Niko Porjo
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Famous math problem solved!

Ten days ago, Maryna Viasovska showed how to pack spheres in 8 dimensions as tightly as possible. In this arrangement the spheres occupy about 25.367% of the space. That looks like a strange number - but it's actually a wonderful number, as shown here.

People had guessed the answer to this problem for a long time. If you try to get as many equal-sized spheres to touch a sphere in 8 dimensions, there's exactly one way to do it - unlike in 3 dimensions, where there's a lot of wiggle room! And if you keep doing this, on and on, you're forced into a unique arrangement, called the E8 lattice. So this pattern is an obvious candidate for the densest sphere packing in 8 dimensions. But none of this proves it's the best!

In 2001, Henry Cohn and Noam Elkies showed that no sphere packing in 8 dimensions could be more than 1.000001 times as dense than E8. Close... but no cigar.

Now Maryna Viasovska has used the same technique, but pushed it further. Now we know: nothing can beat E8 in 8 dimensions!

Viasovska is an expert on the math of "modular forms", and that's what she used to crack this problem. But when she's not working on modular forms, she writes papers on physics! Serious stuff, like "Symmetry and disorder of the vitreous vortex lattice in an overdoped BaFe_{2-x}Co_x As_2 superconductor."

After coming up with her new ideas, Viaskovska teamed up with other experts including Henry Cohn and proved that another lattice, the Leech lattice, gives the densest sphere packing in 24 dimensions.

Different dimensions have very different personalities. Dimensions 8 and 24 are special. You may have heard that string theory works best in 10 and 26 dimensions - two more than 8 and 24. That's not a coincidence.

The densest sphere packings of spheres are only known in dimensions 0, 1, 2, 3, and now 8 and 24. Good candidates are known in many other low dimensions: the problem is proving things - and in particular, ruling out the huge unruly mob of non-lattice packings.

For example, in 3 dimensions there are uncountably many non-periodic packings of spheres that are just as dense as the densest lattice packing!

In fact, the sphere packing problem is harder in 3 dimensions than 8. It was only solved earlier because it was more famous, and one man - Thomas Hales - had the nearly insane persistence required to crack it.

His original proof was 250 pages long, together with 3 gigabytes of computer programs, data and results. He subsequently verified it using a computerized proof assistant, in a project that required 12 years and many people.

By contrast, Viasovska's proof is extremely elegant. It boils down to finding a function whose Fourier transform has a simple and surprising property! For details on that, try my blog article:

https://golem.ph.utexas.edu/category/2016/03/e8_is_the_best.html

#geometry

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Think your browser is protected from tracking? Visit Panopticlick 2.0 to find out...
https://panopticlick.eff.org
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Love this science and engineering twist on this favorite childhood story!

from
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HEI! Pyytäisin teilta apua, ideoita tai vinkkiä. Opiskeluni restonomi YAMK-linjalla on loppusuoralla opinnäytetyötä vaille valmis.Kaikki kurssit tein jo keväällä tarkoituksena valmistua nyt syksyllä,jotta olisin taas työnhaussa vähän paremmissa asemissa. Opinnäytetyön toimeksiantajan haku on ollut vaikeaa. Olen laittanut eri yrityksiin sähköposteja ja kysellyt Twitterissä jo puoli vuotta. Tämä on vähän sama kuin kotiäidillä, joka siivoaa ennenkuin päästää siivoojan asuntoonsa eli yrityksille on ollut vaikea ottaa tarjoustani vastaan, koska "ollaan somessa niin aloittelijoita" tai sitten kenelläkään ei ole ollut tarjota yhteyshenkilöä, kun kaikilla on niin kiire. Olen itse hyvin itsenäinen työskentelijä enkä tarvitse yrityksen puolelta aktiivisuutta välttämättä ellei muuten ole halua vaikuttaa kovinkaan paljon mulloin kuin työn alkuvaiheessa tutkimuskysymysten määrittelyssä.Jouduin tässä luopumaan taas kerran yhdestä toimeksiantajasta, sillä hän ei kolmen viikon aikana ehtinyt vastata tarkentaviin kysymyksiini.Ohjaava opettajani ehdotti nyt toimeksiannon saannin hankaluuden takia sisältöanalyysiä valmiista tutkimuksista.Niitä on vaikea löytää, sillä toivomani aihe liittyy sosiaaliseen mediaa ja asiakaspalveluun ja viestintään (olen jo kirjoittanut teoriaa aiheiden merkeissä melko paljon). Olen itse kiinnostunut sosiaalisen median palveluista palvelualojen, koulutuksen tai työnhaun kannalta. Toivoisin jopa joskus löytäväni sen merkeissä työn itselleni, joten opinnäytetyöni pitäisi hyödyntää minua tai jotain muuta tahoa oikeasti eikä vain opparia sen takia,että se pitää tehdä. Otan mielelläni kommentteja vastaan täällä tai YV.
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Following the path of 'The Martian' – video generated using images acquired by the Mars Express orbiter

Watney is ' ' in the film of the same name (release date in Germany: 8 October) who, in a not too distant future, finds himself stranded on the Red Planet.

Scientists from #DLR – who specialise in producing highly accurate topographical maps of #Mars – reconstructed Watney's route using stereo image data acquired by the High Resolution Stereo Camera on board 's #MarsExpress spacecraft. They then compiled this data into a 3D film that shows the spectacular landscape that the protagonist would see 'in the future'.