Thread by @dwilliams999, John Conway died today from #Coronavirus. He was a central figure in [...]

John Conway died today from #Coronavirus. He was a central figure in one of the deepest and most compelling mysteries in mathematics, physics and computer science: the Monster Vertex Super-algebra and the Moonshine Conjecture. A thread 1/ https://www.quantamagazine.org/mathematicians-chase-moonshine-string-theory-connections-20150312/

Mathematicians Chase Moonshine, String Theory Connections | Quanta Magazine

Researchers are on the trail of a mysterious connection between number theory, algebra and string theory.

https://www.quantamagazine.org/mathematicians-chase-moonshine-string-theory-connections-20150312/

This story begins with Hamming in 1947 at the Bell Telephone Labs: “Two weekends in a row I came in and found that all my stuff had been dumped [so] I said, 'Damn it, if the machine can detect an error, why can't it locate [it] and correct it?'” 2/

https://en.m.wikipedia.org/wiki/Hamming_code

Digital communication over a noisy channel may lead to corrupt bits and messages, voices will be garbled. To correct for this, we send redundant copies of the message for cross referencing. We must also be economical, we can’t send so many copies of the data over the network 3/

Following Hamming’s lead in self-correcting codes, Marcel Golay published his encoding in 1949 which is now known as the binary Golay code, a perfect 23 bit message encoding which is highly efficient. Adding a parity bit yields a 24 bit encoding. 4/

http://mathworld.wolfram.com/GolayCode.html

Golay codes are so efficient that @NASA used them to communicate with the Voyager 1 & 2 space crafts because “memory constraints dictated offloading data virtually instantly leaving no second chances”, sending back images of Jupiter and Saturn. 5/

https://en.m.wikipedia.org/wiki/Binary_Golay_code

Viewing the Golay codes as 24 dimensional binary vectors, we can use them to create a 24 dimensional lattice. This is called the Leech Lattice. 6/ https://cp4space.wordpress.com/2013/09/12/leech-lattice/

Leech lattice

I’ve been intending to write about the Leech lattice for a while now. I wanted to ensure that I could actually do it justice, and in particular make it strictly more comprehensive than the Wi…

https://cp4space.wordpress.com/2013/09/12/leech-lattice/

We also see a deep connection between self correcting codes and sphere packing: “If we center Hamming spheres at each code word, we see the number of errors a code corrects is the largest radius the spheres can have while remaining disjoint.” 7/

http://www.math.ups.edu/~bryans/Current/Journal_Spring_2006/ARoberts_LeechLattice.pdf

In turn, the Leech Lattice is one of the best solutions to the kissing number problem for packing spheres in 24 dimensions. The efficient error correction capabilities of the Golay codes leads directly to dense lattices of tangential spheres. 8/

https://en.m.wikipedia.org/wiki/Kissing_number

“Each sphere is tangent to 196,560 [neighboring spheres], and this is known to be the largest number of non-overlapping 24-dimensional unit balls that can simultaneously touch a single unit ball.” 9/

https://en.m.wikipedia.org/wiki/Leech_lattice#Symmetries

This is where Conway enters the story, by looking at symmetry in the Leech Lattice. Symmetry is core to the human experience. We all know what it looks like. Take a square and rotate 90 degrees, or flip it on an axis, and it looks the same. 10/

“The group of [symmetrical] rotations that preserves the Leech lattice is called the Conway group. It has ~ 8,315 x 10 ^ 15 elements. [It is not simple, but] if you mod out the Conway group by its center [Z2] you get a finite simple group Co_1.” 11/

https://golem.ph.utexas.edu/category/2019/08/the_conway_2groups.html

The history of abstract algebra is far too large for a Twitter thread, however, we can say a few things here: finite group theory is the study of patterns within a finite number permutations and rotations of an object, e.g. the Rubik’s cube. 12/

https://en.m.wikipedia.org/wiki/Rubik%27s_Cube_group

The classification of finite simple groups was the greatest mathematical undertaking in history, spanning decades for hundreds of researchers, including Conway. It is solved in that we now have a complete taxonomy of core finite algebraic patterns. 13/

https://en.m.wikipedia.org/wiki/Classification_of_finite_simple_groups

Finite simple groups are organized into 18 descriptive families, including modular arithmetic, and 26 sporadic members which do not belong to any of the families. The largest of these one-off sporadic groups is called the Monster Group. 14/

The Monster Group can itself be constructed from 3 of the sporadic simple groups: the Fischer group Fi24, the Baby Monster group B, and the Conway group Co1. 15/

https://en.m.wikipedia.org/wiki/Monster_group

In 1978, John McKay noticed that the coefficients of the Fourier expansion of Klein’s J-Function were in fact sums of the dimensions of the irreducible representations of the Monster Group. 16/

https://en.m.wikipedia.org/wiki/J-invariant

In 1979, Conway and Norton named this discovery “Monstrous Moonshine”. Norton is quoted as saying, “I can explain what Monstrous Moonshine is in one sentence, it is the voice of God.” 17/

https://en.m.wikipedia.org/wiki/Monstrous_moonshine

Monstrous Moonshine was proven between 1986 and 1988 using string theory and conformal quantum field theory. The Monster Vertex Algebra was described using 24 bosons compactified on the torus induced by the Leech lattice and orbifolded by reflection. 18/

https://en.m.wikipedia.org/wiki/Monster_vertex_algebra

Interviewer: “Do you have any unfinished business?”

Conway: “There’s a beautiful thing called the Monster Group, and I would just like to know what it’s all about. The one thing I’d like to know before I die, is why the Monster Group exists.” 19/

It’s safe to say that John Conway made an incredible impact during his time here on Earth, from 2-D Cellular Automata which would inspire @stephen_wolfram’s New Kind of Science, to the deepest esoteric quest to know the fabric of our reality. 20/

https://en.m.wikipedia.org/wiki/Cellular_automaton

We can hope that Conway’s spirit will live on in new ideas, like playful creations and continuous automata such as Lenia. I also hope we learn more about the “why” of the sporadic groups and the role the Monster plays in the fabric of our Universe.

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