# Talk:Platonic solid

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## Symmetry groups

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)

## 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)


## Platonic solids

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

## Proof

The following was proven by Descartes and Leonhard Euler.

${\displaystyle V-E+F=2\,\!}$ (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.

${\displaystyle nF=2E\,\!}$ (Eq.2)
${\displaystyle rV=2E\,\!}$ (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

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

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

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

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)

It's an octahedron. See Timaeus, sec. 55a-e. —The Doctahedron, 68.173.113.106 (talk) 22:39, 22 November 2011 (UTC)

## 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)

## 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)

## 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 ${\displaystyle pf=2e=vq}$ and that ${\displaystyle v-e+f=2}$. Multiplying the latter equation by ${\displaystyle pq}$ we obtain ${\displaystyle pqv-pqe+pqf=2pq}$. Substutiting from the first equation we have ${\displaystyle 2ep-pqe+2eq=2pq}$, which implies that ${\displaystyle e(2p-pq+2q)=2pq}$. Now ${\displaystyle e}$ and ${\displaystyle 2pq}$ are positive, so ${\displaystyle 2p-pq+2q}$ is as well. Since ${\displaystyle p}$ and ${\displaystyle q}$ must be at least ${\displaystyle 3}$, it is easy to see that the only possible values of ${\displaystyle (p,q)}$ are ${\displaystyle (3,3),(3,4),(4,3),(3,5),(5,3)}$. 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)

## 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)

## 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)

## 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)

## 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)

Those are truncated icosahedrons. —The Doctahedron, 68.173.113.106 (talk) 22:57, 23 November 2011 (UTC)

## 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.

## 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)

The Atiyah and Sutcliffe reference is known to be wrong, however. See http://math.ucr.edu/home/baez/icosahedron/ for example. If you actually examine the stone balls, you will not find an icosahedron. — Preceding unsigned comment added by 69.116.209.152 (talk) 16:39, 13 November 2011 (UTC)

## 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)

## 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)

## Plato's categorisation of the dodecahedron

In the article it states "The fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven"." I believe he related the twelve faces to the Zodiac, though I can't seem to dig up a reference. If indeed he did so, it would hardly be obscure. I'll keep digging for a reference. Nazlfrag (talk) 09:54, 1 July 2008 (UTC)

Plato writes, in the Timaeus: "There still remained a fifth construction, which the god used for embroidering the constellations on the whole heaven." Plato's statement is vague, and he gives no further explanation. Later Greek philosophers assign the dodecahedron to the ether or heaven or the cosmos. The dodecahedron has 12 faces, and our number symbolism associates 12 with the zodiac. (Copied from [2]).--John Wheater (talk) 11:20, 10 February 2010 (UTC)

I would like to eprsonally apologize. That was my class at school using laptops. You will see at one point the same IP who tried to stop the vandalism was vandalizing, that was because we all were on the same network.--NicholasHopkinzTalk! 01:01, 7 December 2008 (UTC)

## Category:Polyhedra vs. Category:Platonic solids

Category:Platonic solids is itself a category within Category:Polyhedra. — Robert Greer (talk) 03:06, 11 May 2009 (UTC)

## Definition

I hope I'm writing in the right place. I think it should be made clear what the simple, clear, definition of 'platonic solid' is. It's annoying having to read between the lines and reword the definition yourself. —Preceding unsigned comment added by 85.210.124.67 (talk) 19:03, 5 October 2009 (UTC)

(I moved that comment from above.) To what "simple, clear definition" are you referring? In your opinion, how could the intro be made clearer? (talk) 22:06, 5 October 2009 (UTC)
This article is an attractive (to me, maybe not to Mgnbar) mixture of mathematics and humanities. Maybe the intro is a bit heavy for the non-mathematician. Perhaps "Platonic solids are a particular group which exhibit extraordinary regularity and symmetry; their attributes were the subject of much discussion by ancient philosophers, whose ideas are still studied. They are also much discussed by modern mathematicians, and this aspect governs the bulk of this article". I'll maybe add this if no objection. --John Wheater (talk) 12:05, 10 February 2010 (UTC)
This issue arises in every math article (at least, every one that draws any attention from non-mathematicians). There is a specific Manual of Style entry for it. Yes, the intro would be better if it didn't hit the reader with so much jargon so quickly. So let's make changes. While we do, consider:
1. The intro shows pictures of all five Platonic solids, so the casual reader can get an idea of what they are without reading any text at all.
2. The intro already mentions their historical/philosophical relevance.
3. The intro can easily give the precise, succinct mathematical definition as well as an informal one.
4. To my knowledge the Platonic solids are not much studied by mathematicians after about 1900, so we should be clear in any use of the phrase "modern mathematicians".
In geometry, the Platonic solids are the five solids pictured here:
 Tetrahedron Cube (or hexahedron) Octahedron Dodecahedron Icosahedron (Animation) (Animation) (Animation) (Animation) (Animation)
Each of these solids exhibits extraordinary regularity and symmetry, in that its faces are all congruent, its edges are all of the same length, at each vertex the same number of edges meet producing angles of equal measure, etc. In precise mathematical language, a Platonic solid is a convex regular polyhedron.
The beauty and symmetry of the Platonic solids have made them a favorite subject of geometers for thousands of years. Theaetetus around ??? BCE proved that the five Platonic solids pictured above are the only solids satisfying the definition (of convexity and regularity). His contemporary Plato employed these solids in his philosophy, equating them with the elements of nature. The solids are now named after him.

## dual or polar (see dual)

JohnWheater (talk · contribs) is concerned that the text ought to suggest somehow that the word polar, as a synonym for dual, is clarified in Dual polyhedron; somehow he finds it inadequate that polar is mentioned explicitly as a synonym and that Dual polyhedron is explicitly linked in the preceding sentence. (See our personal talk pages.) I find this concern a little bit silly; consider it poor style to link Dual polyhedron a second time and even worse style to say "(see dual polyhedron)" without linking; and if I shared his concern I'd find his solution(s) unsatisfying. What language can make us both happy? —Tamfang (talk) 17:53, 13 May 2010 (UTC)

FWIW, I've never even heard of polar being used in this sense until I saw the recent edits. I've always known the concept by the name dual.—Tetracube (talk) 19:25, 13 May 2010 (UTC)

## Euler formula

For whatever it's worth, I agree with Glenn L (talk · contribs) about reverting the change by NA3349 (talk · contribs) to the expression of Euler's formula. Since Glenn gave no reason for doing so I'd like to put one on the record. I prefer V-E+F over F+V-E because it puts the elements in order of their dimension, and illustrates the beginning of a general (n-dimensional) pattern: the signs alternate. —Tamfang (talk) 22:15, 20 June 2010 (UTC)

## Removal of "medieval" elements, why?

I added the corresponding elements to the platonic solids in the table of "combinatorial properties", and now some intelligent person has decided to remove this relevant information. Do you think only mathematicians use this page of platonic solids?? I use it for meditation on the properties of the solids and the elements, when i first tried to see the solids and its elements, i had to browse up and down up and down to the relevant - difficult to find - sentence in the history section, and i decided to improve the article by making it easier to read!

if it wasnt for philosophers you wouldnt know what platonic solids is. mathematics and creation (elements) are all interconnected, if you havent reached the maturity to realize that, stop imposing your own lack of need for enlightenment on others, and let others use the articles for purposes more serious than you realize exists. i am now reverting the elements as "historically corresponding element". —Preceding unsigned comment added by 188.59.189.29 (talk) 18:58, 12 November 2010 (UTC)

I like having them there; they don't crowd the page. But I'm not in love with the title Historically corresponding element. —Tamfang (talk) 19:40, 12 November 2010 (UTC)
Currently these elemental labels are in the "Combinatorial properties" section, which doesn't make sense. What if we moved them to the table in the lead section? Mgnbar (talk) 22:44, 12 November 2010 (UTC)
Okay, but then the chart would need headings, which would spoil its elegance somewhat. —Tamfang (talk) 23:12, 12 November 2010 (UTC)
The header used to be just "Element" but it wasn't liked, so i added "historically corresponding" to it to make it more justifiable somehow, because someone didn't see it as relevant information. I see no problem with the elements being in "combinatorial properties" section. The numbers are though relevant to some, in regards to the elements. after all, "plato"nic solids and his school of philosophy is also about fundamental meanings of numbers, if im not mistaken. In that sense the corresponding element is in direct relationship to the numbers that the shapes represent and the other way around. I am happy the way it is, i was even happier before when the heading was just "Elements". If you want, you can perhaps change "combinatorial properties" to just "properties", and/or "h. c. element" to "element". —Preceding unsigned comment added by 188.58.27.116 (talk) 02:01, 15 November 2010 (UTC)
The term "combinatorics" (or "combinatorial") is a predominantly mathematical term, rather than a term in general English usage, right? In mathematics it has a fairly specific meaning — more specific than "relating to numbers", anyway — and I don't feel that the elements fit into that meaning. For example, if the History section is correct, then the association between the solids and the elements is largely due to how pointy the solids are. Pointiness is a geometric property, not a combinatorial one. Almost everything in the "Combinatorial properties" section is actually combinatorial, except for the elements. So I still vote that the elements be moved out of that section. Mgnbar (talk) 19:42, 15 November 2010 (UTC)

## sacred geometry

Shouldn't the article mention somewhere the relationship to sacred geometry? Jeiki Rebirth (talk) 22:14, 8 February 2011 (UTC)

Which specific relationship, and what source for it? AnonMoos (talk) 03:43, 9 February 2011 (UTC)

## 24-cell and hexagon

KirbyRider says that the 24-cell and regular hexagon are both "Truncation of a simplex-faceted polytope that has simplices for ridges and is self-dual". What simplex-faceted polytope is truncated to derive the 24-cell? —Tamfang (talk) 06:05, 9 June 2011 (UTC)

A 24-cell is a rectified 16-cell, but not a truncation. Tom Ruen (talk) 05:07, 10 June 2011 (UTC)

## Duality, truncation, snubbing?

The abstract reads "With Duality, Truncation, and snubbing, the Tetrahedron first forms the other triangular platonic solids, then duality makes the two other ones." It's unclear what this is talking about, as truncation is only mentioned once elsewhere in the article and duality and snubbing are not mentioned at all. The irregular capitalization is also jarring. Should this sentence be clarified or removed? Qartar (talk) 03:14, 14 June 2011 (UTC)

I'm not sure that belongs here, meaning show in table at Uniform_polyhedron#Summary_tables, octahedron is a rectified tetrahedron, and icosahedron is a snub icosahedron. Conway polyhedron notation also contains these relations and more, so constructionally all 5 can be generated from the tetrahedron: (T=T, O=aT, C=daT, I=sT, D=dsT) Tom Ruen (talk) 03:41, 14 June 2011 (UTC)

I created a new image for possible use on Wikipedia. It shows the relationships between the Platonic solids by truncation, duality, and snubbing.

—The Doctahedron, 68.173.113.106 (talk) 22:50, 23 November 2011 (UTC)

It is more accurate to say the octahedron is a rectification (geometry) of the tetrahedron, while a truncated tetrahedron is something a bit different. Tom Ruen (talk) 22:57, 23 November 2011 (UTC)
It's essentially the same process. Rectification, truncation, whatever. Except that rectification cuts off more of the solid than regular truncation. My point is, is this image suitable for use on Wikipedia, other than the fact that it needs to be vectorized and transparentized? —The Doctahedron (talk) 19:00, 24 November 2011 (UTC)

## Similarity classes

A recent edit changed the statement "there are five Platonic solids" to the statement "there are five similarity classes of Platonic solids". While technically correct, this statement is counterproductive. It unnecessarily complicates the statement of the classification, by bringing in technical issues such as equivalence relation. I vote that it be changed back. Mgnbar (talk) 21:26, 10 December 2011 (UTC)

I belatedly agree, and like the shortened statement. Ridcully Jack (talk) 00:40, 12 December 2011 (UTC)
Thanks, “up to similarity” is elegant and precise, much better than my complicated formulation. --Chricho ∀ (talk) 21:47, 13 December 2011 (UTC)
"Up to similarity", however, is not proper English. Mangoe (talk) 22:43, 16 April 2012 (UTC)
I utterly disagree. This construction is found in countless math textbooks. Whether or not you have heard it before now, it is idiomatic to mathematics writing, and it is correct and appropriate for this domain of discourse. Mgnbar (talk) 22:48, 16 April 2012 (UTC)
I, as a mathematician, find it awkward and obscuring for people who aren't mathematicians; we are not committed to use of our jargon. Similarity (geometry) does not even use the phrase, nor do I see it used in (for instance) Archimedean solid. Personally, I would prefer to stick to "there are five Platonic solids" given that people intuit that size does not matter here. Mangoe (talk) 01:50, 17 April 2012 (UTC)
It's okay with me, to omit the phrase. As you can see from reading this section of the talk page, it was added by someone other than me. You're probably right, that people understand that size doesn't matter. Mgnbar (talk) 02:34, 17 April 2012 (UTC)

## Moon model

I've moved the sentence about the "Moon Model" for electron shells spaced at nested Platonic solids from the introductory paragraph. There are dozens of facts more important than this in the main body.

For now it is in the more appropriate "History" section. It's there because this is the other place where other spurious models of the real world which are not supported by the scientific community sit. There's nothing at electron shell model about the "Moon model", which is telling. Ridcully Jack (talk) 09:34, 28 March 2012 (UTC)

## Intro section

Recently, there has been great activity in modifying the intro section. This is one of many math articles that are not esoteric and interesting only to mathematicians, but which ordinary people may read. It is essential that the intro remain clear, concise, and correct. If you wish to propose a modification to the intro, then please do so here, so that we can form consensus.

In particular, the most recent edits have employed convoluted sentence structure and unexplained jargon ("superposable") that I don't understand, even with an advanced degree in geometry. That's exactly what we need to avoid. Mgnbar (talk) 20:50, 16 April 2012 (UTC)

Why do people keep making controversial edits to the intro, even after I've asked that we discuss such edits here? Please, please let's talk about it, instead of just reverting each other. Mgnbar (talk) 12:15, 17 April 2012 (UTC)
I've tried a much less technical version, given that this isn't just an article for mathematicians. Mangoe (talk) 13:24, 17 April 2012 (UTC)
And there's no need to say "In Euclidean geometry", as there aren't some other Platonic solids in some other field. Mangoe (talk) 16:28, 17 April 2012 (UTC)
That's true. It's just that many Wikipedia articles begin with an explicit indication, of which discipline they belong to: "In mathematics", "in geometry", etc. Mgnbar (talk) 17:29, 17 April 2012 (UTC)
In what sense is "a Platonic solid is a polyhedron that is regular and convex" superior to "a Platonic solid is a regular, convex polyhedron"? I don't understand why Aughost is aggressively pushing the former. The latter expresses the same content, more succinctly. Mgnbar (talk) 23:26, 17 April 2012 (UTC)
It isn't. Mangoe (talk) 23:49, 17 April 2012 (UTC)
There might be some mention of the synonymous term "perfect solid" in the intro. Some non-mathematicians may search for that (dated) terminology. 75.150.168.6 (talk) 20:33, 26 August 2013 (UTC)

why there are only 5 platonic solids need to be prove in a reasonable way — Preceding unsigned comment added by 111.194.118.16 (talk) 09:42, 27 March 2013 (UTC)

This talk page is for improving the article, not for giving mathematics help. Try asking at Wikipedia:Reference desk/Mathematics. Also, there is a proof in the article. Also, this sounds like a school assignment. Mgnbar (talk) 14:03, 27 March 2013 (UTC)

## Geodesic Sphere

The current "mathematical" definition is incorrect as an infinite variety of solids meet that categorisation can someone write something better without the illusion of mathematical purity. So for example the Platonic solids are more fundamental, perhaps the 5 solids are the most fundamental varieties. It might be more accurate to mention Plato also and his theories of archetypes. DarkShroom (talk) 20:16, 31 December 2013 (UTC)

Geodesic spheres use both hexagons and pentagons, so it is not true that all of their faces are congruent. In addition, for high-order geodesic spheres, not all the hexagons are regular. (Similarly, the geodesic spheres made out of triangles, with the exception of the icosahedron, contain triangles that are not equilateral and in general are not congruent to each other.) Therefore, they do not contradict the information in this article. —David Eppstein (talk) 21:34, 31 December 2013 (UTC)

## Proposals of more data in the article

The article collects info on Platonic solid in a very nice way, however some more info would be useful. Here is what I mean:

1. A chart with coordinates of all 5 solids in the same place (now the info is to be found in separate articles on the solids).
2. The edge central angle = the vertex-centre-vertex angle (with both vertices of the same edge) - no info so far anywhere, except on the tetrahedron (in the article on it). The values are: π − arc cos (1/3), arc cos (1/3), π/2, arc cos (√5/3), arc cos (1/√5) = arc tg 2 for the five solids respectively (which means 109.471221°, 70.528779°, 90°, 41.810315°, 63.434949° approx.).
3. The face-vertex-edge angle (except the octahedron) - no info so far anywhere, except on the tetrahedron (in the article on it).
4. The radius of exspheres. As for now, the info is available for the tetrahedron in the article on it, and for the cube and the icosahedron in the article on the exsphere. No info on the octahedron and the dodecahedron anywhere.
5. The distance to exsphere center from a vertex.
6. The solid diagonals (if any): their number and length.

Note that there is extensive info in the article on the tetrahedron - but not in articles on other Platonic solids. So, adding the information listed above would mean just levelling details in various articles.

31.11.242.188 (talk) 12:37, 29 December 2014 (UTC)

I added a new section and table of coordinates, and some pictures that might help. 20:46, 29 December 2014 (UTC)

## Explicit construction

Please comment on these edits regarding Platonic solid#Classification:

• 4 August 2013 changed the first of the following to the second:
• That all five actually exist is a separate question – one that can be answered easily by an explicit construction.
• ...positively demonstrating the existence of any given solid is a separate question – one that an explicit construction cannot easily answer
• 4 May 2015 changed the last clause of the above to:
• one that can be answered easily by an explicit construction

I reverted the last edit in the belief that an explicit construction would only show an approximate solution. I have copied a message that was posted on my talk, and have asked for thoughts at WT:WikiProject Mathematics#Platonic solid - Classification. Johnuniq (talk) 06:53, 5 May 2015 (UTC)

Copied from User talk:Johnuniq#Platonic solid - Classification.
I'm afraid that you've revoked my recent modification in error. I restored only an erroneous modification made in revision 567073300 by Duxwing.
The original sentence (that was restored by me) is:

That all five actually exist is a separate question – one that can be answered easily by an explicit construction.

Duxwing's sentence (that was restored by you) is:

positively demonstrating the existence of any given solid is a separate question – one that an explicit construction cannot easily answer.

I think the point is that a construction can only be shown to be *approximately* correct

This isn't true. For example, a cube is the ${\displaystyle \{(x,y,z):0\leq x,y,z\leq 1\}}$ set. This is absolutely exact. There is no approximation. So please reconsider this.
Thanks, 89.135.19.75 (talk) 05:22, 5 May 2015 (UTC)
I tend to agree with 89.135.19.75. In what sense is a construction "explicit" if it doesn't actually show that these solids exist? And coordinates for the cube, tetrahedron, and octahedron are very easy to construct and prove regular. The dodecahedron and icosahedron are a bit more work (not so much writing down the coordinates — our artcles have formulas for those that do not involve any approximation — but checking that they do in fact define regular polytopes) but still very explicit. —David Eppstein (talk) 07:32, 5 May 2015 (UTC)
Yes, I see what you mean, and I now see that a list of coordinates is an example of an "explicit construction". I was confused over what exactly that last phrase meant. I think some improvement of the wording is needed because "but positively demonstrating" only works with the "cannot easily answer" text. Instead of:
Two common arguments below demonstrate no more than five Platonic solids can exist, but positively demonstrating the existence of any given solid is a separate question – one that can be answered easily by an explicit construction.
It should be something like:
Two common arguments below demonstrate no more than five Platonic solids can exist, and each of the five possible solids can easily be shown to exist.
Johnuniq (talk) 08:37, 5 May 2015 (UTC)

Presumably there is a general way to see that a polygonal net determines a polyhedron. Could this be illustrated in the article (perhaps for the dodecahedron)? I imagine that would settle the issue of whether there exist explicit constructions of the five solids. Sławomir Biały (talk) 11:39, 5 May 2015 (UTC)

• I don't think there's an issue to settle. I'll also point out that the 2013 edit in question was marked "minor" and summarized "copyedit". I'm guessing the copyeditor simply misread the sentence and did not intend to reverse its meaning. --GodMadeTheIntegers (talk) 14:41, 5 May 2015 (UTC)
Duxwing (who seems not to have edited in the past few months) was a singularly incompetent editor whose "copyediting" generally made text harder to understand and regularly destroyed the meaning of technical language. This kind of error is totally typical. --JBL (talk) 15:35, 5 May 2015 (UTC)

(edit conflict)It also occurs to me that since we are referencing Euclid's proof, "construction" may intend a "synthetic" ruler-and-compass construction in the same style. So e.g.: equilateral triangles are constructible, their centres are constructible, perpendiculars through a plane are constructible, ${\displaystyle {\sqrt {2 \over 3}}}$ is a constructible number. So, mark that point above the centre of an equilateral triangle, and prove that all four points are equidistant. You therefore have a regular tetrahedron. Might be worth adding. --GodMadeTheIntegers (talk) 15:39, 5 May 2015 (UTC)

Seeing consensus at https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics#Platonic_solid_-_Classification, I boldly corrected the statement to “positively demonstrating the existence of any given solid is a separate question—one that requires an explicit construction”. Incidentally, I believe that the question of how to determine whether a given net actually determines a polyhedron is still open. I recall there is a discussion in O'Rourke How to Fold It: The Mathematics of Linkages, Origami, and Polyhedra. —Mark Dominus (talk) 15:05, 7 May 2015 (UTC)
Every net that looks like it determines a convex polyhedron actually does, although figuring out which polyhedron it determines is nontrivial. This is Alexandrov's uniqueness theorem. The open question is whether every convex polyhedron has a planar net. —David Eppstein (talk) 15:42, 7 May 2015 (UTC)
O'Rourke also claims (p. 139) that characterizing the polygons that fold to convex polyhedra is open. (If I understand correctly, Alexandrov's theorem gives an equivalent condition but not an algorithm for determining if a given net actually has a gluing of the needed form.) --JBL (talk) 15:50, 7 May 2015 (UTC)
Though it is worth noting that his notion of "net" does not come with predetermined fold lines. --JBL (talk) 15:50, 7 May 2015 (UTC)

## Why so many dodecahedron vs. icosahedron edits?

Can someone explain why a particular kind of bad edit happens so frequently on this page: switching the meanings of dodecahedron and icosahedron? Is there something in the text that confuses editors? Or is this a bizarrely persistent case of vandalism, from multiple accounts and IPs? Mgnbar (talk) 19:45, 21 July 2016 (UTC)

The confusion in the lower table is caused by the fact that it lists number of vertices and not faces, probably. --JBL (talk) 21:58, 21 July 2016 (UTC)
My confusion was based on a bad/vague understanding of latin, assuming that "dodeca" surely meant "twice ten" and thus was the name of the twenty sided shape. Curiousepic (talk) 13:29, 30 August 2016 (UTC)

## Octahedron vertex angles

A table lists the "vertex angle" of an octahedron as "60°, 90°." According to the vertex angle article, "a vertex angle is an angle formed by two edges of the polyhedron that both belong to a common two-dimensional face of the polyhedron." The only 90° angle at a vertex of an octahedron is formed by two edges that DO NOT belong to the same face. ALL the angles formed by edges belonging to the same face are 60°. Unless someone can refute this, I'll make the change soon, or someone else can. Thank you. Holy (talk) 17:17, 3 January 2017 (UTC)

It is confusing. The vertex angle has no source on the definition on a polyhedra. I traced the edit to an anonymous editor in 2009, [3]. I'd say remove the entire column, UNLESS there's a clear reference for the definition on a polyhedra AND a source the describes these on the platonic solids. Tom Ruen (talk) 11:49, 4 January 2017 (UTC)
I removed the column, not finding any sources. Tom Ruen (talk) 05:11, 5 January 2017 (UTC)