A Quasiperiodic Tiling With 12-Fold Rotational Symmetry and Inflation Factor 1 + Sqrt(3)
AA Quasiperiodic Tiling With 12-Fold RotationalSymmetry and Inflation Factor 1 + √ Theo P. Schaad and Peter Stampfli Rue de Lausanne 1, 1580 Avenches, Switzerland; [email protected]
Abstract
We show how we found substitution rules for a quasiperiodic tiling with local12-fold rotational symmetry and inflation factor 1 + √
3. The base tiles are asquare, a rhomb with an acute angle of 30 degrees, and equilateral triangles thatare cut in half. These half-triangles follow three different substitution rules andcan be recombined into equilateral triangles in nine different ways to make minorvariations of the tiling. The tiling contains quasiperiodically repeated 12-foldrosettes. A central rosette can be enlarged to make an arbitrarily large tilingwith 12-fold rotational symmetry. An online computer program is provided thatallows the user to explore the tiling[1].
About Tilings With 12-Fold Rotational Symmetry
We use the substitution method for creating arbitrarily large tilings. It consists offinding base tiles that can be enlarged and then subdivided with the same base tileswithout any gaps or overlaps. The enlargement of their edges is called the inflationfactor. Each repetition of enlargement and subdivision creates a new generation oftiles. For finite patches of a tiling this increases the number of tiles, whereas aninfinite tiling remains unchanged.Since 1973, many quasiperiodic tilings with n -fold rotational symmetry havebeen discovered, of which the 5-fold Penrose and the 8-fold Ammann-Beenker tilingswere the first[10]. We consider a special 12-fold quasiperiodic tiling. It exhibitslocal 12-fold rotational symmetry that is repeated throughout the tiling withoutany periodicity. Peter Stampfli came up with a particularly simple substitution rulethat he described on his blog ”Geometry in Color”[3] and is also listed in the TilingEncyclopedia of the University of Bielefeld[2]. It serves as an introduction here andshows that it is much easier to find a substitution scheme with an inflation factorof 2 + √ √
3, which is the subject of this paper. A good startingpoint for any n -fold tiling is an n -fold star or rosette of rhombs with acute anglesof 2 π/n as shown in the upper part of Figure 1. The star can then be surroundedwith other tiles until an n -fold regular polygon is obtained, in this case a dodecagon.1 a r X i v : . [ m a t h . HO ] F e b igure 1: A dodecagon made of two different rhombs and squares. The 12 skinnyrhombs form a rosette with local 12-fold rotational symmetry. The rhombs with ◦ acute angles are further divided into two equilateral triangles, foreshadowing the roleit has in this paper. Some quite natural subdivisions of tiles in the dodecagon areshown. They define a substitution scheme for squares, rhombs, and triangles thatare inflated by √ . All sides of the inflated tiles have the same mirror-symmetricgeometry and can be attached to each other.
2n our example, this requires two different rhombs and a square. Peter Stampflifurther subdivided the rhombs with 60 ◦ angles into two equilateral triangles. Heconsidered the resulting three shapes to be base tiles of the tiling, namely, a skinnyrhomb, a square, and an equilateral triangle. Some quite natural subdivisions withinthe dodecagon of Figure 1 lead to a substitution scheme with an inflation factorof 2 + √
3. The edges comprise two side lengths of the rhombs and two heights ofequilateral triangles. They are conveniently laid out such that all sides of the inflatedtiles have the same mirror symmetric pattern of halves of equilateral triangles. Thesehalves meet whenever two tiles are joined edge-to-edge. Thus, any layout using thesebase tiles can be inflated into an arbitrarily large nonperiodic tiling of equilateraltriangles, rhombs and squares.Using halves of equilateral triangles has many advantages. This is more efficientand each iteration gives a patch of the same shape as before, except for inflation.It is nice to be able to create square and triangular patches. Using instead fullequilateral triangles would give patches that have fractal borders similar to Kochsnowflakes because both result from similar iterative methods. Iteration schemeswith larger inflation ratios n + m ∗ √ n ≥
2, are just as easy to find. If n and m are both odd then we have to split squares at the tile edges.Peter Stampfli further noted that the combination of a triangle with a square canbe substituted by another triangle with two skinny rhombs as shown in Figure 2.This leads to many tilings that sometimes look drastically different[3], even thoughthey are only variations. In some cases, the rhombs or the squares can be elimi-nated and the appearance of local 12-fold rotational symmetry as well[9]. The TilingEncyclopedia[2] contains several tilings with 12-fold rotational symmetry and infla-tion factor 2 + √
3. Such tilings are presented by J. Socolar[11], M. Schlottmann[12],and Y. Watanabe et al[13]. J. Socolar’s first tiling was discovered in 1987 and usedregular hexagons, which can also be viewed as three rhombs. This pattern exhibitsdodecagonal features, such as dodecagons filled with other tiles, but no outright12-fold rotational symmetry. The “Shield” tiling by F. G¨ahler[14] has the smallestinflation factor of (2 + √ / . The base tiles consist of an irregular hexagon, asquare, and two equilateral triangles with different matching rules. A larger patchexhibits dodecagonal features but no local 12-fold symmetries.The ”Rorschach” tiling discovered by D. Frettl¨oh[5, 15] has great relevance to our12-fold tiling. The inflation factor 1 + √ π/ igure 2: Different dissections of an irregular hexagon into triangles andrhombs or triangles and squares. They can be exchanged in substitutionrules, resulting in different tilings. Further, these are all possible substitu-tions at the sides of tiles (dotted lines) for a self-similarity ratio of √ . six sets of parallel lines of different orientation, also called a hexa-grid of parallellines. It was first explored by J. Socolar[16] in 1987.A year earlier, Peter Stampfli found another method for creating a dodecagonaltiling[17]. His grid uses periodic tilings of hexagons like a honeycomb. They have aperiodic tiling of equilateral triangles as their dual. He discovered that two hexagongrids rotated by 30 degrees to each other have as dual a 12-fold tiling with skinnyrhombs, squares, and equilateral triangles. However, this transformation requiredfairly complicated mathematics. By 2012, he not only posted the results on hisblog[9], but he also made the computer code freely available. It took several moreyears to disseminate, but it made the front cover of Science[18] in 2018 to illus-trate Graphene Quasicrystals made by twisted bilayers of the hexagonal graphene.This 12-fold quasiperiodic tiling does not lend itself to a substitution scheme, but itspurred his interest into finding others.The work of J. Socolar was generalized in 1994 by D. Haussler, H.U. Nissen,and R. Lueck[19] in a paper called ”Dodecagonal Tilings Derived as Duals fromQuasiperiodic Ammann-Grids”. They discovered 12 tilings made of rhombs andregular or irregular hexagons but without triangles. The rhombs can be decoratedto make new magnified copies of themselves, all with a characteristic deflation factorof 2 + √
3. The results cannot be reproduced by substitution alone. W. Steurer andS. Deloudi[8] also discuss the grid method and show that a tiling with a deflationratio of 1 + √ igure 3: From top left to bottom right: Dissection of a dodecagon into rhombs andtriangles. Substitution rules for squares, rhombs and triangles used in the tiling of1998. The Tiling of 1998
In 1998, Theo Schaad tried to find substitution rules for a quasiperiodic tiling with12-fold rotational symmetry. For the ratio between the edges of an inflated tile andthe base tiles he chose 1 + √ √ √ igure 4: Drawing of 1998: The green outline is the dodecagon of Figure 3, dividedinto 4 squares, 4 rhombs, and triangles (green lines). The green squares and rhombswere then further divided (orange lines) in the manner shown at the bottom. Theorange shapes were then filled again with squares, rhombs, and triangles using therules of Figure 3.
6o correctly choose triangle substitutions, one must check that the tiles of thenext generation match. Theo Schaad accomplished this in Figure 4. First, he drewa large dodecagon and subdivided it with green lines into 4 squares, 4 rhombs,and equilateral triangles. This would be the first generation of substitutions as inFigure 3. He then repeated the iteration with orange lines. This is now the secondgeneration. Careful examination shows that triangles in an inflated square are of thesecond type with two rhombs at the top and a square base. It was now possible tomove to the third generation, at least partially, by filling all the orange squares andrhombs with base tiles. Spontaneously, a rosette of orange rhombs had appeared atthe center. It was obvious that the triangles that completed the rosette had to be ofthe third type in Figure 3 with two rhombs pointing towards a square at the base.This filled almost the entire tiling with only a few orange triangles left unfilled. Notsurprisingly, Theo Schaad tried to find homes for the 3-fold symmetric triangles with3 squares. A few could be completed that way, leaving the rest with 2 squares and arhomb, a half-triangle of type 1 joined by a half-triangle of type 2. Hence, a coherentsubstitution rule with only the tiles of Figure 3 was not possible. The result was,nevertheless, a large finite tiling of nonperiodic 12-fold rotational symmetry with aninflation ratio of 1 + √
3. It consists of thin rhombs, equilateral triangles and squares.The substitution puzzle was not solved for another 22 years but was greatly helpedby the discovery of another tiling by Peter Stampfli in 2012.
The Tiling of 2012
Independently, Peter Stampfli[7] created a 12-fold tiling with the same properties in2012. He encountered similar issues trying to find substitutions for triangles thathad to match either an adjacent square or rhomb. The end result was quite similarbut not exactly the same. He actually found one of the nine variations presented inthe next section.Since Peter Stampfli worked with a computer program, he could not just fill intriangles as they fit. He had to define an algorithm for a specific iteration scheme.He could then use the program to find out if his scheme made a good tiling. At first,he noted that there are three choices to lay out the edges of an enlarged triangleas shown in Figure 2. Potentially, there could be 27 different triangles. Some aresimply rotations of others, but there are still 13 left. He chose the same substitutionrule for squares as in Figure 3 and observed that a triangle next to a square hasto have a square and two triangles as substitution at its corresponding side, see thecenter of Figure 2. In other cases, when two triangles met, this substitution wasalso the best symmetric solution. Therefore, every triangle had to have at least onesquare in its substitution, which left 9 different possibilities. This could be reorderedin a much simpler way. He cut the triangles in half and used these half-trianglesas base tiles for the substitution scheme. There were only three left and they couldbe combined in nine different ways to make equilateral triangles. He kept track ofthem with letters, see Figure 5. This simplification had another advantage. By7 igure 5: The substitutions of the tiling of 2012. Note the similarity to Figure 3. igure 6: A patch of Peter Stampfli’s 12-fold tiling resulting from the substitutionsof Figure 5. √
3. The triangleshad to be chosen such that half-triangles always met together to give equilateraltriangles in the tiling. As Theo Schaad had found too, some substitutions wereobvious, e.g., the triangles in the substitution of the squares had to be B triangles(or of the second type in Figure 3). Likewise, the inflated rhombs had to be filledwith C and A triangles. This finally left only very few choices for the triangles.Peter Stampfli found the substitution rules shown in Figure 5 and created the patchof Figure 6. The result was a potentially infinite quasiperiodic 12-fold tiling withthree tile shapes that were themselves composites of smaller base tiles (half-trianglesand quarter-squares). Peter Stampfli noted in his blog that the result looked rathercomplex, joining others that found 12-fold tilings somewhat chaotic. But it was alsoa choice of presentation. Inflated portions resulting from central 12-fold rosettes orof other rotational symmetries are quite aesthetic.
The New Substitution Rules
We decided to follow Peter Stampfli’s choice of base tiles with a rhomb of unit sidelength, a square with sides of half unit length, and right triangles with a hypotenuseof unit length. Further research gave the substitution rules of Figure 7. Insteadof labeling the triangles with letters, we preferred different colors. This makes itconsiderably easier to follow repeated substitutions as shown in the example givenin Figure 8. We see that in higher generations two-colored equilateral trianglesappear. Here gray and yellow halves are combined. To regain the simplicity ofa tiling with only three shapes (rhomb, square and equilateral triangle with unitside lengths) we could draw all triangles using the same color. Comparing the newrules with the tiling of 2012, see Figures 5 and 6, the triangle A corresponds to agray triangle and follows the second variation, and the triangle B follows the firstvariation of the yellow triangle. We also discovered that the substitutions of the grayand blue triangles have to use a yellow triangle at the apex because yellow trianglesare substituted with rhombs at the apex. This rule showed that there could notbe a triangle substitution with only gray triangles as proposed in the 1998 Tiling.One final caveat: whenever two triangles are joined along an edge, it is still possibleto exchange at this edge the square and triangles by two rhombs and triangles.However, it would invalidate our goal of making inflated tiles that are similar to thebase tiles.
Results
The reader can explore this tiling in his browser using a computer program writtenby P. Stampfli[1]. Here, we present some of its results. If we use only one colorfor the three triangle substitutions, the resulting equilateral triangles will be indis-tinguishable from each other and the final tiling will have only three apparent base10 igure 7: The new substitution rules for a quasiperiodic 12-fold tiling with an infla-tion factor √ . For the gray and yellow triangles we can choose between threesubstitutions leading to nine different tilings. igure 8: The second generation of one the substitution rules of Figure 7. Here,using the second choice in Figure 7 yellow triangles are put at the ◦ corner of grayand yellow triangles. This leads to rhombs in the lower corners of the triangles,ultimately generating many rosettes in a larger patch. igure 9: The substitution rules with the choice of Figure 8 applied four times to asquare tile. All triangles are shown in one color. This patch can be compared to thetiling in Figure 4, except that it was iterated to a fourth generation. igure 10: Applying three times the substitution rules with the choice of Figure 8 toa rosette of twelve rhombs and triangles. This creates a tiling with 12-fold rotationalsymmetry. Summary
We presented a quasiperiodic 12-fold tiling of squares, equal sided triangles, and30 degree rhombs. It is defined and built by its substitution rules. An inflatedpatch of the tiling does not change its shape. Only its size is increased by the self-similarity ratio of 1 + √
3. The tiling is edge-to-edge without gaps or overlaps andthe substitution process can be repeated to make an arbitrarily large tiling. Theauthors would like to thank Robert Ingalls, known for his work on octagonal anddecagonal tilings, for providing the source material of 12-fold patterns by J. Socolarand D. Haussler et al.
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