A set of tiles for covering a surface is composed of three types of tiles. The base shape of each one of them is a diamond or rhombus, with acute angles of π/7, 2π/7, and 3π/7 radians (A, B, and C tiles). Such a set is called 7-fold, as the B tiles can be arranged in 7-fold rotational symmetry forming a 7-pointed star. A substitution scheme is discovered wherein a number of proto-tiles are arranged to fill larger similar tiles that have the same matching characteristics as the base tiles. This process can then be repeated to tile an arbitrarily large surface. 7-fold sets have been discovered before, but they require a large number of base tiles, at least 41 tiles or more. The set described here is thought to be the most minimal or smallest set wherein only 11 base tiles are needed to start the process.
Background:
Tile sets with n-fold symmetry have been studied extensively since the discovery of the Penrose 5-fold aperiodic pattern 1 . The 5-fold pattern is aesthetically pleasing because it incorporates the golden ratio, a number that appears in nature, architecture, and in the Fibonacci series. In 5fold stars made with 5 rhombs (with unit side lengths), the long diagonal of the rhomb is the golden ratio. In similar fashion, the number 7, the 7-pointed star, and other heptagrams have significant religious and occult symbolism and have been widely used in heraldry and logos. The equivalent 7-fold magic number Φ (Phi) is the long diagonal of the B tile that can be tiled into a 7-pointed star. Finding 7-fold patterns could therefore be a worthwhile endeavor, especially if it incorporates 7-pointed stars. However, finding such patterns is far from trivial.
There are two known schemes to find large (infinite) tilings. The first one was pioneered by Penrose and involves substitution and matching rules to generate larger tiles that have the same characteristics as the base tiles. This process is called inflation and can be repeated as a recurring series. The other method is a multi-grid method developed by de Bruijn 2 that leads to a potentially infinite variety of solutions. P. Steinhardt 3 used a hepta-grid to find a particularly attractive tiling with 7-pointed stars. A reconstructed patch is shown in Fig. 1 to point out some general features about 7-fold tilings. 1. It is obvious that the three base tiles are not only sufficient to cover a large surface, all three are necessary. 2. In the Steinhardt patch are some straight lines that follow edges and diagonals of base tiles. This is promising for the substitution method, as one tries to place entire base tiles, or at least half-tiles along the straight edges of the inflated tile. 3. The Steinhardt tiling does not contain straight sequences that lend themselves to a substitution scheme. An impressive 7-fold tiling with substitution and matching rules was discovered by J. Socolar 4, 5 . The substitution requires 129 base tiles. Fig. 2 shows another substitution tiling that requires 353 base tiles discovered by the author. Socolar had noted that some combinations of tiles can easily be interchanged (are weakly matched). The advantage of large substitution schemes is that certain features, like 7-pointed stars, can be enhanced.
On the other hand, 7-fold tilings with a low inflation factor are much more constrained. An example was discovered by Chaim Goodman-Strauss 6 . A minor variation of his tiling is shown in Fig. 3 with added notches to specify the matching directions. It takes 41 base tiles to cover the three inflated tiles. The sides of the enlarged tiles follow the inflation sequence δ = 1 + Φ + 1, where Φ is the 7-fold magic number mentioned above. The inflation factor is δ = 3.802… It is noted that there are no 7-pointed stars in the Goodman-Strauss tiling. Fig. 4 is a patch of the tiling derived from Fig. 3 arranged as a rosette or Mandala in circular or rotational symmetry (a Mandala is a geometric configuration of symbols arranged by a division of a circle). A Minimal 7-fold Rhombic Tiling:
One goal of this study of 7-fold tilings was to find the most minimal or smallest set of base tiles that can be used to generate an infinite tessellation by substitution and matching rules. Such an example is shown in Fig. 5 . Substitution rules set the number of tiles needed to tile the three larger tiles. In addition, matching rules are used to specify the orientation of the base tiles and how they can be adjoined with other tiles.
The substitution rule of this minimal tiling is as follows.
This can be conveniently written as
Where M is the substitution matrix. In this case, In this minimal tiling set, the sides of the inflated rhombs are enlarged by the inflation factor δ, which is the long diagonal of the A tile (denoted by Γ). The base tiles have sides of unit length.
Hence the inflation factor of this minimal tiling is δ = 1.950…
Surprisingly, M 2 is the same substitution matrix as the Goodman-Strauss tiling of Fig. 4 which has an inflation factor of δ = Φ + 2 (= 3.802…). From this, one can derive one of many relationships between diagonals, areas, and area ratios:
Figs. 5-8 show the first four generations of the minimal set. A variation is shown in Figs. 9-12, wherein the base tiles are taken from the first generation of the previous set. The A0 tile ("shield") is composed of three and the B0 ("skate") and C0 tiles ("kite") are composed of four proto-rhombs. The name "kite" is in analogy to the kites and darts in the Penrose tiling, a moniker originally coined by John Horton Conway 7 . Another variation is shown in Figs. 14-16 nicknamed "minimal polka dot 7-fold tiling". It consists as before of only three proto-rhombs, but the coloring brings out a new set of tiles, two trapezoids, and one triangle.
Magic and Impossible Substitutions: Fig. 18 demonstrates that it is physically possible (with paper and scissors) to tile inflated tiles that are enlarged by the 7-fold magic number Φ. However, some additional cuts are necessary to create inflated rhombs. There are no known matching rules to make an infinite pattern. But the substitution matrix is perfectly valid:
Φ is the long diagonal of the B tile and is the inflation factor of this "magic" substitution: Φ = 2 cos(π/7) (= 1.802…) Surprisingly, the "magic" substitution matrix has a square root with integer value:
Potentially, it should be possible to make inflated tiles with only 5 proto-tiles with an inflation factor of δ = Φ ½ (= 1.342…). But how can this be achieved? This is akin to asking, how can a circle be tiled with square tiles? The first substitution (top row of the matrix) is inflating the A0 tile to the size of B0. The ratio of the two areas is B0/A0 = Φ, an irrational number. There is no way to tile it with only one proto-tile as it is impossible to tile a circle with a finite number of square tiles.
The "square root of Φ" matrix leads to an interesting cubic equation. From the second row we get B1 = A0 + C0. The ratio of the areas B/A = Φ (a summary of various relationships is given at the end of the paper). The area inflation is Φ, i.e., B1 = ΦB0, or B1 = Φ 2 A0. From the second row, divided by A0, Φ 2 = 1 + C0/A0. The ratio of the areas C/
Rearranging all terms leads to the cubic equation:
This really defines the magic number Φ in the context of 7-fold tessellations. The equation has three real roots and Φ = 1.802… is one of them. This can be easily checked numerically. The equation is hard to solve, and I leave it as a challenge to the reader.
The General Form of the 7-fold Substitution Matrix:
The A0 tile could also be inflated to the size of the C0 tile. Again, this is physically impossible (with paper and scissors) but makes perfect sense mathematically. The substitution matrix is:
It has an inflation factor of δ = (Φ 2 -1) ½ (= 1.499…). This can be extended to any integers a,b,c. It suffices to know the seed values (a,b,c) to tile A1 to predict how many tiles are needed to tile B1 and C1. (a,b,c) can be any integer combination, but only a few will lead to a substitution and matching scheme that tiles an infinite area.
The Magic Number Φ in an Infinite Series:
In a large 7-fold tiling, the number of tiles NA, NB, NC are in proportion of their areas, i.e.,
Looking at just one of the inflated tiles in the nth generation, An, it is composed of An = nA A0 + nB B0 + nC C0
The ratio of nB tiles to nA tiles approaches lim(n -> ∞) nB/nA = Φ Any substitution matrix with an arbitrary (a0,b0,c0) seed could lead to an infinite series that generates nA and nB. Starting with the general substitution matrix M (above) A1 = a0 A0 + b0 B0 + c0 C0
Performing another inflation with M results in A2 = b0 A0 + (a0 + c0) B0 + (b0 + c0) C0
Which can be the start of another 7-fold tiling with A2 = a1 A0 + b1 B0 + c1 C0
And so on. This can be given as the following infinite series 8 :
Let (a0,b0,c0) be any integer seed values A good starting seed is a0 = 5, b0 = 9, c0 = 11 (Φ ~ a0/b0), but any other seed will also converge.
The 7-fold Magic Number Φ:
During this study of 7-folds, it became obvious that areas, area ratios, and diagonals of the three proto-rhombs are all related to Φ. Rhomb edges are normalized to unit length.
Areas: Summary:
In the following table, some of the 7-fold tilings mentioned in this paper are summarized and grouped by the inflation factor. The substitution matrix can be calculated from the seeds a,b,c where A1= a A0 + b B0 + c C0 is the substitution of the smallest tile which determines the number of tiles needed for B1 and C1. The inflation factor δ is expressed in relation to the magic number Φ. The # Tiles is the total number of base tiles needed to tile the next generation. The label Pattern # is arbitrary and was roughly in the sequence of studying the particular pattern. Epilogue:
This study of 7-fold tilings started on Valentine's Day 2020. Within weeks, the author was in quarantine for 3 months with his family due to the Coronavirus pandemic. It was not quite planned that way but cutting out hundreds of tiles and gluing them into patterns was a perfect pastime. The author expects that many other tiling enthusiasts have also tried their hands with 7-folds. In self-isolation, this work was independent, but the hope is to give credit where credit is due. I had started working on Pattern 1 and 2 in 1983 but did not get anywhere. I was surprised how difficult it was to find new patterns by substitution. But then again, every time I found a new 7-fold tessellation, I was amazed that it existed.
Sadly, the pandemic took its toll on John Horton Conway 7 who passed away in April. He made contributions to many branches of recreational mathematics and played an important role in the early days of the Penrose 5-fold tessellation and in solutions for Rubik's Cube. 353 base tiles are used to tile three similar larger tiles. The inflation sequence (sides of the inflated tiles) is δ = 1 + ψ + Φ + γ + 1 + Φ + ψ + 1 + Φ, where Φ is the long diagonal of the B tile, ψ is the short diagonal of the C tile, and γ is the short diagonal of the A tile. The inflation factor is δ = 11.345… Substitution and matching rules for a 7-fold tiling. This pattern was found independently but is essentially the same as the one discovered by Chaim Goodman-Strauss. It takes 41 base tiles to make the three inflated tiles. In addition, notches were added to specify the matching directions. The inflation sequence (sides of the larger tiles) is δ = 1 + Φ + 1, and the inflation factor δ = 3.802… The inflation factor is δ = 2.802… However, there are no known matching rules that allow an inflation sequence that can be repeated. Fig. 18 : The base tiles can fit into inflated tiles that are enlarged by the 7-fold magic number Φ with the help of some additional cuts. The inflation factor is δ = 1.802… This could be the smallest substitution set (9 tiles), but there are no known matching rules to repeat the inflation scheme. 