Tübingen generates quasiperiodic tilings by projecting selected triangles from the An lattice onto a plane.

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An is an n-dimensional lattice, but it can most easily be described as the set of points in a space of one more dimension: those points in Rn+1 whose coordinates are all integers, and whose sum is zero. The condition that the coordinates sum to zero means that the points lie in an n-dimensional subspace of Rn+1.

We can think of each lattice point L in An as being surrounded by a Voronoi cell, which consists of all points in the subspace that are closer to L than to any other lattice point. The Voronoi cells of An are all identical convex polytopes that pack together to fill the subspace.

We can also define Delaunay cells associated with the lattice, as follows: a vertex, V, of any Voronoi cell will be shared by the Voronoi cells of several lattice points (much as each corner of a cell in a honeycomb is shared by several cells); we define the Delaunay cell “dual to” V to be the convex hull of all the lattice points whose Voronoi cells share V. The Delaunay cells are convex polytopes that pack together to fill the subspace, but unlike the Voronoi cells they are not all identical.

Now, we can extend the duality between Delaunay cells and Voronoi vertices to one between Delauny (n–m)-boundaries and Voronoi m-boundaries. A 0-boundary of a polytope is a vertex, a 1-boundary is an edge, a 2-boundary is a face, and so on. Any Voronoi m-boundary will be shared by the Voronoi cells of several lattice points; the convex hull of those lattice points will be an (n–m)-boundary of some Delaunay cell.

To produce these quasiperiodic tilings, we pick a plane P that lies in the same subspace of Rn+1 as An, and whenever P intersects a Voronoi (n–2)-boundary, we project the dual Delaunay 2-boundary, which is a triangle, onto P.

Confused? The complete mathematical details are given in this companion page.

Reference: Peter Kramer, “Dual Canonical Projections”.

Applets Gallery / Tübingen / created Sunday, 14 December 2008 / revised Thursday, 25 December 2008