Point Lattice

Author

Eric W. Weisstein
July 26, 2008

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

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

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

Additional Authors

Text and graphics by
Todd Rowland
Wolfram Research, Inc.

List

LatticeData[]

LatticeData["Classes"]

LatticeData["Properties"]

TableForm[Sort[{LatticeData[#,"Dimension"],LatticeData[#,"KissingNumber"],LatticeData[#,"AutomorphismGroupOrder"],#}&/@LatticeData[]],TableDepth->2]//TraditionalForm

Proprties

AutomorphismGroupCount

Select[LatticeData[],Head[LatticeData[#,"AutomorphismGroupOrder"]]===Missing&]

Images

Select[LatticeData[],Head[LatticeData[#,"Image"]]===Missing&]

LatticeData["HexagonalLattice","Image"]

LatticeData["SquareLattice","Image"]

ThetaFunctionSeries

l=Select[LatticeData[],Head[LatticeData[#,"ThetaSeriesFunction"]]=!=Missing&]

TableForm[{#,LatticeData[#,"ThetaSeriesFunction"][Log[q]/(I Pi)]}&/@l,TableDepth->2]//TraditionalForm

TableForm[{#,Series[LatticeData[#,"ThetaSeriesFunction"][Log[q]/(I Pi)],{q,0,10}]}&/@l,TableDepth->2]//TraditionalForm

Explanation

The Point Lattice entry is missing a formal definition.  I include this in the one-variable notebook because of its relevance to the theory of elliptic curves and elliptic functions. Besides, it is pretty much a topic without a well-defined branch of mathematics, though important to many areas.

The animation shows that the subgroup generated by 1 and 1/(Sqrt[2]) is not a lattice, by showing a+b/(Sqrt[2]) for successive b's in [0,1].

(*n+m2^(-1/2) is not a lattice*)
Clear[pts];
pts[n_Integer]:=Graphics[{PointSize[.03],Purple,Table[Point[{Mod[ 2^(-1/2)i,1],0}],{i,0,n}],
Point[{1,0}]
}]

movie=Table[Show[pts[k],Axes->{True,False},Ticks->{{0,.2,.5,2^(-1/2),1},{}},PlotRange->{{-.015,1.015},{-1,1}}],{k,50}];

<<Utilities`GIF`;

Export[gifs<>"pointlat.gif",movie]

Show[GraphicsArray[Partition[Table[Graphics[pts[k][[1]],Axes->{True,False},Ticks->{{0,.2,.5,2^(-1/2),1},{}},PlotRange->{{0,1},{-1,1}}],{k,1,51,10}],2]],
GraphicsSpacing->{.1,-.5}]