# On the Notions of Symmetry and Aperiodicity for Delone Sets

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## Abstract

**:**

## 1. Introduction

## 2. General Notions and Definitions

## 3. Rotations and Reflections of Individual Delone Sets

**Lemma**

**1**

**Figure 1.**Sketch of the incompatibility between points of fivefold rotational symmetry at minimal distance (black line) and periodicity in the Euclidean plane.

**Proposition**

**1**

**Figure 2.**Inflation rule for the (essentially undecorated) Ammann–Beenker tiling (left), and a patch (right) obtained by three inflation steps from a square-shaped patch (consisting of two triangles) in the centre; see text for details. The patch has no rotation symmetry, but is reflection symmetric in the diagonal.

## 4. Symmetries of LI Classes and Hulls

**Lemma**

**2**

**Proposition**

**2**

## 5. Non-Periodicity versus Aperiodicity

**Definition**

**1**

**Proposition**

**3**

**Definition**

**2**

**Figure 7.**Inflation rule (left) and patch (right) of the pinwheel tiling. The prototiles are right triangles with side length 1, 2 and $\sqrt{5}$. The dots represent control points of an equivalent Delone set. Starting from a single triangle with the control point in the origin leads to a fixed point tiling with circular symmetry.

## 6. Probabilistic Extensions and Outlook

**Definition**

**3**

## Acknowledgment

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Baake, M.; Grimm, U.
On the Notions of Symmetry and Aperiodicity for Delone Sets. *Symmetry* **2012**, *4*, 566-580.
https://doi.org/10.3390/sym4040566

**AMA Style**

Baake M, Grimm U.
On the Notions of Symmetry and Aperiodicity for Delone Sets. *Symmetry*. 2012; 4(4):566-580.
https://doi.org/10.3390/sym4040566

**Chicago/Turabian Style**

Baake, Michael, and Uwe Grimm.
2012. "On the Notions of Symmetry and Aperiodicity for Delone Sets" *Symmetry* 4, no. 4: 566-580.
https://doi.org/10.3390/sym4040566