Abstract
Discrete Algorithms A method is described for constructing, with computer assistance, planar substitution tilings that have n-fold rotational symmetry. This method uses as prototiles the set of rhombs with angles that are integer multiples of pi/n, and includes various special cases that have already been constructed by hand for low values of n. An example constructed by this method for n = 11 is exhibited; this is the first substitution tiling with elevenfold symmetry appearing in the literature.
Highlights
Rotational symmetry is one of the most distinctive qualities that aperiodic planar tilings can have
The most famous aperiodic family of tilings is the family of Penrose tilings [16], which possess fivefold rotational symmetry
Aperiodic tilings are often used as models of quasicrystals; when Dan Shechtman and his co-authors made the Nobel prize-winning discovery of quasicrystals in [19], the first sign by which they knew that they had found something special was the presence of tenfold rotational symmetry in the X-ray diffraction patterns
Summary
Rotational symmetry is one of the most distinctive qualities that aperiodic planar tilings can have. Since the substitution behaves in the same way for two tiles with the same label, this guarantees that the image of the patch after several substitutions will be rotation-invariant This condition is not met in [15, Remark 6.3], in which there are patches consisting of fourteen isosceles triangles that appear to have fourteenfold rotational symmetry, but that only have twofold symmetry. A common approach to this problem of scale has been to take a known substitution rule with fivefold or sevenfold symmetry and to generalise it to higher n in some way This has led to the discovery of various infinite families of tiling spaces, but in all cases the desirable property of having an individual tiling with n-fold symmetry is lost upon passage to higher n. These ingredients have been known for many years, during which many researchers have tried without success to produce n-fold symmetric substitution tilings for large n; the fact that this has been achieved here is evidence that the combination of these ingredients in this particular way is a significant development
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
