This is the
talk page for discussing improvements to the
Platonic solid article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
This
level-5 vital article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||||||||||||||||||
|
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 McKay 24.200.80.227 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
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 |
The following was proven by Descartes and Leonhard Euler.
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.
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
If we divide both sides of this equation by 2E, we have
Charles Matthews 07:48, 21 Sep 2004 (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)
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)
why dodecahedrom is randomly bold in the list?
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)
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 and that . Multiplying the latter equation by we obtain . Substutiting from the first equation we have , which implies that . Now and are positive, so is as well. Since and must be at least , it is easy to see that the only possible values of are . Joshua Davis 14:08, 2 June 2006 (UTC)
Should that be mentioned?-- user talk:hillgentleman 08:52, 22 November 2006 (UTC)
(to User:Rsholmes) I am well aware not every dodecahedron is regular, but the place to discuss that is in the dodecahedron 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 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:
{{
cite book}}
: Cite has empty unknown parameter: |coauthors=
(
help)sloth_monkey 09:47, 28 December 2006 (UTC)
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)
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? Thanks 159.91.19.3 22:34, 29 March 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.
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)
Joshua, Thanks for that. Rosser 10:08, 18 May 2007 (UTC)
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)
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)
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)
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.-- NicholasHopkinz Talk! 01:01, 7 December 2008 (UTC)
Category:Platonic solids is itself a category within Category:Polyhedra. — Robert Greer ( talk) 03:06, 11 May 2009 (UTC)
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)
Tetrahedron |
Cube (or hexahedron) |
Octahedron | Dodecahedron | Icosahedron |
( Animation) |
( Animation) |
( Animation) |
( Animation) |
( Animation) |
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)
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)
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)
Shouldn't the article mention somewhere the relationship to sacred geometry? Jeiki Rebirth ( talk) 22:14, 8 February 2011 (UTC)
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)
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 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)
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'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)
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 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)
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)
The article collects info on Platonic solid in a very nice way, however some more info would be useful. Here is what I mean:
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)
Please comment on these edits regarding Platonic solid#Classification:
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)
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.
I think the point is that a construction can only be shown to be *approximately* correct
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)
(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, 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)
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)
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 might be interesting to add a reference to a concept called "Metatrons Cube". See explanation here: https://www.youtube.com/watch?v=YhSKFVCa3V0 — Preceding unsigned comment added by 2.244.3.69 ( talk • contribs) 21:43, 23 March 2020 (UTC)
all platonic solids have the property of being able to be inscribed in succession, as follows: Icosahedron contains dodecahedron, that contains hexahedron, that contains tetrahedron, that contains octahedron. Can that be indicated within the article? — Preceding unsigned comment added by 181.64.192.44 ( talk) 22:16, 30 October 2020 (UTC)
It should be worth mentioning that Plato is a very great writer and stylist in purely literary terms. If I could find the place, the late RGM Nisbet called him the greatest prose stylist there had ever been, with Cicero second. We schoolboys and undergraduates had to try to write like him. (I was Seadowns until my name was changed through a muddle over passwords.) Esedowns ( talk) 08:52, 12 September 2021 (UTC)
Why isn't the below an improvement? Don't just delete it without stating your reasons. "Oh my goodness" is not a reason. It is very elucidating to know the basis for the solids which is nowhere else stated. It might actually help you to understand the philosophy behind it. I note the Group order reconditely alludes to the basis, but it is not clear.
The text is referenced and reasonable and logical.
Basis for the Platonic Solids
The basis for all the Platonic solids are the isosceles and equilateral triangles. Further, the isosceles triangle is called a semi-square and half of the equilateral triangle, which is bisected by a perpendicular from the vertex to the base is known as a semi-triangle. This is elaborated by Simplicius, Proclus and Taylor:
" They supposed two primogenial right-angled triangles, the one isosceles, but the other scalene, having the greater side the double in length of the less, and which they call a semi-triangle, because it is the half of the equilateral triangle, which is bisected by a perpendicular from the vertex to the base. And from the isosceles triangle, which Timaeus calls a semi-square, four such having their right angles conjoined in one centre, a square is formed. The semi-triangle, however, constitutes the pyramid, the octaedron, and the icosaedron, which are distributed to fire, air, and water.
And the pyramid, indeed, consists of four equilateral triangles, each of which composes six semi-triangles. But the octaedron consists of eight equilateral triangles, and forty-eight semi-triangles; and the icosaedron is formed from twenty equilateral triangles, but one hundred and twenty semi-triangles. Hence, these three, deriving their composition from one element, viz. the semi-triangle, are naturally adapted, according to the Pythagoreans and Plato, to be changed into each other; but earth, as deriving its composition from another triangle specifically different, can neither be resolved into the other three bodies, nor be composed from them."
Hence the triangular pyramid [tetrahedron], or the element of fire is comprised of 24 semi-triangles; the octahedron, or the element of air is comprised of 48 semi-triangles; and the icosahedron, or the element of water is comprised of 120 semi-triangles. The cube, or the element of earth is comprised of 6 squares or 12 semi-squares [ 24 according to the Timaeus which says 4 semi-squares make a square.]
— Preceding unsigned comment added by Darylprasad ( talk • contribs)
The semi-triangle, however, constitutes the pyramid, the octaedron, and the icosaedron, which are distributed to fire, air, and wateretc.), it's via modern historical work commenting on the Greek views. We already have such a discussion in the History section of this article (it even mentions Proclus). It's totally possible that it could be improved, but additions of long quotes from primary sources are not a step in the right direction. -- JBL ( talk) 20:57, 19 October 2021 (UTC)
As a post postscript I would like to leave a copy of the final text, in square brackets, should anyone see fit to put it somewhere in Wiki, either in this article, or "Classical elements"
[Historically, the basis for all the Platonic solids was an isosceles triangle and an equilateral triangle. Further, the isosceles triangle was called a semi-square and half of the equilateral triangle, which is bisected by a perpendicular from the vertex to the base was known as a semi-triangle. This is elaborated by Simplicius, Proclus and Taylor as follows:
"They [the Pythagoreans and Plato] supposed two primogenial right-angled triangles, the one isosceles, but the other scalene, having the greater side the double in length of the less, and which they call a semi-triangle, because it is the half of the equilateral triangle, which is bisected by a perpendicular from the vertex to the base. And from the isosceles triangle, which Timaeus calls a semi-square, four such having their right angles conjoined in one centre, a square is formed. The semi-triangle, however, constitutes the pyramid, the octaedron, and the icosaedron, which are distributed to fire, air, and water.
And the pyramid, indeed, consists of four equilateral triangles, each of which composes six semi-triangles. But the octaedron consists of eight equilateral triangles, and forty-eight semi-triangles; and the icosaedron is formed from twenty equilateral triangles, but one hundred and twenty semi-triangles. Hence, these three, deriving their composition from one element, viz. the semi-triangle, are naturally adapted, according to the Pythagoreans and Plato, to be changed into each other; but earth, as deriving its composition from another triangle specifically different, can neither be resolved into the other three bodies, nor be composed from them."
Hence the triangular pyramid [tetrahedron], or the element of fire is comprised of 24 semi-triangles; the octahedron, or the element of air is comprised of 48 semi-triangles; and the icosahedron, or the element of water is comprised of 120 semi-triangles. The cube, or the element of earth is comprised of 6 squares or 12 semi-squares [24 according to the Timaeus which says 4 semi-squares make a square.]
And when Proclus says, "that in the dissolution of water into air, when fire resolves it, two parts of air are generated, and one part of fire" the calculation of water to air from fire is made as follows: the element water = 120 semi-triangles = (2 x 48 semi-triangles) + (1 x 24 semi-triangles) = two parts of the element air and one part of the element fire..]
with the reference being: "The Fragments That Remain of the Lost Writings of Proclus" translated by Thomas Taylor 1825, Published by Black, Young, and Young, Tavistock-street, Covent Garden London, pp.11-12 with the scholiast, 14 id: ark:/13960/t1bk58w3b Darylprasad ( talk) 22:05, 19 October 2021 (UTC)
There is a move discussion in progress on Talk:Kepler–Poinsot polyhedron which affects this page. Please participate on that page and not in this talk page section. Thank you. — RMCD bot 23:16, 24 October 2021 (UTC)
Is there a reason to have the backgrounds of the diagrams in Platonic_solid#Stereographic_projection saturated yellow and red? It hurts the eyes and doesn't seem to add any value. Cheers, cmɢʟee⎆ τaʟκ 03:43, 7 December 2022 (UTC)
I suggest to remove the section. As the stereographic projection projects a line to a line, the images cannot be the stereographic projection of the edges of the platonic solid. It is probably (I have not verified the details) the stereographic projection of the central projection of the edges on a circumscribed sphere. I have fixed the sentence of the article, but the section may remain confusing for many reader. So, I strongly recommend to remove the section that adds nothing to the understanding of the article subject. D.Lazard ( talk) 20:33, 7 December 2022 (UTC)
An editor has identified a potential problem with the redirect Where's the d10? and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2023 January 14 § Where's the d10? until a consensus is reached, and readers of this page are welcome to contribute to the discussion. ✠ SunDawn ✠ (contact) 10:19, 14 January 2023 (UTC)
This article says that the reason Platonic solids are used to make dice is that they can be made fair. That's not the reason. Plenty of other shapes could be made fair.
It also implies (if not states) that they're the only shapes commonly used to make dice, which is not true. Games that use Platonic solids also usually use 10 sided dice, which aren't Platonic solids.
I fixed these errors, but for some reason someone reverted my fix.
From the section above this one, it looks like there's been some discussion of this in the past. Although some of that was deleted, so it's hard to tell what happened. - Burner89751654 ( talk) 01:08, 10 April 2023 (UTC)
Could a sphere be an honorary platonic solid? It is made up of congruent (honorary) regular polygons (circles) with the same number of faces at each vertex (0) 31.94.9.193 ( talk) 12:24, 6 July 2023 (UTC)
Maybe this had been discussed somewhere else, but why was (hexahedron) removed? I am not a math guy, and English is not my first language, so sorry for any inaccurate terminologies. From what I read, these *hedrons means a 3d shape with * faces (maybe the definition is stricter, but that is not my point), and there are regular and irregular *hedrons. The use of the word "cube" implies that you are talking about a regular hexahedron, but according to this logic, why are the other solids not referred to as "regular *hedron"? In my opinion, it would be better to use "hexahedron (cube)", that way it is more aligned with the other solids, but of course "hexahedron" or "cube (hexehedron)" are also reasonable. Changing all instances of "cube" would not be necessary, I suggest changing only the instances in the tables, and perhaps the first one or two use of "cube" in the paragraphs. At the very least, let the word "hexahedron" appear once in the article, I find it strange that the current page does not even have one instance of that word. Sohryu Asuka Langley Not Shikinami ( talk) 01:55, 27 April 2024 (UTC)
This is the
talk page for discussing improvements to the
Platonic solid article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
This
level-5 vital article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||||||||||||||||||
|
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 McKay 24.200.80.227 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
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 |
The following was proven by Descartes and Leonhard Euler.
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.
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
If we divide both sides of this equation by 2E, we have
Charles Matthews 07:48, 21 Sep 2004 (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)
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)
why dodecahedrom is randomly bold in the list?
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)
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 and that . Multiplying the latter equation by we obtain . Substutiting from the first equation we have , which implies that . Now and are positive, so is as well. Since and must be at least , it is easy to see that the only possible values of are . Joshua Davis 14:08, 2 June 2006 (UTC)
Should that be mentioned?-- user talk:hillgentleman 08:52, 22 November 2006 (UTC)
(to User:Rsholmes) I am well aware not every dodecahedron is regular, but the place to discuss that is in the dodecahedron 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 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:
{{
cite book}}
: Cite has empty unknown parameter: |coauthors=
(
help)sloth_monkey 09:47, 28 December 2006 (UTC)
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)
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? Thanks 159.91.19.3 22:34, 29 March 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.
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)
Joshua, Thanks for that. Rosser 10:08, 18 May 2007 (UTC)
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)
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)
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)
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.-- NicholasHopkinz Talk! 01:01, 7 December 2008 (UTC)
Category:Platonic solids is itself a category within Category:Polyhedra. — Robert Greer ( talk) 03:06, 11 May 2009 (UTC)
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)
Tetrahedron |
Cube (or hexahedron) |
Octahedron | Dodecahedron | Icosahedron |
( Animation) |
( Animation) |
( Animation) |
( Animation) |
( Animation) |
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)
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)
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)
Shouldn't the article mention somewhere the relationship to sacred geometry? Jeiki Rebirth ( talk) 22:14, 8 February 2011 (UTC)
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)
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 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)
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'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)
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 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)
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)
The article collects info on Platonic solid in a very nice way, however some more info would be useful. Here is what I mean:
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)
Please comment on these edits regarding Platonic solid#Classification:
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)
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.
I think the point is that a construction can only be shown to be *approximately* correct
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)
(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, 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)
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)
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 might be interesting to add a reference to a concept called "Metatrons Cube". See explanation here: https://www.youtube.com/watch?v=YhSKFVCa3V0 — Preceding unsigned comment added by 2.244.3.69 ( talk • contribs) 21:43, 23 March 2020 (UTC)
all platonic solids have the property of being able to be inscribed in succession, as follows: Icosahedron contains dodecahedron, that contains hexahedron, that contains tetrahedron, that contains octahedron. Can that be indicated within the article? — Preceding unsigned comment added by 181.64.192.44 ( talk) 22:16, 30 October 2020 (UTC)
It should be worth mentioning that Plato is a very great writer and stylist in purely literary terms. If I could find the place, the late RGM Nisbet called him the greatest prose stylist there had ever been, with Cicero second. We schoolboys and undergraduates had to try to write like him. (I was Seadowns until my name was changed through a muddle over passwords.) Esedowns ( talk) 08:52, 12 September 2021 (UTC)
Why isn't the below an improvement? Don't just delete it without stating your reasons. "Oh my goodness" is not a reason. It is very elucidating to know the basis for the solids which is nowhere else stated. It might actually help you to understand the philosophy behind it. I note the Group order reconditely alludes to the basis, but it is not clear.
The text is referenced and reasonable and logical.
Basis for the Platonic Solids
The basis for all the Platonic solids are the isosceles and equilateral triangles. Further, the isosceles triangle is called a semi-square and half of the equilateral triangle, which is bisected by a perpendicular from the vertex to the base is known as a semi-triangle. This is elaborated by Simplicius, Proclus and Taylor:
" They supposed two primogenial right-angled triangles, the one isosceles, but the other scalene, having the greater side the double in length of the less, and which they call a semi-triangle, because it is the half of the equilateral triangle, which is bisected by a perpendicular from the vertex to the base. And from the isosceles triangle, which Timaeus calls a semi-square, four such having their right angles conjoined in one centre, a square is formed. The semi-triangle, however, constitutes the pyramid, the octaedron, and the icosaedron, which are distributed to fire, air, and water.
And the pyramid, indeed, consists of four equilateral triangles, each of which composes six semi-triangles. But the octaedron consists of eight equilateral triangles, and forty-eight semi-triangles; and the icosaedron is formed from twenty equilateral triangles, but one hundred and twenty semi-triangles. Hence, these three, deriving their composition from one element, viz. the semi-triangle, are naturally adapted, according to the Pythagoreans and Plato, to be changed into each other; but earth, as deriving its composition from another triangle specifically different, can neither be resolved into the other three bodies, nor be composed from them."
Hence the triangular pyramid [tetrahedron], or the element of fire is comprised of 24 semi-triangles; the octahedron, or the element of air is comprised of 48 semi-triangles; and the icosahedron, or the element of water is comprised of 120 semi-triangles. The cube, or the element of earth is comprised of 6 squares or 12 semi-squares [ 24 according to the Timaeus which says 4 semi-squares make a square.]
— Preceding unsigned comment added by Darylprasad ( talk • contribs)
The semi-triangle, however, constitutes the pyramid, the octaedron, and the icosaedron, which are distributed to fire, air, and wateretc.), it's via modern historical work commenting on the Greek views. We already have such a discussion in the History section of this article (it even mentions Proclus). It's totally possible that it could be improved, but additions of long quotes from primary sources are not a step in the right direction. -- JBL ( talk) 20:57, 19 October 2021 (UTC)
As a post postscript I would like to leave a copy of the final text, in square brackets, should anyone see fit to put it somewhere in Wiki, either in this article, or "Classical elements"
[Historically, the basis for all the Platonic solids was an isosceles triangle and an equilateral triangle. Further, the isosceles triangle was called a semi-square and half of the equilateral triangle, which is bisected by a perpendicular from the vertex to the base was known as a semi-triangle. This is elaborated by Simplicius, Proclus and Taylor as follows:
"They [the Pythagoreans and Plato] supposed two primogenial right-angled triangles, the one isosceles, but the other scalene, having the greater side the double in length of the less, and which they call a semi-triangle, because it is the half of the equilateral triangle, which is bisected by a perpendicular from the vertex to the base. And from the isosceles triangle, which Timaeus calls a semi-square, four such having their right angles conjoined in one centre, a square is formed. The semi-triangle, however, constitutes the pyramid, the octaedron, and the icosaedron, which are distributed to fire, air, and water.
And the pyramid, indeed, consists of four equilateral triangles, each of which composes six semi-triangles. But the octaedron consists of eight equilateral triangles, and forty-eight semi-triangles; and the icosaedron is formed from twenty equilateral triangles, but one hundred and twenty semi-triangles. Hence, these three, deriving their composition from one element, viz. the semi-triangle, are naturally adapted, according to the Pythagoreans and Plato, to be changed into each other; but earth, as deriving its composition from another triangle specifically different, can neither be resolved into the other three bodies, nor be composed from them."
Hence the triangular pyramid [tetrahedron], or the element of fire is comprised of 24 semi-triangles; the octahedron, or the element of air is comprised of 48 semi-triangles; and the icosahedron, or the element of water is comprised of 120 semi-triangles. The cube, or the element of earth is comprised of 6 squares or 12 semi-squares [24 according to the Timaeus which says 4 semi-squares make a square.]
And when Proclus says, "that in the dissolution of water into air, when fire resolves it, two parts of air are generated, and one part of fire" the calculation of water to air from fire is made as follows: the element water = 120 semi-triangles = (2 x 48 semi-triangles) + (1 x 24 semi-triangles) = two parts of the element air and one part of the element fire..]
with the reference being: "The Fragments That Remain of the Lost Writings of Proclus" translated by Thomas Taylor 1825, Published by Black, Young, and Young, Tavistock-street, Covent Garden London, pp.11-12 with the scholiast, 14 id: ark:/13960/t1bk58w3b Darylprasad ( talk) 22:05, 19 October 2021 (UTC)
There is a move discussion in progress on Talk:Kepler–Poinsot polyhedron which affects this page. Please participate on that page and not in this talk page section. Thank you. — RMCD bot 23:16, 24 October 2021 (UTC)
Is there a reason to have the backgrounds of the diagrams in Platonic_solid#Stereographic_projection saturated yellow and red? It hurts the eyes and doesn't seem to add any value. Cheers, cmɢʟee⎆ τaʟκ 03:43, 7 December 2022 (UTC)
I suggest to remove the section. As the stereographic projection projects a line to a line, the images cannot be the stereographic projection of the edges of the platonic solid. It is probably (I have not verified the details) the stereographic projection of the central projection of the edges on a circumscribed sphere. I have fixed the sentence of the article, but the section may remain confusing for many reader. So, I strongly recommend to remove the section that adds nothing to the understanding of the article subject. D.Lazard ( talk) 20:33, 7 December 2022 (UTC)
An editor has identified a potential problem with the redirect Where's the d10? and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2023 January 14 § Where's the d10? until a consensus is reached, and readers of this page are welcome to contribute to the discussion. ✠ SunDawn ✠ (contact) 10:19, 14 January 2023 (UTC)
This article says that the reason Platonic solids are used to make dice is that they can be made fair. That's not the reason. Plenty of other shapes could be made fair.
It also implies (if not states) that they're the only shapes commonly used to make dice, which is not true. Games that use Platonic solids also usually use 10 sided dice, which aren't Platonic solids.
I fixed these errors, but for some reason someone reverted my fix.
From the section above this one, it looks like there's been some discussion of this in the past. Although some of that was deleted, so it's hard to tell what happened. - Burner89751654 ( talk) 01:08, 10 April 2023 (UTC)
Could a sphere be an honorary platonic solid? It is made up of congruent (honorary) regular polygons (circles) with the same number of faces at each vertex (0) 31.94.9.193 ( talk) 12:24, 6 July 2023 (UTC)
Maybe this had been discussed somewhere else, but why was (hexahedron) removed? I am not a math guy, and English is not my first language, so sorry for any inaccurate terminologies. From what I read, these *hedrons means a 3d shape with * faces (maybe the definition is stricter, but that is not my point), and there are regular and irregular *hedrons. The use of the word "cube" implies that you are talking about a regular hexahedron, but according to this logic, why are the other solids not referred to as "regular *hedron"? In my opinion, it would be better to use "hexahedron (cube)", that way it is more aligned with the other solids, but of course "hexahedron" or "cube (hexehedron)" are also reasonable. Changing all instances of "cube" would not be necessary, I suggest changing only the instances in the tables, and perhaps the first one or two use of "cube" in the paragraphs. At the very least, let the word "hexahedron" appear once in the article, I find it strange that the current page does not even have one instance of that word. Sohryu Asuka Langley Not Shikinami ( talk) 01:55, 27 April 2024 (UTC)