Blog:All of the regular skew polyhedra in 4D space

A while ago I came across this paper called Four-Dimensional Regular Polyhedra published by Peter McMullen in 2007 (Peter McMullen also authored the initial paper proving the list of all 48 regular polyhedra complete!). As it turns out, it enumerates all of the finite regular polyhedra in 4D space (the enumeration of all the apeirohedra is still an open question), however the paper does not actually list all of the polyhedra out, which is why I decided to make this blog post that does just that. Almost all the information here is taken directly from the paper itself.

Why I need your help
Although I was able to figure out a lot of things about these polyhedra, there are still many things about these shapes that I haven't been able to determine yet. I have been able to figure out how many of these polyhedra there are, but as you will see a lot of polytopes with information missing.

This is the other reason I'm making this post: so that you can help. I have a goal of finding out everything we can about these polyhedra, and I need your help to make it a reality. Feel free to comment your findings in this blog post. All missing information I have marked with question marks ?. Empty areas mean that information is not applicable for that polyhedron.

Currently, the dimension vectors that have missing information are:
 * (1,2,3): nearly everything is missing
 * (2,2,3): nearly everything is missing
 * (2,3,2): element counts are missing
 * (2,2,2): most element counts and most Schläfli symbols are missing

I will also soon format the polyhedra in this list that are currently only in plaintext like {5/(1,2),3:10/(1,3)}.

Schlafli symbols
The paper uses Schlafli symbols differently than other papers so far. First, instead of polygons being written using blending they are defined using the following shorthand: $$\left\{\frac{a}{b,c}\right\}=\left\{\frac{a}{b}\right\} \# \left\{\frac{a}{c}\right\}$$. This is helpful because the resulting blend will always be an $$\{a\}$$-sided polygon. Blending with a segment is actually the same as blending with a digon, so those skew polygons are represented with this notation.

$$\{a,b:c\}$$ is the polyhedron with $$\{a\}$$-gonal faces, a $$\{b\}$$-gonal vertex figure, and $$\{c\}$$-gonal Petrie polygons. a, b, and c can also be skew polygons when the above a/(b,c) notation is used. This is essentially regular map notation but with the addition of skew polygons as inputs.

$$\{a,b|c\}$$ is like the above notation but instead of c-gonal Petrie polygons we have c-gonal holes. An important thing to note is that if a vertex figure is a skew polygon, it must be written down as a skew polygon using our new skew polygon notation. For example, $$\{4,6|4\}$$ would be invalid, but $$\left\{4,\frac{6}{1,3}|4\right\}$$ is valid because $$\left\{\frac{6}{1,3}\right\} = \left\{\frac{6}{1}\right\} \# \left\{\frac{6}{3}\right\} = \{6\} \# \{2\}$$, which is a skew hexagon and therefore the vertex figure of the mucube.

Operations
The paper defines 6 different kinds of operations on polytopes:
 * 1) The dual operation δ (defined the same as in 3D, an involution when it works)
 * 2) The Petrial operation π: take the set of all Petrie polygons of a polyhedron to get a new one (defined the same as in 3D, an involution)
 * 3) The Kappa operation κ: Replace the first generating mirror with its orthogonal complement (an involution)
 * 4) The halving operation η: alternate a polyhedron with square faces to get another regular polyhedron (defined the same as in 3D, one-way)
 * 5) The facetting operation ϕk: take the set of all k-holes of a polyhedron to get a new one (only works sometimes)
 * 6) The skewing operation σ: Equivalent to πδηπδ, these operations can be performed abstractly so the intermediate polyhedra don't have to be valid for the skewing operation to be (only works sometimes, one-way). There is one example of this operation in 3D: the skew muoctahedron.

I am the most unsure about how the kappa operation works, so any polyhedron defined as the kappa of another polyhedron I have put no information on.

Dimension Vectors
A final piece of notation I want to introduce are dimension vectors. Polyhedra in 4D space are generated using mirror reflections in the same way that Coxeter-Dynkin diagrams do, with one difference. The mirrors can be a hyperplane of any dimension, not just the dimension one below the rank of the polytope. The dimension vector (a,b,c) means that the first mirror (hyper)plane is a-dimensional, the second b-dimensional, and the third c-dimensional.

Now without further ado, let's look at all of = The regular polyhedra in 4D space = I'll go through each possible type of dimension vector one by one. If a dimension vector is missing from this list it means that there are no polyhedra with that dimension vector.

Dimension Vector (2,3,3)
These are the blends of the Platonic solids and Kepler-Poinsot polyhedra with a segment. None of these polyhedra have valid duals, halvings, kappas, facettings, or skewings (at least, they weren't mentioned in the paper). The blended cube can be halved to get the tetrahedron, but the tetrahedron is 3D and so this is not a relation between 4D polyhedra. The blended cube is also unique because it is the only polyhedron that doesn't have double the edges and vertices compared to its unblended counterpart.

Dimension Vector (1,3,3)
These polyhedra are the Petrials of the previous class. They are also the segment blends of the finite regular Petrial polyhedra in 3D space. Just like the previous class, these do not have duals, halvings, facettings, skewings, or kappas (or at least these relations were not mentioned in the paper).

Dimension Vector (3,2,3)
This is the first class with 'interesting' polyhedra that aren't easily derived from 3D regulars. The first set of polyhedra is actually our first infinite set: the duocombs {4,4|n} formed from the comb product of a planar regular polygon with itself. These are toroids and the hole size can even be a star polygon, however it can't be a skew polygon. All of them are also self-dual. We then have the other 'mu-like' that are related to {4,6|3} and {4,8|3}. They all have triangular holes. The first four are the only polyhedra out of the eight without self-intersections. They can also all be formed by faceting certain uniform polychora: these polychora are spid, deca, cont, spic, siddic, giddic, quippic, and gic respectively. Their verfs contain regular skew polygons within them, these are the vertex figures of the above 8 polyhedra.

The final set of (3,2,3) polyhedra are facetings of the triangular and pentagonal rectates. They can be formed by first taking any regular polychoron other than hex, then taking the Petrial of all its cells, and then taking the dual of every face. I like to call these 'petrirectates' but that's a very unofficial name. There are fifteen of these in total.

The first seven of these are facetings of the triangular rectates, and they all have skew triangle vertex figures. None of these polyhedra have valid duals, and are facetings of firp, firt, frico, firsashi, firgaghi, fry, and firgogishi respectively.

Finally we have the last eight polyhedra that are facetings of the pentagonal rectates. Their vertex figures are either skew pentagons or skew pentagrams. None of these polyhedra have duals either, and these polyhedra are facetings of sophi, papvixhi, gippapivady, spapivady, quiphi, prap vixhi, sprapivady, and giprapivady respectively.

Not including the duocombs, there are a total of 8+15=23 regular polyhedra with this dimension vector.

Dimension Vector (1,2,3)
These are the kappas of the (3,2,3) polyhedra. We first obtain an infinite set of Kappa duocombs: We then have the kappas of the other 23 polyhedra. As you can see, I currently know next to nothing about this class of polyhedra.

Dimension Vector (2,2,3)
These are the Petrials of the previous two classes. We first have the two infinite sets: We then have the petrials of the 23 (3,2,3) polyhedra: Finally we have the petrial kappas of the (3,2,3) polyhedra. Not including the two infinite sets, there are 23+23=46 polyhedra with this dimension vector.

Dimension Vector (2,3,2)
Next we have dimension vector (2,3,2). These polyhedra can be formed by halving (η) the duocombs, {4,6|3}, and {4,8|3}. The other polyhedra in this class are formed by applying other operations to those. These polyhedra use the {a,b:c} Petrie polygon notation instead of the hole-based notation used in the (3,2,3) class.

First we have three infinite sets of polyhedra derived from the halved duocombs. The first infinite set is the set of alternated duocombs. Halving a duocomb twice gives you a duocomb again, and the final infinite set is self-Petrie. The second infinite set only has a valid dual when t is odd.

Next we have the halved {4,6|3} and its Petrial: The first polyhedron is also self-dual.

We then have a final set of 10 polyhedra derived from the halved {4,8|3}. Two of these polyhedra are actually halvings of other polyhedra: {8/(1,4),8:6/(2,3)} is the halving of {8,4|3} and {8/(3,4),8/3:6/(2,3)} is the halving of {8/3,4|3}.

Not counting the infinite sets, there are 2+10 = 12 polyhedra with this dimension vector.

Dimension Vector (2,2,2)
Our final class of polyhedra has dimension vector (2,2,2), and it is by far the most esoteric. McMullen used quaternions to figure out how many polyhedra there are in this class, although he doesn't list some of them individually in the paper, just mentions their existence. This is why there is a lot of missing information in this category, it is only beaten by (1,2,3) and (2,2,3) which only have a paragraph dedicated to them in the paper.

The groups of these polyhedra are constructed from two component 3D rotation groups: these can be either the chiral dihedral, chiral octahedral, or chiral icosahedral groups.

The first case is when both component groups are dihedral. We get an infinite set from this that is self-Petrie and self-kappa. Next we have one component group be dihedral and the other octahedral. We get 8 polyhedra here that are made up of Petrial pairs. The four polyhedra with triangular verfs have a symmetry group of order 576, and the four polyhedra with square verfs have a symmetry group of order 384.

We then have the possibility where one group is dihedral and the other is icosahedral. We get 16 polyhedra here made up of Petrial pairs. The first eight have triangular verfs and a symmetry group of order 1440, and the other eight have a symmetry group of 2400. The next possible case is when one group is octahedral and the other is icosahedral. Some of the symbols here are defined using the ⋈ operation, which I don't fully understand and so can't convert into the {a,b:c} notation. There are 16 polyhedra in total made up of Petrial pairs. There are only two polyhedra that both have octahedral component groups, they are both self-Petrie and a kappa pair. They can also be formed by skewing (σ) {8,4|3} and {8/3,4|3} respectively. Finally, we have a set of 8 polyhedra where both component groups are icosahedral. The first three polyhedra are actually embeddings of some other familiar polyhedra. {5/(1,2),3:10/(1,3)} is an embedding of the dodecahedron in 4D space, {10/(1,3),3:5/(1,2)} is an embedding of the petrial dodecahedron in 4D space, and {5/(1,2),3:5/(1,2)} is actually an embedding of the hemi-dodecahedron in 4D space (the hemi-dodecahedron is self-Petrie).

Not including the infinite set, there are 8+16+16+2+8=50 regular polyhedra with this dimension vector. = Conclusion = In total, there are 172 (finite) regular polyhedra and 8 infinite sets of (finite) regular polyhedra in 4D Euclidean space:
 * 9 polyhedra from (2,3,3)
 * 9 polyhedra from (1,3,3)
 * 23 polyhedra and 1 infinite set from (3,2,3)
 * 23 polyhedra and 1 infintie set from (1,2,3)
 * 46 polyhedra and 2 infinite sets from (2,2,3)
 * 12 polyhedra and 3 infinite sets from (2,3,2)
 * 50 polyhedra and 1 infinite set from (2,2,2)

The total number of regular skew apeirohedra in 4D space or regular skew polychora in 4D space is still unknown.

So in conclusion, there are 172+8∞ regular skew polyhedra in 4D space.

Also, please take a look at Four-Dimensional Regular Polyhedra, as it was where I took almost all of my information from and also proved this set of polyhedra complete. It gives a much more in-depth explanation on all the different polyhedra mentioned in this post.