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.

• Click to increase n, SHIFT-click to reduce n, CTRL-click to get a new random tiling with the same n.
• You can display this applet in a large window (suitable for taking screen shots to use as desktop wallpaper) by hitting the W key. You might have to select the applet first by clicking on it (you can use SHIFT-CTRL-click to select the applet without changing the tiling).
• To close the large window, you can use the close gadget on its frame (if there is one), or hit either the C key or the ESC key. Some browsers, alas, will not pass any keystrokes to the large window, even after you've clicked on it, so as an EMERGENCY EXIT you can RIGHT-click the mouse to close the large window.
• Hitting the SPACE BAR will cause the applet to stop/start automatically scrolling across the tiling. Hitting the PAGE UP and PAGE DOWN keys scrolls a full page; the UP ARROW and DOWN ARROW keys scroll by one tenth of a page.
• Some browsers have a bug that will cause the applet to ignore these key commands after you've opened a large window then closed it. If you find the applet ignoring these commands, even after you click on it, you will probably need to reload the page.

Reference: N.G. deBruijn, “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.

Applets Gallery / deBruijn / created Tuesday, 15 February 2000 / revised Thursday, 25 December 2008