# Hexagonal Inflation Tilings and Planar Monotiles

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Penrose’s Aperiodic Hexagon Tiling and Related Patterns

**Proposition**

**1**

**Figure 4.**A patch of the double hexagon tiling, exactly corresponding to that of Figure 3.

**Figure 5.**Rules for the mutual local derivation between $(1+\epsilon +{\epsilon}^{2})$-tilings and double hexagon tilings.

**Proposition**

**2**

**Figure 6.**Parity pattern of a $(1+\epsilon +{\epsilon}^{2})$-tiling, as derived from a fixed point tiling of the inflation rule of Figure 2. This particular pattern (also the infinite one) possesses an almost colour reflection symmetry with respect to the indicated lines; see text for details.

**Theorem**

**1**

**Remark**

**1**

**Proposition**

**3**

**Corollary**

**1**

## 3. Taylor’s Inflation Tiling

**Lemma**

**1**

**Theorem**

**2**

**Figure 7.**The primitive inflation rule of Taylor’s half-hex inflation (top). The central patch of a fixed point tiling is shown on the lower left panel, with its parity pattern to the right.

**Figure 8.**Patch of a llama tiling, as derived from a fixed point tiling of the inflation rule of Figure 7. This particular pattern (and its infinite extension) possesses an almost colour reflection symmetry with respect to the indicated lines; see text for details.

**Theorem**

**3**

**Figure 9.**Correspondence between fully decorated (half) hexagons, with central hexagon of type C, and their parity patterns.

- R1.
- The hexagons must match at common edges in the sense that the decoration lines do not jump on crossing the common edge;
- R2.
- The point markers must satisfy the edge transfer rule sketched in the middle panel of Figure 10, as indicated by the arrow, for any pair of hexagons separated by a single edge. The two points at the corners adjacent to that edge have to be in the same position, and this rule applies irrespective of the chirality types of the tiles;
- R3.
- No vertex configuration is allowed to have adjacent points in a threefold symmetric arrangement, such as the one shown in the right panel of Figure 10, or its rotated and reflected versions.

**Figure 10.**Taylor’s functional monotile (left), a sketch of the edge transfer rule R2 (central) and the forbidden threefold vertex seed (right); see text for details.

**Theorem**

**4**

**Corollary**

**2**

## 4. Topological Invariants and the Structure of the Hulls

**Figure 11.**Hexagon (pseudo) inflation for the half-hex (left) and for two (locally equivalent) variants of the arrowed half-hex (centre and right). If only the shaded hexagons are retained, a standard inflation for a wedge with opening angle $\frac{2\pi}{3}$ is obtained; see text for details.

**Corollary**

**3**

**Figure 12.**Patch of the arrowed half-hex tiling. The blue decoration lines form an infinite hierarchy of triangles of all sizes. The blue triangles of a given size and orientation form a triangular lattice.

**Figure 13.**Patch of the $(1+\epsilon +{\epsilon}^{2})$-tiling in the variant with line decorations. In addition to the hierarchy of blue triangles from the half-hex tiling, there is now also a hierarchy of red hexagons. Hexagons of each size form a lattice periodic array, with lattices of different densities.

**Figure 14.**Inflation rules for $(1+\epsilon +{\epsilon}^{2})$-tiling in the variant with line decorations. If a hexagon is replaced by 7 hexagons, a pseudo inflation is obtained. Taking only the 4 shaded hexagons results in a standard inflation rule (for one sector).

**Figure 15.**Schematic representation of the hulls of the different hexagon tilings via a (closed) toral slice of the underlying solenoid. The projection fails to be 1-to-1 on the lines as follows. For the half-hex, the projection is 3-to-1 precisely at the points marked by a hexagon, and 1-to-1 elsewhere. For the arrowed half-hex, the 1d sub-solenoids where the projection is 2-to-1 are shown as blue lines. At the hexagon points, three blue sub-solenoids intersect, and the projection is 6-to-1. For the Taylor and the Penrose hexagon tilings, there are three additional sub-solenoids (red lines) along which the projection is 2-to-1. These intersect at two inequivalent points (marked by triangles) where the projection is 6-to-1 (at such points, three infinite order supertiles meet). The projection is 12-to-1 at the hexagon points (which are centres of infinite-order supertiles). Note that, at the points marked by diamonds, not only two but in fact all six sub-solenoids intersect. This is due to the dyadic structure of the hull. As these points have half-integer coordinates with respect to the hexagonal lattice, they are equivalent (in the solenoid) to the points marked by hexagons.

**Theorem**

**5**

**Theorem**

**6**

**Corollary**

**4**

**Corollary**

**5**

## 5. Outlook and Open Problems

## Acknowledgments

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Baake, M.; Gähler, F.; Grimm, U.
Hexagonal Inflation Tilings and Planar Monotiles. *Symmetry* **2012**, *4*, 581-602.
https://doi.org/10.3390/sym4040581

**AMA Style**

Baake M, Gähler F, Grimm U.
Hexagonal Inflation Tilings and Planar Monotiles. *Symmetry*. 2012; 4(4):581-602.
https://doi.org/10.3390/sym4040581

**Chicago/Turabian Style**

Baake, Michael, Franz Gähler, and Uwe Grimm.
2012. "Hexagonal Inflation Tilings and Planar Monotiles" *Symmetry* 4, no. 4: 581-602.
https://doi.org/10.3390/sym4040581