# The Curled Up Dimension in Quasicrystals

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Quasicrystals and the Perpendicular Space in the Cut-and-Project Approach

## 3. Periodic Order in the Curled Up Perpendicular Space

## 4. “Frustration” in the ${\mathbf{P}}_{\mathbf{\perp}}+{\mathbf{P}}_{\mathbf{\Vert}}$ Space and Shear as a Solution

**d**. This frustration breaks the periodicity in ${P}_{\Vert}$.

## 5. Summary and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

## References

- Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J.W. Metallic Phase with Long-Range Orientational Order and No Translational Symmetry. Phys. Rev. Lett.
**1984**, 53, 1951–1953. [Google Scholar] [CrossRef] [Green Version] - Levine, D.; Steinhardt, P.J. Quasicrystals: A New Class of Ordered Structures. Phys. Rev. Lett.
**1984**, 53, 2477–2480. [Google Scholar] [CrossRef] [Green Version] - Janot, C. Quasicrystals: A Primer; Oxford University Press: Oxford, UK, 2012. [Google Scholar]
- Levine, D.; Steinhardt, P.J. Quasicrystals. I. Definition and structure. Phys. Rev. B
**1986**, 34, 596. [Google Scholar] [CrossRef] [PubMed] - Baake, M. A guide to mathematical quasicrystals. In Quasicrystals; Suck, J.B., Schreiber, M., Häussler, P., Eds.; Springer Series in Materials Science; Springer: Berlin/Heidelberg, Germany, 2002; Volume 55. [Google Scholar] [CrossRef] [Green Version]
- de Boissieu, M. Phonons, phasons and atomic dynamics in quasicrystals. Chem. Soc. Rev.
**2012**, 41, 6778–6786. [Google Scholar] [CrossRef] - Baggioli, M.; Landry, M. Effective Field Theory for Quasicrystals and Phasons Dynamics. SciPost Phys.
**2020**, 9, 62. [Google Scholar] [CrossRef] - Janssen, T.; Janner, A. Aperiodic crystals and superspace concepts. Acta Crystallogr. B
**2014**, 70, 617–651. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sadoc, J.F.; Mosseri, R. Geometric Frustration; Cambridge University Press: Cambridge, MA, USA, 1999. [Google Scholar]
- Fang, F.; Clawson, R.; Irwin, K. Closing Gaps in Geometrically Frustrated Symmetric Clusters: Local Equivalence between Discrete Curvature and Twist Transformations. Mathematics
**2018**, 6, 89. [Google Scholar] [CrossRef] [Green Version] - Kleinert, H. Gauge Fields in Condensed Matter; World Scientific: Singapore, 1989. [Google Scholar]
- Senechal, M.J. Quasicrystals and Geometry; Cambridge University Press: Cambridge, MA, USA, 1995. [Google Scholar]
- Socolar, J.; Steinhardt, P.J. Quasicrystals. II. Definition and structure. Phys. Rev. B
**1986**, 34, 617. [Google Scholar] [CrossRef] [PubMed] - Baake, M.; Grimm, U. Aperiodic Order Volume I; Cambridge University Press: Cambridge, MA, USA, 2013. [Google Scholar]
- Burdik, Č.; Frougny, C.; Gazeau, J.P.; Krejcar, R. Beta-integers as natural counting systems for quasicrystals. J. Phys. A Math. Gen.
**1998**, 31, 6449–6472. [Google Scholar] [CrossRef] [Green Version] - Baake, M.; Klitzing, R.; Schlottmann, M. Fractally shaped acceptance domains of quasiperiodic square-triangle tilings with dedecagonal symmetry. Phys. A Stat. Mech. Its Appl.
**1992**, 191, 554–558. [Google Scholar] [CrossRef] - Lapidus, M.; Hung, L. Self-similar P-adic fractal strings and their complex dimensions. P-Adic Num. Ultrametr. Anal. Appl.
**2009**, 1, 167–180. [Google Scholar] [CrossRef] - Strogatz, S.H. Nonlinear Dynamics and Chaos; Perseus Books, Reading: Cambridge, MA, USA, 1994. [Google Scholar]
- Grassberger, P.; Procaccia, I. Measuring the strangeness of strange attractors. Phys. D Nonlinear Phenom.
**1983**, 9, 189–208. [Google Scholar] [CrossRef]

**Figure 1.**Cut-and-project scheme for Fibonacci chain. (

**a**) ${\mathbb{Z}}_{2}$ parent lattice with lattice vectors l

_{1}and l

_{2}, showing ${P}_{\perp}$, ${P}_{\Vert}$, and cut region bounded by dashed lines $tan\theta =\mathsf{\Phi}$, golden ratio. (

**b**) Detail of parent lattice (rotated so ${P}_{\Vert}$ is horizontal) showing distances between points as projected in ${P}_{\perp}$.

**Figure 2.**(

**a**) Cut region in ${\mathbb{Z}}_{2}$ parent lattice (region bounded by top and bottom dashed lines) and projection in ${P}_{\Vert}$ (on solid line). (

**b**) Periodicity after identifying top and bottom boundaries of ${P}_{\perp}$, showing equal increments in ${P}_{\perp}$ (examples ${\Delta}_{1}$, ${\Delta}_{2b}$, and ${\Delta}_{2t}$ labeled). (

**c**) Same periodicity illustrated after wrapping ${P}_{\perp}$ into a loop. Dashed circle is ${P}_{\perp}$; colors match points and increments in (

**a**,

**b**) with points and arcs in (

**c**). (

**d**) Displaying the successive arc segments and arrow tips with increasing radius spreads the circle into an annulus, making the self-similarity of the structure more visually apparent.

**Figure 3.**Self similarity of point distribution on the ${P}_{\perp}$ loop. In a given step, there are n long arcs (L) of angle $\frac{2\pi}{{\mathsf{\Phi}}^{n}}$, short arcs (S) having angle $\frac{2\pi}{{\mathsf{\Phi}}^{n+1}}$. In next step, n new points are added by incrementing the angle in steps of $\frac{2\pi}{{\mathsf{\Phi}}^{2}}$. These all land in L arcs, dividing them into $LS$ arcs at new scale, with new L having angle $\frac{2\pi}{{\mathsf{\Phi}}^{n+1}}$. The number of L arcs in successive steps are Fibonnaci numbers 1, 2, 3, 5….

**Figure 4.**Cut region (

**a1**) wrapped into a cylinder (

**b1**), with an exaggerated view to better see mismatch (

**c1**). When introducing shear, cut region (

**a2**) becomes a parallelogram and mismatches in cylinder close to form a true helix (

**b2**,

**c2**).

**Figure 5.**When the cut region is curved into a cylinder, different choices of shear connect different sections to make different helices. n helices are created when the shear connects each sections with its ${n}^{\mathrm{th}}$ neighbors, rather than just its immediate neighbors.

**Figure 6.**Comparing cut windows of different thickness: (

**a**) thinner window requires a smaller shear correction d to connect a section with its first neighbors and make a single helix; (

**b**) thicker window requires a larger shear correction ${d}^{\prime}$ for a single helix.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fang, F.; Clawson, R.; Irwin, K.
The Curled Up Dimension in Quasicrystals. *Crystals* **2021**, *11*, 1238.
https://doi.org/10.3390/cryst11101238

**AMA Style**

Fang F, Clawson R, Irwin K.
The Curled Up Dimension in Quasicrystals. *Crystals*. 2021; 11(10):1238.
https://doi.org/10.3390/cryst11101238

**Chicago/Turabian Style**

Fang, Fang, Richard Clawson, and Klee Irwin.
2021. "The Curled Up Dimension in Quasicrystals" *Crystals* 11, no. 10: 1238.
https://doi.org/10.3390/cryst11101238