
Sphere Packing Solved in Higher Dimensions - digital55
https://www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions/
======
foota
This is a really excellent walkthrough of the applications of sphere packing
to information theory: [http://www.ams.org/programs/students/wwtbam/cohn-
slides-2014...](http://www.ams.org/programs/students/wwtbam/cohn-
slides-2014-web.pdf) as well as the answer to a question I asked earlier in
the thread.

edit: that's funny, I just realized the author of this link is quoted in the
OP's article.

~~~
kchoudhu
Whoa.

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cperciva
Can we have s/Higher/8 and 24/ in the title? As written it suggests that
sphere packing had been solved in _all_ higher dimensions, which would be far
more dramatic.

~~~
cmrx64
Seriously, I was very excited when I saw the title, and very let down by the
actual result.

------
Xophmeister
It would be nice to see a graph of packing density against dimensionality.

EDIT: Looks like I spoke too soon! There's one in the link posted by @foota in
these comments (p33 of [1])

[1] [http://www.ams.org/programs/students/wwtbam/cohn-
slides-2014...](http://www.ams.org/programs/students/wwtbam/cohn-
slides-2014-web.pdf)

------
sigil
_It’s possible to build an analogue of the pyramidal orange stacking in every
dimension, but as the dimensions get higher, the gaps between the high-
dimensional oranges grow._

In case like me you're wondering just _how_ densely you can pack spheres in
the highly favorable 24th dimension: the best you can do is about 0.2% of the
total volume.

[https://en.wikipedia.org/wiki/Leech_lattice#Geometry](https://en.wikipedia.org/wiki/Leech_lattice#Geometry)

~~~
foota
It would be interesting to look at the "believed optimal" schemes for packing
spheres in each dimension and analyze how the packing density changes.

~~~
ecma
MathWorld has a really good article [0] with density numbers from R2 through
R8 if you're interested.

[0]
[http://mathworld.wolfram.com/HyperspherePacking.html](http://mathworld.wolfram.com/HyperspherePacking.html)

~~~
foota
Thanks! You might also be interested in this thing, sort of a slideshow, that
I posted up above: [http://www.ams.org/programs/students/wwtbam/cohn-
slides-2014...](http://www.ams.org/programs/students/wwtbam/cohn-
slides-2014-web.pdf)

------
dekhn
Way back in college I wanted to learn more about robots, so I took a class
called 'Cybernetics' taught by David Huffman (who invented Huffman coding).
The class wasn't about robots (I misunderstood; it was about information
theory and control systems).

In the very first lecture Huffman concluded the class with the statement "It
turns out, sphere packing in 7 dimensions is easy! And this has applications
to telephone routing..." (he worked for Bell Labs).

I thought it was interesting how he spent nearly an hour explaining something
that seemed totally pure math and it ended up helping AT&T scale phone
networks.

------
giomasce
Nice, although it would be nice to have an idea of the proofs behind.

~~~
ecma
The paper [0] was linked, the article gave context to the problem and outlined
the approach. What more do you want?

[0] [http://arxiv.org/abs/1603.04246](http://arxiv.org/abs/1603.04246)

~~~
iheartmemcache
This is more grokable for the average geek than most, but I think he was
looking for a less mathematically jargon-y translation. E.g., something
like[1] this written to translate from the "math" domain into the "average
dude" domain, done by the co-founder of Julia. R_8 / R_24 is still impressive,
haters. ;)

We both might know enough about math to know what a Lebesgue measure is, but
Real Analysis wasn't a requirement for most CS majors posting on here I'd
imagine. (I agree with my uncle/aunt poster, though-- this is way more
accessible to the general mathematician than a Wiles on Fermat or Perelman on
Poincare. It's self-contained enough to clearly define * weakly-holomorphic
modular form* [which I could take a stab at, but not with too high of a
confidence factor]. Spending a few minutes to throw (19) in there was kind of
the author to do, so those from other disciplines of mathematics could
continue reading instead of just presuming what \gamma was.)

------
Daneel_
Fascinating article. I wonder if this is indeed the same E8 as the Lie group
that Garrett Lisi suggested (incorrectly) as a possible structure for a theory
of everything?

~~~
andyjohnson0
Lisi's proposal was discussed 8 years ago at [0]. A quick scan of [1] and [2]
leads me to suspect that it is the same E8 - but I'm not a mathematician so
don't go and bet any money on it...

[0]
[https://news.ycombinator.com/item?id=258198](https://news.ycombinator.com/item?id=258198)

[1]
[https://en.wikipedia.org/wiki/E8_(mathematics)](https://en.wikipedia.org/wiki/E8_\(mathematics\))

[2]
[https://en.wikipedia.org/wiki/Leech_lattice](https://en.wikipedia.org/wiki/Leech_lattice)

~~~
gjm11
Not the exact same mathematical object, but closely related -- it's not at all
coincidental that both are called E8.

The lattice is really easy to describe. It's a set of points in 8-dimensional
space, so each point is specified by 8 numbers. A point is in the lattice if
the following conditions hold. (1) Either all its coordinates are integers, or
all are half-integers. (2) The sum of the coordinates is an even integer.

(And then the densest packing of spheres in 8-dimensional space puts the
centre of a sphere at each point. The radius of the spheres is then
1/sqrt(2).)

The E8 that features in Garrett Lisi's attempted "theory of everything" is a
so-called "Lie group". I'll attempt to sketch the relationship between
lattices like the E8 lattice, and "Lie algebras", and Lie groups, and particle
physics, but note that all of what follows is really handwavy and sometimes
just plain wrong. Sometimes on purpose, probably sometimes by mistake. I'm
sorry about any mistakes, but the deliberate lies and sketchiness are
necessary to keep this shorter than textbook length. I hope it's at least
semi-comprehensible.

(Although it's shorter than a textbook, it's also too long for a single HN
comment. I'll break it into pieces and make each a reply to its predecessor.
HN mods, let me know if this is considered bad form and I'll refrain from
doing it in the future.)

OK, here goes.

PHYSICS AND SYMMETRY

OK, let's begin at the physics end. Here are some facts about plain old
Newtonian physics. (1) The physical laws make no reference to particular
positions in space; equivalently, if we e.g. replace (x,y,z) with
(x+a,y+b,z+c) everywhere, for constant a,b,c, then the laws don't change. (2)
If you have a system of particles interacting only with one another, the total
momentum (sum of mass times velocity) is constant. It turns out that #1 and #2
are sort of equivalent: for a broad class of physical theories, _symmetries_
(like #1, the fact that adding a constant to the coordinates changes nothing)
are equivalent to _conservation laws_ (like #2, conservation of momentum).
Here's a subtler one: the laws don't pick out preferred times any more than
preferred positions, and that fact is equivalent to conservation of energy.

That was all in Newtonian physics, but it works more or less the same in
relativity and in quantum mechanics.

There are some much subtler kinds of symmetry present in our physical laws,
and they give rise to subtler conservation laws. For instance, you may be
familiar with the notion of "phase" in optics; in a classical picture this
corresponds to the exact location/timing of the electromagnetic waves that
make up light; in quantum mechanics everything is described by a
"wavefunction" whose value at every point in the relevant configuration space
is a complex number, and the optical phase of light turns out to correspond to
the phase of the photons' wavefunctions. Changing all the phases by the same
constant amount leaves every possible observation you could make unaltered,
which means it's a symmetry of the physical laws; and this symmetry turns out
to correspond to _conservation of electric charge_.

This picture gets more complicated when you throw in the "weak nuclear force",
which turns out actually to be kinda the same thing as electricity and
magnetism, and hairier still when you throw in the "strong nuclear force". You
get symmetries that, e.g., describe exchanging different kinds of particle
with one another (kinda), and corresponding conservation laws.

GROUPS AND LIE GROUPS

If you look at a high enough level of abstraction at what different kinds of
symmetry have in common, you get the pure-mathematical notion of a _group_.
The idea, roughly, is that if you have some thing (e.g., a block of metal or a
physical theory) and some kind of symmetry (e.g., ways you can move the block
of metal around and end up with it looking exactly the same as before; ways
you can change some things in the laws of physics and end up with equivalent
laws), then _all the symmetries of the thing_ together have a particular kind
of mathematical structure. This basically comes down to these rather obvious
observations: if you have two operations that leave your thing looking the
same then _doing one and then the other_ also leaves your thing looking the
same; and _undoing_ one such operation also leaves your thing doing the same.
And a mathematical structure that supports "doing one thing and then another",
and "undoing a thing", is a group.

These symmetries in physics have an important feature: they don't occur in
isolation but come in continuously-varying families. E.g., translation-
invariance, the (x+a,y+b,z+c) thing, allows _any_ value of a,b,c; you can vary
them continuously and get the same results all the time as you do so. This
means that the _symmetry groups_ that describe them mathematically have this
continuously-varying property, and the mathematical formalization of this
gives you the notion of "Lie groups". (Named after a mathematician called
Sophus Lie.)

Some Lie groups can be built out of other simpler ones, in something a bit
like the way that some numbers can be written as the product of other smaller
ones. If you break them down as far as possible, you get to the so-called
"simple Lie groups", which are kinda like the prime numbers: every Lie group
can be built out of simple Lie groups.

And it turns out that we can figure out what all the possible simple Lie
groups are. There are four infinite families of them called A,B,C,D, and a few
special ones that don't fit into the families. The most complicated of these
special ones is called E8.

~~~
gjm11
(Second part. There will be one more.)

A BIT MORE PHYSICS

Garrett Lisi's attempted "theory of everything" was based around the idea that
the pattern of symmetries of the laws of physics is that of the group E8. We
don't see all those symmetries in the laws we know, but that isn't necessarily
a problem because of a thing called _spontaneous symmetry breaking_. Imagine a
big tank of water. Within that tank (at least away from its edges) no
direction is different from any other -- the laws of physics don't have
preferred directions, and neither does a bunch of water molecules bouncing off
one another at random. This symmetry corresponds to _conservation of angular
momentum_ , so e.g. movements of water within the tank should obey that law.
OK; now what if we reduce the temperature and the water freezes? Ice crystals,
unlike liquid water, have a regular structure and they _do_ have a preferred
direction; so some things that happen in our tank full of ice may fail to
conserve angular momentum.

(Let me be careful about this. Of course the overall laws of physics still
satisfy that law, so if we consider the whole contents of the tank C of AM
will still hold. But if we treat the ice in the tank as "background" and
consider, I dunno, electrons moving around within the ice or something,
_their_ total angular momentum may fail to be constant, because of angle-
dependent interactions with the ice.)

The same kind of thing can happen with the more "abstract" kinds of symmetry:
as the universe cools -- and note that the universe now is much much cooler
than shortly after the big bang -- it may settle into some less-symmetrical
configuration (like our tank of ice), and the asymmetries in the universe may
look to use like asymmetries in the laws, even though the real fundamental
laws are symmetrical.

So, anyway, Lisi's theory is probably wrong, but the mere fact that E8 is
bigger than the group of symmetries we actually observe in the universe isn't
in itself good reason to disbelieve it. Now let's do (or at least gesture
vaguely towards) some more mathematics, with the hope of ending up with some
understanding of how the group E8 relates to the lattice E8.

LIE ALGEBRAS

To describe the relationship between the Lie group E8 and the lattice E8, I
need to go via another kind of mathematical object called a Lie algebra. These
are in some sense simpler cousins of the Lie groups, but they're harder to
describe and harder to explain why you'd care about them. Still, here goes.

Let's look at one of the very simplest Lie groups. Take the ordinary euclidean
plane -- imagine an infinitely large perfectly flat piece of paper. Fix a
point in it; call it O. And consider _rotations around O_. These form a Lie
group; it's sometimes called U(1), never mind why. We can describe it
geometrically: it's a circle. (Pick some point that isn't O; call it P.
Describe any rotation by saying where P goes to. That point is always on the
circle through P with centre O. Conversely, if we pick such a point, there's a
rotation that takes P there.)

A circle is pretty simple, but you know what's even simpler? A straight line.
And if you look only at very small portions of the circle, they look a lot
like very small portions of a straight line. This is the fundamental idea
behind calculus: all nice things look linear if you look at a small enough
scale. And linear things are easier to work with, mathematically. So: suppose
we have a Lie group -- that is, a continuously-varying set of symmetries of
something. Take a tiny region of it near to the origin; that is, near to the
trivial symmetry that does nothing. This tiny region looks like a piece of
euclidean space in some number of dimensions, but there's some extra structure
on it that describes the relationships between those symmetries -- what
happens when you do one followed by another.

What does that extra structure look like? It turns out that it's best
described by what's called a _bracket operation_ , meaning that given two
elements x,y we have another written [x,y] which you should think of as
meaning xy-yx. Imagine e.g. that x,y are actually matrices. Matrix
multiplication usually doesn't commute, so generally xy-yx is not zero. Here,
in some sense the multiplication operation corresponds to the "do one symmetry
followed by another" operation on the original Lie group; that no longer
exists in our "infinitesimal" view, but the bracket describing what changes
when you switch the order of the symmetries still does. Kinda.

The bracket operation obeys a number of laws like the so-called Jacobi
identity [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0. The details aren't important
right now. Anyway, the idea that we end up with when we follow this path and
_do_ attend to the details is that of a thing called a "Lie algebra", which
describes what small pieces of a Lie group look like, and which in general
looks like a euclidean space of some dimension together with a "bracket
operation".

Corresponding to the Lie group called E8 there is, therefore, a Lie algebra
also called E8.

~~~
gjm11
(Third and final part.)

REPRESENTATIONS, WEIGHTS, AND ROOTS

Suppose we have a Lie group and its corresponding Lie algebra which, remember,
is kinda like a blown-up copy of an infinitesimally tiny bit of the original
group. Then there's a natural way for the group to "act on" the algebra, which
roughly speaking corresponds to the ordinary "do one symmetry and then
another" operation in the group.

But the Lie algebra is, among other things, a vector space (what I've
described above as a euclidean space), and it turns out that each symmetry in
the group gives us a linear transformation of that space. Another way to say
this is that we're taking our abstract symmetry group, and exhibiting its
elements as symmetries of a particular object, namely the Lie algebra. This
situation, where you have a group and you find a way to turn its elements into
linear transformations of a vector space, is called a _representation_ of the
group.

We can also have the Lie algebra act on itself in a basically-equivalent way,
and then what we get is a representation of the Lie algebra.

(In quantum physics, the state of the universe is described by a vector in an
enormous vector space, symmetries of the laws of physics correspond to linear
transformations on that vector space, and so representations arise quite
naturally. If instead of asking about the whole universe we ask about some
much smaller configuration of particles, we get a much smaller vector space;
if the symmetries we're interested in leave that configuration of particles
alone, we get a representation on the smaller vector space.)

The next bit is kinda unmotivated. I'm sorry.

So, suppose you have a representation of a Lie algebra -- a manifestation of
its elements as linear transformations on a vector space. Consider what's
called a "Cartan subalgebra", which is to say as large as possible a subspace
of the Lie algebra with the property that within it _all the brackets are
zero_. (Meaning that the corresponding things in the Lie group _commute_ ,
meaning that it doesn't matter whether you do A then B or B then A.) Let's
give names to some key objects here. G consists of the linear transformations
corresponding (under our representation) to the original Lie algebra. H
consists of the ones coming from the Cartan subalgebra; so it's just a portion
of G, usually a lot smaller than G itself.

Now, suppose we have g in G and h in H. We can apply our bracket operation to
them -- in fact, since we're doing all this via the representation, the
bracket is always just gh-hg in terms of matrix (= linear transformation)
multiplication. If we're lucky, the following amazing thing might happen: this
might (1) always be a scalar multiple of g, where (2) the actual multiple only
depends on h. That is, we might have [g,h] = f(h)g for some function f, which
in fact will always have to be linear. That would be a hell of a coincidence,
but perhaps a smaller coincidence might happen: the same might hold but only
for some subspace of G.

The linear functions on H for which this _does_ happen for a nontrivial
subspace of G are called _weights_ for the representation. (It's a pretty
terrible name. I'm sorry.)

There's a particularly important representation called the "adjoint
representation", which is the one I described above immediately under the
heading "REPRESENTATIONS, WEIGHTS, AND ROOTS". The weights of the adjoint
representation are called "roots", which is also a terrible name.

What happens for E8? Well, the Cartan subalgebra is pretty small; it's
8-dimensional (versus 248-dimensional, as it happens, for E8 itself). To
describe a linear function that maps an 8-dimensional space onto numbers, you
need 8 numbers -- e.g., the first says what number (1,0,0,0,0,0,0,0) maps to.
So you can think of its roots as also living in an 8-dimensional space.

It turns out that there are exactly 240 roots for E8; they are as follows.
First, there are all the 8-dimensional integer vectors of length sqrt(2); that
is, the vectors of 8 integers consisting of six 0s and two +-1s. There are 112
of these. Second, there are the 8-dimensional vectors whose entries are all
+1/2 or -1/2, where we have an even number of +1/2 and an even number of
-1/2\. There are 128 of these.

THE ROOT LATTICE

Finally: the _root lattice_ for a Lie algebra consists of all integral
combinations of its roots. So, e.g., for E8 we have the 240 roots listed
above; call them r1,r2,...,r240. The root lattice consists of everything you
can get by picking integers a1,a2,...,a240 and computing a1.r1 + ... +
a240.r240.

And lo, the root lattice is exactly the E8 lattice I described way up above.

~~~
rperce
That was one hell of a ride. Great explanations, really enjoyed it!

------
cabirum
So how many golf balls fit in 7-dimensional school bus?

~~~
pavel_lishin
If I'm solving 7-dimensional brain teasers during interviews, I better be
getting a 7-dimensional paycheck if I'm hired.

~~~
knodi123
Of course you will. Now the trick is to find a 7-dimensional bank to cash it.

------
hellbanner
Relevant: How to turn a sphere inside out:
[https://www.youtube.com/watch?v=R_w4HYXuo9M](https://www.youtube.com/watch?v=R_w4HYXuo9M)

Perfect Shapes in Higher Dimensions:
[https://www.youtube.com/watch?v=2s4TqVAbfz4](https://www.youtube.com/watch?v=2s4TqVAbfz4)

~~~
theoh
I think the sphere eversion is actually not a related problem in mathematical
terms.

In any case, that video is part of the excellent longer film "Outside In" made
at the Geometry Center in the 90s:

[http://www.geom.uiuc.edu/docs/outreach/oi/](http://www.geom.uiuc.edu/docs/outreach/oi/)

------
Kinnard
I wonder if this has impact on pCells, the "holy grail of networking":
[http://www.cnet.com/news/is-pcell-the-holy-grail-of-
wireless...](http://www.cnet.com/news/is-pcell-the-holy-grail-of-wireless-
networking/)

~~~
josh2600
There are lots of holy grails in networking. PCells are possible today but
there's basically zero business cases for rolling this tech. Consider that
users don't want to pay more for faster access.

------
leecarraher
although the practical implication of proving leech and E8 are optimal is
somewhat unsurprising (as both have been known to be remarkably close to the
feasible optimal), the theoretical framework using modular forms could shed
some light on the connection with the monster. but then again i tend to be a
bit of an optimist hoping john conway's desire for a connection get fulfilled.

------
pitchka
It's just amazing what mathematics can do.

What was once attacked with NP-hard algorithms now has a constructive one.

Reminds me a little bit of the lottery problem - one has to find the minimum
sized set of tickets that guarantee a match of at least K numbers on any of
the tickets. Statistics can easily get you the expected number of tickets you
have to buy but finding that minimum set can be mapped to a minimum set cover
(where you're trying to cover all tickets with your set so that tickets in the
set match all possible tickets in at least K numbers).

Then, all of a sudden maths can put bounds and direct formulas of how many
tickets your set has to have (not trivial).

Leaving you again with a constructive polynomial algorithm.

~~~
sibrahim
For anyone else that's curious, the lottery problem is called the
Transylvanian lottery:
[https://en.wikipedia.org/wiki/Transylvania_lottery](https://en.wikipedia.org/wiki/Transylvania_lottery)

The primary observation is if you want to match k numbers on tickets where you
pick n numbers, each ticket simultaneously covers n choose k different
k-tuples. The Translyvanian lottery is designed so that finite projective
planes give you the minimal result (no pair appears on more than one ticket).

This sort of combinatorial design is rumored to have been used by the MIT
syndicate that gamed the Massachusetts lottery (notably, they manually and
laboriously picked their numbers while the other two syndicates were using
random quick picks).

(I've had to learn a lot about lotteries in the last two years :P)

On a side note, finding a minimal set of tickets to cover a set of outcomes is
exactly analogous to testing software with pairwise testing:
[https://en.wikipedia.org/wiki/All-
pairs_testing](https://en.wikipedia.org/wiki/All-pairs_testing)

~~~
pitchka
General problem - having ticket with k numbers where possible numbers are from
1 to n, find the minimal set of tickets that will cover all of the possible
ones so that at least one ticket in your set has at least m equal numbers on
the ticket that is drawn (m <= k). (Trivial example. To win the lottery -
guarantee k matches - one has to buy all of the tickets.)

It is still unsolved although maths has provided some nice bounds for large
amounts of (n,k,m) triplets, and solutions to (n,k,2). Although, for the
(n,k,2) a simple greedy set cover finds the optimal solution always.

------
dschiptsov
Higher dimensions like in Hegel?)

