I made use of this astonishing (to me) fact a couple of years ago, in an article I co-authored on hypothetical physical theories [0].
We considered theories where the "Bloch sphere" [1] of a primitive system took the form of a regular convex polytope, and showed that this led to highly restricted multi-system dynamics. We used the fact that every facet was diametrically opposite to another facet, except in the case of odd-sided polygons, for which we had to drive the same thing in a different way. It wasn't a huge or surprising result, but it was a lot of fun to prove something that applied to those polytopes in all dimensions.
So, what's being talked about here are finite Coxeter diagrams and finite Dynkin diagrams. Or rather irreducible (i.e. connected) such diagrams. "Finite" here doesn't mean that the diagram itself is finite -- the diagrams themselves are always finite -- but rather that the things they represent are finite, and there's only a few way those can happen.
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.)
* 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.
* 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. :)
> Which means a sphere cannot be divided perfectly into more than 20 sections (the number of faces of the icosahedron)! any division scheme beyond that number is doomed to be imperfect.
Define "perfect". You can divide the sphere into 120 identical triangles, and that's just the most you can get using a Catalan solid. There's no upper limit if you allow general isohedral divisions.
This isn't pure nitpicking. The article talks says "Let’s say we wanted to divide earth into approximately equal pieces of land and give one to every human, what can we do?" and continues with a discussion of approximations. There's no need to approximate.
(When I say a sphere is divided into triangles, I mean that all the vertices are on the sphere.)
If you want to go down that road, you'd want to divide by value.
(And to make it more efficient and objective, charge a (global) land value tax and use the proceeds to finance a (global) basic income. The average person will have a net impact of zero, but anyone using less land by value than the average will be reimbursed for that generosity.)
Subjectivity of value is a challenge to be solved, not an obstacle to capitulate before.
Mathematics helps here, too. Game theory to be more precise. Of course, lots of the solutions to fair distribution problems have been discovered before mathematicians got a chance to invent game theory. For two people, the you-cut/I-choose procedure works.
For lots of people introducing money is more efficient: just auction off all the plots to the highest bidders, and distribute the proceeds equally. Auctions are a great way to convert subjective valuations into fair numerical valuations.
(Kind of off-topic, sorry) Is this URL permanent/static? i.e. does it change when a new comment is added at the bottom? Haven't seen an IPFS URL in the wild yet...
Aren't they served by Gitalk? There's a username dropdown, and inside there's a link "Gitalk1.1.4" —> https://github.com/gitalk/gitalk. Also, doesn't really look like Disqus to me.
> This also means you cannot put more than 20 points on a sphere without some being closer to each other’s neighbors than others.
There's an important limitation missing here: you must also require that the angles formed between a point and successive neighbors are all equal. As written, the Archimedean solids also space points out equally distant from their neighbors, but the angles between neighbors are different. (Not sure if all the Archimedean solids inscribe a sphere, but there are 13 of them.)
And the "n-cube" is the "measure-polytope" because you can use squares to measure 2-space, cubes to measure 3-space, hypercubes for 4-space and so on...
Is this sequence also part of the On-Line Encyclopedia of Integer Sequences (OEIS)? I didn't find it at my first attempt, because apparently one needs more items from the sequence to identify it correctly. Also, it is not clear to me with which value "infinity" should be replaced (perhaps 0? or better -1?)
I don't. For one, it's broken on Android (figures floating over text, lines cut in half etc.). But the worst flaw is in the way they made scrolling change the 2D polygon: if you leave the scroll bar in some positions, it no longer actually shows a regular polygon. For a math text that's a deadly sin.
Android aside, the flaw with scrolling caught me initially, but once you get your head around it, it is actually a very intuitive way of illustrating the points being made.
It's not just the 2D polygon, all of the illustrations use the same technique. The gradient descent works especially well.
Yes, the same could be achieved with a slider, but I personally don't think I'd have tried it when scanning through.
I'm not condoning this technique for general use..
> These are solids where all the sides are equal and all the faces are the same regular polygon.
There is, for instance, rhombic triacontahedron[1] which fulfil the above definition while not being a platonic solid.
By wiki[2] ,
> It is constructed by congruent (identical in shape and size) regular (all angles equal and all sides equal) polygonal faces with the same number of faces meeting at each vertex.
The faces there are rhombuses, which are not regular polygons, right? The definition of a regular polygon includes equality of angles, not just of sides.
Thanks for pointing out. You are right! It looks silly to have mistaken such a simple concept but let's leave it here in case someone has same thought as me
”platonic solids are solids where all the sides are equal and all the faces are the same regular polygon” is true, but the reverse isn’t.
I think you also want all vertices to lie on a single sphere.
Without that, you can have various stellated polyhedra, constructions of 6 or 10 equilateral triangles, an infinite number of constructs that only contain squares, and probably others.
I also think “all the faces are the same regular polygon” implies “all the sides are equal” for the ‘normal’ definitions of “polyhedron”, but that definition is surprisingly hard. One typically wants to exclude faces crossing each other and constricts that aren’t connected, such as “two cubes”, so who knows?
By the way, because it's helpful to have a name if you want to do a literature dive, the optimisation problem this is solving is called the Thomson problem. It was originally formulated in the context of the lowest energy configuration of electrons restricted to a sphere under electrostatic repulsion.
So I was chatting to my 8 yo about this and he made the analogy that as in 2D we have infinite numbers of polygons meaning that at the limit a circle would be a polygon of infinite sides(?), to 3D where a sphere would be a polyhedron with infinite faces ... I can't really see a flaw there except through definition.
Or do we say that the circle itself is not in the set of polygons?
In other words why isn't a sphere a platonic solid?
It doesn't actually work in 2D for infinity, just for arbitrarily large integers (and there is an infinite amount of them).
If it actually were infinite, three "neighbouring" points wouldn't actually describe a face but just a point, since they all fall onto each other. You could look at the Banach–Tarski paradox for a better understanding. "Half" the set of points is still enough to describe a sphere since there are infinitely many.
Surely then there's arbitrarily large numbers of polygons but not infinite, because if they're infinite then the spacing between points will be zero?
In Banach-Tarski In thought that the assemblage of points relied on creating, eg, a new pair of spheres that worked under different non-Euclidean measures. But I'll need to review it, it's outside my domain (heh!).
For each integer n >= 3, we can create a perfect n-sided polygon in two dimensions. That polygon will have some nonzero spacing between the points, roughly a distance like 2pi/n.
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.
> Surely then there's arbitrarily large numbers of polygons but not infinite, because if they're infinite then the spacing between points will be zero?
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).
That's kinda my point, if there are aleph_0 polygons, then surely the length of a side has to tend to 0. As you increase the number of vertices then the difference between the points occupied by the vertices and those of a coincidental circle reduces, ultimately the vertices must describe a circle if they form a set of size aleph_0?
If you want the sides of the polygons to have a length then the polygons, to me, seem necessarily to be unable to have an infinite number of vertices, they could only have a very large number but one necessarily less than infinity.
Perhaps it's my misunderstanding how a curve works? If you form a second set from an infinite number of equally spaced points between two arbitrary points on a continuous curve (the first set) then it seems, to me [ie intuitively, always a risk in maths], that the second set must also be a continuous curve? The corollary of that would seem to be that if you have an infinite number of vertices for your polygon it also forms a circle, to not form a circle it simply has to have a number of sides less than infinite.
My contention would then be that either the set of polytopes in 2D is < aleph_0 or the set in 3D is 6 (ie applying the same rules makes circles and spheres both parts of the set of polytopes in the particular n-dimensional space).
I was trying to convey that a sphere-sphere is an infinite amount of points with some unintuitive properties, like half the points are still an infinite amount.
For a set of faces defined by an arbitrarily large but finite subset of points of the sphere, the important properties no longer hold.
That's what the article is about, you can divide a circle in an infinite number of polygons, but, somewhat unintuitively, you can't do that with spheres, only 5 divisions are possible ! If you try with hexagons, for example, you'll find that you can't join them without distorting them.
I only got as far as 'Approximations' before we started chatting. I can entirely understand it not working for polygons "tiling" the sphere up until they become infinitesimal, at which point they'd occupy all points on the sphere.
Um, I thought it didn't matter, that's like how long is the point that you make a curve out of when you're integrating to determine the length of a functions curve. But we could say square (or any shape that will tile a 2D plane), because at the limit as the side approaches zero, on the surface of a sphere, any region looks like a flat 2D plane?
I'm flying by the seat of my pants a bit here [in case you couldn't tell], it's a long time since I did any formal maths.
What is the name for this sequence notation making use of an infinity sign? I don't know how to read that symbol; is it some kind of operator between finite sequences to generate an infinite sequence?
In the context of the sequence in the article I think "infinite value" is confusing. Infinity is a better choice. Using infinity signifies that that number of regular polytopes is not finite. The ∞ in this context just means that there is no bound on the number of regular polytopes and not meant to be taken as an actual number. When you say infinite value it makes me think, "Which infinite value?" There are infinitely many infinite cardinal numbers.
As ∞ is not the same type of object as 1, 5, 6 3, etc; the sequence isn't technically well defined. Actually ∞ itself isn't well defined as it could be one of an infinite number of cardinal numbers.
Something along the lines of "1 1 \aleph _{0} 5 6 3 3 3...", with each element being a well-defined cardinal number [1], might be technically closer to being correct, but it is still confusing.
From context all the numbers are cardinal numbers since we are talking about the size of a set (i.e. the set of different regular polytypes in each dimension). Also from context the infinity here must be countable, so indeed it must be the sequence "1, 1, \aleph _{0}, 5, 6, 3, 3, 3...". You don't need to spell everything out explicitly as long as it is clear from the context.
If we want to get really formal, then what we have is a mapping from the natural numbers into equivalence classes of sets under the bijection equivalence relation.
Well, the argument is that for any natural number n we can make an n-gon. So we have a one-to-one correspondence between natural numbers and dimension 2 polytypes, so by definition the cardinality of that set is aleph zero
We considered theories where the "Bloch sphere" [1] of a primitive system took the form of a regular convex polytope, and showed that this led to highly restricted multi-system dynamics. We used the fact that every facet was diametrically opposite to another facet, except in the case of odd-sided polygons, for which we had to drive the same thing in a different way. It wasn't a huge or surprising result, but it was a lot of fun to prove something that applied to those polytopes in all dimensions.
[0] https://arxiv.org/abs/1508.03491
[1] https://en.wikipedia.org/wiki/Bloch_sphere