The sequence 1 1 ∞ 5 6 3 3 3

I'm not going to give a full explanation here -- it would take too long, this is a big subject and I'm not an expert in it -- but you can go look up a bunch of this on Wikipedia (well, or you can try; last I checked the articles on this subject were still fairly confusing). But, Coxeter diagrams, and Dynkin diagrams, represent root systems -- particular configurations of vectors in space. What space? Well, for our purposes it will be Euclidean space, because finite root systems always fit in Euclidean space. For infinite root systems this won't work -- you need something like Minkowski space -- but we're ignoring those here. (Indeed the finite root systems are precisely the ones fitting in Euclidean space.) The key thing is that a lot of things in mathematics end up being controlled by root systems -- particularly by finite root systems -- and thus by the diagrams that represent them.

Coxeter diagrams represent root systems; Dynkin diagrams represent crystallographic root systems. (Warning: The terminology here varies a bit. This is the terminology I'll use, though.) Not every root system is crystallographic, so not every Coxeter diagram can be made into a Dynkin diagram. But every Dynkin diagram has an underlying Coxeter diagram. The Dynkin diagram can contain more information than the underlying Coxeter diagram, however, because you could have two crystallographic root systems which, though not equivalent as crystallographic root systems, become equivalent if just treated as root systems, with the crystallographic aspect ignored.

Importantly, though, any root system can be decomposed into irreducible root systems; at the diagram level, that corresponds to taking connected components. Conversely if you have two diagrams you can take their disjoint union to get another. So we're just going to focus on the irreducible root systems, the ones whose diagrams are connected.

I'll skip for now what these diagrams actually mean -- I'll put a bit about that at the end if you want -- but the point is, the finite irreducible Coxeter diagrams and Dynkin diagrams fall into just a few infinite families and a few irregular exceptions. I'll include links so you can see the actual diagrams. But to just list the names here, they are:

(Note, I'm going to avoid listing some redundancies. Like you could define D_3, but it would just be the same as A_3.)

(Crystallographic -- see https://en.wikipedia.org/wiki/File:Finite_Dynkin_diagrams.sv... )

* A_n (infinite family; defined for n>=1)

* B_n (infinite family; defined for n>=2)

* C_n (infinite family; defined for n>=2, but C_2=B_2, so sometimes restricted to n>=3)

* D_n (infinite family; defined for n>=4)

* E_6, E_7, E_8

* F_4

* G_2

These names can refer to Dynkin diagrams or to the underlying Coxeter diagrams. However, considered as Coxeter diagrams, B_n=C_n, so if you're listing Coxeter diagrams, you can skip C.

(Non-crystallographic -- see https://commons.wikimedia.org/wiki/File:Finite_coxeter.svg)

* H_3, H_4

* I_2(p) (infinite family; defined for p>=3; but I_2(3)=A_2, I_2(4)=B_2=C_2, and I_2(6)=G_2)

...so note that if we're listing finite Coxeter diagrams, as opposed to finite Dynkin diagrams, we don't actually need to include G, as G is just a special case of I. That's why I said above that my list is redundant, because I included G as well as I.

(You may notice that the file I linked refers to as I_n what I called I_2(p)... I guess notation varies...)

Some things to note here, that will explain what I said above:

1. As mentioned above, some things are controlled by finite Dynkin diagrams, others by finite Coxeter diagrams.

2. A, D, and E are the families with only single-edges (or in the Coxeter context, with only unlabeled edges). These diagrams are called "simply-laced". A number of things are controlled specifically by these; that's why crasshopper above mentioned "ADE-theory".

3. Regular polytopes, however, are controlled not by simply-laced Dynkin diagrams, but rather by straight-line Coxeter diagrams. Or rather, straight-line Coxeter diagrams together with an order to read them in (of course in some cases the diagram is symmetric and both ways get you the same thing). These are also called Schläfli symbols, where you just list out in order the labels on the edges. (Unlabeled edges are implicitly a 3; sorry, I didn't explain what the diagrams actually mean, I'll get to that below.)) So H_3 read one way, as (5,3), gets you the dodecahedron, but read the other way, as (3,5), gets you the icosahedron.

Which is why I listed all the families except D and E, the ones that branch. Because the original subject was regular polytopes, and those are controlled by straight-line Coxeter diagrams, rather than ADE.

So what do the diagrams actually mean? Well, I'm not going to give a full explanation, but every root system has what's called a base, a set of vectors that in some sense generate all the others. The vertices in the diagram correspond to the vectors in the base, and the edges encode the angles between them. The angle between two vectors in a root system is always of the form θ=π-π/n. (Note that the number of vectors in the base is always equal to the dimension of the space. So you can easily tell from the diagram what dimension a given root system is in; just count the vertices.)

For a Coxeter diagram, basically, you just put down all the vectors, draw an edge between each pair, and label each one with the appropriate "n". Except not actually -- the key thing is, if the two vectors are perpendicular, i.e. n=2, you don't draw an edge. Also, 3s are so common that we don't bother to explicitly label the 3s; an unlabeled edge is a 3. But that's just a convention. Not drawing n=2 edges is important, though. An example of why is the decomposition fact I mentioned above -- if your diagram is disconnected, aha, your root system decomposes.

(Actually, it's also possible to have n=∞, representing an "angle" of π... sort of; remember we're not in Euclidean space anymore once we're talking infinite root systems. But that doesn't occur in finite root systems, so I'm going to just gloss over the matter. Note that any diagram drawn like this -- take a graph, label some of the edges with whole numbers greater than 3, or possibly infinity -- gets you a valid Coxeter diagram, it's just that most likely it'll represent an infinite root system; the finite ones are all listed above. Or rather the irreducible finite ones are all listed above; more generally, you have to check the connected components against the above list.)

For a Dynkin diagram, well, first off, not all those angles are legal anymore. The only legal n now (aside from n=2, i.e. no edge) will be 3, 4, 6, and ∞. (Although again that last one will only occur in infinite Dynkin diagrams.) What you may notice about these specific values of n is the value of 4cos^2(θ); specifically, n=2 gets you 0, n=3 gets you 1, n=4 gets you 2, n=6 gets you 3, and (if we're allowing infinite root systems) n=∞ gets you 4.

But now also for a Dynkin diagram we have to encode some information that wasn't relevant when we were just talking about Coxeter diagrams, namely, the relative length of the vectors. (And so now B and C become different.)

For n=3, the two vectors are always equal in length, so we just draw a single edge, like before.

For n=4, we draw a double edge. One of the two vectors will be √2 the length of the other, so we orient the edge pointing at the shorter of the two.

For n=6, we draw a triple edge. One of the two vectors will be √3 the length of the other; again, we orient the edge pointing at the shorter of the two.

If we're allowing infinite root systems, then for n=∞, there are two possibilities. One of the two vectors could be twice (i.e. √4) the length of the other; in this case we draw a quadruple edge pointing at the shorter of the two. Or they could be equal in length; in this case we draw an undirected double edge. (Why double rather than quadruple? Well, I have an idea as to why that might be, but no definitive answer; like I said, I'm not an expert in this subject. But again, neither of these types of edges occur in finite Dynkin diagrams.)

(Why are there two possibilities here? Well, I'm not going to give a detailed explanation, but ultimately it's because 4 is composite. You can factor it either as 4⋅1, yielding the first case, or as 2⋅2, yielding the second.)

(For n=2, where there's no edge, the relative lengths could be anything. But that's OK, you don't need that information in this case.)

So, if you draw a diagram like this, you'll get a Dynkin diagram of some root system. But again the only ones that give you finite root systems are the ones listed above. (Or, once again, disjoint unions of them, since I only listed the connected ones.)

Anyway that is my brief introduction to Coxeter and Dynkin diagrams which hopefully at least gives a rough idea of what I was talking about! Like I said it's a really big subject and I'm not an expert, but luckily crasshopper gave a bunch of links! So I don't know, maybe start there. :)

For a set of faces defined by an arbitrarily large but finite subset of points of the sphere, the important properties no longer hold.

If we describe the set of these polgons as P, then |P| = |N|, for N referring to the Natural Numbers.

This is usually called "abzählbar unendlich" (countable infinite).

Okay, now that is true for every n >= 3. This means that there are an infinite number of n for which it is true, i.e. an infinite number of polygons. However, each particular polygon has a nonzero length between points. However again, by picking n as large as you want, you can find polygons where this length is as small as you want, as long as it's strictly greater than 0. None of them are zero.