Talk:Semiregular polyhedron

This article contains a summary description of convex semiregular polyhedra. Semiregular polyhedra are a subset of Uniform polyhedrons which include concave forms. I find it useful to keep this more classical "convex" only article.

I admit this article could use some work - cutting and adding.

Tom Ruen 06:28, 16 October 2005 (UTC)

Reworked

I made a bold reduction of this article. Since most of the content was repeated elsewhere, I figured a short opening with links was good. The unique content, enumerating existence by vertex figure was retained, even if I'm unsure if it belongs here (Since it applies to regular polyhedra, and uniform tilings as well.)

On the bad side, I also removed most of the picture! But easy to add back from either polyhedron or uniform polyhedron!

One 'alternative approach is to move symmetry information to here from the section Uniform_polyhedron#Convex_forms_and_fundamental_vertex_arrangments. This is more interesting than the vertex figure enumerations!

Any other ideas are welcome, including votes to delete (or move existence) section.

Tom Ruen 09:52, 14 March 2006 (UTC)

I agree there's a lot of unnecessary repetition in polytope articles. Kudos for taking that bull by the horns. —Tamfang 23:42, 14 March 2006 (UTC)

Article title

Is there any objection to moving this article (back) to Semiregular polyhedron? I see it was moved earlier this year by MathBot, but it now violates the "singular title" guideline and is now the only polyhedron article with "-hedra" in the title rather than "-hedron". If there was no express purpose in changing the name I suggest changing it back. Any objections?

I'll wait a few days and if no one objects I will make the change. andersonpd 01:22, 3 August 2006 (UTC)

Support. Tom Ruen 02:14, 3 August 2006 (UTC)

Yes, obviously. Tamfang 04:26, 3 August 2006 (UTC)

Sure, let me make the change, then tell me about the nine-bazillion links to "Semiregular polyhedra". Stupid prisms and antiprisms! But seriously, I apparently misrepresented the move by MathBot -- that move was from "Semi-regular polyhedra" to "Semiregular polyhedra", not from "Semiregular polyhedron". My bad. andersonpd 05:32, 4 August 2006 (UTC)

Appreciate it, and now I know better to singular naming, although I think list of uniform polyhedra ought to stay plural, right?! Tom Ruen 05:56, 4 August 2006 (UTC)

What you mean? "List" are singular. —Tamfang 06:10, 4 August 2006 (UTC)

You're probably making funs of me. Tom Ruen 07:00, 4 August 2006 (UTC)

Definition

I find the definition given here confusing, especially in light of comparison with the text of Uniform polyhedron and Regular polyhedron. Here it says

A semiregular polyhedron is a geometric shape constructed from a finite number of regular polygon faces with every face edge shared by one other face, and with every vertex containing the same sequence of faces, and, moreover, for every two vertices there is an isometry mapping one into the other.

Nothing is said about convexivity, but it goes on to enumerate only convex polyhedra and says

These 4 sets compose the convex polyhedra, along with a set of 53 nonconvex forms compose the larger set of uniform polyhedra.

(which aside from anything else is ungrammatical). The implication seems to be that convex polyhedra satisfying the above requirements are semiregular, and these plus the nonconvex polyhedra satisfying those requirements are uniform. (Convex) regular polyhedra are included among the semiregular polyhedra.

But in Uniform polyhedron it says

A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. Furthermore, for every two vertices there is an isometry mapping one into the other, so there is a high degree of reflectional and rotational symmetry. Uniform polyhedra are regular or semi-regular but the faces and vertices need not be convex.

The definition seems to be the same as for semiregular polyhedra, other than that nothing is said about how many faces may share an edge. The implication seems to be that semiregular polyhedra are those (convex or not) which are uniform but not regular although this is not stated categorically.

Meanwhile in Regular polyhedron, convexivity is not required, and they are not said to be a subset of the semiregular polyhedra.

This kind of confusion seems not limited to Wikipedia. MathWorld says

A polyhedron or plane tessellation is called semiregular if its faces are all regular polygons and its corners are alike (Walsh 1972; Coxeter 1973, pp. 4 and 58; Holden 1991, p. 41). The usual name for a semiregular polyhedron is an Archimedean solid, of which there are exactly 13. In addition, a prism or antiprism is considered semiregular if all its faces are regular polygons.

which first suggests the term is synonymous with Archimedean solid, and then undermines that suggestion by adding in prisms and antiprisms, but not nonconvex polyhedra. And here it says

The uniform polyhedra are polyhedra with identical polyhedron vertices. Badoureau discovered 37 nonconvex uniform polyhedra in the late nineteenth century, many previously unknown (Wenninger 1983, p. 55). The uniform polyhedra include the Platonic solids and Kepler-Poinsot solids.

Coxeter et al. (1954) conjectured that there are 75 such polyhedra in which only two faces are allowed to meet at an polyhedron edge, and this was subsequently proven. In addition, there are two polyhedra in which four faces meet at an edge, the great complex icosidodecahedron and small complex icosidodecahedron (both of which are compounds of two uniform polyhedra). Finally, the five pentagonal prisms can also be considered uniform polyhedra.

which seems utterly garbled -- nothing is said about regularity of faces, and the pentagonal prisms, but no other prisms, are thrown in without explanation.

Many writers seem to regard "semiregular polyhedra" as a synonym for "Archimedean solid". Encyclopedia Britannica's article on Archimedes (subscription required) refers to

the 13 semiregular (Archimedean) polyhedra (those bodies bounded by regular polygons, not necessarily all of the same type, that can be inscribed in a sphere)

with no other references to semiregular or uniform polyhedra that I can find. Of course the above is wrong: the Archimedean polyhedra are not the only regular-faced polyhedra (even limiting to convex) that can be inscribed in a sphere.

In "Uniform Polyhedra" (JSTOR subscription required) (Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 246, No. 916. (May 13, 1954), pp. 401-450) Coxeter et al. say

A polyhedron is said to be uniform if its faces are regular while its vertices are all alike. By this we mean that one vertex can be transformed into any other by a symmetry operation.

and then, rather surprisingly, semiregular polyhedra seem to be defined as those with Wythoff symbol of the form p q | r, encompassing prisms but not antiprisms, only 7 of the Archimedean solids, and numerous nonconvex polyhedra. I don't think I've seen that definition used anywhere else, but one can't disregard Coxeter on the subject of polyhedra, can one?

In summary, from what I've read, it looks as though "semiregular polyhedra" does not have a commonly accepted definition, unless it's that it's a synonym for Archimedean solid. If I've gotten the wrong impression I'd be happy to learn otherwise, but I'm thinking this article at the very least should (1) clarify its definition and be consistent with the Uniform polyhedra article and (2) acknowledge other definitions. -- Rsholmes 15:22, 14 December 2006 (UTC)

Yet another definition, from Cromwell, Peter R. (1997). Polyhedra. Cambridge: Cambridge University Press. p. 149.: "in current terminology, 'semiregular polyhedra' refers to the Archimedean and Catalan solids" (! -- emphasis mine). -- Rsholmes 13:57, 16 December 2006 (UTC)

I appreciate this investigation, wish I could add to it, but I'll see what I can find too, but not quickly I'm afraid. Tom Ruen 20:12, 16 December 2006 (UTC)

Rewrite

I have boldly rewritten this article from scratch to reflect the apparent lack of an agreed definition in the literature. -- Rsholmes 20:55, 19 December 2006 (UTC)

I can hold appreciation for boldness even if annoying. :) I rewrote the intro. I think it is key to say these polyhedra have regular faces and are vertex uniform, whatever Cromwell says about the Catalan solids. I don't understand the Coxeter reference, although I have the paper and can look at it. Linking to uniform polyhedron seems vital as well. Tom Ruen 22:52, 19 December 2006 (UTC)

What reasons do you have for disregarding Cromwell's assertion? He is so far as I am aware a respected and knowledgeable geometer. If we are to dismiss what he says (and I do find his definition odd, but I have no good ammunition against it), we need better authority than "I think..." And absent such authority, we cannot assert uniformity. -- Rsholmes 03:09, 20 December 2006 (UTC)

Reverting without discussion isn't very nice. There's some SMALL differences in definitions, but the definition I gave IS the definition of the Archimedean solid. The differences are whether the prisms, or antiprisms are included. Tom Ruen 23:30, 19 December 2006 (UTC)

The definition you give is not that of an Archimedean solid. The latter, in addition to two or more regular polygons as faces, requires convexivity and higher than dihedral symmetry. But that's beside the point, because this isn't the Archimedean solid article. It is the semiregular polyhedra article, and there does not seem to be agreement that "semiregular polyhedron" is a synonym for "Archimedean solid", or for "Archimedean solids plus prisms plus antiprisms", or for "Archimedean solids plus prisms plus antiprisms plus nonconvex, nonregular uniform polyhedra", or for "Archimedean solids plus Catalan solids", or for "uniform polyhedra with Wythoff symbol of the form p q | r". This is the point: that it is incorrect to lead off this article with "semiregular polyhedra are <definition>", because there isn't an agreed upon definition. At least not so far as I have been able to find; and the differences between the above definitions are not small. If you believe the disagreements I've found are just aberrations and 98% of mathematicians do agree on a definition, then there need to be some sources cited to support that proposition. The sources I've found simply disagree. -- Rsholmes 03:02, 20 December 2006 (UTC)

I removed this example since it stands alone without reasonable explanation. I've looked at the paper. The term is ONLY used as a section header and doesn't have any explanation, so I imagine Coxeter just wanted a title, and his reasoning is lost. Tom Ruen 00:43, 20 December 2006 (UTC)

• In a 1954 paper, H. S. M. Coxeter uses the term "semiregular polyhedra" to classify uniform polyhedra with Wythoff symbol of the form p q | r, a definition encompassing only six of the Archimedean solids, as well as the regular prisms (but not the regular antiprisms) and numerous nonconvex solids.

Yes, it is used as a section header; and looking at the paper as a whole, it's clear that the headers of these sections are Coxeter's classification (at least for purposes of that paper). The polyhedra discussed in that section are the ones he is classifying as semiregular. The fact that he doesn't explain his choice of terminology is once again beside the point, which is that he (probably the 20th century's foremost authority on polyhedra) chose, at least on this occasion, to use the term "semiregular polyhedra" to mean something other than what other people have used it to mean. This is significant in demonstrating lack of agreement on the definition, which is why I included it. Sweeping it under the rug doesn't solve anything. I am putting this back. -- Rsholmes 03:02, 20 December 2006 (UTC)

Ah... having said all that, now I notice you listed some "external references" at the end. I'll need to look at them; perhaps they answer my objections to a single definition. Meanwhile I've done some rearranging and copy editing. -- Rsholmes 03:23, 20 December 2006 (UTC)

OK, I've looked at them. My comments:

George Hart: Archimedean Semi-regular Polyhedra: The term "semi-regular" appears only in the page title, nowhere in the text. The implication is that the Archimedean solids are semiregular, but no one seems to disagree on that! There is discussion of the Archimedeans + prisms + antiprisms, but no name is assigned to this category.

David Darling: semi-regular polyhedron: Explicitly includes the nonconvex uniform polyhedra (it doesn't even exclude the Kepler-Poinsot solids, though I suspect they aren't intended to be included).

polyhedra.mathmos.net: Semi-Regular Polyhedron: I find this a poor reference. It's full of typos and punctuation errors, for one thing, and for another its definition is inconsistent with its enumeration: the definition does not require convexivity or vertex transitivity, yet the enumeration excludes nonconvex solids and the elongated square gyrobicupola. Then it calls the nonconvex uniform polyhedra "stellated Archimedeans", which is incorrect, isn't it? Frankly, I'd recommend deleting this link.

Encyclopaedia of Mathematics: Semi-regular polyhedra, uniform polyhedra, Archimedean solids: What a very strange article -- it says "Semi-regular polyhedra" at the top, and nowhere else; the article defines and enumerates the convex uniform polyhedra.

So the first reference implies support for Archimedeans+prisms+antiprisms, but doesn't state such a definition explicitly; the second disagrees by including nonconvex polyhedra; the third is a mess; and the fourth doesn't actually discuss semiregular polyhedra at all. Altogether I don't think they settle much of anything. Though the Coxeter papers cited in the last one would be worth looking at. -- Rsholmes 03:59, 20 December 2006 (UTC)

This rather lunacy to call small differences as central to an article. EVERYONE agrees Archimedean solids are semiregular. NO ONE is excluding prisms and antiprisms except through laziness. Including Archimedean duals as semiregular is just silly. And however much I respect Coxeter, I'LL WAIT until we actually have information to defend his meaning. Tom Ruen 04:12, 20 December 2006 (UTC)

I've remove this again, until we have SOME EXPLANATION FOR IT. Tom Ruen 04:13, 20 December 2006 (UTC)

• In a 1954 paper, H. S. M. Coxeter uses the term "semiregular polyhedra" to classify uniform polyhedra with Wythoff symbol of the form p q | r, a definition encompassing only six of the Archimedean solids, as well as the regular prisms (but not the regular antiprisms) and numerous nonconvex solids.

Coxeter is dead; I don't think an explanation will be forthcoming. But again, you miss the point, which is that the term has been used for many different meanings, and this illustrates that.

I really, really hate edit wars. But this is going back in, because it is relevant and Coxeter is authoritative. Feel free to add words to this passage if you feel it cannot stand as it is, but please do not remove it. Removal of relevant facts simply because you do not understand them is not proper Wikipedia etiquette. -- Rsholmes 04:28, 20 December 2006 (UTC)

You're the one who started the reverting. I don't believe MOVING a statement from an article to a talk page until its context can be determined is improper wikipedia etiquette. Tom Ruen 04:35, 20 December 2006 (UTC)

Versi-regular polyhedra

Mathworld references Versi-regular polyhedra (from Norman Johnson) as the subclass of uniform polyhedra with (p q | r) or (p.h.q.h vertex figures). [1] Tom Ruen 06:04, 20 December 2006 (UTC)

Johnson, N. W. Uniform Polytopes. Cambridge, England: Cambridge University Press, 2000.

Johnson's book, and the accompanying terminology, remain unpublished. Best to ignore this term for now. Steelpillow 14:31, 26 December 2006 (UTC)

References

Catalan, Cromwell, and Coxeter

Results of some further digging:

I wrote to Cromwell to ask about the surprising "Archimedeans and their duals" definition. In his reply he made two main points. First, Catalan himself referred to the Archimedeans, prisms, and antiprisms as "semiregular of the first kind" and their duals as "semiregular of the second kind". Cromwell acknowledges that this usage is "not often" seen today, though some do use it. Secondly, he says he avoids using the term "semiregular" (the footnote I quoted is apparently its only use in his book) precisely because it is defined so differently by different people.

Cromwell does use the term elsewhere. On Page 80 he describes the thirteen Archimedeans as semiregular. On Pages 367 ff. he discusses the Catalans and their relationship to the 'semiregular' Archimedeans. By implication he treats the Catalans as not semiregular. So Cromwell is effectively contradicting his own usage. Steelpillow 22:21, 21 December 2006 (UTC)

One person who apparently did use Catalan's terminology, or at least something like it, in 1940, was Coxeter. Citation: Coxeter, H.S.M. (1940). "Regular and Semi-Regular Polytopes I". Math. Zeit. 46: 380–407. He writes, "Kepler showed that the uniform polyhedra consist of (a) the five regular solids, (b) the thirteen Archimedean solids (or semi-regular solids of the first kind), (c) the prism..., (d) the antiprism..." (In this article Coxeter is explicitly considering only convex polyhedra.) Clearly it was not due to "laziness" that Coxeter here did not include the prisms and antiprisms as semiregular solids.

>>But he used the term "semi-regular solids of the first kind"! Did he envisage s-r-s "of the second kind"? <<

Coxeter may have shared Cromwell's aversion to the term "semiregular"; despite its presence in the title of this paper (and its two sequels) the above is the only use of the word I can find in their texts. Nor does he make much use of it in Regular Polytopes (citation: Coxeter, H.S.M. (1948). Regular Polytopes. London: Methuen. p. 162.) in which he quotes Gosset's definition of a semi-regular polytope as follows: "having regular cells (of several kinds) and a symmetry group that is transitive on the vertices". In three dimensions that would mean two or more kinds of regular polygonal faces plus vertex transitivity. On the next page he speaks of polytopes that are "not semi-regular in Gosset's sense" (emphasis mine). On page 210 he refers to "Elte's rather artificial definition of 'semi-regular'" (but he doesn't seem to quote it).

Finally, here is a quote confirming that the idea of Archimedean duals as semiregular polyhedra was not dead as of 15 years ago. Citation: Hazewinkel, M. (ed.) (1991). Encyclopedia of Mathematics. Vol. 7. Dordrecht: Kluwer Academic Publishers. p. 226. {{cite book}}: |first= has generic name (help):

"Isogons and isohedra are convex polygons all polyhedral angles of which are equal (isogons) or all faces are equal (isohedra); moreover, the group of rotations (with reflections) of an isogon (isohedron) around the centre of gravity maps any vertex (face) into any other vertex (face). Each isohedron can be realized so that all its faces are regular polygons. The polyhedra thus obtained are called semi-regular polyhedra (Archimedean solids and their duals)."

I have noticed that people are often a little woolly on whether the semiregular solids include the quasiregular ones, and whether the "Archimedean" solids include the prisms and antiprisms. People tend to use whatever form of words is handy even if it contradicts their usage elsewhere in the same work. Coxeter and Cromwell are both guilty of this. It looks to me as if we can add the current question, as to whether the Catalans are semiregular, to this list of confusions - at least for Cromwell, as I noted above.Steelpillow 22:21, 21 December 2006 (UTC)

So odd as it may seem, apparently Cromwell is correct in saying some mathematicians regard semiregular polyhedra as including the Catalan solids. -- Rsholmes 20:00, 21 December 2006 (UTC)

As for the Wiki page, I think that tbe bulleted list approach is not very helpful. I think it would be better if a basic definition were given pride of place, viz. polyhedra having regular faces and transitivity on their vertices. The various options and confusions should be dealt with afterwards in text paragraphs. Just my two penn'orth.Steelpillow 22:21, 21 December 2006 (UTC)

More soul-searching

The French Wiki says, "Un polyèdre est semi-régulier si ses faces sont constituées de plusieurs sortes de polygones réguliers, et que tous ses sommets sont identiques. Ainsi sont par exemple les solides archimédéens, les prismes et les antiprismes réguliers. La terminologie ne paraît pas tout à fait arrêtée. On parle parfois de solides semi-réguliers de la première espèce pour désigner ceux de ces solides qui sont convexes, et de solides uniformes pour le cas général. Les polyèdres de Catalan ne sont pas semi-réguliers, mais ont des faces identiques et des sommets réguliers. On dit parfois de tels polyèdres qu'ils sont semi-réguliers de la seconde espèce."

Which translates roughly to, "A polyhedron is semi-regular if its faces are regular polygons and all its vertices are congruent. Examples include the Archimedean solids, the prisms and the regular-faced antiprisms. The terminology does not appear to be entirely settled. People sometimes say the semi-regular solids of the first kind to indicate the convex ones, and the uniform solids in general. The Catalan solids are not semi-regular, but have congruent faces and regular vertices. People sometimes say of such polyhdra that they are semi-regular of the second kind.

Translator's note:The French often seem to use phrases along the lines of "of the first/second/etc. kind" to distinguish classes of figure, at least until a proper naming system comes along. Their subtle get-out does not work in English, where we are not accustomed to it.

Meanwhile, Cundy & Rollett observe that the Archimedeans, prisms and anti-prisms are facially regular while the Catalans are vertically-regular (c.f. Johnson's 'versi regular' discussed above). On one page they say that the prisms and anti-prisms are among the Archimedeans, on another they contradict this and observe that, like the Archimedeans, the prisms and antiprisms meet the definition of being semi-regular. More chaos, sigh.

I think I know how to write something sensible. Hope I can keep finding the time to do it. Steelpillow 20:46, 26 December 2006 (UTC)

Ta-daa!

Well, I have done my best. Added my thoughts and rearranged based on the idea of working forwards from Gossett's original definition. Hope others can add/update their favourite bits that I didn't have time to read through in the above discussions. Yeah, and if you hate it all then you can always revert back.... Best wishes for the New Year. Steelpillow 12:30, 27 December 2006 (UTC)

Very nice. Thank you! Tom Ruen 23:58, 27 December 2006 (UTC)

Elte's semiregular polytopes

I enumerated (and translated) Elte's semiregular polytope summary table on a userpage. I'll convert to an article after I check it more. It would be good to have more on Elte, but I have nothing except the book title and date of publishing. Coxeter has limited references, like [2] Tom Ruen (talk) 02:47, 15 June 2010 (UTC)

User:Tomruen/Emanuel Lodewijk Elte

And Gosset's list, which included honeycombs are listed at:

User:Tomruen/semiregular polytopes