deBruijn generates tilings by choosing a plane in n-dimensional space (where n is from 4 to 12) and projecting selected points from a hypercubic lattice onto the plane. The points selected are those that lie inside any hypercube, oriented parallel to the lattice, whose centre lies on the plane. Square faces defined by the lattice are projected down to tiles if all four of their vertices are selected. For n=5, the result is one of Roger Penrose’s famous quasiperiodic tilings. For n=4, it’s the Ammann-Beenker tiling.

The projections of the n coordinate axes and their opposites form a symmetrical 2n-pointed star; this is achieved by choosing a plane spanned by:

(1, –cos(π/n), cos(2π/n) … (–1)(n–1)cos((n–1)π/n))
(0, –sin(π/n), sin(2π/n) … (–1)(n–1)sin((n–1)π/n))

For odd values of n, the plane is orthogonal to one of the lattice diagonals, (1,1,1,1,...1), but this is not the case for even n; if it were, the projections of the 2n vectors would overlap each other to form an n-pointed star.

More details of the construction are given in this companion page. A generalisation of this method from the hypercubic lattice to the An lattice is used by the Tübingen applet.

Reference: N.G. de Bruijn, “Algebraic theory of Penrose’s nonperiodic tilings of the plane, I, II”, Nederl. Akad. Wetensch. Indag. Math. 43 (1981) 39–52, 53–66. (Available online as a PDF.) I learnt about this method from the documentation for Eugenio Durand’s program Quasitiler, which generates the same kind of tilings interactively via forms submitted to a server.

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Applets Gallery / deBruijn / created Tuesday, 15 February 2000 / revised Tuesday, 9 February 2021
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