What are subscripts after a Schläfli symbol and where do they originate?

Question

I'm reading Abstract Regular Polytopes by Schulte and McMullen.

In particular I'm reading the section on projective polytopes. The authors use extended Schläfli symbols to represent these polytopes. The hemipolytope is given by first its Schläfli type and then a subscript for half the number of edges of that shape's Petrie polygon.

So for example, the Hemicube is $\{4,3\}_3$, $\{4,3\}$ for the cube and the subscript because the Petrie polygon of the cube has 6 sides.

This notation hints at some sort of more general notion than is being used here. They could just do something like $\{4,3\}_h$ with the subscript h just meaning "take the hemi of that polytope". But instead they choose a number which encodes extra information, but they decide to halve that number seemingly because it can only ever be even.

But I don't know what the notion is. So my question is: What is the motivation for this notation and where does it come from?

What I know so far

So I went looking. The wikipedium for Schläfli symbols has a section on extensions but it is poorly written, unsourced and doesn't seem to address the notation I am looking for.

There is a wiki for polytopes which also has a section on extensions which does mention these but just repeats the info from ARP. Poking around on that wiki I find that this notation is in fact being used for things beyond what is described in ARP.

They give the "Petrial great stellated dodecahedron" the symbol $\{10/3,3\}_{5/2}$ which has a fractional subscript. This is also half of some identifying boundary, as the Petrie polygon of the "Great stellated dodecahedron" is $\{10/2\}$. The Euler characteristic (-4) shows this can't be a projective polytope, but my geometric reasoning isn't good enough to build a fundamental domain for this shape.

They give the "Petrial cube" the symbol $\{6,3\}_4$ which is interesting because the Petrial cube is not a hemi-polytope. It's a toroid. The number 4 is curious to me though, because I cannot seem to find an octagonal fundamental domain for the Petrial cube. I can find fundamental domains with 14 and 18 sides pretty easily, but 8 seems impossible.

The "Petrial octahedron" is similar. The symbol is $\{6,4\}_3$ and the smallest fundamental domain I can find has 16 sides. (Although that might just be my difficulty visualizing hyperbolic space.)

Answer 1

According to [1]:

An effective method for the construction of regular maps of type $\{p,q\}$ is the identitfication of those pairs of vertices of the regular tessellation $\{p,q\}$ which are separated by $r$ steps along a Petrie-polygon this identification process gives in fact a regular map denoted by $\{p,q\}_r$.

We can see that this definition generalizes the definition given in Abstract Regular Polytopes. In the case where $r$ is half the number of vertices in the Petrie polygon this identifies opposite points on the polytope, and thus gives the hemipolytope.

However it is more general. If we start with the hexagonal tiling $\{6,3\}$ the Petrie polygons have an infinite number of vertices and thus there is no way to identify opposite vertices. Under the new definition we can make sense of $\{6,3\}_4$ as associating points along the infinite zigzags. And this gives us the "Petrial cube". A quick verification shows that this also holds true for the symbol given for the "Petrial octahedron".

The key mistake in the framing of the question was thinking about the boundary of a fundamental domain. It coincides for the case of projective polytopes but it generalizes very poorly, as the fundamental domain of a polytope in its universal cover does not have to be unique nor is it determined by its number of sides.

Now [1] is unlikely the origin of this notation, it cites [2], which although I don't have access to verify, I believe is likely the origin of this notation.

The last thing troubling is the symbol given by the wiki for the "Petrial great stellated dodecahedron". This has a non-integer subscript and thus cannot be said to be a number of steps in the usual sense. What I believe has happened here is that the someone observed that in the above definition the subscript basically gives the number of edges in the Petrie polygon. So an alternative way of thinking about it is that these symbols specify the faces, the degree of the vertices and the Petrie polygon of the polyhedron. So by the subscript they mean that the Petrie polygon of the "Petrial great stellated dodecahedron" is a pentagram or $\{5/2\}$ and gave it that symbol. I can't find any reference for this sort of thing, it may be an invention of the editors of that wiki.

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1. Wills, J.M. The combinatorially regular polyhedra of index 2. Aeq. Math. 34, 206–220 (1987). https://doi.org/10.1007/BF01830672
2. Coxeter, H.S.M., Moser, W.O.J. (1972). Regular Maps. In: Generators and Relations for Discrete Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21946-1_8

Answer 2

There are different types of such Schläfli symbol extensions, both going back onto Coxeter, who wrote them either in the form $\{P, Q | R\}$ (regular skew polyhedra) and $\{P, Q\}_{(r,s)}$ (regular maps), which however, depending on the various subsequent authors both sometimes get written by subindexes - thence a detailed reading of the respective definition of usage by a given author is necessary always!

For according details e.g. cf. to regular maps according to Coxeter-Moser and skew polytopes according to Coxeter-Petrie respectively. Using these keywords you even can find some details within wikipedia too.

--- rk