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
nRatioValue
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 means x_n. The basic triangle I has edges of length a, x_n, 1. Inflation by a factor x_n gives a congruent 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 identical to the basic triangle I and another one, II, with edge lengths ax_n, nx_n, a. If tile II is on its turn 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 the figure below for n=2. Note that the division can be done in more than one way, even for n=1.

metallic mean triangle construction for n=2.

The tiles for n=1 are closely related to the famous Penrose tiles, because x_1 = 1/\tau = 1.618 is the golden ratio. For a=1/\tau the triangles I and II are the prototiles for the Penrose-Robinson Tiling or the TübingenTriangle Tiling, while for a=1 the Robinson tiles are obtained.

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

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

, which is related to the famous Danzer ABCK tiles. The relationship is most easily seen by considering the octahedral set of prototiles as shown in the next picture. 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).

Danzer ABCK octahedra

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

Danzer golden mean tiles.