# The Curled Up Dimension in Quasicrystals

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## 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

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**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.

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**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