Consider a solid with the following properties -

  1. It is composed of congruent, regular polygons.
  2. At each vertex, the same number of edges and faces meet.

This is the same as the requirement for the Platonic solids, but the solid need not be convex. Of course, the five Platonic solids will satisfy these conditions, but are there any others?

EDIT: consider the topological proof given in the Wikipedia article on Platonic solids - https://en.wikipedia.org/wiki/Platonic_solid

If we require the Euler characteristic to be 1 instead of 2 (as in the proof) we get -

$$\frac{1}{p} + \frac{1}{q} = \frac{1}{2} + \frac{1}{2E}$$

This still keeps open the possibility (using the same argument as for the Platonic solids given in that proof) of five such solids with Euler characteristic 1 (so they won't be convex). Question is, do these solids exist?

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    $\begingroup$ If you drop the regularity requirement, you get Kepler-Poinsot polyhedra. $\endgroup$ – Noah Schweber Dec 11 '17 at 2:49
  • $\begingroup$ @NoahSchweber - these are interesting, but I don't think they satisfy the requirement that each vertex should have the same number of faces and edges intersecting at it. $\endgroup$ – Rohit Pandey Dec 11 '17 at 2:53
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    $\begingroup$ They do, actually… the faces or vertex arrangements are regular star polyhedra. $\endgroup$ – Parcly Taxel Dec 11 '17 at 2:55
  • $\begingroup$ @ParclyTaxel can you provide an example of such a polyhedron? $\endgroup$ – Rohit Pandey Dec 11 '17 at 3:01
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    $\begingroup$ The Kepler-Poinsot polyhedra do satisfy the requirements because the non-convex "vertices" are not considered vertices of the polyhedra, they merely happen to be points where three faces intersect. For example, the great dodecahedron is considered to have only 12 vertices and 12 pentagonal faces. However, since the faces pass through each other, I suppose you cannot really call it a "solid". It does not have a clearly defined interior, it is merely an arrangement of 12 pentagons with nice regularity properties. $\endgroup$ – user856 Dec 11 '17 at 3:16

The central question here was if there exist solids that satisfy all requirements of the Platonic solids, but aren't convex. On thinking about this, at least one such solid exists (which proves they are possible). When constructing an Icosahedron, we take five triangles and form a bowl out of them. Then, we put an intermediate "ring" of triangles on top of this and finally, another bowl composed of five triangles is put on the very top. Now, imagine putting the top bowl upside down. This leads to a concave version of the Icosahedron. Note that the Euler characteristic of this non-convex Icosahedron is still 2. And based on the edit to my question, there is the possibility of a solid with the same vertices, edges and faces as an Icosahedron (or any other Platonic solid) but having Euler characteristic 1. I'm interested in how we might find these solids as well, but I'll save that for another question.

enter image description here

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    $\begingroup$ If you required only that the number of faces and edges meeting at each vertex be the same, than you could construct quite a few "Platonic" polyhedra satisfying these rules; at least, you could invert just one vertex of a regular icosahedron, or two or three vertices that are not opposite. But these solids don't seem very "regular," which is why usually one requires at least that the region around each vertex be congruent, beyond just having the same number of faces. Then even without the convexity requirement, you only get the five, but I think you need Euler's formula to prove it. $\endgroup$ – Glen Whitney Sep 20 '18 at 21:08
  • $\begingroup$ This polyhedron is not regular and so is not one of the Platonic solids. A regular polyhedron has all its vertices within the same symmetry orbit, i.e. they lie evenly distributed on its circumsphere. $\endgroup$ – Guy Inchbald Mar 9 '20 at 7:04

There are four regular non-convex or "star" polyhedra, known collectively as the Kepler-Poinsot polyhedra after their discoverers. Cayley named them:

  • Small stellated dodecahedron
  • Great dodecahedron
  • Great stellated dodecahedron
  • Great icosahedron

But they are not "Platonic", as this description is reserved for the five convex regular solids, which were first listed by Plato.

Two of these polyhedra do not obey the usual Euler formula $V - E + F = 2$, which caused much consternation until the formula was generalized for toroids.

All four figures self-intersect. There are no regular polyhedra which are non-convex but do not self-intersect.


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