# Why isn't a sphere a Platonic Solid? **URL:** \ **Category:** Factual Questions\ **Created:** [July 7, 2017, 2:47pm UTC](https://boards.straightdope.com/t/why-isnt-a-sphere-a-platonic-solid/790692 "2017-07-07T14:47:00Z")\ **Posts on this page:** 9\ **Page:** 3
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [July 13, 2017, 8:11pm UTC](https://boards.straightdope.com/t/why-isnt-a-sphere-a-platonic-solid/790692/41 "2017-07-13T20:11:54Z")
> [@Chronos](#): > > I think I understand what you’re saying, but some parts of your phrasing are puzzling. It sounds like you’re saying to make such bands along meridian lines, taking each band as close to the poles as it can get without overlapping other bands. But what do you mean by “From any polygon n, where n is an odd number…”? By that I mean I had a weird brain fart. A weird and wrong brain fart. > [@](#): > > I’m also not convinced that this method (if that is indeed what you’re proposing) would approach total coverage in the limit of small triangles. If your triangle size is chosen such that there are no gaps between bands at the equator, then as soon as you get off of the equator, half of your bands will be excluded, and you’ll have wedge-shaped gaps extending all the way up (and down) to ±60º. And at the same point where those gaps close, you need to end even more bands, and so have another range of latitudes with gaps, and so on. I think that this actually works out to 50% coverage overall. I think you’re right. :smack: ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [July 13, 2017, 8:32pm UTC](https://boards.straightdope.com/t/why-isnt-a-sphere-a-platonic-solid/790692/42 "2017-07-13T20:32:08Z")
“Tessellating” a sphere with same sized equilateral triangles is just not going to work. Look at the [section](https://en.wikipedia.org/wiki/Platonic_solid#Geometric_proof) of the Wikipedia article where it talks about angle deficit. A vertex is either going to have 5 or less triangles meeting there and the surface angle is fixed and will never approach 0. Or it has 6 triangles meeting and the surface is perfectly flat. Not locally flat, but flat for the entire object. The later could be used to argue something weird about the plane being a Platonic solid. (Did I just type that?) You could give up on having the vertices of the triangles not meet in general and hope something nice happens in the limit, but then the edges won’t line up right, etc. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [July 13, 2017, 10:17pm UTC](https://boards.straightdope.com/t/why-isnt-a-sphere-a-platonic-solid/790692/43 "2017-07-13T22:17:45Z")
> [@ftg](#): > > A vertex is either going to have 5 or less triangles meeting there and the surface angle is fixed and will never approach 0. Or it has 6 triangles meeting and the surface is perfectly flat. Not locally flat, but flat for the entire object. You can never jam together more than 5 at a vertex, but the “surface angle” (actually I am thinking of spherical triangles) _will_ approach zero as the triangles get smaller. Equilateral triangles of different sizes on a sphere have different angles… > [@](#): > > You could give up on having the vertices of the triangles not meet in general and hope something nice happens in the limit, but then the edges won’t line up right, etc. Limit of what, though? An explicit construction would be one way to solve the problem, assuming it is possible. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 14, 2017, 1:33am UTC](https://boards.straightdope.com/t/why-isnt-a-sphere-a-platonic-solid/790692/44 "2017-07-14T01:33:59Z")
> [@ftg](#): > > The later could be used to argue something weird about the plane being a Platonic solid. (Did I just type that?) Well, yes, the plane _is_ a Platonic solid, on all the most natural generalizations beyond the traditional concept. If there’s to be any generalizing at all, you’ll soon want to include the plane. Basically all the mathematics of Platonic solids (e.g., the theory of Coxeter/triangle groups) applies to regular planar tilings as well; indeed, the theory of Platonic solids might as well be the theory of planar tilings, just for different kinds of “plane”. In more detail, to each pair (n, m) of integers \> 1, there is a freely generated regular tiling of n-gons, m of which meet at each vertex; all that changes is the curvature of the space it lives in. If 1/n + 1/m \> 1/2, the angles of the faces around a vertex sum to less than 360 degrees, and it lives in elliptic space (i.e., as a tessellation of a sphere, wrapping around on itself finitely; i.e., corresponds to a familiar sort of solid); if 1/n + 1/m = 1/2, the angles of the faces around a vertex sum to precisely 360 degrees, and it lives in Euclidean space (i.e., as a tessellation of the infinite Euclidean plane, and can also be quotiented into a finite torus at various scales); and if if 1/n + 1/m \< 1/2, the angles of the faces around a vertex sum to more than 360 degrees, and it lives in hyperbolic space (i.e., as a tessellation of the infinite hyperbolic plane, and can also be quotiented into various exotic finite structures). Yep, you could even have 7 or more equilateral triangles meeting at each vertex; it just takes you into a hyperbolic plane (much like having 5 or less takes you out of the familiar Euclidean plane and into an elliptic plane; i.e., sphere). See [Order-7 triangular tiling - Wikipedia](https://en.wikipedia.org/wiki/Order-7_triangular_tiling). ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 14, 2017, 1:40am UTC](https://boards.straightdope.com/t/why-isnt-a-sphere-a-platonic-solid/790692/45 "2017-07-14T01:40:46Z")
(Well, strictly, the sphere double covers the elliptic plane and is not identical to it, but putting that niggle to the side…) ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [July 14, 2017, 11:09am UTC](https://boards.straightdope.com/t/why-isnt-a-sphere-a-platonic-solid/790692/46 "2017-07-14T11:09:23Z")
> [@DPRK](#): > > You can never jam together more than 5 at a vertex, but the “surface angle” (actually I am thinking of spherical triangles) _will_ approach zero as the triangles get smaller. Equilateral triangles of different sizes on a sphere have different angles… The phrase “spherical triangles” takes you completely out of the topic of Platonic solids. Those involve flat regular polygons only. The phrase “of different sizes” takes you completely out of the topic of Platonic solids. Those involve identical polygons only. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 14, 2017, 2:38pm UTC](https://boards.straightdope.com/t/why-isnt-a-sphere-a-platonic-solid/790692/47 "2017-07-14T14:38:09Z")
> [@](#): > > Quoth **Indistinguishable** : > > Well, yes, the plane is a Platonic solid, on all the most natural generalizations beyond the traditional concept. More precisely, the plane is three different Platonic solids, one each with triangular, square, and hexagonal faces. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [July 14, 2017, 5:49pm UTC](https://boards.straightdope.com/t/why-isnt-a-sphere-a-platonic-solid/790692/48 "2017-07-14T17:49:36Z")
> [@ftg](#): > > The phrase “spherical triangles” takes you completely out of the topic of Platonic solids. Those involve flat regular polygons only. I wouldn’t say completely out of the topic; any classical Platonic solid can be inscribed in a sphere and is inextricably related to a regular tessellation of the sphere (just join the vertices by spherical geodesics rather than in Euclidean space). You should be amenable to this point of view if you want the plane to be a Platonic solid as well, and also regular tessellations of the hyperbolic plane. There is also something to be said for everything happening on the sphere rather than in some ambient space. > [@](#): > > The phrase “of different sizes” takes you completely out of the topic of Platonic solids. Those involve identical polygons only. True, the packing problem is a bit of a sidetrack, but not because of polygons of different sizes. Consider a sphere of radius r, and fix a number x. Now try to pack the sphere with many identical equilateral triangles of area 4πr[sup]2[/sup]/x. How many can you fit? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 14, 2017, 8:09pm UTC](https://boards.straightdope.com/t/why-isnt-a-sphere-a-platonic-solid/790692/49 "2017-07-14T20:09:58Z")
> [@Chronos](#): > > More precisely, the plane is three different Platonic solids, one each with triangular, square, and hexagonal faces. Good point! [Previous page](https://boards.straightdope.com/t/why-isnt-a-sphere-a-platonic-solid/790692.md?page=2)