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arxiv: 1906.10392 · v1 · pith:SYBRIFZTnew · submitted 2019-06-25 · 🧮 math-ph · math.MP

Quasicrystals

Uwe Grimm , Peter Kramer This is my paper

Pith reviewed 2026-05-25 16:27 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords quasicrystalsnon-periodic tilingsPenrose tilingAmmann-Beenker tilingquasiperiodic functionsforbidden symmetrieslong-range orderquasiperiodic coverings
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The pith

Quasicrystals combine long-range atomic order with rotational symmetries forbidden to any periodic crystal, and their structures are modeled by non-periodic tilings built from quasiperiodic functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Quasicrystals are materials whose atoms maintain order over long distances yet display symmetries, such as five-fold rotation, that cannot occur in any repeating three-dimensional lattice. The review shows that these structures correspond to non-periodic tilings of space, with the Penrose tiling in the plane and the Ammann-Beenker tiling in three dimensions serving as concrete examples. The underlying mathematics begins with Bohr's quasiperiodic functions, which generate the tilings and also permit an equivalent description in which space is covered by overlapping clusters arranged quasiperiodically. This framework accounts for the observed diffraction patterns and explains why certain intermetallic alloys adopt these arrangements rather than ordinary crystal structures.

Core claim

Quasicrystals are characterised by the coexistence of long-range atomic order and 'forbidden' symmetries which are incompatible with periodic arrangements in three-dimensional space. Their structures relate directly to non-periodic tilings of space, illustrated by the Penrose and Ammann-Beenker examples. A general construction starts from the theory of quasiperiodic functions and yields both the tilings themselves and an alternative picture of quasiperiodic coverings by overlapping clusters.

What carries the argument

Non-periodic tilings of space generated from quasiperiodic functions, such as the Penrose and Ammann-Beenker tilings, which serve as models for the atomic arrangements observed in quasicrystals.

If this is right

  • The observed 'forbidden' symmetries arise as direct geometric consequences of the quasiperiodic construction rather than from periodicity.
  • Diffraction patterns of quasicrystals match those predicted by the Fourier transforms of the corresponding tilings.
  • An equivalent description exists in which space is filled by overlapping clusters whose centers follow quasiperiodic rules.
  • The same mathematical framework applies to any material whose order is long-range but non-periodic.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar non-periodic order might be engineered in other classes of materials by enforcing the same quasiperiodic constraints.
  • The covering description could simplify the analysis of local atomic environments compared with global tiling rules.
  • Numerical simulations of alloy formation could test whether energy minima correspond to the known Penrose or Ammann-Beenker geometries.

Load-bearing premise

The atomic structures found in real quasicrystal alloys can be captured accurately by non-periodic tilings derived from quasiperiodic functions.

What would settle it

A measured atomic structure or diffraction pattern from an intermetallic quasicrystal that cannot be reproduced, even approximately, by any tiling or covering constructed from quasiperiodic functions.

Figures

Figures reproduced from arXiv: 1906.10392 by Peter Kramer, Uwe Grimm.

Figure 1
Figure 1. Figure 1: The icosahedron (left) and dodecahedron (right). Symmetry axes of order two, three and five are indicated. There are 15 axes of order 2 (half the number of edges in both cases), 10 axes of order 3 (half the number of faces in the icosahedron or vertices in the dodecahedron) and 6 axes of order 5 (half the number of vertices in the icosahedron or faces in the dodecahedron). It thus came as a surprise when i… view at source ↗
Figure 2
Figure 2. Figure 2: The first published evidence of icosahedral crystals is this selected area elec￾tron diffraction pattern obtained by Dan Shechtman [25] from a a rapidly cooled aluminium manganese alloy. Angles are measured with respect to a fivefold axis, and the observed 2-, 6- and 10-fold symmetries in the diffraction patterns match the ori￾entation of 2-, 3- and 5-fold axes of icosahedral symmetry, compare [PITH_FULL_… view at source ↗
Figure 3
Figure 3. Figure 3: A holmium magnesium zinc ‘single’ quasicrystal [10]. It shows perfect do￾decagonal morphology, compare [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Planar periodic tilings with 2-, 3-, 4- and 6-fold rotational symmetry. A simple argument goes as follows. Assume for a moment that you had a periodic lattice with fivefold symmetry. This means that you can rotate the lattice by multiples of 72◦ about any of its lattice points, and obtain the same lattice again. We shall now show that this is impossible. Start with two lattice points which have minimal dis… view at source ↗
Figure 5
Figure 5. Figure 5: Two starting points (dark), and the two sets of 4 points obtained by rotating one about the other by multiples of 72◦ . The highlighted pair of rotated points is closer than the pair of rotation centres, leading to the contradiction. 60o 60o 45o 45o [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The same as [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The tiles of the rhombic Penrose tiling with arrow decorations. Clearly, ignoring the arrows these tiles can give rise to periodic tilings of the plane – as an example just take the tiling made by repeating one of the two tiles periodically. To obtain the Penrose tiling, tiles are assembled subject to the constraint that tiles in the tiling are edge-to-edge and such that the arrow decorations on adjacent e… view at source ↗
Figure 8
Figure 8. Figure 8: A legal patch of a Penrose tiling. rules to preferred local arrangement of atoms, this has sparked discussions about how quasicrystals grow, a topic that is still not completely understood, see [11] for a review. The Ammann-Beenker tiling, again built from two different tiles, has eight￾fold rotational symmetry. It also possesses matching rules; however, in this case decorations on edges and vertices are n… view at source ↗
Figure 9
Figure 9. Figure 9: A legal patch of an Ammann-Beenker tiling. dissection of the triangle breaks the reflection symmetry of the unmarked tiles – you need to consider the orientation of the triangle, marked by the arrow on the hypotenuse. The dissection of a triangle of the opposite orientation is the mirror image of the one shown here. Now, you can build an Ammann-Beenker tiling by repeated application of this rule, starting … view at source ↗
Figure 10
Figure 10. Figure 10: Inflation of the Ammann-Beenker tiling. this patch will occur in the infinite tiling over and over again. This property is called repetitivity (not to be confused with periodicity). You can deduce this property by applying the inverse transformation, deflation, to the tiling with your chosen patch. Eventually, after a finite number of steps, your patch will be mapped to a single tile. Since the two tiles … view at source ↗
Figure 11
Figure 11. Figure 11: Planar projections of a packing of cubes, showing different surfaces. The projected planar tiling consist of three rhombic tiles, corresponding to the projec￾tions of the three visible faces of the cube. Onlye for special surfaces, such as shown on the left and in the centre, the rhombs form a periodic tiling; in general, the rhombic pattern on a surface is non-periodic. This is the main idea behind the c… view at source ↗
Figure 12
Figure 12. Figure 12: The vectors ak, k = 1, . . . 4 spanning the Z-module Λ in ‘physical’ space, and the corresponding vectors a ∗ k spanning the ‘internal’ space [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The Ammann-Beenker tiling as a cut and project set. On the left, the projected points VAB of the hypercubic lattice in physical space are shown, with lines connecting points of unit distance. On the right, the corresponding projections V ∗ AB in internal space are shown, which fall into a regular octagon of unit edge length. Explicitly, for the Ammann-Beenker tiling the cut and project approach can be imp… view at source ↗
Figure 14
Figure 14. Figure 14: Calculated diffraction image of the Ammann-Beenker tiling. It can be shown that, under rather general assumptions, cut and project sets defined in this way possess a pure point diffraction measure. As an exam￾ple, the calculated diffraction pattern of scatters positioned on the vertices of an Ammann-Beenker tiling is shown in [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Voronoi and dual Delone complex for the root lattice A2. The Voronoi cells are hexagons centred at lattice points (black squares), the Delone cells triangles centred at two inequivalent types of holes (black and white cycles). Windows can also be defined for vertices, for holes, and for covering clusters discussed below. The dual Voronoi and Delone cell complexes allow for a second construc￾tion with the … view at source ↗
Figure 16
Figure 16. Figure 16: Construction of the Fibonacci tiling. Two new squares A, B for Λ = Z 2 (lattice points black squares, holes white circles) play the role of the polytopes T ∗ in Eq. (14). The squares are constructed by projection from dual pairs of 1-boundaries and tile E 2 periodically. The irrational horizontal Fibonacci line E 1 k runs through this periodic tiling. Its intersections display the tiles Ak, Bk of the quas… view at source ↗
Figure 17
Figure 17. Figure 17: The Penrose rhombus tiling (T , A4). It consists of two types of rhombus tiles (thin lines), projections of 2-boundaries from the Voronoi complex. Any rhom￾bus vertex is the projection of a hole from Λ = A4. In addition we show a covering of the Penrose tiling by overlapping decagons (heavy lines) [13]. Each decagon covers 10 rhombus tiles and is centred at the projection of a lattice point (black square)… view at source ↗
Figure 18
Figure 18. Figure 18: The T¨ubingen triangle tiling (T , A4). The tiles (thin lines) are two tri￾angular projections of Delone 2-boundaries from the Delone complex. Any triangle vertex is the projection of a lattice point. In addition we show a covering of the triangular tiling by two types of overlapping pentagons (heavy lines). compare [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The 3D icosahedral tiling (T ∗ , D6). Top: The three Delone windows D a ⊥, Dc ⊥, Db ⊥ ∈ E⊥ of the tiling (T ∗ , D6). Bottom: The six tiles of this tiling are four pyramids on a rhombus base and the two rhombohedra known from the prim￾itive tiling. The holes a, c are marked by black and white circles, the holes b by a double circle. each consisting of 10 rhombus tiles. This covering is shown in [PITH_FULL… view at source ↗
Figure 18
Figure 18. Figure 18: Acknowledgement. UG acknowledges support by EPSRC via Grant EP/D058465. The authors thank D. Shechtman and P.C. Canfield, and the American Physical Society, for granting permission to reproduce Figs. 2 and 3 in this article [PITH_FULL_IMAGE:figures/full_fig_p022_18.png] view at source ↗
read the original abstract

Mathematicians have been interested in non-periodic tilings of space for decades; however, it was the unexpected discovery of non-periodically ordered structures in intermetallic alloys which brought this subject into the limelight. These fascinating materials, now called quasicrystals, are characterised by the coexistence of long-range atomic order and 'forbidden' symmetries which are incompatible with periodic arrangements in three-dimensional space. In the first part of this review, we summarise the main properties of quasicrystals, and describe how their structures relate to non-periodic tilings of space. The celebrated Penrose and Ammann-Beenker tilings are introduced as illustrative examples. The second part provides a closer look at the underlying mathematics. Starting from Bohr's theory of quasiperiodic functions, a general framework for constructing non-periodic tilings of space is described, and an alternative description as quasiperiodic coverings by overlapping clusters is discussed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript is a review that first summarizes the main properties of quasicrystals (long-range atomic order coexisting with forbidden symmetries incompatible with periodicity in 3D), their relation to non-periodic tilings, and introduces the Penrose and Ammann-Beenker tilings as examples. The second part develops the mathematics via Bohr's theory of quasiperiodic functions, a general framework for constructing non-periodic tilings, and an alternative view as quasiperiodic coverings by overlapping clusters.

Significance. As a review of established facts with no new theorems or empirical claims, the paper has modest significance. It correctly restates the standard characterization of quasicrystals and their modeling by quasiperiodic structures and tilings; its value would lie in providing a concise entry point for readers, provided the exposition is accurate and well-referenced.

minor comments (2)
  1. The abstract and introduction could more explicitly state the intended audience (mathematicians vs. physicists) and the level of mathematical detail provided in §2.
  2. Figure captions for the Penrose and Ammann-Beenker tilings should include a brief note on the inflation rules or matching conditions used to generate them.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation to accept. The referee's summary accurately reflects the scope and content of our review.

Circularity Check

0 steps flagged

No significant circularity; review of established concepts

full rationale

This is a review paper summarizing known properties of quasicrystals, their relation to non-periodic tilings (Penrose, Ammann-Beenker), and mathematical frameworks from Bohr's quasiperiodic functions. No original derivations, equations, fitted parameters, or load-bearing claims are advanced that reduce to self-citation or input by construction. The text explicitly positions itself as an overview without new theorems or predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper; it introduces no new free parameters, axioms, or invented entities beyond referencing standard mathematical concepts such as quasiperiodic functions.

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