Leech Lattice

Author

Eric W. Weisstein
July 26, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Lattices/LeechLattice.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LeechLattice.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Properties

MatrixPlot[LatticeData["Leech"]]

LatticeData["Leech","Image"]

LatticeData["Leech","AutomorphismGroupOrder"]

FactorInteger[%]

LatticeData["Leech","KissingNumber"]

LatticeData["Leech","Dimension"]

Theta Series

Table[(65520/691)(DivisorSigma[11,n]-RamanujanTau[n]),{n,5}]

(((1+240Sum[DivisorSigma[3,m](q^(2m)),{m,∞}]))^3)-720(q^2)((QPochhammer[(q^2),(q^2)])^24)//FullSimplify

FunctionExpand[%]

Series[(((1+240Sum[DivisorSigma[3,m](q^(2m)),{m,100}]))^3)-720(q^2)((QPochhammer[(q^2),(q^2)])^24),{q,0,10}]

(Subscript[E, 4])==240 (∑_m^∞)(q^(2 m)) (Subscript[σ, 3])(m)+1==(1/2) ((((Subscript[ϑ, 2])(0,q))^8)+(((Subscript[ϑ, 3])(0,q))^8)+(((Subscript[ϑ, 4])(0,q))^8))

1+240Sum[DivisorSigma[3,m](q^(2m)),{m,∞}]//InputForm

Series[1+240Sum[DivisorSigma[3,m](q^(2m)),{m,100}],{q,0,10}]

(1/2)Series[((EllipticTheta[2,0,q])^8)+((EllipticTheta[3,0,q])^8)+((EllipticTheta[4,0,q])^8),{q,0,10}]

(∏_m^∞)(((1-(q^(2 m))))^24)==((QPochhammer2[(q^2), (q^2)])^24)

Product[(((1-(q^(2m))))^24),{m,∞}]

Built-In

(ser=LatticeData["Leech","ThetaSeriesFunction"][Log[q]/(I Pi)])//TraditionalForm

SeriesCoefficient[ser,{q,0,n}]//InputForm

Table[Switch[n,0,1,
_?OddQ,0,_,
(65520/691)(DivisorSigma[11,n/2]-RamanujanTau[n/2])],{n,0,10}]

Series[ser,{q,0,10}]

CoefficientList[%118,q]

Series[720(q^2)((QPochhammer[(q^2),(q^2)])^24),{q,0,10}]

Series[(45/16) ((EllipticTheta[2,0,q])^8) ((EllipticTheta[3,0,q])^8) ((EllipticTheta[4,0,q])^8),{q,0,10}]

FullSimplify[(45/16) ((EllipticTheta[2,0,q])^8) ((EllipticTheta[3,0,q])^8) ((EllipticTheta[4,0,q])^8)]

720 16/45

expr1=EllipticTheta[2,0,q]^8 EllipticTheta[3,0,q]^8 EllipticTheta[4,0,q]^8;

expr2=256 q^2 QPochhammer[q^2,q^2]^24;

FullSimplify[expr1]

FullSimplify[expr1-expr2]

Series[%,{q,0,100}]

Graphs

(2, (Sqrt[6]), (Sqrt[6])) construction of "HigmanSimsGraph"

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

https://en.wikipedia.org/wiki/Higman–Sims_graph#Inside_the_Leech_lattice

Take three lattice points that form the vertices of a triangle with side lengths of {2, √6, √6}. There are 100 lattice points of minimal distance 2 from all three of these three vertices. Consider these 100 points, and join two points with an edge if the distance is √6. The result is the Higman-Sims graph on 100 vertices

(2, 2, (Sqrt[6])) construction of "McLaughlinGraph"

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

We can also obtain the 275-vertex McLaughlin graph, by applying the same construction to a {2, 2, √6} triangle instead of a {2, √6, √6} triangle.

2300-Conway graph

16624 points from Michael

Length[pts=]