# softmatter:Quasicrystal (QC)

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## Contents |

## Introduction to Quasicrystallinity

Crystals are substances which exhibit perfect translational symmetry. That is, provided with a lattice and a basis, one can theoretically extend a crystal structure without limit. These materials have been thoroughly studied and are examined in any [solid state physics] text. Of critical importance in the description of a crystal structure is the notion of symmetry. In particular, two-, three-, four-, and six-fold rotations are permitted in traditional crystal structures. Five-fold and rotations of order seven and above are not typically observed and are called forbidden rotations. This fact follows from the inability to tile space with a polygon whose vertex angles are equal to 2pi/5, 2pi/7, 2pi/8 ,etc...

Quasiperiodic structures or quasicrystals (QC) are materials that exhibit perfect long range order but *without* three-dimensional periodicity. Furthermore, the diffraction patterns of quasicrystals exhibit forbidden symmetries. It is this order without periodicity that makes quasicrystals so radically different from the traditional crystal.

Quasicrystals were first publicized by Shechtman et. al when they observed icosahedral 10-fold symmetry in a metallic solid (Al-14 at. %-Mn), which is inconsistent with crystal lattice translations.

## Types of Quasicrystals

Octagonal quasicrystals have 8-fold symmetry.

Decagonal quasicrystals have 10-fold symmetry.

Dodecagonal quasicrystals have 12-fold symmetry.

- View example of a dodecagonal quasicrystal on the MATDL Repository.

## Quasicrystals and high dimensional spaces

Projection procedure: (needed)

Cut procedure: (needed)

## References

1. Kittel, Charles. *Introduction to Solid State Physics*.

2. Janot, Christian. *Quasicrystals: A Primer*

3. D. Shechtman, I. Blech, D. Gratias, J.W. Cahn. *Metallic Phase with Long-Range Orientational Order and No Translational Symmetry*, Phys. Rev. Lett. **53**, 20 (1984).