of them aren't Platonic solids, and the most famous set of Platonic solids among them has a hoaxed octahedron and icosahedron. Double sharp (talk) 06:51, 31 May 2012 (UTC)

(Another paper about these abstract regular polyhedra of index greater than unity: https://link.springer.com/article/10.1007/BF03322410) Double sharp (talk) 05:16, 18 September 2018 (UTC)

flag-transitive.

Just by its definition, flag transititivity implies edge/vertex/face transitivity. But the converse is at least not obvious: flag transitivity means e/v/f-transitivity with one simultaneous mapping. --Boobarkee (talk) 23:38, 6 December 2011 (UTC)

That is correct. However I think that Cromwell (Polyhedra, CUP, 1997) seems to have missed it. If that is the case, then until one of us can find a cast-iron reference, correcting the article would fly in the face of a widely-known authority. :roll: — Cheers, Steelpillow (Talk) 19:52, 7 December 2011 (UTC)

I'm not even sure it's right, imagine the quotient of the square tiling by the subgroup generated by the translations (+2,+2) and (+3,-3), where the edge length is one unit, and the edges are parallel to the x- and y-axes. This is basically a tiling of the torus with squares in a "diamond" orientation. The symmetry group of this abstract polytope acts transitively on the vertices, on the edges, and on the faces of the polytope, but not on its flags (for any face-vertex pair, it could be aligned the "short" way or the "long" way). Also, its facets and vertex figures are all squares. Now maybe there's no way to make any similar polytope regular (in the sense defined in this article) in Euclidean space, but since this article covers abstract polytopes as well, maybe the more general "flag-transitivity" definition should be used instead of the current "congruent regular facets and congruent vertex figures" one. 173.227.48.5 (talk) 17:46, 15 April 2013 (UTC)

Sorry, my group theoretic skills are poor. Can you explain what you mean by "the quotient of the square tiling by the subgroup generated by the translations (+2,+2) and (+3,-3),"? — Cheers, Steelpillow (Talk) 18:59, 15 April 2013 (UTC)

Identify two vertices/faces/edges if one is two up and two right of the other, or three down and three right.

Another way to construct it is to take a 2 by 3 square tiling of the torus in the "natural" orientation (edges make circles running around the torus for curves of constant theta or constant phi with the most common parametrization) and rectify it to get them in a "diamond" orientation (put vertices at the center of each edge of the original polyhedron and connect them by edges to the four new vertices that are on edges of the old polyhedron that share a face and vertex with the edge of the old polyhedron). You get 12 square faces, 12 vertices, and 24 edges. There are 96 flags in two orbits of 48 each. The example works for any two unequal integers greater than or equal to two.

Picture it like this:

/ \ / \ / \

\ / \ / \ /

/ \ / \ / \

\ / \ / \ /

Where if you go off the right, you come back on the left side at the same height, and similarly with the top and bottom. Some of the square faces are split in the diagram above so they appear on both sides. One of the faces is split into quarters (one quarter of the face in each corner).

173.227.48.5 (talk) 20:05, 15 April 2013 (UTC)

Well, I hope this isn't controversial, but I'll rephrase the intro slightly to be more general. 173.227.48.5 (talk) 16:57, 16 April 2013 (UTC)

Ah, but are these not rhombs 4 wide x 6 high, and not squares? If you squash the dimensions to make them squares, then the tiling becomes transitive on its flags. Or, if you are concerned with abstract form, then the +2/+2 and +3/-3 metrics have no relevance? — Cheers, Steelpillow (Talk) 19:02, 16 April 2013 (UTC)

I'm looking at it as an abstract polytope. But you can tile a torus with squares this way too, to make it more concrete (if you do this you need to take an appropriate metric on the torus, of course). They're only rhombi in the diagram because it's ASCII (or Unicode or whatever) art. The flags are in two orbits. For any vertex-face pair (the edge doesn't affect the orbit) it's either oriented the "long" way or the "short" way. If you "hop" from each face to the one that meets it across the vertex, then take the opposite vertex of that face for the next "hop", it will take either two or three "hops" to get back to where you started. There is no symmetry that maps a "two-hop" face-vertex pair to a "three-hop" face-vertex pair. There are reflection symmetries across the diagonals of the faces, but not along the edges or across the lines connecting midpoints of opposite edges (the faces have reflection symmetries across those midlines, but the overall polytope doesn't). The (+2,+2) and (+3,-3) translations are the "circumferences" of the torus, the edges are length 1 and parallel to either the x- or y-axis. These measurements only matter when looking at it as a tiling of a torus. I think the diagram above is the best way to express it, the slashes are the edges. Just glue the left to the right and the top to the bottom in a way that makes a torus. This is the same person as the other IP, by the way, I'm on a different device. 166.137.191.46 (talk) 19:18, 16 April 2013 (UTC)

Just to try to clarify the quotient construction, you start with a square tiling, then draw a diagonal across two squares, then an orthogonal diagonal across three squares. These are two of the edges of a rectangle. Complete the rectangle, and cut it out to get a rectangular strip, which you then glue to itself to make a torus. The vertices and edges are carried over from the tiling.

Another construction is start with the rectangle with vertices (0,0) (0,2) (3,0) and (3,2). Cut it out and glue it together as a torus, but keeping the coordinate grid as a way of naming points. Now put vertices on each point (x,y) where x is 0, 1, or 2, and y is 0 or 2, OR where x is 1/2, 3/2, or 5/2, and y is 1/2, or 3/2. Now draw edges between vertices where the coordinates each differ by 1/2 (considering 0 to differ from 5/2 by 1/2 in the x direction and from 3/2 by 1/2 in the y direction, since we have identified the points (0,0) (0,2) (3,0) and (3,2) in making the torus). The faces are the square regions bounded by the edges. 173.227.48.5 (talk) 21:45, 16 April 2013 (UTC)

Ah, I get the construction now. The abstract symmetries of this kind of object are new to me. Thank you for your time and trouble in explaining it all so carefully and clearly, I really appreciate it. — Cheers, Steelpillow (Talk) 08:46, 17 April 2013 (UTC)

regular star polyhedra

A regular polyhedron is said to be vertex transitive, but the regular star polyhedra aren't (the number of edges meeting in a vertex is not independent of the vertex), so there is a notational problem there. — Preceding unsigned comment added by 161.116.84.155 (talk) 15:43, 15 March 2012 (UTC)

Can you give an example of a regular star polyhedron where the number of edges meeting in a vertex is dependent on the vertex ? Are you sure you are only considering true vertices, and not including the "false" vertices where sides intersect but do not form a vertex ? Gandalf61 (talk) 15:49, 15 March 2012 (UTC)

I replied here Talk:Kepler–Poinsot_polyhedron#Regular polyhedra, explaining the "false vertices" also. Tom Ruen (talk) 19:24, 15 March 2012 (UTC)

Inconsistent treatment of the regular polyhedra

There is currently no consistency across our articles in the treatment of the regular polyhedra. For example:

• We are inconsistent over whether a star polyhedron can be a "solid". For example regular solid redirects to Platonic solid, while we sometimes describe the regular star polyhedra as solids.
• There is no consistent recognition that the regular varieties of the dodecahedron and the icosahedron are not all convex but include the four regular star solids.
• we have a separate article for the regular icosahedron but not the regular dodecahedron which currently redirects to dodecahedron.

The first two of these are easy enough to fix, we just appeal to reliable sources (WP:RS). There we find:

• The Kepler-Poinsot figures are consistently described as "regular" and included in full enumerations. Where the author omits them, it is only because the author's focus is on convex figures.
• Where the author treats all polyhedra as solid, stars are also described as "solid" and, naturally, vice versa. Some authors get muddled, but nowhere does any author consistently treat the convex ones as solids but the star ones as surfaces.

I'd suggest that the third issue is best tackled with a top-down process, in accordance with WP:PAGEDECIDE. We begin with the present article on Regular polyhedron and add all we reasonably can to it. If any topic area cannot be squeezed in fully, then we create a separate article for it, otherwise we just create the topic page as a redirect. And so on with any sub-topics. This may spoil somebody's fun maintaining unnecessary pages, but that's Wikipedia's policies and guidelines WP:PG for you.

Any comments? — Cheers, Steelpillow (Talk) 11:51, 3 January 2015 (UTC)

On #1, we should certainly go with the sources and allow all polyhedra, convex, concave, or star, to be either solids or surfaces.

I'm not so sure about #2 as it applies to article titles. While the unqualified "icosahedron" alone can often mean "the convex regular icosahedron" (the context will certainly be specified first), I'm not sure it can ever mean "great icosahedron". In fact, I'm not even sure the semi-qualified "regular icosahedron" is most usually taken to mean both of them: but how do we check this? (It's not very simple to check, as we'd have to take into account the presence or absence of contextual cues, which may be several pages away...)

Agreed on #3. In practice, though, the five Platonic solids should definitely have their own articles. They would naturally be renamed regular tetrahedron, cube, regular octahedron, (convex) regular dodecahedron, and (convex) regular icosahedron: we'll need to do some source-checking to see if "convex" is necessary for the last two. Not as sure if the Kepler-Poinsots deserve their own articles: there's certainly enough to cover all of them together, but I'm not sure if there's enough to cover them one at a time. The generalized examples probably would do better merged here, or perhaps without individual articles but merged into individual classes (so one article about regular skew polyhedra, one about abstract regulars, one about projective regulars, etc., but individual abstract or projective regulars don't get articles). Double sharp (talk) 15:08, 3 January 2015 (UTC)

Regular Tilings

Planar apeirohedra aka regular tilings should go on the page, especially since Regular skew apeirohedra are already here. — Preceding unsigned comment added by 2A01:E35:2FB0:2D10:A9F0:4CDC:F195:40A1 (talk) 11:07, 18 November 2020 (UTC)

• Why has nobody mentioned the dual of the petrial halved mucube???????it's a reference don't kill me casualdejekyll (talk) 02:44, 22 June 2021 (UTC)

  Do you count 48 regular polyhedra on this page? Because I sure do not. MrMasterGamer0 (talk) 01:32, 25 October 2022 (UTC)

jpg images

@Watchduck: I was wondering why the SVG images of the Platonic solids were inferior to the JPEG. I didn't see anything in WP:IUP#FORMAT that would seem to be the reason. If there's something wrong with the SVGs, I might try to fix them. AquitaneHungerForce (talk) 13:15, 15 January 2024 (UTC)

To see which image is superior, your have to look at the images – not at some page about file formats in general. Files like File:Tetrahedron.jpg have 3D edges, while those like File:Tetrahedron.svg have flat lines as edges. The JPGs are good quality, and the SVGs are poor imitations. No file format has any inherent advantage for illustrating an article. So the images with better quality should be used. --Watchduck (quack) 17:52, 16 February 2024 (UTC)

Thanks. That's hard too see. I will make versions of the SVGs with 3D edges and adjust them in the article. AquitaneHungerForce (talk) 17:57, 16 February 2024 (UTC)

That is not what SVG is for, and the result will be be inferior to a raster graphic. Why would you want to replace the JPGs, when they are good? Did anyone convince you, that SVG is somehow the best file format? I suggest not to believe that kind of thing. --Watchduck (quack) 18:58, 16 February 2024 (UTC)

I'd like to replace them because they look bad. They have a annoying white square around them and (on my screen) visible artifacting. At least in my opinion the SVGs already look better, but it would be nice to have an image that looks nice for everyone. Is there something else wrong with an SVG that would make it inferior to a high quality raster? AquitaneHungerForce (talk) 19:53, 16 February 2024 (UTC)

Cube with 3D edges

Here's the 3D version of the cube. If you have any suggested changes I can adjust it and then complete the set. AquitaneHungerForce (talk) 20:54, 16 February 2024 (UTC)

Yes, the white background is bad. Raytraced images should be PNG, not JPG. Your SVG looks better than I expected. I suggest, that you overwrite the current SVGs with your improved versions. Overwriting is sometimes abused, but this is exactly what it is for. (BTW, maybe you could upload a little screencast as a tutorial? The tetrahedron would be ideal for that.) --Watchduck (quack) 21:21, 16 February 2024 (UTC)

Here's a screencast: File:3D Edges in Inkscape demonstration.webm. However I ended up not being satisfied with the tetrahedron in the video so I redid that from scratch. I'll finish and upload when I have some more time. AquitaneHungerForce (talk) 22:35, 24 February 2024 (UTC)