9 ms·

The sequence 1 1 ∞ 5 6 3 3 3

305 points by quickthrower2 9y ago

deleted 9y ago
[deleted]
mholt 9y ago
(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...
- viraptor 9y ago
  Ipfs address depends on the content, so it's going to stay the same. Not sure if that public gateway supports registered names, but if it does, there would be ipns in the url. (https://dweb-primer.ipfs.io/classical-web/lessons/public-gateways.html https://dweb-primer.ipfs.io/classical-web/lessons/public-gat...) (Comments are loaded in by JavaScript, so they're not included)
- daurnimator 9y ago
  Yes it's a permanent url. No the comments are not static: for this article they are served via disqus.
  - KajMagnus 9y ago
    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 https://github.com/gitalk/gitalk. Also, doesn't really look like Disqus to me.
    - icebraining 9y ago
      Yeah, it uses Github issues: https://github.com/bmabsout/blog-comments/issues/5 https://github.com/bmabsout/blog-comments/issues/5
    - stiangrindvoll 9y ago
      How can we make this happen with ipfs itself? :)
      - lgierth 9y ago
        CRDTs! https://ipfs.io/blog/30-js-ipfs-crdts.md https://ipfs.io/blog/30-js-ipfs-crdts.md
philipov 9y ago
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?
- chrisrhoden 9y ago
  As explained in the linked page, the third value in the sequence is infinite. The fourth in the sequence is five.
  - laughinghan 9y ago
    Nitpick: I think you meant "infinity" not "infinite". Infinite is an adjective, infinity is a noun.
    - chrisrhoden 9y ago
      I'm not a mathematician but I tend to avoid using the word infinity and prefer "infinite value" as I used above.
      - yequalsx 9y ago
        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. Am mathematician.
        deleted 9y ago
        [deleted]
- johnhenry 9y ago
  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. (Also, sorry for the TeX). 1. https://en.wikipedia.org/wiki/Cardinal_number https://en.wikipedia.org/wiki/Cardinal_number
amluto 9y ago
> 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.)
- deleted 9y ago
  [deleted]
- nabla9 9y ago
  >Define "perfect" identical equiangular and equilateral polygons.
- guywaffle 9y ago
  A sphere is the wrong shape in the first place since the majority of Earth is water. Divide the liveable land equally instead.
  - PhasmaFelis 9y ago
    It's a metaphor.
    - guywaffle 9y ago
      I don’t know what it’s phor(for)
  - eru 9y ago
    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.) But let's get back to the math.
crasshopper 9y ago
For those who want to know more, this topic links nicely into A-D-E theory, Dynkin diagrams, and Coxeter groups. https://www.youtube.com/playlist?list=PL-XzhVrXIVeSVcV9iRJ4S9WAH1ryq4hTQ https://www.youtube.com/playlist?list=PL-XzhVrXIVeSVcV9iRJ4S... https://www.youtube.com/playlist?list=PL-XzhVrXIVeQ298S6uCyoDGWNActWwnzZ https://www.youtube.com/playlist?list=PL-XzhVrXIVeQ298S6uCyo... http://math.ucr.edu/home/baez/week230.html http://math.ucr.edu/home/baez/week230.html https://unapologetic.wordpress.com/category/geometry/root-systems/ https://unapologetic.wordpress.com/category/geometry/root-sy... http://www.math.harvard.edu/~lurie/papers/thesis.pdf http://www.math.harvard.edu/~lurie/papers/thesis.pdf Beautiful world this author is peeking into. Enjoy.
- Sniffnoy 9y ago
  A-D-E? Surely you mean A-B-C-F-G-H-I? :) (Yes, I know that list is redundant...)
  - quickthrower2 9y ago
    Can you explain the joke?
    - Sniffnoy 9y ago
      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.) (Crystallographic -- see https://en.wikipedia.org/wiki/File:Finite_Dynkin_diagrams.svg 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 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. :)
colehasson 9y ago
I like the frontend design for this work. Nice job
- semi-extrinsic 9y ago
  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.
  - _puk 9y ago
    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..
alanbernstein 9y ago
Interesting article. Just based on the title, I was expecting some intuitive explanation of the higher dimensions stabilizing at 3 polytopes. I guess the n-tetrahedron, n-cube and n-octahedron generalize to any dimension, but that's it?
- veeedz 9y ago
  There's a neat numberphile video on the subject! [1] [1] https://www.youtube.com/watch?v=2s4TqVAbfz4 https://www.youtube.com/watch?v=2s4TqVAbfz4
- nhaehnle 9y ago
  Yes. The "n-tetrahedron" is usually called the simplex, and the "n-octahedron" is usually called the cross-polytope.
  - grkvlt 9y ago
    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...
klodolph 9y ago
> 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.)
a_c 9y ago
> 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. [1] https://en.wikipedia.org/wiki/Rhombic_triacontahedron https://en.wikipedia.org/wiki/Rhombic_triacontahedron [2] https://en.wikipedia.org/wiki/Platonic_solid https://en.wikipedia.org/wiki/Platonic_solid
- bzbarsky 9y ago
  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.
  - a_c 9y ago
    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
- Someone 9y ago
  ”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?
n4r9 9y ago
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. [0] https://arxiv.org/abs/1508.03491 https://arxiv.org/abs/1508.03491 [1] https://en.wikipedia.org/wiki/Bloch_sphere https://en.wikipedia.org/wiki/Bloch_sphere
vog 9y ago
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?) https://oeis.org/ https://oeis.org/
- deleted 9y ago
  [deleted]
- palotasb 9y ago
  https://oeis.org/A060296 https://oeis.org/A060296 I searched for the part after infinity, apparently you can find based on sub ranges too: 5, 6, 3, 3, ...
jwilk 9y ago
Please use the original title ("Spheres and points").
- justinhj 9y ago
  this title was more clickbaity but in a good way. made me click after spending some time (fruitlessly) trying to figure it out
pbhjpbhj 9y ago
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?
- ajnin 9y ago
  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.
  - pbhjpbhj 9y ago
    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.
    - kybernetikos 9y ago
      But what shape are you using for your tiles?
      - pbhjpbhj 9y ago
        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.
- 9y ago
jordigh 9y ago
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. https://en.wikipedia.org/wiki/Thomson_problem https://en.wikipedia.org/wiki/Thomson_problem
tbodt 9y ago
This is the first time I've seen IPFS-hosted content on HN for a reason other than that it is IPFS-hosted.

305 points by quickthrower2 9y ago

deleted 9y ago

[deleted]

(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...

viraptor 9y ago
Ipfs address depends on the content, so it's going to stay the same. Not sure if that public gateway supports registered names, but if it does, there would be ipns in the url. (https://dweb-primer.ipfs.io/classical-web/lessons/public-gateways.html https://dweb-primer.ipfs.io/classical-web/lessons/public-gat...) (Comments are loaded in by JavaScript, so they're not included)
daurnimator 9y ago
Yes it's a permanent url. No the comments are not static: for this article they are served via disqus.
- KajMagnus 9y ago
  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 https://github.com/gitalk/gitalk. Also, doesn't really look like Disqus to me.
  - icebraining 9y ago
    Yeah, it uses Github issues: https://github.com/bmabsout/blog-comments/issues/5 https://github.com/bmabsout/blog-comments/issues/5
  - stiangrindvoll 9y ago
    How can we make this happen with ipfs itself? :)
    - lgierth 9y ago
      CRDTs! https://ipfs.io/blog/30-js-ipfs-crdts.md https://ipfs.io/blog/30-js-ipfs-crdts.md

philipov 9y ago

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?

chrisrhoden 9y ago
As explained in the linked page, the third value in the sequence is infinite. The fourth in the sequence is five.
- laughinghan 9y ago
  Nitpick: I think you meant "infinity" not "infinite". Infinite is an adjective, infinity is a noun.
  - chrisrhoden 9y ago
    I'm not a mathematician but I tend to avoid using the word infinity and prefer "infinite value" as I used above.
    - yequalsx 9y ago
      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. Am mathematician.
      - deleted 9y ago
        [deleted]
johnhenry 9y ago
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. (Also, sorry for the TeX). 1. https://en.wikipedia.org/wiki/Cardinal_number https://en.wikipedia.org/wiki/Cardinal_number

amluto 9y ago

> 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.)

deleted 9y ago
[deleted]
nabla9 9y ago
>Define "perfect" identical equiangular and equilateral polygons.
guywaffle 9y ago
A sphere is the wrong shape in the first place since the majority of Earth is water. Divide the liveable land equally instead.
- PhasmaFelis 9y ago
  It's a metaphor.
  - guywaffle 9y ago
    I don’t know what it’s phor(for)
- eru 9y ago
  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.) But let's get back to the math.

crasshopper 9y ago

For those who want to know more, this topic links nicely into A-D-E theory, Dynkin diagrams, and Coxeter groups. https://www.youtube.com/playlist?list=PL-XzhVrXIVeSVcV9iRJ4S9WAH1ryq4hTQ https://www.youtube.com/playlist?list=PL-XzhVrXIVeSVcV9iRJ4S... https://www.youtube.com/playlist?list=PL-XzhVrXIVeQ298S6uCyoDGWNActWwnzZ https://www.youtube.com/playlist?list=PL-XzhVrXIVeQ298S6uCyo... http://math.ucr.edu/home/baez/week230.html http://math.ucr.edu/home/baez/week230.html https://unapologetic.wordpress.com/category/geometry/root-systems/ https://unapologetic.wordpress.com/category/geometry/root-sy... http://www.math.harvard.edu/~lurie/papers/thesis.pdf http://www.math.harvard.edu/~lurie/papers/thesis.pdf Beautiful world this author is peeking into. Enjoy.

Sniffnoy 9y ago
A-D-E? Surely you mean A-B-C-F-G-H-I? :) (Yes, I know that list is redundant...)
- quickthrower2 9y ago
  Can you explain the joke?
  - Sniffnoy 9y ago
    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.) (Crystallographic -- see https://en.wikipedia.org/wiki/File:Finite_Dynkin_diagrams.svg 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 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. :)

colehasson 9y ago

I like the frontend design for this work. Nice job

semi-extrinsic 9y ago
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.
- _puk 9y ago
  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..

alanbernstein 9y ago

Interesting article. Just based on the title, I was expecting some intuitive explanation of the higher dimensions stabilizing at 3 polytopes. I guess the n-tetrahedron, n-cube and n-octahedron generalize to any dimension, but that's it?

veeedz 9y ago
There's a neat numberphile video on the subject! [1] [1] https://www.youtube.com/watch?v=2s4TqVAbfz4 https://www.youtube.com/watch?v=2s4TqVAbfz4
nhaehnle 9y ago
Yes. The "n-tetrahedron" is usually called the simplex, and the "n-octahedron" is usually called the cross-polytope.
- grkvlt 9y ago
  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...

klodolph 9y ago

> 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.)

a_c 9y ago

> 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. [1] https://en.wikipedia.org/wiki/Rhombic_triacontahedron https://en.wikipedia.org/wiki/Rhombic_triacontahedron [2] https://en.wikipedia.org/wiki/Platonic_solid https://en.wikipedia.org/wiki/Platonic_solid

bzbarsky 9y ago
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.
- a_c 9y ago
  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
Someone 9y ago
”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?

n4r9 9y ago

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. [0] https://arxiv.org/abs/1508.03491 https://arxiv.org/abs/1508.03491 [1] https://en.wikipedia.org/wiki/Bloch_sphere https://en.wikipedia.org/wiki/Bloch_sphere

vog 9y ago

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?) https://oeis.org/ https://oeis.org/

deleted 9y ago
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palotasb 9y ago
https://oeis.org/A060296 https://oeis.org/A060296 I searched for the part after infinity, apparently you can find based on sub ranges too: 5, 6, 3, 3, ...

jwilk 9y ago

Please use the original title ("Spheres and points").

justinhj 9y ago
this title was more clickbaity but in a good way. made me click after spending some time (fruitlessly) trying to figure it out

pbhjpbhj 9y ago

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?

ajnin 9y ago
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.
- pbhjpbhj 9y ago
  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.
  - kybernetikos 9y ago
    But what shape are you using for your tiles?
    - pbhjpbhj 9y ago
      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.
9y ago

jordigh 9y ago

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. https://en.wikipedia.org/wiki/Thomson_problem https://en.wikipedia.org/wiki/Thomson_problem

tbodt 9y ago

This is the first time I've seen IPFS-hosted content on HN for a reason other than that it is IPFS-hosted.