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I heard there are 48 regular polyhedrons. With what Jan Misali calls regular polyhedrons, are there any more?

Assumptions:

  1. A polyhedron must lie in 3D Euclidean space.

  2. It must be a single connected shape.

  3. It's invalid for two vertices edges or faces to have the exact same location while remaining distinct.

If there are only 48 polyhedrons, what about 4D polytopes?

Watch this video if this info isn't full enough here: https://www.youtube.com/watch?v=_hjRvZYkAgA

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    $\begingroup$ As far as I know, there are only $9$ regular polyhedra, of whoch $5$ are convex (the platonic solids). $\endgroup$ – Bernard Aug 5 '20 at 15:30
  • $\begingroup$ @Bernard have you seen the video? you might be right my question is how many with his definition? what if you use skew polygons. and allow polyhedrons to be infintie $\endgroup$ – Harrison Aug 5 '20 at 15:40
  • $\begingroup$ Do you have a reason to doubt the source Jan Misali provided? $\endgroup$ – Mark S. Aug 5 '20 at 15:51
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    $\begingroup$ I remember another paper talking about this about 4 years ago and it said 49 but didn't have pictures so I'm not sure if that paper was wrong of he was but I can't find the paper anymore tho. @MarkS. $\endgroup$ – Harrison Aug 5 '20 at 15:53
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    $\begingroup$ This is a generalisation. Unfortunately, I know about nothing on this subject. $\endgroup$ – Bernard Aug 5 '20 at 16:07
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With the assumptions you give, there are in fact infinitely many other regular polyhedra. Take any rational $m$, $n$ with $m, n>2$ and $$(m-2)(n-2)<4.$$ It turns out that you can create a configuration of $\{m\}$ polygons around a vertex, creating an $\{n\}$ vertex figure. Here's an example with $m=n=\frac{5}{2}$ with a pentagrammic face highlighted:

The vertex figure of {5/2, 5/2}.

This configuration will always have a circumscribed sphere. We can thus uniquely repeat this construction at each of the new vertices we create countably infinitely many times, at each step preserving the circumscribed sphere, and we'll either end up with a Platonic solid or a Kepler–Poinsot solid after finitely many steps, or end up with a regular apeirohedron $\{m,n\}$.

In reality, McMullen and Schulte's considered polyhedra excludes this sort of construction. The rules you mention are correct, but incomplete. The most basic rule is of course the basic rule that defines polyhedra:

  1. A regular polyhedron must, of course, satisfy the properties of a polyhedron. That is, each edge must be adjacent to exactly two vertices and to exactly two faces.

McMullen and Schulte then add the restrictions you had already mentioned:

  1. A regular polyhedron must be embedded in 3D Euclidean space.

  2. A regular polyhedron must be connected, which means that every two vertices are connected by a path of edges.

  3. No two vertices, edges or faces of a regular polyhedron can occupy the same position in space.

Of course, these restrictions are often the same implicit restrictions used to describe all polyhedra. We still haven't gotten to defining regularity. This is the standard definition, which we add as a further rule:

  1. A regular polyhedron must be flag transitive.

A flag is just a set of a face, an edge, and a vertex, all incident to each other. Flag transitivity means that for any two flags, there exists a symmetry of the polyhedron (a rotation, reflection, translation or glide reflection) that sends the former to the latter, while preserving the overall shape of the polyhedron.

One can verify that the $\{m,n\}$ apeirohedra I previously mentioned do in fact satisfy requirements 1–5. However, McMullen and Shulte give one further requirement to narrow down the set of regular polyhedra to 48.

  1. The symmetry group of a regular polyhedron must be discrete.

For our purposes, the symmetry group of a figure is nothing more than the set of all symmetries of such figure. A symmetry group is called discrete when it has a smallest nonzero rotation and translation. Intuitively, you can think of the discreteness requirement as prohibiting "small enough" rotations or translations from being symmetries.

Thus, the $\{m,n\}$ apeirohedra earlier described do not in fact satisfy condition 6, as it turns out that their symmetry group contains arbitrarily small rotations. There are, however, 48 regular polyhedra that satisfy rules 1–6, and these are the regular polyhedra described in Jan Misali's video.

Jan Misali's main source contains the proof of this fact (I do warn that it's quite technical). I'm not aware of any similar results in 4D or beyond.

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    $\begingroup$ As a check on my understanding, would it be correct to say that circles and discs are commonly seen examples of shapes with non-discrete symmetry groups? $\endgroup$ – realityChemist Sep 8 '20 at 16:02
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    $\begingroup$ @realityChemist Precisely. Here's another example: the set of rational numbers, seen as a set of points, has a symmetry group that contains translations by any rational number, and is therefore also non-discrete. $\endgroup$ – URL Sep 8 '20 at 16:05

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