Talk:Platonic solid

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The symmetry groups are given by <x,y,z: x^a = y^b = z^c = xyz >, 1/a+1/b+1/c > 1. There is a pretty article on the finite ones in Tensor - by Conway, Coxeter, and Shephard(?sp).

No-one mentions the additional regular maps obtained by projecting on to the surface of an escribing sphere. These are allowed to have digons as faces. This completes the five types (A_n, D_n, E6,E7,E8} in terms of Lie notation. [John McKay24.200.80.227 02:57, 19 October 2007 (UTC)]

I think we should explain the symmetry group part a bit.

And we should add the number of edges for each solid.

The number of edges for each solid is half number of vertices times the number of faces meeting at each vertex.

Euler established that it's the sum of the number of faces and number of vertices minus two. This applies to any other polyhedron that has no hollowed out spaces and no holes. --- Karl Palmen

I'd also like to see the correspondence between these solids and the classical elements. I remember this from way, way back when, so I don't remember to whom it's attributed, or which solid goes with which element, or I'd do this myself. I do remember fire being the tetrahedron, and I think aether was the icosahedron. And I came here hoping the article would tell me which was which, after all these years, so that's why I'm asking now. Please? -- John Owens 10:54 Apr 28, 2003 (UTC)

Could someone add images to this page? It would be nice to visualise these objects. -- Astudent

Images added (before reading the above request). كسيپ Cyp 22:23 30 May 2003 (UTC)

[edit] Categorization Geometric zoology

I heard this term from professor V.Zalgaler, and it seems to be used before a lot for classification of different types of polyheda. I am thinking, maybe we should add such subcategory into "Category:Discrete geometry" and put there all kinds of related articles?

Tosha 14:31, 14 Jun 2004 (UTC)

Contents

This is the text from platonic solids, now redirected here:

The Platonic Solids, The Five Pythagorean Solids, or The Five Regular Solids

Combinatorics of Regular Polyhedra
n r F E V
Tetrahedron 3 3 4 6 4
Octahedron 3 4 8 12 6
Icosahedron 3 5 20 30 12
Hexahedron 4 3 6 12 8
Dodecahedron 5 3 12 30 20

[edit] Proof

The following was proven by Descartes and Leonhard Euler.

V - E + F = 2 \,\! (Eq.1)

where F is the number of faces, E is the number of edges, and V is the number of corners or vertices of a regular solid.

nF = 2E \,\! (Eq.2)
rV = 2E \,\! (Eq.3)

where r is how many edges meet at each vertex.

Substituting for V and F in Eq.1 from Eq.3 and Eq.4, we find

\frac{2E}{r} - E + \frac{2E}{n} = 2 \,\! (Eq.4)

If we divide both sides of this equation by 2E, we have

\frac{1}{n} + \frac{1}{r} = \frac{1}{2} + \frac{1}{E} \,\! (Eq.5)
\frac{1}{r} = \frac{1}{E} + \frac{1}{6} \,\! (Eq.6)
\frac{1}{n} = \frac{1}{E} + \frac{1}{6} \,\! (Eq.7)

Charles Matthews 07:48, 21 Sep 2004 (UTC)

I've put this back in, without showing the details of the algebra, since it is a good example of how topology is sometimes adequate to solve geometric problems. I did not attribute it to Descartes and Euler, since I don't have a reference. Joshuardavis 15:48, 2 March 2006 (UTC)

[edit] external link missing

Sorry that I don't have the time to edit the page properly, but the foldable paper models page is not there anymore. Anyone who know where it went please change the link.

Somebody should fix the "Ancient Symbolism" section:

This concept linked fire with the tetrahedron, earth with the cube, air with the octahedron and water with the icosahedron. There was logical reasoning behind these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the most spherical solid, the dodecahedron; its minuscule components are so smooth that one can barely feel it.

So is air an octahedron or a dodecahedron? Transfinite 19:58, 18 Nov 2004 (UTC)

[edit] Fluorite

On this page there's a link to the non-existant page [[calcium floride]. Through some searches I found the page Fluorite, which mentions octohedrons and dodecahedrons in the opening paragraph. Is this the mineral that was supposed to be linked to? --Spug 12:11, 19 Nov 2004 (UTC)

[edit] will somebody explain

why dodecahedrom is randomly bold in the list?

Presumably as the highest percentage. I've changed it. Charles Matthews 10:06, 22 Mar 2005 (UTC)

There are four classical elements, not five. Right???

can some 1 help me with my question? i need to know what are the faces of 1 or more faces in a hexagon? (comment from IP address 24/1/06)

Could you rephrase the question I'm not quite sure what your asking. --Salix alba (talk) 00:03, 25 January 2006 (UTC)

[edit] Topological proof

The reorganization of 2 June 2006 was well done, Fropuff. Thanks. I have just one complaint. In the Classification section you give two versions of the same proof. Both hinge, in my view, on the same fact: Item #2 in the first version, i.e. the "elementary result" in the second.

I vote that we replace one of these versions with a purely topological proof using Euler's formula (which you've already introduced) as the linchpin. To me this is a good example of how seemingly geometric facts are sometimes determined purely by topology. The proof was already in the earlier versions (put in by me — with details left out because it's a common exercise for students):

We know that pf = 2e = vq and that v − e + f = 2. Multiplying the latter equation by pq we obtain pqv − pqe + pqf = 2pq. Substutiting from the first equation we have 2ep − pqe + 2eq = 2pq, which implies that e(2p − pq + 2q) = 2pq. Now e and 2pq are positive, so 2p − pq + 2q is as well. Since p and q must be at least 3, it is easy to see that the only possible values of (p,q) are (3,3),(3,4),(4,3),(3,5),(5,3). Joshua Davis 14:08, 2 June 2006 (UTC)

Yes, I somehow missed the point of that paragraph in the previous version. I've inserted a varation of the topological proof into the article (actually more akin to Charles's version above). Thanks for the comment. -- Fropuff 18:38, 2 June 2006 (UTC)

[edit] Discrete subgroups of SU(2)

Should that be mentioned?--user talk:hillgentleman 08:52, 22 November 2006 (UTC)

Okay! —Tamfang 04:24, 24 November 2006 (UTC)
I had debated mentioning the discrete subgroups of SU(2) back when I did the rewrite of this article. Unfortunately, we don't have a good discussion of this topic elsewhere on Wikipedia, and it would take too many words to describe the concept here. The section on the symmetry groups is already rather long. If a coherent treatment is given elsewhere it would be worthwhile to mention the relationship in this article and provide a link. -- Fropuff 03:46, 30 November 2006 (UTC)

[edit] Regularity

(to User:Rsholmes) I am well aware not every dodecahedron is regular, but the place to discuss that is in thedodecahedron article not in the intro to an article on Platonic solids. In a context where one is only talking about regular solids, calling everything regular is distracting and unnecessarily pedantic. It is also a rather technical point to be discussing in the article lead. If we must mention the distinction than I insist we put it in a footnote or somewhere more suitable than the lead. -- Fropuff 01:21, 23 December 2006 (UTC)

I agree. There's lots of places there's ambiguity of language, but here there's no debate on regularity. Tom Ruen 01:45, 23 December 2006 (UTC)
I strongly disagree. To say that "the dodecahedron (or icosahedron, or whatever) is one of the Platonic solids" is to state a falsehood -- unless "dodecahedron" is understood to mean "regular dodecahedron" in that context. And so it should be; but a naive reader doesn't know that, unless it's stated. (They may understand that you're discussing regular polyhedra, but not that "dodecahedron" can refer also to non regular solids.) Of course the article should not say "regular dodecahedron" every time; that would indeed be distracting and pedantic -- but for us to be understood correctly by those who are not already famlliar with the subject, it's necessary to let them know the shorthand we're using. It's one additional sentence up front, and it makes the meaning far more clear. A similar point is made in the articles for each of the regular solids, and it should be made here. I'm shocked that you expect the readers to figure this out on their own -- its obfuscation for its own sake. I do, however, have no objection to moving the point into a footnote. In fact I'll do that. -- Rsholmes 23:12, 24 December 2006 (UTC)

[edit] Leonardo da Vinci

I believe someone should add to this article information regarding Leonardo da Vinci and his study of the platonic solids; particularly in relation to the Flower of Life. The following sources may conatin relevant information:

sloth_monkey 09:47, 28 December 2006 (UTC)

[edit] Icosahedral water clusters

Someone may want to add to this article information regarding water molecules in relation to the icosahedron platonic solid.

sloth_monkey 11:06, 28 December 2006 (UTC)

[edit] Saving Animations for powerpoint

I would like to use the animations for the solids in a power point presentation. Anyone know if this is allowed? And also, how I can copy the files? Thanks159.91.19.3 22:34, 29 March 2007 (UTC)

I don't use powerpoint, but you can save images from a web browser. In Internet Explorer, I would right-click on the image and select "Save Target as..." from the popup menu. On being allowed, I believe you generally just need to attribute the source (Wikipedia). Tom Ruen 02:26, 30 March 2007 (UTC)
Well, seems the right-click doesn't work the same in IE for animated gifs. But if you go directly to the image, you can select File/Save AS... Like:[1] Tom Ruen 02:29, 30 March 2007 (UTC)


I tried that (the only option given was "Save PAGE as" and it simply saved the .gif image. I really would like the animations though. Is there a way? Perhaps the person who created them would be kind enough to email me a copy?Ags412 20:20, 30 March 2007 (UTC)
The GIF file contains the animation. There's nothin' else. Tom Ruen 02:13, 18 April 2007 (UTC)
I saved the .gif file. When I opened it, it did not animate. Maybe you are opening it in a program I don't have? What program are you opening it?
As of now, it is not animating when I open the .gif file I saved. It only shows a still image - and nothin' else.Ags412 04:09, 18 April 2007 (UTC)

Similar animations of the Platonic solids in animated .gif format are available here: http://www.3quarks.com/GIF-Animations/PlatonicSolids/ The author says, on that page, that the images can be downloaded and used freely with attribution to him or to that Web page.

[edit] Level of Proof and supposed vandalism

I added the point that the Carved Stone Balls from the Neolithic exist in at least 9 categories and not the five you suggest.

The fact that five of them fit in with your Platonic solids is likely to be pure chance - it is therefore unlikely that they were deliberately manufactured with some insight into your topic. I put it to you that your comment is very unscientific and is not worthy of the standards that WIKI requires.

Lets not forget that the manufacture of these balls was carried out in different places, by different groups and over hundreds of years.

Please alter your article to reflect your comment about Neolithic people as being a miscellany, a point of trivia only. Love the animation. Rosser 12:06, 17 May 2007 (UTC)

Hi, Rosser. I do not understand your comment fully. It's not clear who the "you" in your comment is. I cannot find any mention of nine solids in your edits to the article. Also, I don't see what "Level of Proof" means, or why you're talking about vandalism.
Nonetheless, per your request I have softened the language in this article somewhat, since the carved stone balls obviously include many non-regular polyhedra. However, we need to keep in mind that original research is not allowed. All of Wikipedia's factual claims should be backed up with cited references; they should not arise from your or my inferences about what neolithic people did and not know, however, reasonable. I suspect that this is why your edit was reverted. If the Atiyah and Sutcliffe reference says that the neolithic Scots didn't know Joshua, what they were doing, then we can cite it. Joshua R. Davis 15:07, 17 May 2007 (UTC)

Joshua, Thanks for that. Rosser 10:08, 18 May 2007 (UTC)


[edit] Solid Angle

The description of solid angle is not clear, especially for someone who isn't already very familiar with platonic solids and geometry. Is there anyway we can elaborate on it? It is especially confusing given that the article named Solid Angle mentions, "the solid angle subtended at the center of a cube by one of its sides is one-sixth of that, or 2π/3 sr." Whereas the solid angle of a cube in the Platonic Solids table is bluntly listed as π/2, with no real distinction or explanation being made as to exactly what the table is referring to, in comparison to the statements of the other article. My main concern is that not enough context is provided for people to understand what it means to say, "the solid angle of a polyhedron..." Firth m (talk) 03:42, 25 February 2008 (UTC)

[edit] Vandalism or just a faullty edit?

Hi all. I just happened to visit this page for the first time and very soon noticed the weird content of the broken link underneath the last solid, the icosahedron. At the time of writing, the text reads: ([[:imahallo yall
(which to me signals either a case of misplaced typing or a deliberate change like spray-tagging on a wall).
I am not very familiar with reading the page history and, furthermore, there is no mention on the talk page that this is a desired change, so I will try my best to rectify the link.
For those of you more familiar with reading the tracking history of the page, would you be so kind as to point out to me when and which edit brought this about? --TrondBK (talk) 00:39, 17 March 2008 (UTC)

Going backward in the comparisons one by one, I find that it was done on March 15 by User:74.255.70.210. —Tamfang (talk) 23:02, 18 March 2008 (UTC)