Images of the Ammann-Beenker Tiling

Year: 2007 Authors: Edmund Harriss

Core claim

The Ammann-Beenker tiling can be generated by substitution or by projection from a higher-dimensional lattice, and its geometry yields striking artistic images.

Topics

aperiodic tilings, substitution rules, higher-dimensional projection, hierarchical structure, Ammann bars

Domains

tiling theory, geometry, quasicrystals, mathematical art, visualization, pattern design

Methods

substitution construction, lattice projection, coloring by directions, hierarchical overlays

Media

rhombs and squares, digital images, colored tiling diagrams

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Edmund Harriss

Department of Mathematics

Imperial College London

LONDON, SW7 2AZ, United Kingdom

E-mail: edmund.harriss@mathematicians.org.uk

The Ammann-Beenker tiling is the eight-fold sibling of the more famous, five-fold Penrose rhomb tiling. It was discovered independently by R. Ammann [AGS92] and F. Beenker [Bee82]. It shares many properties with the Penrose tiling. In particular it shares two particular constructions. The first by a substitution rule ${}^1$ , and the second as a slice of a higher dimensional lattice. Also it shares the important property with the Penrose that it is a strikingly beautiful tiling.

R. Ammann’s discovery of the tiling was at the same time as Penrose’s. In fact he sent in several examples in response to Martin Gardener’s Scientific America article that announced the Penrose tiling [Gar77]. Though it was only later that something was published more formally, as Ammann was very detached from the mathematical community[Sen04]. Like Penrose (and nearly all the early aperiodic tilings) he used a substitution rule to construct the tiling.

A substitution rule has two phases. First the tiles are expanded. Then the new larger tiles are replaced by copies of the original tiles. By repeating this process larger and larger patches of tiles can be generated and, at the infinite limit, tilings of the whole plane. The substitution rule for the Ammann tiling is shown in Figure 1. In this case the expansion is multiplication by $1+\sqrt{2}$ .

Figure 1: The substitution rule for the Ammann-Beenker Tiling

Figure 2: A portion of the Ammann Tiling showing the Ammann bars.

In 1981, the Dutch mathematician N.G. de Bruijn found an alternative construction for the Penrose tiling [dB81]. His student N. Beenker followed his method for the eight-fold case and found another tiling with substitution rule ${}^2$ . This method is related to the way a computer draws a line on the monitor. As the monitor is made up of square pixels it is not possible just to draw the line. Instead one must choose a sequence of pixels that stay close to the line. Looking at the top or bottom of these pixels gives a staircase, with vertical and horizontal lines. We may then project the staircase to the original line giving a tiling with two tiles, corresponding to the projection of the vertical and horizontal lines. In a similar way one may construct a plane out of hypercubes in five (for the Penrose) or four (for the Ammann-Beenker) dimensions, and project one face of this to a

Figure 3: Ammann Scaling

Figure 4: Ammann Squares

plane. If the plane is positioned correctly the tilings produced by this are precisely the Penrose Rhomb, or

Ammann-Beenker tilings. As shown in Figure 2, one consequence of this is that the tiles line up into “worms”, lying along parallel sets of lines at 45 deg angles, called Ammann bars.

Figures 3 and 4 show artistic (or at least pretty) images that explore the two construction methods of the tiling. Figure 3 overlays several layer of the hierarchical structure of the tiling, thus illustrating how the substitution rule generates larger patches of tiling. Figure 4 has tiles on the same line with the same colour for three of the four directions (as every tile lies on two lines, after colouring three of the directions, every tile is coloured).

References

[AGS92] R Ammann, B Grünbaum, and G C Shephard, Aperiodic tiles, Discrete Comput. Geom. 8 (1992), no. 1, 1-25. MR 93g:52015 [Bee82] F P M Beenker, Algebraic theory of non-periodic tilings of the plane by two simple building blocks: a square and a rhombus, TH-Report 82-WSK04, Eindhoven University of Technology, 1982. [dB81] N G de Bruijn, Algebraic theory of Penrose’s nonperiodic tilings of the plane. I, II, Nederl. Akad. Wetensch. Indag. Math. 43 (1981), no. 1, 39-52, 53-66. MR 82e:05055 [Gar77] M Gardner, Extraordinary nonperiodic tiling that enriches the theory of tiles, Sci. Am. November (1977), 110-119. [Har03] E O Harriss, On canonical substitution tilings, Ph.D. thesis, Imperial College London, 2003. [Sen04] Marjorie Senechal, The mysterious Mr. Ammann, Math. Intelligencer 26 (2004), no. 4, 10-21. MR MR2104463 (2005j:52026) [Soc89] J E S Socolar, Simple octagonal and dodecagonal quasicrystals, Phys. Rev. B (3) 39 (1989), no. 15, 519-551. MR 90h:52019