非周期的な構造のフーリエ展開：準結晶と概周期性
秋山茂樹, 筑波大学数理物質系

2015 2 月 10 日 第二回数理連携サロン– Typeset by FoilTEX –In 1982 Shechtman et al.
[5]found Bragg peaks with 10-th foldsymmetry in rapidly cooled Mn-Al
alloy, which is later called ‘quasicrystal’. Shechtman won 2011 Novel
prize by this discovery.
Recently
a lot of quasi-crystals are found inmuch larger scale. Mathematics ofAperiodic Order is a new branch ofmathematics, which aims at giving
mathematical models of them.

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Figure 1: Dodecagonal tiling in mesoporous silica
Nature, Vol 487: 2012, Xiao, Fujita, Miyasaka, Sakamoto,– Typeset by FoilTEX –2Terasaki

Figure 2: Mosaic two-lengthscale quasicrystals
Nature, Vol 506: 2014, Dotera, Oshiro & Ziherl– Typeset by FoilTEX –3

In number theory, Riemann zeta function play a crucial role:111
ζ(s) = s + s + s +
123
1
1111
1= (1 + s + 2s + . . . )(1 + s + 2s + . . . )(1 + s + 2s + . . . ).
223355We consider s = σ +– Typeset by FoilTEX –√−1t to be a complex variable. Then we4

can formally write:

ζ(s) = 1 + exp(−s log 2) + exp(−s log 3) + . . .
∞∑√=exp(−σ log n − t log n −1)
=n=1∞
∑

√

an exp(2π −1t/λn)n=1√with λn = −2π/ log n. Then exp(2πt −1/λn) has a period
λn. In this way, we may think ζ is a general type of Fourier– Typeset by FoilTEX –5

series:f (t) ∼∞∑√an exp(2π −1nt).

n=0

Motivated by Riemann zeta function, Bohr constructed atheory of generalized Fourier expansion, called almost periodicfunction.
A key observation is that such f which is convergent anduniformly bounded are characterized by almost periodicity:
For any ϵ > 0 and there exists L’s that |f (t)−f (t−L)| < ϵfor any t and such L appears with bounded gaps.
A point set X is almost periodic if for any ε > 0 the set of– Typeset by FoilTEX –6

almost period t, that satisﬁesdensX \ (X − t + B(0, ε)) < εis relatively dense in Rd. Here relatively dense means that tappears with bounded gaps.
This deﬁnition is in fact equivalent to pure point
diﬀractiveness of point sets. Thus we may use this as amathematical deﬁnition of quasicrystal.
Almost periodic Delone set is the mathematical model of
QC (Baake-Moody[2], Gouéré [4]).
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Cut and Project set (Model set) is generated by cut andproject scheme, i.e., the set generated by a projection of higher
dimensional lattice (LCA group) points stay in some irrationalband.

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A well known example of aperiodic tiles is due to Penrose
which consists of two kinds of tiles: kites and darts withmatching rules as in Figure 3:

Figure 3: Penrose Tile
– Typeset by FoilTEX –9that the circular markings must match across the boundary
like:– Typeset by FoilTEX –10

Figure 4: Penrose tiling– Typeset by FoilTEX –11

To show that Penrose tiles are aperiodic, we have to show
two things.
• They admit a tiling.
• Each tiling generated by kite and dart has no period.
de Bruijn [3] showed that Penrose tiling is really a goodmodel of quasi-crystal by showing that its reference point set
is understood by cut and projection. More precisely such
Delone set is the projection of 5-dim lattice points which lies insome irrational band. This assures that the diﬀraction patternof Penrose tiling is pure diﬀractive.
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Figure 5: Penrose Diﬀraction– Typeset by FoilTEX –13

How can we check that tiling gives a quasicrystal
structure ?
Short Answer: Use my Mathematica program to checkoverlap coincidence.
http://math.tsukuba.ac.jp/~akiyama/Coincidence/Coincidence.html– Typeset by FoilTEX –14

Self aﬃne tiling is made of prototiles A = {T1, . . . , Tm}
which satisﬁes a set equation ( substitution rule ):∪QTj =Ti + Dij
iwhere Q is an expanding matrix and Dij is a ﬁnite set oftranslations.
We use spectral theory of tiling dynamical system. Fromthese data Q and Dij we can check that this gives a quasicrystal
structure. Akiyama-J.-Y.Lee [1] gave a practical algorithm
on overlap coincidence and implemented into Mathematica
program.
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Summary

Quasi-crystal is explained better by an aperiodic tile set
which produces almost periodic points. For self-aﬃne tilings,?

• Bragg peak ↔ Non weak mixing ⇐⇒ X: Meyer set ⇐⇒ Q:
Pisot family• Pure pointed diﬀraction ⇐⇒ Pure discrete spectrum ⇐⇒Overlap Coincidence ⇐⇒ Cut and projection• Overlap coincidence can be conﬁrmed by Mathematica.
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References[1] S. Akiyama and J.-Y. Lee, Algorithm for determining purepointedness of self-aﬃne tilings, Adv. Math. 226 (2011),
no. 4, 2855–2883.
[2] M. Baake and R.V. Moody, Weighted dirac combs withpure point diﬀraction, J. Reine Angew. Math. 573 (2004),
61–94.
[3] N. G. de Bruijn, Algebraic theory of Penrose’s nonperiodic
tilings of the plane. I, II, Nederl. Akad. Wetensch. Indag.
Math. 43 (1981), no. 1, 39–52, 53–66.
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[4] J.-B. Gouéré, Quasicrystals and almost periodicity, Comm.
Math. Phys. 255 (2005), no. 3, 655–681.
[5] D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn,
Metallic phase with long-range orientational order and notranslational symmetry, Phys. Rev. Lett. 53 (1984), no. 20,1951–1953.

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