Aperiodic Metallic Mean Tilings.

Metallic means or ratios (https://en.wikipedia.org/wiki/Metallic_mean) are the largest roots of the polynomial

(1)   \begin{equation*}x^2-nx-1=0\end{equation*}

where n is a nonnegative integer, i.e.

(2)   \begin{equation*}x_n=\frac{1}{2}(n+\sqrt(n^2+4))\end{equation*}

The first four values are given in the table below. The ones for n=1 and n=2 are the well-known Golden and Silver means.

Metallic means
1: (1 + √5)/2 1.618033989 Golden
2: (2 + √8)/2= 1 + √2 2.414213562 Silver
3: (3 + √13)/2 3.302775638 (Bronze)
4: (4 + √20)/2= 2 + √5 4.236067978
Metallic means up to n=4

In the following we wil show that equation 1 may be used to construct pairs of triangles tiling the plane aperiodically by inflation, the inflation factor being the metallic mean x_n. The basic triangle I has edges of length a, x_n, 1. Inflation by a factor x_n gives a similar tile with edges of length ax_n, x_n^2,  x_n. Using eq. 1, the second edge may be split into line segments of length nx_n and 1, which devide the triangle into two triangles, one congruent to the basic triangle I and another one, II, with edge lengths ax_n, nx_n, a. If tile II on its turn is inflated by a factor x_n, a triangle with edge lengths ax_n^2=anx_n+a, nx_n^2=n^2x_n+n and ax is obtained. This inflated type II triangle may be divided into n^2 prototiles I and n^2+1 prototiles II, as shown in Fig. 1.

Fig. 1a. Metal mean tiles. Prototiles are two triangles, one with edges x_n, 1 and a, and one with edge lengths nx_n, a and ax_n. The inflation factor is x_n, the largest root of the polynomial x^2-nx-1=0. a is a free parameter in the interval \tau-1,\tau+1. Tiles III and IV are multiples of tiles I and II by a factor of n.
Fig. 1b. Golden (n=1) and Silver (n=2) mean tiles.

The tiles for n=1 are closely related to the famous Penrose tiles, because x_1 = \tau = 1.61803 is the golden ratio. For a=1 the triangles I and II are the prototiles for the TübingenTriangle Tiling, while for a=\tau the Robinson triangles are obtained. (Note that the Penrose triangle substitution tiling does not fit into the above general substitution scheme although the prototiles are similar). Another special case is the Golden Triangle tiling. The value of a is \sqrt\tau in that case. (Fig. 2)

Fig. 2. Four different Golden Mean Tiles.

Because a is a free parameter, the “golden mean triangles” are a generalization of these Penrose type of tiles, with \tau - 1 < a < \tau + 1. Apart from the ones cited above, there is yet another special set of tiles, not mentioned in literature as such, i.e. the one for a =a_D= \gamma_D\tau, where

(3)   \begin{equation*}$\gamma_D =  cos(\pi/6)/cos(\pi/10)\tau = \sqrt(6/(5+\sqrt(5)) = 0.91059 $\end{equation*}

These tiles are also shown in fig. 2 and denoted as Danzer Golden Mean tiles. The relationship with the famous Danzer ABCK tiles is most easily seen by considering the octahedral set of prototiles as shown in Fig. 3 . The octahedra have been constructed using the red and yellow Zometool elements. The red struts have lengths proportional to \tau^n\cos(\pi/10), n= 0, 1 and the yellow struts have lengths proportional to \tau^n\cos(\pi/6), n= 1, 2 (https://mathworld.wolfram.com/Zome.html).

Fig. 3. Danzer ABCK octahedra

These octahedra have three different types of faces, which are similar to the pair of golden mean prototiles with a=a_D. In Fig.4 these “Danzer golden mean triangles” are shown as Zometool assemblies. The three octahedral faces are congruent to the two golden mean prototiles and one of the first inflated tiles, i.e. the smallest one at the lower left of fig.4

Fig. 4. Danzer golden mean tiles using Zometool elements.

In Chapter 1 of Volume 2 of the book on Aperiodic Order edited by Baake and Grimm, Dirk Frettloeh discusses another very similar free parameter family of triangular inflation tilings originally introduced by Danzer. Its inflation factor is the largest root of the polynomial x^3-x-1=0, i.e. \lambda=1.3247, the socalled plastic number. Because the polynomial is of the third degree, a third intermediate sized prototile is needed.

Alternative for the Danzer ABCK Tiling.

The simplest tetrahedra having exclusively Danzer golden mean tiles as faces are the ones shown in fig.5. They may be connected to form a pyramid with a parallelogram as its base. The base is made up of three Danzer golden mean triangles. Because the height of the three tetrahedra is the same, their volume is proportional to the area of the bases of the individual tetrahedra, i.e. their volume ratios are 1:\tau:\tau^2.

Fig. 5. Three simple tetrahedra with Danzer golden mean faces.

These DEF tetrahedra are alternative prototiles for a tiling of space. To avoid confusion with the ABCK tiles of Danzer we will denote them by D, E and F. The third generation of substitution tiles may be constructed by a combination of second and first generation tiles as shown in fig. 6. The volumetric relationship is

(4)   \begin{equation*}D_3=2D_2+3F_1+E_1=2\tau^3D_1+3F_1+E_1\end{equation*}

(5)   \begin{equation*}E_3=2D_2+4F_1+5E_1+2D_1=2(\tau^3+1)D_1+4F_1+5E_1\end{equation*}

(6)   \begin{equation*}F_3=2F_2+3E_2+D_2=\tau^3(2F_1+3E_1+D_1)\end{equation*}

Filling in the volume ratios of D_1, E_1 and F_1 , the volumes of D_3, E_3 and F_3 are calculated to be proportional to \tau^6,\tau^7 and \tau^8 respectively, as they should.

We have chosen the sets of prototiles in such a way that the substitution tiles have the same point group symmetry as their corresponding prototiles, i.e. C_2 for tiles F and D and D_2 for tile E. The faces of the substitution tiles are all similar planar aperiodic Danzer golden mean tilings. Note that the faces of the Danzer octahedra are very different because they also contain dissimilar faces of the ABCK tetrahedra.