5 ms·

Sphere Packing Solved in Higher Dimensions

218 points by digital55 11y ago

giomasce 11y ago
Nice, although it would be nice to have an idea of the proofs behind.
- ecma 11y ago
  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 11y ago
    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.)
- hyperbovine 11y ago
  The paper is linked in the article. I encourage you to take a look--for a breakthrough result it's unusually accessible, verging on self-contained.
sigil 11y ago
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 11y ago
  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 11y ago
    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 11y ago
      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-web.pdf http://www.ams.org/programs/students/wwtbam/cohn-slides-2014...
- pwm 11y ago
  Maybe this is why we perceive so little matter in our universe. Maybe our universe is packed to its densest in its natural higher dimension :))
Daneel_ 11y ago
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 11y ago
  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 11y ago
    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.
foota 11y ago
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-web.pdf http://www.ams.org/programs/students/wwtbam/cohn-slides-2014... 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 11y ago
  Whoa.
Xophmeister 11y ago
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-web.pdf http://www.ams.org/programs/students/wwtbam/cohn-slides-2014...
dschiptsov 11y ago
Higher dimensions like in Hegel?)
cperciva 11y ago
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 11y ago
  Seriously, I was very excited when I saw the title, and very let down by the actual result.
deleted 11y ago
[deleted]
cabirum 11y ago
So how many golf balls fit in 7-dimensional school bus?
- hyperbovine 11y ago
  Uncountably many.
  - ajuc 11y ago
    ℶ^4 I think?
    - ubercow13 11y ago
      Is it not just ℶ^1?
      - ajuc 11y ago
        I messed up I meant continuum but the symbol is beth, so it should be beth1 or that weird small c. And you're right the power doesn't matter because continuum^n = continuum.
- pavel_lishin 11y ago
  If I'm solving 7-dimensional brain teasers during interviews, I better be getting a 7-dimensional paycheck if I'm hired.
  - knodi123 11y ago
    Of course you will. Now the trick is to find a 7-dimensional bank to cash it.
pitchka 11y ago
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 11y ago
  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 11y ago
    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.
hellbanner 11y ago
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 11y ago
  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/
leecarraher 11y ago
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.
dekhn 11y ago
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.
Kinnard 11y ago
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-networking/ http://www.cnet.com/news/is-pcell-the-holy-grail-of-wireless...
- josh2600 11y ago
  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.

218 points by digital55 11y ago

giomasce 11y ago

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

ecma 11y ago
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 11y ago
  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.)
hyperbovine 11y ago
The paper is linked in the article. I encourage you to take a look--for a breakthrough result it's unusually accessible, verging on self-contained.

sigil 11y ago

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 11y ago
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 11y ago
  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 11y ago
    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-web.pdf http://www.ams.org/programs/students/wwtbam/cohn-slides-2014...
pwm 11y ago
Maybe this is why we perceive so little matter in our universe. Maybe our universe is packed to its densest in its natural higher dimension :))

Daneel_ 11y ago

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 11y ago
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 11y ago
  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.

foota 11y ago

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-web.pdf http://www.ams.org/programs/students/wwtbam/cohn-slides-2014... 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 11y ago
Whoa.

Xophmeister 11y ago

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-web.pdf http://www.ams.org/programs/students/wwtbam/cohn-slides-2014...

dschiptsov 11y ago

Higher dimensions like in Hegel?)

cperciva 11y ago

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 11y ago
Seriously, I was very excited when I saw the title, and very let down by the actual result.

deleted 11y ago

[deleted]

cabirum 11y ago

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

hyperbovine 11y ago
Uncountably many.
- ajuc 11y ago
  ℶ^4 I think?
  - ubercow13 11y ago
    Is it not just ℶ^1?
    - ajuc 11y ago
      I messed up I meant continuum but the symbol is beth, so it should be beth1 or that weird small c. And you're right the power doesn't matter because continuum^n = continuum.
pavel_lishin 11y ago
If I'm solving 7-dimensional brain teasers during interviews, I better be getting a 7-dimensional paycheck if I'm hired.
- knodi123 11y ago
  Of course you will. Now the trick is to find a 7-dimensional bank to cash it.

pitchka 11y ago

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 11y ago
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 11y ago
  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.

hellbanner 11y ago

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 11y ago
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/

leecarraher 11y ago

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.

dekhn 11y ago

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.

Kinnard 11y ago

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-networking/ http://www.cnet.com/news/is-pcell-the-holy-grail-of-wireless...

josh2600 11y ago
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.