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On aperiodic tilings by the projection method (New Aspects of Analytic Number Theory)

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On aperiodic tilings by the projection method

高知大学・理学部 小松和志(Kazushi Komatsu)

Faculty of Science,

Kochi Univ. In

1982

quasi-crystals with icosahedral symmetry were discovered, (published in 1984). It had been

axiomatic

that the structure of a crystal was periodic, like awallpaper pattern. Periodicty is another

name

for translational symmetry. Icosahedral symmetry is incompatible with translational symmetry and therefore quasi-crystals

are

not peri-odic. Most famous

2-dimensional mathematical

model for aquasi-crystal is aPenrose tiling of the plane. In

1981

de Bruijn introduced projection methods to construct aperiodic tilings such as Penrose tilings.

We recall the definition oftilings by the projection method.

$L$ : alattice i$\mathrm{n}$

$\mathrm{R}^{d}$ with abasis $\{b:|i=1,2, \cdots,d\}$

.

$E$ : a$p$-dimensional subspaceof$\mathrm{R}^{d}$,

$E^{[perp]}:$ its orthogonal complement.

$\pi:\mathrm{R}^{d}arrow E$, $\pi^{[perp]}:$ $\mathrm{R}^{d}arrow E^{[perp]}:$ the orthogonal projections.

$A$ : aVoronoi cell of$L$

For any $x\in \mathrm{R}^{d}$ we put

$W_{x}=\pi^{[perp]}(x)+\pi^{[perp]}(A)=\{\pi^{[perp]}(x)+u|u\in\pi^{[perp]}(A)\}$

$\Lambda(x)=\pi((W_{x}\mathrm{x}E)\cap L)$

.

The Voronoi cell of apoint $v\in\Lambda(x)$

$V(v)=\{u\in \mathrm{R}^{n}||v-u|\leq|y -u|,\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}11y\in\Lambda(x)\}$

.

$\mathcal{V}(x)$ : the Voronoi tiling induced by $\Lambda(x)$, which consists ofthe Voronoi

cells of$\Lambda(x)$

.

For avertex $v$ in $V(x)$

$S(v)=\cup\{P\in \mathcal{V}(x)|v\in P\}$

.

Thetiling$T(x)$ given by the projection methodisdefined asthecollection

of tiles Conv $(S(v)\cap\Lambda(x))$, where Conv (B) denotes the convex hull of

aset $B$

.

Note that $\Lambda(x)$ is the set of the vertices of $T(x)$

.

数理解析研究所講究録 1274 巻 2002 年 174-176

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Inorderto statetheorems werecallseveraldefinitions. The dual lattice

$L^{*}$ is defined by the set of vectors $y\in \mathrm{R}^{d}$ s$\mathrm{u}\mathrm{c}\mathrm{h}$ that

$\langle y, x\rangle\in \mathrm{Z}$ for all

$x\in L$, where $\langle$ , $\rangle$ denotes standard inner product. Alattice $L$ is called

integral if all its vectors satisfy that $\langle x, y\rangle\in \mathrm{Z}$ for all $x$,$y\in L$

.

The

standard lattice is both integral and selfdual.

For $L=\mathrm{Z}^{d}$, C. Hillman characterized the number of periods of the

tilings. He also constructed periods for given tilings.

One

of Hillman’s results is extended to the case that $L$ is integral.

Theorem. Let $T(x)$ be the tiling by the projection method and assume

that $L$ is integ$ml$

.

Then, rank $\mathrm{K}\mathrm{e}\mathrm{r}(\pi^{[perp]}|L)$ is equal to the dimension

of

the linear space

of

the periods

of

$T(x)$

.

For the general lattices Theorem is not true. We have the following example;

$L$ : alattice in $\mathrm{R}^{2}$ with abasis

$\{(1, \sqrt{2}), (1, -1)\}$,

$E$ : the $x$-axis of R.

In this

case

it is easy to see that all tilings in $\mathrm{R}^{1}$ obtained by the

projection method are periodic and rank $\mathrm{K}\mathrm{e}\mathrm{r}(\pi^{[perp]}|L)=0$

.

The following property is analogous to classical uniformdistribution of sequences.

Theorem (de Bruijn and Senechal, 1995)

Assume that $\pi^{[perp]}(L)$ is dense in $E^{[perp]}$

.

$K_{1}$,$K_{2}:(d-p)$-dimensional cubes in $E^{[perp]}$

$J\subset E:ap$-dimensional cube centered at the origin.

For anypositive real number $\lambda$, we set

$P_{\lambda}^{1}=K_{1}\cross\lambda J$,$P_{\lambda}^{2}=K_{2}\cross\lambda J$

.

Then,

$\lim_{\lambdaarrow\infty}\frac{\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}P_{\lambda}^{1}\cap L}{\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}P_{\lambda}^{2}\cap L}=\frac{Vol(K_{1})}{Vol(K_{2})}$

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Atiling space $\mathcal{T}(E)$ is defined by aspace of tilings consisting of all

translates by $E=\mathrm{R}^{p}$ofthe tilings $T(x)$ for all $x\in E$

.

Tiling spaces are

topological dynamical systems, with

acontinuous

$\mathrm{R}^{p}$ translation action

and atopology defined by atiling

metric

on tilings of$\mathrm{R}^{\mathrm{p}}$

Let Orb(T(x)) denote theorbit of$T(x)$ in $\mathcal{T}(E)$ by the $\mathrm{R}^{p}$ translation

action and span(A) denote the $\mathrm{R}$-linear span of aset $A$

.

Uniform distribution of the

projection

method is closelyrelated to the ergodicity of the tiling space.

Theorem Let $\mathcal{T}(E)$ be the tiling by the projection method in tems

of

a $p$

-dimernsional

subspace $E$

of

$\mathrm{R}^{d}$ and $p’$ : $E^{[perp]}arrow \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(L^{*}\cap E^{[perp]})$

be the orthogonal prvyjection.

Define

$p$ : $Larrow \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(L^{*}\cap E^{[perp]})$ by $p=$ $p’\mathrm{o}(\pi^{[perp]}|L)$

.

We take a basis $x_{1},\ldots$ ,$x_{k}$

of

the direct summand $K$ such

that$L=p^{-1}(\{0\}\underline{)\oplus K.\mathfrak{M}en\mathcal{T}(E)decom}\mu ses$ into a$k$ parameter family

of

orbit closures $\mathrm{O}\mathrm{r}\mathrm{b}(T(t_{1}x_{1}+\cdots+t_{k}x_{k}))$

for

$t_{1}$,$\ldots,t_{k}\in \mathrm{R}$

.

In particular, we obtain that $k$ is equal to rank $(L^{*}\cap E^{[perp]})$

.

Note that $\pi^{[perp]}(L)$ is dense in $E^{[perp]}$ if and only if $E^{[perp]}\cap L^{*}=\{0\}$

.

A.

Hof(1988) proved that $E^{[perp]}\cap L^{*}=\{0\}$ if and only if$\mathcal{T}(E)=\mathrm{O}\mathrm{r}\mathrm{b}(T(0))$

.

Assume that $L$ is integral. Then we see that rank $(L^{*}\cap E^{[perp]})=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}$ $(L\cap$

$E^{[perp]})=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}$$\mathrm{K}\mathrm{e}\mathrm{r}(\pi|L)$ because $L\subset L^{*}$ and $L^{*}/L$ isfinite. The number of

independent periodsofthe tiling space$\mathcal{T}(E^{[perp]})$ is equaltorank $\mathrm{K}\mathrm{e}\mathrm{r}(\pi|L)$

.

We immediately obtain the following theorem in the casethat $L$ is inte

gral:

Theorem Let$\mathcal{T}(E)$ (resp. $\mathcal{T}(E^{[perp]})$) be the tiling space by the projection

methodin terms

of

a$p$-dimensionalsubspace $E$ (resp. $(d-p)$-dimensional

subspace $E^{[perp]}$)

of

$\mathrm{R}^{d}$ and

assume

that $L$ is an integral lattice. Then$\mathcal{T}(E)$

decomposes into a $k$ parameter family

of

orbit closures, where $k$ is equal

to the number

of

independent periods

of

the tiling space $\mathcal{T}(E^{[perp]})$

.

参照

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