
The sequence 1 1 ∞ 5 6 3 3 3 - quickthrower2
https://ipfs.io/ipfs/QmfN5DojVnEsf1Une3DFwfUiFpfWnQf31f61qgybiXVeQE/blog/spheres-and-points/
======
n4r9
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)

------
crasshopper
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-
XzhVrXIVeSVcV9iRJ4S...](https://www.youtube.com/playlist?list=PL-
XzhVrXIVeSVcV9iRJ4S9WAH1ryq4hTQ)

[https://www.youtube.com/playlist?list=PL-
XzhVrXIVeQ298S6uCyo...](https://www.youtube.com/playlist?list=PL-
XzhVrXIVeQ298S6uCyoDGWNActWwnzZ)

[http://math.ucr.edu/home/baez/week230.html](http://math.ucr.edu/home/baez/week230.html)

[https://unapologetic.wordpress.com/category/geometry/root-
sy...](https://unapologetic.wordpress.com/category/geometry/root-systems/)

[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
A-D-E? Surely you mean A-B-C-F-G-H-I? :) (Yes, I know that list is
redundant...)

~~~
quickthrower2
Can you explain the joke?

~~~
Sniffnoy
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.sv...](https://en.wikipedia.org/wiki/File:Finite_Dynkin_diagrams.svg)
)

* 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. :)

~~~
quickthrower2
I remember a Simpsons episode where Bart cheats his way into a special school.
The teacher tells a maths joke and he is the one not to get it.

This is a whole another level though.

Thanks for the detailed reply!

------
amluto
> 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.)

~~~
guywaffle
A sphere is the wrong shape in the first place since the majority of Earth is
water. Divide the liveable land equally instead.

~~~
eru
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.

~~~
guywaffle
Value is subjective, area of land is not.

~~~
eru
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.

See eg things like
[https://en.wikipedia.org/wiki/Shapley_value](https://en.wikipedia.org/wiki/Shapley_value)
or
[https://en.wikipedia.org/wiki/Fair_division](https://en.wikipedia.org/wiki/Fair_division)

~~~
guywaffle
Subjectivity can’t be solved from a subject point of view (mortal) since the
person would be introducing biases from her/his subjective experiences.

------
mholt
(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...

~~~
daurnimator
Yes it's a permanent url.

No the comments are not static: for this article they are served via disqus.

~~~
KajMagnus
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.

~~~
stiangrindvoll
How can we make this happen with ipfs itself? :)

~~~
lgierth
CRDTs! [https://ipfs.io/blog/30-js-ipfs-crdts.md](https://ipfs.io/blog/30-js-
ipfs-crdts.md)

------
klodolph
> 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.)

------
alanbernstein
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?

~~~
nhaehnle
Yes. The "n-tetrahedron" is usually called the simplex, and the "n-octahedron"
is usually called the cross-polytope.

~~~
grkvlt
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...

------
vog
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/)

~~~
palotasb
[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, ...

------
colehasson
I like the frontend design for this work. Nice job

~~~
semi-extrinsic
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
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..

------
a_c
> 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
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
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

------
jordigh
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)

------
pbhjpbhj
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?

~~~
maweki
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.

~~~
pbhjpbhj
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!).

~~~
kuschku
> 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).

~~~
pbhjpbhj
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).

~~~
kuschku
The problem seems to be your definition of infinity – the definiton as |N| is
exactly defined the way you said it. An arbitrarily large number.

------
philipov
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?

~~~
johnhenry
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)

~~~
rcfox
> As ∞ is not the same type of object as 1, 5, 6 3,

Technically they are, if you consider the set of IEEE 754 floating point
numbers.

~~~
posterboy
but those are not the intended type of object, because MAX_FLOAT plus n and
1.0/0.0 are ambiguous, unintended values.

~~~
Retr0spectrum
Why would that matter?

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

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

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

