# Talk:Polyhedron

Well, I've had a bash at the history. More to do though.

But what's the problem with the references? The important ones are in there. Steelpillow 20:05, 15 April 2007 (UTC)

## Quasi-regular

Could someone please improve this article? Towards the begining it states that quasi-regular polyhedra are edge- and vertex-uniform, but then later it makes a statement about how the icosidodecahedron cuboctahedron are quasi-uniform with the additional property of being edge-uniform. This seems contradictory. Please help. superscienceman 18:26, 3 October 2006 (UTC)

I changed the quasiregular definition to only edge-uniform, since there's two classes vertex-uniform and face-uniform polyhedra, which are actually duals of each other. I agree it needs further work. Tom Ruen 22:28, 3 October 2006 (UTC)
I also removed constraint that semiregular can't be edge-uniform, since all of the Archimedean solids, prisms, antiprisms can be considered semiregular. Tom Ruen 22:33, 3 October 2006 (UTC)
I have created a stub page for quasiregular polyhedron. I am also asking around about the definition of "quasiregular" - it's yet another of those messy areas. Steelpillow 12:13, 8 January 2007 (UTC)
I'd say that these are quasiregular:

They are the rectified tetrahedron/tetrahedron, cube/octahedron, dodecahedron/icosahedron, great stellated dodecahedron/great icosahedron, small stellated dodecahedron/great dodecahedron. --116.14.72.74 (talk) 04:24, 26 July 2009 (UTC)

## Archimedean stellations

I changed the following sentence to avoid the problematic term "semiregular", but I think there are other issues -- or at least, I don't understand it:

Of the 39 non-convex, non-prismatic uniform polyhedra, 17 are stellations of Archimedean solids.

There are 75 non-prismatic uniform polyhedra. Of these 18 are convex -- the Platonic and Archimedean solids. That leaves 57 nonconvex, nonprismatic uniform polyhedra (either all of which, or none of which, are semiregular by most definitions of the term.) Where does the number 39 come from? Also, which 17 are stellations of Archimedean solids (admittedly that enumeration doesn't belong in this article) and what's the source of this fact (that does belong!) -- Rsholmes 16:24, 22 December 2006 (UTC)

No one's come to the defense of this statement, which I believe is either wrong or badly stated, so I'm deleting it. -- Rsholmes 00:05, 8 January 2007 (UTC)

## Bit of a shake-up

I've just rearranged the headings a bit to make a more ordered hierarchy, also tidied up a little of the semi-regular confusion and a few other bits. The main section is still pretty unwieldly. I think some of the lists of names should go (eg various uniform polyhedra), just leaving the thumbnails behind. Also, I think it better to list each type of uniform and its duals in the same sub-section, but I don't have time right now.

Well, I've had another bash today. :-) Much more to be done though :-( Steelpillow 13:08, 8 January 2007 (UTC)

Finally, does anybody know how to archive the older bits of this Discussion page? It's getting a bit top-heavy.

Cheers, Steelpillow 22:11, 7 January 2007 (UTC)

I've done little but cleanup myself in this article in the past, although I suppose I did get the image zoo going here. I moved the pre-July talk to an archive subpage.
I finally got a Wythoff symbol article going this weekend, not great but a fair start, and pretty spherical polyhedra images! Tom Ruen 01:06, 8 January 2007 (UTC)

### Planned changes

Thinking of a few more big changes (as I find the time):

• Replace the phrase 'classical polyhedron' with 'traditional polyhedron'. Somehow 'classical' suggests an accepted terminology (as in say 'classical architecture') which is not so.
• Shrink the info on the various uniform polyhedra, as this should all be covered by the Uniform polyhedra page.
• As mentioned above, combine the Uniforms and their Duals in a single strand.

Any objections? Steelpillow 09:39, 12 January 2007 (UTC)

Sounds good to me. Tom Ruen 10:19, 12 January 2007 (UTC)
Phew! All done in a rush. Any better? Still some more shrinking to do. Steelpillow 20:07, 12 January 2007 (UTC)
Well, done some more of that now! Have I pruned too much info from "Symmetrical polyhedra"? It was all duplicated on other pages. Still needs more tidying, tho. Steelpillow 20:17, 27 January 2007 (UTC)

### Definition

I don't like the new definition section much. Saying it's a 3-dimensional polytope doesn't help anyone unless they already know what a polytope is -- in which case they probably already know what a polyhedron is. Certainly by reading onward one can get a sense of what a polyhedron is, but really there should be an accessible, helpful definition right up at the top. Under the heading "Definition"! -- Rsholmes 20:09, 12 January 2007 (UTC)

Better now? Steelpillow 14:57, 13 January 2007 (UTC)
Yes it is, thanks. I still think there's room for improvement though. My feeling is this article is mostly going to be read by lay people who don't need to hear right from the outset about nonplanar faces, nonlinear edges, and nullities. This is not to say they can't be mentioned close to the top, but I'd advocate starting off with a (one of the!) traditional definition of a polyhedron -- e.g. a volume of 3-dimensional space bounded by polygons that meet at their edges. Give a paragraph about that, and then say "More generally, ..." et cetera. -- Rsholmes 17:30, 13 January 2007 (UTC)
OK. Try this...
BTW, anybody got a suitable pretty picture of one to put up there at the start? Steelpillow 20:48, 13 January 2007 (UTC)
I'd probably drop nullity definition, not referenced anywhere else in the article. Tom Ruen 19:16, 18 January 2007 (UTC)
The definition is incomplete without it, but I don't think it's worth discussing any deeper on this page. I've linked it to the Abstrct Polyhedra bit. Steelpillow 18:16, 19 January 2007 (UTC)

## More pictures of polyhedrons at the top of the page?

Is the one picture, that of the dodecahedron, sufficient? kabbelen 04:23, 1 March 2007 (UTC)

More would be good, perhaps a row of small-ish ones. Do you have time to put together some nice ones? Steelpillow 19:54, 1 March 2007 (UTC)
I have time, but I don't know how to put together some nice ones. Is it possible to put a column of images next to the table of contents? kabbelen 18:21, 3 March 2007 (UTC)
I expect that it is possible, but I do not know how. Anybody else...? Steelpillow 21:35, 3 March 2007 (UTC)
Okay, I tried a column of test images in a table. Feel free to change or whatever. Tom Ruen 23:09, 3 March 2007 (UTC)
Nice one, Tom. Tell you what - if one of us gets time, how about a bit more variety in colour and style? Steelpillow 10:50, 4 March 2007 (UTC)
Okay one more improvement. Two columns, added a descriptive category following each. A fair representation, always room for more! :) —The preceding unsigned comment was added by Tomruen (talkcontribs) 03:20, 7 March 2007 (UTC).
I like that a lot. Just one small thought. In my browser (Firefox 1.5) the table border is right up against the introductory text above the Table of Contents. Is there any way to put a few pixels' clear space between them? Steelpillow 20:17, 7 March 2007 (UTC)
Images can be made smaller. It's impossible to decide how big things should be since people run different screen resolutions. My screen is 1280 pixels wide so lots of room. The new IE also does nice automatic shrinking for printing - previously it would just cut things off. Tom Ruen 20:38, 7 March 2007 (UTC)
It's not the size of the images, it's the way Firefox runs the text right up to the table border. Nice to hear that IE is at last learning lessons from Firefox about usability. Steelpillow 10:12, 8 March 2007 (UTC)
Fixed it with a nested table. Steelpillow 17:46, 18 March 2007 (UTC)

## Books on polyhedra

I added this new section. Please help! Especially if you think I've put a book in the wrong subsection. Steelpillow 17:46, 18 March 2007 (UTC)

Discussion moved to Talk:List of books about polyhedra.

## Polyhedron - Surface or Solid?

Isn't the word "polyhedron, -hedra" intended to describe a Surface (and its associated Area) rather than a Solid (and its Volume)71.116.178.98 09:30, 8 May 2007 (UTC)? Even the Greek root of the word seems to suggest this. Shouldn't those famous figures be called "Platonic Surfaces" rather than "Platonic Solids"? Is there a word-ending which would convey this idea? "Polyhedroid" might work, except that the "-oid" ending as currently used seems to convey the feeling of "almost" or "-like". Any comments or opinions? Ed Frank

The oldest known polyhedra are definitely solid - solid stone! Modern thought is that the Greeks considered polyhedra as solids - I have no idea if this is correct. Leonardo da Vinci made skeletal (stick) models, and Kepler drew the seven regular polyhedra he knew of as thin-walled "surfaces". Most recently, Grünbaum has advocated a theory based on partially-ordered pairs of points. There are also purely abstract definitions. So in this respect a "polyhedron" is pretty much what you want it to be at the time. Some of the above stuff is written up on the page, but I think it needs the rest adding. Maybe I'll find the time. Hopefully someone else will (grin). BTW the term "polyhedroid" has been used for various polyhedron-like things from time to time, but never seems to have stuck. HTH -- Steelpillow 17:28, 8 May 2007 (UTC)
Polyhedra which enclose a subset of space have a volume, or could also be said to divide space into an interior and exterior volume. The infinite skew polyhedrons can divide space into two disjoint volumes, both infinite. Self-intersecting star polyhedrons also divide space if they are orientable, although definition of interior is ambiguous. Lastly nonorientable polyhedra have only one side and don't enclose any volume. Lastly I guess you can consider - is an edge "two points" or "a line segment" - it has an interior length at least. And similarly are polyhedron faces "solid" or "empty", but they have to be solid to have a surface area. As well, the Convex uniform honeycombs are made of polyhedron cells, which have interior volumes. I guess there's no right answer - if you say a polyhedron is a surface, then you could say its faces are only perimeter, and then it also has no "surface area" and is just a wire frame, or if edges have no interior length, then it's just a set of points in space! Tom Ruen 00:59, 9 May 2007 (UTC)

## Use of 'kai' system for naming polyhedra and polygons DOESN'T WORK

Some might argue that the article "tetradecahedron" should be renamed tetrakaidecahedron, chaing the initial paragraph to this: "A tetrakaidecahedron is a polyhedron with 14 faces. No tetrakaidecahedron is regular. The term tetradecahedron (without the kai) is also sometimes used, though strictly this should be reserved for polyhedra having four (tetra) faces of one kind and ten (deca) of another kind." [1]

I'm inclined to let this trip-you-in-the-middle-of-the-word affectation die.

Mathworld omits the 'kai' for ALL polyhedra.[2] "kai" simply means "and".

I can't find many pages on the WWW that even mention the 'kai'. But here they are, arguing against me:

On Math Forum, here's a page that likes the 'kai'. (24-gon == icosikaitetragon): [3]

The most important of the reasons which make me prefer the 'kai' forms is that they permit these prefixes to be unambiguously parsed even when concatenated, as they are in Kepler's names for certain polyhedra; for example, the icosidodecahedron or (20,12)-hedron, so called because it has 20 faces of one type and 12 of another. Kepler said 'this particular triacontakaidihedron I call the icosidodecahedron', a remark showing that he also preferred the kai forms.

However, the linked-to page, on the same site, gives kai no need or significance at all. (24-gon == icositetragon)[4]

Another page on Math Forum (not linked, but found) posts a follow-up explanation on the first note. Explains that ONE professor with a Greek historian tried to work out the scheme with 'kai'. It seems to agree with some usage by Kepler. Even the responder criticizes the scheme: "You use 'kai' when you mean one number, and leave it out when you mean two. I might have preferred to do it the other way around, ..." [5]

Math.com give kai no significance below 20 (take it or leave it), but by example shows it used always above 20 sides.[6]

This guy thinks -kai- works the opposite way! (or is it a typo?): [7]

In earlier versions of this table, I used the syllable kai to join numerical syllables into complete prefixes, as in "icosikaitetrachoron" for the 24-cell. The kai is equivalent to "and" in the number "four-and-twenty." Norman Johnson has persuaded me that this is redundant and can safely be omitted to save space, except in certain cases where using it avoids confusion. (For example, the hecatonicosachoron has 120 congruent dodecahedral cells, but a hecatonkaiicosachoron might have 100 cells of one kind and 20 cells of another.)

This interesting page shows stranger variations. While 13 and 14 keep the 'kai' (13 = triskaideca- or trideca-; 14 = tetrakaideca- or tetradeca-), 12 and 15+ do not. (dodeca-; pendeca-, hexadeca-, heptadeca-, octodeca-, ...) [http://phrontistery.info/numbers.html}

Interesting, on Google Books, tetradecahedron gives 53 hits dated 1836-2006; tetrakaidecahedron gives 260 hits, dated 1880 to 2007. The people who say tetrakaidecahedron probably have no idea why there are saying it. Whiner01 09:33, 29 October 2007 (UTC)

The 'kai' system seems at first to be self-consistent and logical. But it is an elaborate system that few seem to explain. And it leads to contradictions.

If the 'kai' system were accepted and required, then a 12-sided solid would have to be called a dokaidecahedron, because dodecahedron would imply a solid with 10 faces of one kind and two faces of another kind. (These instances might be the first mention EVER of dokaidecahedron -- try searching on it.)

Some would argue that calling a 14-sided solid a tetradecahedron implies a solid having four facets of one shape and ten of a different shape, and so it must be called a tetrakaidecahedron. Unfortunately, tetrakaidecahedron implies 14 sides of the same shape -- in which case tetrakaidecahedron can only refer to the "triaugmented triangular prism" and "heptagonal dipyramid", which have 14 triangles. Every 14-sided shape would need a different name:

• "triaugmented triangular prism" and "heptagonal dipyramid" (14 triangles) would become or remain tetrakaidecahedron.
• cuboctahedron and truncated cube (both 8 triangles, 6 squares) and truncated octahedron (6 squares, 8 hexagons) would all become octohexahedron or hexaoctohedron. (two unique coinages)
• hexagonal antiprism (12 triangles, 2 hexagons), dodecagonal prism (12 squares, 2 dodecagons), and (no name found) (12 pentagons, 2 hexagons) would become dodecadohedron (another unique coinage), or dododecahedron (it exists, but only as a joke on "dodo").
• 13-sided pyramid (13 triangles, 1 tridecagon) would become triskaidec[a]un[a]hedron or un[a]triskaidecahedron (six more unique coinages).

Clearly we're better off keeping tetradecahedron for a generic 14-sided solid.

Clearly the 'kai' system cannot be taken as a strict system to be applied everywhere. It kind of evaporates when you shine a light on it. I think Mathworld and others got it right -- ignore "kai" and omit "kai" because it simply means "and". I wonder whether they accidentally glossed over the background. If they studied and decided, I wish they had published their explanation for ignoring the "kai" system.

I didn't even know the tetrakaidecahedron article existed. I've not (much?) seen kai used with polyhedra, BUT mostly because the n-hedron naming scheme is much less useful names since there's so many topological variations with the same number of faces. I'm not in a position to want to defend or criticize such names. If the names are used in references, good to include them as alternates in wiki articles. Tom Ruen 21:17, 29 October 200 7 (UTC)
The kai system seems to have been introduced fairly recently by mathematicians who ran into trouble when naming some new kinds of polyhedra. For example the regular octahedron has cubic symmetry. But if you colour alternate faces black and white then it has tetrahedral symmetry - and further, in the same way that the cuboctahedron can be thought of as the core of the regular compound oa cube and an octahedron, so this coloured octahedron can be thought of as the core of a regular compound of one black tetrahedron and one white one. Hence its mathematical name - the tetratetrahedron. But when we consider say a "tetradecahedron" there is now an ambiguity, do we mean the core of a compound of a tetrahedron with a decahedron, or just any polyhedron with 14 faces? The kai was apparently used for some numbers in some Greek writings, and so was adopted as a way of resolving this ambiguity by indicating any polyhedron with 14 sides. It does actually work very well, despite User:Whiner01's criticisms. For example Weaire and Phelan refer to their 14-sided bubble as a tetrakaidecahedron.
This was the basis on which I set up the "tertakaidecahedron" page. Since then I have discovered that this is still the exception rather than the rule, and it is more common to do things the way User:Whiner01 has set them up.
P.S. Don't ever take mathworld's word for anything polyhedral. They really don't know what they are talking about, and those pages are riddled with false folklore and other errors.
HTH -- Steelpillow 09:37, 30 October 2007 (UTC)

### Why the kai- forms exist for 13 to 19 (and not 11 or 12)

This is not some arbitrary or dubious system invented by modern (or even medieval) mathematicians. The numbers from 10 to 20 in the language of the ancient Greek mathematicians were: 10 déka, 11 héndeka, 12 dṓdeka, 13 tr(e)iskaídeka, 14 tessareskaídeka or tetrakaideka- (in compounds only), 15 pentekaídeka, 16 hekkaídeka, 17 heptakaídeka, 18 oktōkaídeka, 19 enneakaídeka, 20 eíkosi. No forms without -kai- are recorded for 13 to 19, and they can safely be assumed not to have existed; certainly not to have been passed on to succeeding generations of mathematicians.

The names of the polygons and polyhedra were also created by ancient Greek mathematicians (not by later Western mathematicians who had learned Greek), and are perfectly natural formations in the language. The geometrical terms dekágōnon, hendekágōnon, dōdekágōnon, tr(e)iskaidekágōnon, tessareskaidekágōnon, pentekaidekágōnon, hekkaidekágōnon, eikoságōnon, dōdekáedron, tetrakaidekáedron (or tessareskaidekáedron), hekkaidekáedron and eikosáedron are all found in Liddell & Scott's Greek-English Lexicon. (H doesn't occur in the middle of Greek words). If words had existed for 17 to 19-sided polygons, or 13, 15, or 17 to 19-faced solids, the -kai- forms would unquestionably have been used. Kai is a normal part of the names of these numbers.

The argument then is about whether terminology well over 2000 years old needs to be revised. The names were natural and obvious ones when they were created, and simple to the Greeks who used them. We are merely paying the price for the old English convention of drawing on Latin and Greek for our technical vocabulary. Koro Neil

Thanks for that. Woodhouse seems to agree, at least for fourteen. Modern Greek is rather different, and I wonder whether the Greek dialects of Byzantine and Islamic scholars leading up to the Renaissance might have been different again. Anyway, the reasons why modern English usage developed as it did are not really relevant, en.wikipedia.org needs to follow it regardless. Some of us use kai, some don't. If there is a single originating authority in a given context (e.g. Weaire-Phelan bubbles) then I would be inclined to follow their usage in the article. For the present article I'd expect to see adequate referencing for whichever position is eventually taken. — Cheers, Steelpillow (Talk) 20:02, 3 September 2010 (UTC)

Hello all. I would like to point out that while the most common usage of the word "polyhedron" means one of those 3-d shapes, another common definition is "an intersection of half-spaces in n-space".

Hi, thanks for the comments. See the subsection on General polyhedra. This definition corresponds to what one would normally call a convex polyhedron; sadly, it is all too common for mathematicians to forget what kind of polyhedra they are talking about, and to generalise quite unjustifiably. -- Steelpillow (talk) 20:07, 20 November 2007 (UTC)

I also am not sure whether long lists of various types of polyhedra is useful for this article. Perhaps it should be relegated to a "lists of polyhedra" article (kinda like TV show articles have "lists of episodes" articles).

If you can find any long lists of polyhedra in the article, I'd be interested to know where they are. I (re)moved the ones I could find quite a long time ago. -- Steelpillow (talk)

Finally, the Characteristics section seems quite out of place and utterly inaccessible. In particular, the term "simply connected" is not in common usage as far as I know, and should at least be referenced (if not omitted entirely). In addition, duality should probably not be a little sub-point, and "vertex figure" is a technical notion that has no place in that part of the article. -Yuliya (talk) 00:40, 20 November 2007 (UTC)

The term "simply connected" is being used here in exactly the way it is used in topology: a polyhedron is simply connected if it is path connected and all loops are contractible. Some authors allow a cube with a rectangular hole drilled all the way through it to be a polyhedron, even though it is homeomorphic to a solid torus instead of a ball. For some purposes one even allows the surface of a polyhedron to contain self-intersections. On the other hand, the definition you give agrees with Minkowski, which is a point in its favor. For a history of proposed definitions of "polyhedron", I recommend the book by Lakatos referenced in the article, Proofs and Refutations. Michael Slone (talk) 03:34, 20 November 2007 (UTC)
By "out of place" I take you to mean that it does not seem relevant? I agree that it is badly laid out and terse to the point of obscurity. Please, if you have time, improve it! And I agree that the duality of polyhedra is a very important topic - which is why it has its own page at Dual polyhedron and is merely introduced/linked to on this page. But I disagree about vertex figures: they and their properties are as important to a polyhedron as its faces, and to cut a corner off and examine the cut is really not very technical. They at least need to be introduced, as done here. HTH. -- Steelpillow (talk) 20:07, 20 November 2007 (UTC)

## Googolhedron

Googolhedron redirects here, but is not described in any way in the article. This needs to be fixed. I'd do it myself, but I don't have the time to figure out where it would belong, as I did with Googolgon. The history of googolgon and googolhedron is likely similar. See Talk:polygon#Googolgon for more information about the first. -- trlkly 07:15, 11 May 2008 (UTC)

Generally, see my reply on Talk:Googolgon. Also, this article does not have a long list of number-based names (the number of faces is not such a fundamental identifying characteristic for polyhedra as it is for polygons), so there is no place for a googolhedron as such unless it fits in somewhere else in the article - which, IMHO, it does not. If somebody wants a place for cited googolhedra and the like, I would suggest a separate article, listing polyhedra by the number of faces. But be warned, by the time you get to eight or more faces, you will be facing a gargantuan task - see for example Steve Dutch's enumeration. -- Steelpillow (talk) 12:43, 11 May 2008 (UTC)

## Notable polyhedra

I have noticed a steady trickle of new articles describing individual polyhedra. In general these polyhedra are not especially notable and the articles rather short, for example the First and Second stellation of icosahedron and so on. According to Wikipedia's notability guidelines, such articles should be replaced by redirects and any worthwhile content merged into a more general article (which might need to be created), for example The 59 icosahedra. Any objections? -- Cheers, Steelpillow (Talk) 20:16, 20 July 2008 (UTC)

## Contour Lines

Contour lines are pictures that hold lines which make enclosed figures! —Preceding unsigned comment added by 66.204.139.126 (talk) 15:38, 24 September 2008 (UTC)

## i know i know what a polyhedron is!:)

in simiplier words...... this is a very, very, very misunderstanderable shape. this is coming from a very confused 5th grader. and if you want to.. if you press new section then write in a simiple way what a polyhedron looks likes. thank you if you tryed.—Preceding unsigned comment added by 76.101.128.38 (talk) 22:18, 19 February 2009 (UTC)

## Original sin not relevant?

What does Original sin, a theological concept, have to do with polyhedra?--DThomsen8 (talk) 02:33, 28 May 2009 (UTC)

I went back in the revision history about 7 or 8 steps and couldn't find who added that. Odd. I reverted original sin to origins of Rybu (talk) 09:24, 28 May 2009 (UTC)
I did. It appears in the quoted passage from Grünbaum's seminal paper Polyhedra with hollow faces, which was instrumental in kicking off the work which led to the modern theory of abstract polytopes. It is relevant because if we do not define the objects we are talking about, then it is hard to know whether our mathematical results make any sense. Mathematicians live and die by their logical rigour, hence the description of this appalling and habitual lapse, which was not pointed out until the 1970's, as an "original sin". Grünbaum's colourful language has been quoted many times since. -- Cheers, Steelpillow (Talk) 21:03, 28 May 2009 (UTC)
Ah. Somehow the context wasn't clear, apparently both to myself and Dthomsen8. IMO it might make sense to keep those two paragraphs together. Because it's not a stand-alone thought as a separate paragraph. I'll show you what I mean and you can tweak to your likings. Rybu (talk) 21:35, 28 May 2009 (UTC)
No problem. I have just made minor tweaks to the punctuation and run the following para in too, for consistency. (BTW, the problem was not pointed out until the 1990s - my typo above, oops). -- Cheers, Steelpillow (Talk) 19:35, 29 May 2009 (UTC)

This was added anonymously, doesn't look very useful even if it's correct. Tom Ruen (talk) 02:02, 20 November 2009 (UTC)

Well, it looks vaguely convincing, a bit like the stuff that Don Hatch used to win his \$1,000 for finding a centre of reciprocation, but I don't really know - it's way over my head. I have tried to make sense of the grammar, but not quite made it home there either. -- Cheers, Steelpillow (Talk) 21:51, 20 November 2009 (UTC)
Sleeping on it, I am not sure what definition of a polyhedron is assumed - it might assume convexity, or simplicity (non-self-intersecting), or one of those weird definitions where a "polyhedral surface" need not be closed or finite. I think this needs clarifying before we can restore it to the article. Citation, both for the formulae and for the definition, would be necessary too. (By way of explanation - Grünbaum's book Convex polytopes lies at the core of a remarkable diversity of modern mathematics. But its terminology is often misunderstood or taken out of context. For example he omitted the word "convex" hundreds, probably thousands, of times to avoid repetition - so many people mistakenly assume his results apply to non-convex figures too. He also used "polyhedral" to describe any general surface which has been decomposed piecewise such that the pieces are flat faces, including open surfaces such as (say) a paraboloid - many people mistakenly assume that a "polyhedral surface" must be the surface of a polyhedron, and take their understanding of a polyhedron accordingly. The equations below appear to me to be typical of the kind of mathematical discipline where such mistaken assumptions are rampant - for examples it assumes that the figure in question has an identifiable centroid.) -- Cheers, Steelpillow (Talk) 12:21, 22 November 2009 (UTC)
Following links through via Green's theorem, it seems likely that this applies only to topological spheres - for example Stokes' theorem appears to require orientability. Convexity is not assumed, but I'm not sure about self-intersection. I think this gives enough context to restore the section, so that hopefully some more knowledgeable editor can make any clarifications still necessary. -- Cheers, Steelpillow (Talk) 08:59, 20 December 2009 (UTC)

### Polyhedron volume

The volume of an arbitrary polyhedron can be calculated using the Green-Gauss Theorem

${\displaystyle \int \limits _{\Omega }{div({\vec {F}})d\Omega =}\oint \limits _{S}{{\vec {F}}\bullet d{\vec {S}}}}$

by choosing the function

${\displaystyle {\vec {F}}={\frac {\left({x{\hat {i}}+y{\hat {j}}+z{\hat {k}}}\right)}{3}}}$

where (x,y,z) is the centroid of the surface enclosing the volume under consideration. As we have,

${\displaystyle div({\vec {F}})=1}$

Hence the volume can be calculated as:

${\displaystyle volume=\oint \limits _{S}{{\vec {F}}\bullet {\hat {n}}dS}}$

where the normal of the surface pointing outwards is given by:

${\displaystyle {\hat {n}}=(n_{x}{\hat {i}}+n_{y}{\hat {j}}+n_{z}{\hat {k}})}$

The final expression can be written as

${\displaystyle volume={\frac {1}{3}}\sum \limits _{faces}{\left[{\left({x\times n_{x}+y\times n_{y}+z\times n_{z}}\right)\bullet S}\right]}}$

where S is the surface area.

I think that when F is defined the x,y,z are still just coordinates and only in the last expression should they should be substituted for a different symbol for the centroids of each of the faces. And why the bullet symbol? 131.180.16.252 (talk) 14:45, 21 January 2010 (UTC)
Sorry, the details are beyond me. I restored the material because it is better than nothing. If you or anyone else can make or suggest specific changes, and especially provide a reference, that would be good. -- Cheers, Steelpillow (Talk) 13:24, 23 January 2010 (UTC)
When you integrate the given vector field on the polygon you get the face barycenter.. not "any point on face" --Andrea Tagliasacchi 00:27, 12 January 2013 (UTC)
Are you certain that the barycenter is what you want? We don't want to be assuming that the polyhedron is regular. I would have thought that the correct answer is: the point on the face plane closest to the origin (which might not even be contained within the face itself). —David Eppstein (talk) 00:35, 12 January 2013 (UTC)

For starters, I'd suggest a couple of candidate topics for splitting off as separate articles:

-- Cheers, Steelpillow (Talk) 09:35, 20 December 2009 (UTC)

## Bowers style acronyms

After putting my knowledge of polyhedra on firmer ground, I'd like to ask: should the Bowers style acronyms redirect to the polyhedron names? 4 T C 06:28, 24 January 2010 (UTC)

Since no one has complained, I've done the whole lot - except prisms. 4 T C 10:01, 24 January 2010 (UTC)

Are the Bowers acronyms widely referenced in the peer-reviewed literature? I rather doubt it. I suspect that their presence on formulasearchengine may well constitute WP:OR and they ought to be removed. -- Cheers, Steelpillow (Talk) 15:23, 24 January 2010 (UTC)

## footnotes

Somebody added Template:No footnotes on the basis that the lack of inline citations obscures which references are sourced for each factoid.

Much of the information given here is well known and widely available from many sources. The references listed give a fair selection. wp:cite gives four reasons for citing sources:

Cite sources when
• adding or restoring material that is challenged or likely to be challenged,
• quoting someone, with or without quotation marks, or closely paraphasing a source,
• summarizing source material in your own words, in order to give credit to the source of your ideas,

The third is the only vaguely relevant one here, and in general typical multiple sources are already given in the references. I see no imperative here to go scattering inline citations about the place.

I am sure that things can be improved - Pearce is not a standard reference and inline citation would be better, while sources such as Cundy & Rollett and Lines are standard references and need adding.

Can we remove this template and leave citation requests to specific sections and factoids? — Cheers, Steelpillow (Talk) 10:26, 12 February 2011 (UTC)

## Polytope more prominent

People are going to go to this article looking for the higher-dimensional generalization of polyhedra, that is, polytopes. Polytopes are mentioned in the article, but perhaps they rate their own section under "Generalisations".

The articles on polygon and polyhedron should probably mention in the introduction. "A polygon(polyhedron) is a 2(3)-dimensional polytope." Mohanchous (talk) 18:38, 23 July 2011 (UTC)

Someone removed the polytope link from the lead because it was getting too cluttered. I have now made some space and put it back there. Polytopes are really the generalisation of the sequence point, line segment, polygon, polyhedron, ... rather than just of polyhedra. I feel that the link in the lead is enough. Also the section on generalisations focuses on 3D, to introduce higher polytopes would be out of place as it stands. — Cheers, Steelpillow (Talk) 21:34, 23 July 2011 (UTC)

## Arab or Islamic?

My understanding of mathematics in the European "dark age" is that scholars from many countries such as India, Persia and North Africa contributed new advances and insights. AFAIK pretty much all these scholars were Muslims but not all were Arabs (e.g. in India). Therefore I propose to revert the recent edit to the relevant section title. Any comments? — Cheers, Steelpillow (Talk) 21:11, 11 August 2011 (UTC)

I prefer Islamic to Arab for exactly the reasons you give. (Also, Spain was part of the Islamic world at that time as well.) —David Eppstein (talk) 21:43, 11 August 2011 (UTC)
Pretty much all were Muslim? Thabit ibn Qurra, the very scholar mentioned was not a Muslim. Granted that ethnically he was Assyrian and not an Arab, he nevertheless wrote in Arabic. It is my understanding that the Persian Abū al-Wafā' Būzjānī also wrote in Arabic. I picked Arab, (and perhaps I should have used Arabic), to serve as a common bond since not all were followers of Islam. But, I can see your point because, for better or worse, it is commonplace to speak of "Islamic Civilization" despite the fact that a large number of the conquered people under Muslim domination were not Muslims until much later in history. How about this suggestion - perhaps instead of using "religious" designations, the subheadings could reformulated as regional? There could be a "European," "East Asian," "South Asian," "Middle Eastern," etc. In fact, the whole "History" section seems to need to be better organized (ex. integrating the star polyhedra subsection).

-Emmo827(talk) 11:50, 12 August 2011 (UTC)

Another reason to use "Islamic" here is to be consistent with the title of the article Mathematics in medieval Islam, something that has been hashed out repeatedly there (see Talk:Mathematics in medieval Islam/Archive 1). —David Eppstein (talk) 22:54, 12 August 2011 (UTC)
I take Emmo827's point about personal religion, and the lack of a clear-cut choice. However we often refer to say "Roman" when talking about things that never went anywhere near Rome, just happened somewhere in their civilisation - similarly with "Greek" mathematicians who lived and worked in North Africa. So I think it safest to talk of "Islamic" mathematics as meaning the Islamic civilisation. I'll revert the change. — Cheers, Steelpillow (Talk) 12:03, 13 August 2011 (UTC)

## File:First stellation of dodecahedron.png Nominated for Deletion

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## Volume and vector calculus

The Volume section has a clever general approach to computing the volume of polyhedra. But putting myself in the shoes of an excited youngster or layman wanting to learn about this stuff for the first time, vector calculus is a brick wall. Without disputing the utility of the high level approach, there more elementary, accessible approaches that a newcomer could get their head around. Perhaps a table of volume formulas for the simplest regular polyhedra would be good enough. Or perhaps referrals to the volume sections for the regular polyhedra articles. An illustration of how the final vector formula works for a cube? Comments welcome, --Mark viking (talk) 00:43, 23 February 2013 (UTC)

Just had a go at it. Needs wikifying, but I hope it's in the right direction. — Cheers, Steelpillow (Talk) 12:58, 23 February 2013 (UTC)
The elementary section looks great. I wikified the math to be more consistent with the advanced section and added the cube as an example of the elementary approach. Thanks, --Mark viking (talk) 19:40, 23 February 2013 (UTC)
Great. Thanks more to you than me! — Cheers, Steelpillow (Talk) 21:32, 23 February 2013 (UTC)

## Goldberg Polyhedra

I've just come across an article of Goldberg Polyhedra[8] which I've not heard of before. Wikipedia does not seem to have any coverage of these. They look like they are duals of geodesic domes.--Salix (talk): 22:29, 25 June 2013 (UTC)

There's some closely related information hidden in a chemistry article: Fullerene#Other buckyballs. —David Eppstein (talk) 23:18, 25 June 2013 (UTC)
Fun. George W. Hart has plans for making a Goldberg polyhedron on a 3D printer on this page and he has a paper on them in G. Hart, "Goldberg Variations," Shaping Space, (Marjorie Senechal, ed.) Springer, 2012. --Mark viking (talk) 23:47, 25 June 2013 (UTC)
Stub created at Goldberg polyhedron.--Salix (talk): 07:21, 26 June 2013 (UTC)