Talk:Platonic solid
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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)
Contents |
[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)
This is the text from platonic solids, now redirected here:
The Platonic Solids, The Five Pythagorean Solids, or The Five Regular Solids
| 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.
(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.
(Eq.2)
(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
(Eq.4)
If we divide both sides of this equation by 2E, we have
(Eq.5)
(Eq.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)
