# ## Lattice Theory and Geometry of Numbers

Informally, a lattice is an infinite arrangement of points spaced with sufficient regularity that one can shift any point onto any other point by some symmetry of the arrangement. More formally, a lattice can be defined as a discrete subgroup of a finite-dimensional vector space (the subgroup is often required not to lie within any subspace of the vector space, which can be expressed formally by saying that the quotient of the space by the lattice is compact).

The simplest and most commonly-studied example of a lattice (the "integer grid") is formed by the points all Cartesian coordinates of which are integers. Other types of lattices arise in crystallography and in sphere packing, where they are used to describe the locations of atoms or spheres. Lattices are also particularly important in the theory of periodic tilings, since they describe the set of translational symmetries of a tiling.

• Catalogue of lattices, N. J. A. Sloane, AT&T Labs Research. See also Sloane's sphere-packing and lattice theory publications.

• Connect the dots. Ed Pegg asks how many sides are needed in a (self-crossing) polygon, that passes through every point of an n*n grid. I added a similar puzzle with circular arcs. • Crystallographic topology. C. Johnson and M. Burnett of Oak Ridge National Lab use topological methods to understand and classify the symmetries of the lattice structures formed by crystals. (Somewhat technical.)

• Equilateral triangles. Dan Asimov asks how large a triangle will fit into a square torus; equivalently, the densest packing of equilateral triangles in the pattern of a square lattice. There is only one parameter to optimize, the angle of the triangle to the lattice vectors; my answer is that the densest packing occurs when this angle is 15 or 45 degrees, shown below. (If the lattice doesn't have to be square, it is possible to get density 2/3; apparently this was long known, e.g. see Fáry, Bull. Soc. Math. France 78 (1950) 152.)

Asimov also asks for the smallest triangle that will always cover at least one point of the integer lattice, or equivalently a triangle such that no matter at what angle you place copies of it on an integer lattice, they always cover the plane; my guess is that the worst angle is parallel and 30 degrees to the lattice, giving a triangle with 2-unit sides and contradicting an earlier answer to Asimov's question. • Grid subgraphs. Jan Kristian Haugland looks for sets of lattice points that induce graphs with high degree but no short cycles.

• Mike Kolountzakis' publications include several recent papers on lattice tiling.

• Lattice pentagons. The vertices of a regular pentagon are not the subset of any lattice.

• The no-three-in-line problem. How many points can be placed in an n*n grid with no three on a common line? The solution is known to be between 1.5n and 2n. Achim Flammenkamp discusses some new computational results including bounds on the number of symmetric solutions.

• Polyominoes, figures formed from subsets of the square lattice tiling of the plane. Interesting problems associated with these shapes include finding all of them, determining which ones tile the plane, and dissecting rectangles or other shapes into sets of them. Also includes related material on polyiamonds, polyhexes, and animals.

• Russian math olympiad problem on lattice points. Proof that, for any five lattice points in convex position, another lattice point is on or inside the inner pentagon of the five-point star they form. • Spiral tilings. These similarity tilings are formed by applying the exponential function to a lattice in the complex number plane. • Triangulations with many different areas. Eddie Grove asks for a function t(n) such that any n-vertex convex polygon has a triangulation with at least t(n) distinct triangle areas, and also discusses a special case in which the vertices are points in a lattice.

• Untitled, Forbes Gallery, 1987, Ken Goldberg.

• Voronoi diagrams of lattices. Greg Kuperberg discusses an algorithm for constructing the Voronoi cells in a planar lattice of points. This problem is closely related to some important number theory: Euclid's algorithm for integer GCD's, continued fractions, and good approximations of real numbers by rationals. Higher-dimensional generalizations (in which the Voronoi cells form zonotopes) are much harder -- one can find a basis of short vectors using the well-known LLL algorithm, but this doesn't necessarily find the vectors corresponding to Voronoi adjacencies. (In fact, according to Senechal's Quasicrystals and Geometry, although the set of Voronoi adjacencies of any lattice generates the lattice, it's not known whether this set always contains a basis.)

From the Geometry Junkyard, computational and recreational geometry pointers.
Send email if you know of an appropriate page not listed here.
David Eppstein, Theory Group, ICS, UC Irvine.
Semi-automatically filtered from a common source file.