Growth Forms of Grid Tilings

Year: 2022 Authors: Peter Hilgers; Anton Shutov

Core claim

Any regular grid tiling in 2D or 3D has an explicitly computable growth form given by a projection of a central orthoplex section.

Topics

quasiperiodic tilings, growth forms, grid method, corona limits, orthoplexes

Domains

tiling theory, discrete geometry, convex polytopes, projection geometry, mathematical visualization, Bridges art exhibition, pattern design

Methods

grid tiling construction, projection analysis, corona scaling, combinatorial counting

Media

2D and 3D figures, tiling diagrams, polyhedral models

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2022 Conference Proceedings

Growth Forms of Grid Tilings

Peter Hilgers¹ and Anton Shutov²

¹Hasenbergweg 9, 76275 Ettlingen, Germany; peter.hilgers@posteo.de ²Vladimir State University, Gorky Str. 87, Vladimir, 600000, Russian Federation; a1981@mail.ru

Abstract

Growth forms of tilings are an interesting invariant of tilings. They are fully understood in the periodic case but we know very little in the quasiperiodic case. Here we study this problem for quasiperiodic tilings obtained by the grid method. We show that such tilings have polygonal/polyhedral growth forms that result from projections of central sections of orthoplexes. Finally we study characteristics of the computed growth forms in 2D and 3D cases.

Introduction

A tiling $T$ is a covering of plane or space by finitely many types of polygonal or polyhedral figures which overlap only on their boundaries. There are various methods to construct non-periodic tilings, here we use the grid method which was introduced by de Bruijn [3] in the case of the Penrose tiling. It was widely generalised to obtain 3D quasiperiodic tilings with icosahedral and arbitrary symmetry. The main advantage of the method is that it allows to construct not only point sets, but to find tiles directly. This is beneficial especially in the 3D case.

Grid tilings

We will use the following construction: Let $\{g_1,\dots ,g_N\}$ be a family of $N$ unit vectors in d-dimensional Euclidean space. Choose also $N$ real parameters ${\gamma }_i$ which serve as phase shifts. The $N$-grid $L_N$ is a union of $N$ arrays of equidistant parallel hyperplanes in $R^d$:

Here $(\cdot ,\cdot )$ is a scalar product and $1\le i\le N$. If there is no point where more than $d$ grid hyperplanes intersect, the grid is called regular. Hereafter we always suppose that the grid is regular. Note that the grid will be regular for almost all values of ${\gamma }_i$.

The N-grid $L_N$ defines some tiling of the d-dimensional space, but this tiling is “bad” because the number of tile types is infinite. A key idea of the grid method is to consider the tiling dual to $L_N$ in some sense. Define N functions $K_i$ and function $K$ as follows:

Informally $K_i(x)$ is the index of the hyperplane perpendicular to $g_i$ through $x$. It can be proved that $K(x)$ is constant on tiles of $L_N$. So, $K$ maps the set of tiles of $L_N$ to a discrete set $\Lambda$ in $R^d$. The set $\Lambda$ is a set of vertices of some d-dimensional tiling $Til$ that is called grid tiling. To define this tiling we must describe edges connecting points from $\Lambda$. The rule is as follows: Two points of $\Lambda$ are connected by an edge if and only if corresponding tiles of $L_N$ have a common edge (step 4).

Furthermore, the set of all tiles of $L_N$ sharing some fixed vertex is mapped (under $K$) to the set of all vertices of some tile of the grid tiling. In the 3D case the grid tiles are parallelepipeds.

Hilgers and Shutov

The construction of a 2D tiling is exemplarily illustrated in Figure 1 for $\mathrm{N}=4$ and $\mathrm{d}=2$ :

(a)

(b)

(c) Figure 1: Construction of a grid tiling: (a) grid vectors $g_i$ , (b) 4-grid $L_N$ (step 1) with intersection point $X$ and adjacent tiles $A,B,C,D$ ; dotted line for hyperplane through $X$ and perpendicular to $g_3$ , (c) vertices $A$ of the dual tiling (steps 2, 3) with $A\leftrightarrow a$ , $B\leftrightarrow b$ , $C\leftrightarrow c$ , $D\leftrightarrow d$ , (d) edges of dual tiling (step 4)

(d)

Coronas and Growth forms

A patch $P_0$ is a finite set of tiles in $Til$ . The first coordination shell $P_1$ of $P_0$ consists of all tiles which are adjacent to a tile of $P_0$ . In the n-th coordination shell $P_n$ are all tiles adjacent to $P_{n-1}$ that are not in $P_{n-2}$ . Here we abstain from giving a formal definition of growth form, also known as corona limit, and instead refer to [4]. Informally the definition may be given as “outer contour, scaled by $1/n$ ”, or

if it exists, compare Figure 2.

(a)

(b) Figure 2: Initial patch $P_0$ , growth form of Ammann-Beenker tiling (black line) and (a) $n=5$ corona, (b) $n=15$ corona, (c) $n=30$ corona; all tiles scaled by $1/n$ .

(c)

It is known that all periodic tilings do have a growth form, which is some polygon or polyhedron [2], and we know that the growth form does not depend on the selected initial patch. For non-periodic tilings so far we have only few concrete examples of the calculation of the growth form, such as for the Penrose or Ammann-Beenker tiling [1, 7, 8]. We have not found examples of 3D growth forms in the literature. Many crystallographic applications of growth forms are mentioned in [4]. A complex 3D growth form is the topic of the artwork “Late arrival” [6].

Growth Forms of Grid Tilings

New Theorem and Method to Calculate 2D and 3D Growth Forms

Now consider a regular N-grid tiling $Til$ produced by vectors $\{g_1,\dots ,g_N\}$ . Let $\{e_1,\dots ,e_N\}$ be the standard orthonormal base of N-dimensional space. Let $O_N$ be a boundary of the convex hull of the vectors $\pm e_i$ . The polytope $O_N$ is known as N-dimensional cross-polytope or as N-dimensional orthoplex. Consider the projections

${\pi }_1:R^N\to R^d$ defined as ${\pi }_1(t)={\pi }_1((t_1,\dots ,t_N))={\sum }_{i=1}^N{t}_i{g}_i$ and

${\pi }_2:R^N\to R^{N-d}$ , an orthogonal projection to the (N-d)-dimensional plane ${\pi }_1(t)=0$ .

Let $\mathrm{P}$ be the d-dimensional plane ${\pi }_2(t)=0$ . Then for any regular grid tiling the growth form exists and is given by ${\pi }_1(O_N\cap P)$ . For proof and computational details see [4].

Characteristics of the Growth Forms

The growth forms are centrally symmetric convex polygons or polyhedrons. In the 2D case the number $V$ of vertices and number $E$ of edges is $2N$ each and, if the grid vectors have rotational symmetry, the growth form is a regular $2N$ -gon with radius $(N/2)\tan (\pi /(2N))$ .

For the 3D case we define: A $\mathbf{r}$ -tuple is set of $\mathbf{r}$ grid vectors. A $\mathbf{r}$ -tuple is called flat if all its vectors belong to one plane. A flat $\mathbf{r}$ -tuple is called complete if none of the remaining N-r grid vectors belongs to the plane of the $\mathbf{r}$ -tuple. Let $\mathrm{k}(\mathrm{r})$ be the number of complete $\mathbf{r}$ -tuples. Then we have V, E, F (for faces):

If there are no flat $r$ -tuples for $r\ge 3$ , the sums do not apply. Each complete $r$ -tuple produces a vertex of degree $2r$ , remaining vertices have degree 4. Table 1 gives examples without and with complete 3- tuples.

Table 1: Examples of growth forms with $N=5$ .

Hilgers and Shutov

3D Ammann Tiling

Probably the best known 3D grid tiling is the Ammann tiling with icosahedral symmetry, Figure 3. Its 6 grid vectors are derived from the dodecahedron and the growth of the tiling was the theme in [5]. The form turned up as a surprise in computer experiments. Now it is clear that the visual impression is correct: The growth form is an Archimedian polyhedron, the icosidodecahedron inscribed in a sphere of radius $\sqrt{5-2\sqrt{5}}$ [4].

Figure 3: Grid vectors and growth of the 3D Ammann tiling.

Summary and Conclusions

We can explicitly calculate growth forms for any regular grid tiling in 2D and 3D. An interesting unsolved problem is to find full descriptions of all polyhedra which are growth forms of grid tilings.

References

[1] S. Akiyama and K. Imai. “Cellular Automata and Discrete Complex Systems.” In Lecture Notes in Computer Science, Volume 9664, edited by David Hutchinson et al., Springer, 2016 [2] S. Akiyama, J. Caalim, K. Imai, H. Kaneko. “Corona Limits of Tilings: Periodic Case.” Discrete and Computational Geometry, vol. 61, no. 3, 2019, pp. 626-652 [3] N.G. de Bruijn. “Algebraic theory of Penrose’s non-periodic tilings of the plane.” Kon. Nederl. Akad. Wetensch. Proc. Ser. A, vol 84, 1981, pp.35-48 [4] D. Demski, P. Hilgers, and A. Shutov. “Growth forms of grid tilings.” Acta Crystallographica A, vol. 78, no. 4, 2022 [5] A. Gross and P. Hilgers. “Corona 27.” Bridges Stockholm 2018 Art Exhibition Catalog. http://gallery.bridgesmathart.org/exhibitions/2018-bridges-conference/hilgers [6] P. Hilgers. “Late arrival.” Bridges Aalto 2022 Art Exhibition Catalog. http://gallery.bridgesmathart.org/exhibitions/2022-bridges-conference/hilgers [7] A.V. Shutov and A.V. Maleev. “Study of Penrose Tiling Using Parameterization Method.” Crystallography Reports, vol. 64, no. 3, 2019, pp. 351-361 [8] A.V. Shutov and A.V. Maleev. “Layer-by-Layer Growth of Ammann-Beenker Graph.” Crystallography Reports, vol. 64, no. 6, 2019, pp. 851-856