Planar cross section of leech lattice?

Question

I was wondering what any planar cross sections of the leech lattice would look like. I don't know much about this topic at all, I'm just quite curious. Is there any way to find out?

Answer 1

The first thing to say is that there are very few restrictions on a two-dimensional section of the Leech lattice. I will get to those. The jpeg below is such a section. The intersections of the green lines are lattice points, and you can see each green fundamental parallelogram. The blue hexagons are the Voronoi cells. I drew in a bunch of red segments of equal length, showing how each Voronoi cell is actually inscribed in a circle around its lattice point. That is, in dimension two there are only deep holes, no shallow holes.

The Leech lattice is called "even," meaning that the norm (squared distance from the origin) of any lattice point is an even positive integer. What is unusual about the Leech lattice is the lack of roots: it does not have any points of norm $2.$ So the same thing applies our binary form. Regarded as a binary quadratic form $f(x,y) = 2(a x^2 + b x y + c y^2), $ abbreviated $2 \langle a,b,c \rangle,$ I have arranged that the result be Gauss reduced, that is $$ |b| \leq a \leq c. $$ So $a\neq 1,$ and $ \langle a,b,c \rangle,$ is not the principal form in its discriminant.

As a result, the covering radius (the length of the red line segments) is likely to be larger than the covering radius of the Leech lattice itself, that being $\sqrt 2.$ However, it does contain $2 \langle 2,1,2 \rangle,$ with a covering radius of $ \sqrt {\frac{8}{5}}. $ It also contains $4 \langle 1,1,1 \rangle,$

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It took me a few minutes to copy the Gram matrix from Gabriele Nebe's website, then convert it into the one-line matrix input format for gp-pari. Note that, as the right column and the final row have a number of minus signs, the last line is rather longer than the others, each minus sign takes up an extra character. First, natural:

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8 4 4 4 4 4 4 2 4 4 4 2 4 2 2 2 4 2 2 2 0 0 0 -3
4 4 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 1 1 2 1 0 0 -1
4 2 4 2 2 2 2 2 2 2 2 2 2 1 2 1 2 2 1 1 1 0 0 -1
4 2 2 4 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 1 1 0 0 -1
4 2 2 2 4 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 1 0 0 -1
4 2 2 2 2 4 2 2 2 2 2 1 2 2 1 1 2 1 2 1 0 0 0 -1
4 2 2 2 2 2 4 2 2 2 2 1 2 1 2 1 2 1 1 2 0 0 0 -1
2 2 2 2 2 2 2 4 1 1 1 2 1 2 2 2 1 2 2 2 2 0 0 1
4 2 2 2 2 2 2 1 4 2 2 2 2 2 2 2 2 2 2 2 1 1 1 -1
4 2 2 2 2 2 2 1 2 4 2 2 2 2 1 1 2 2 1 1 0 1 0 -1
4 2 2 2 2 2 2 1 2 2 4 2 2 1 2 1 2 1 2 1 0 0 1 -1
2 2 2 2 1 1 1 2 2 2 2 4 1 2 2 2 1 2 2 2 2 1 1 1
4 2 2 2 2 2 2 1 2 2 2 1 4 2 2 2 2 1 1 1 1 1 1 -1
2 2 1 1 2 2 1 2 2 2 1 2 2 4 2 2 1 2 2 2 2 2 1 1
2 1 2 1 2 1 2 2 2 1 2 2 2 2 4 2 1 2 2 2 2 1 2 1
2 1 1 2 2 1 1 2 2 1 1 2 2 2 2 4 1 2 2 2 2 1 1 1
4 2 2 2 2 2 2 1 2 2 2 1 2 1 1 1 4 2 2 2 1 1 1 -1
2 1 2 1 2 1 1 2 2 2 1 2 1 2 2 2 2 4 2 2 2 2 1 1
2 1 1 2 2 2 1 2 2 1 2 2 1 2 2 2 2 2 4 2 2 1 2 1
2 2 1 1 2 1 2 2 2 1 1 2 1 2 2 2 2 2 2 4 2 1 1 1
0 1 1 1 1 0 0 2 1 0 0 2 1 2 2 2 1 2 2 2 4 2 2 2
0 0 0 0 0 0 0 0 1 1 0 1 1 2 1 1 1 2 1 1 2 4 2 2
0 0 0 0 0 0 0 0 1 0 1 1 1 1 2 1 1 1 2 1 2 2 4 2
-3 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 1 1 1 2 2 2 4 

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jagy@phobeusjunior:~/Desktop/Pete$ gp
Reading GPRC: /etc/gprc ...Done.

                                                        GP/PARI CALCULATOR Version 2.5.0 (released)
                                                    i686 running linux (ix86/GMP kernel) 32-bit version
                                              compiled: Nov 17 2011, gcc-4.6.2 (Ubuntu/Linaro 4.6.2-2ubuntu1) 
                                                       (readline v6.2 enabled, extended help enabled)

                                                           Copyright (C) 2000-2011 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.

parisize = 4000000, primelimit = 500509

Type ?12 for how to get moral (and possibly technical) support.

parisize = 4000000, primelimit = 500509

? L = [8, 4, 4, 4, 4, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 2, 4, 2, 2, 2, 0, 0, 0, -3; 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 0, 0, -1; 4, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 0, 0, -1; 4, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 0, 0, -1; 4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 0, 0, -1; 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 0, 0, 0, -1; 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 0, 0, 0, -1; 2, 2, 2, 2, 2, 2, 2, 4, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 0, 0, 1; 4, 2, 2, 2, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, -1; 4, 2, 2, 2, 2, 2, 2, 1, 2, 4, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 0, 1, 0, -1; 4, 2, 2, 2, 2, 2, 2, 1, 2, 2, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 0, 0, 1, -1; 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 4, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1; 4, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, -1; 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 4, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1; 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 4, 2, 1, 2, 2, 2, 2, 1, 2, 1; 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 4, 1, 2, 2, 2, 2, 1, 1, 1; 4, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 4, 2, 2, 2, 1, 1, 1, -1; 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 1, 1; 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 1, 2, 1; 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 4, 2, 1, 1, 1; 0, 1, 1, 1, 1, 0, 0, 2, 1, 0, 0, 2, 1, 2, 2, 2, 1, 2, 2, 2, 4, 2, 2, 2; 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 2, 2; 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 2; -3, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 2, 2, 2, 4]


? matdet(L)
%3 = 1
? 
? 
? 
? M = L - mattranspose(L)

?  P = M * mattranspose(M)

? 
? charpoly(P)
%8 = x^24
? 
? 

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Note that the line defining the Gram matrix for Pari is some 1759 characters long. If you are unable to grab the text for your own use, just email me. I know that Magma has the lattice as a built in command, I'm not sure about Mathematica, Maple, other items. Probably Sage. Anyway, I took the Gram matrix and simply wrote a C++ program to read that text file and output a Pari-ready version. The disappointing part is that Pari does not display matrices past about the first 20 lines. So i did some checking to confirm that I really had a symmetric matrix, and in any case determined that the determinant was actually $1.$