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Sheel structure ofdodecagonal quasicrystalsassociated with rootsystem F4 and cyclotomicfield Q(ζ12)Jun Morita a & Kuniko Sakamoto aa Institute of Mathematics , University oTsukuba , Tsukuba, Ibaraki, 305-8571, JapanPublished online: 03 Nov 2008.

To cite this article: Jun Morita & Kuniko Sakamoto (2000) Sheel structureof dodecagonal quasicrystals associated with root system F4 andcyclotomic field Q(ζ12), Communications in Algebra, 28:1, 255-263, DOI:10.1080/00927870008826841

To link to this article: http://dx.doi.org/10.1080/00927870008826841

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COMMUNICATIONS IN ALGEBRA, 28(1), 255-263 (2000)

SHELL STRUCTURE OF DODECAGONALQUASICRYSTALS ASSOCIATED WITH ROOTSYSTEM F4 AND CYCLOTOMIC FIELD Q«(12)

Jun Morita and Kuniko Sakamoto

Institute of Mathematics, University of Tsukuba,Tsukuba, Ibaraki, 305-8571, Japan

Abstract

Using a Coxeter transformation of type F 4, we will construct qua-sicrystals of dimension 2 with twelvefold symmetry. Then, such qua-sicrystals will be realized as subsets of a certain cyclotomic field. Andwe will study a typical shell structure algebraically. In particular, we cangive its mathematical meaning in terms of elementary number theory.

1 Dodecagonal quasicrystals

Root system ~ of type F4 can be constructed in a four dimensional Euclideanspace V with an inner product (.,.) as follows:

255

Copyright @ 2000 by Marcel Dekker, Inc. www.dekker.com

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256 MORITA AND SAKAMOTO

~ = { XCi, XCj X ck, ~(XCI X C2 X c3 X C4) I 1 ~ i ~4, 1 ~ j < k ~ 4 },

where { Cl, C2, C3, C4 } is an orthonormal basis of V and we need all possiblesigns (cf. [3]' [8]). Hence, the cardinality of ~ is 48. Then, we can chooseIT = { al, a2, a3, a4 } as simple roots of ~ with

1al = C2 - C3, a2 = C3 - C4, a3 = C4, a4 = -(cl - C2 - C3 - c4).

2

Its root lattice is Q = EBt=IZai. Let W be the Weyl group of~, which is gener-ated by simple reflections rl, r2, r3, r4, where ri is the involutive automorphismof V defined by

( )(ai, v)

ri v =v-2---ai(ai,ai)

for all v E V. They preserve the inner product of V. We put w = rlr3r2r4,which is called a Coxeter transformation.

Let T = w + w-r, called the fundamental inflation map associated with(V,Q,~,w). Then, V is decomposed into the direct sum of a couple of T-eigenspaces. Namely, if we put

and we set

VI al + (1 + J3)a3' V2V~ = al + (1 - J3)a3, V~

a2"- (1 - J3)a4,a2 - (1 + J3)a4,

VJ3 = lRVl EB lRV2, V-v'3 = lRv~ E9lRv~,

then Vv'3 and V-v'3 are T-eigenspaces with eigenvalues J3 and -J3 respec-tively, and we have an orthogonal direct sum decomposition:

with orthogonal projections

Then, for r > 0, we define a quasicrystal

This is invariant under the action of H = (R1, R2 ) with R1 = rlr3 and R2 =r2r4. Hence, EQ has dodecagonal symmetry. For quasicrystals, quasiperiodicpatterns and related structures, we refer [1]' [4 - 7]' [9 - 14]' [16 - 18]' [20] andso on.

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DODECAGONAL QUASICRYSTALS

2 Realization of quasicrystals

257

We put lE= Q( vI3) and JF = lE(i) with i=H. To understand the subspacesV:i:v'3' we introduce a relation with the complex plane. For x, y E C, we definea new product x . y by

1x . y = 2 { xy + xy },

where x is the complex conjugate of x. Then x. x = xx = I X 12. This product

is nothing but the standard inner product if we identify C with ]R2 as lR-vectorspaces by

x = a + b i E C +----+ (a, b) E lR2,

because (a + b i). (e + d i) = ae+ bd. We define an lR-linear map 11"11 of V ontoC by

11"11(Q:I) = -I~v'3 - I+2v'3 i, 11"11 (Q:2) = (1+ vI3) i,( ) 1 2+v'3. ( ) •11"11 Q:3 = 2 - -2- 1, 11"11 Q:4 = 1.

Then, the kernel Ker 11"11 contains v~, v~, hence Ker 7r1l = V-v'3' Therefore, themap 11"11 takes Vv'3 onto C isomorphically. And we see that

for all v E V. Hence, using this scaling of two inner products, we can totallyidentify Vv'3 with C. That is, 11"11 induces an isometry, called 1rlb of Vv'3 onto Cup to scaling. The restrictions of 11"11 and 1r1l to VlE = lE@ Q and V1= PII(VlE

)

are also denoted by the same symbols 11"11 and 1r1l respectively. Similarly wedefine an lR-linear map 11"1- of V onto C satisfying

11"1- (Q:2) = (1 - vI3) i,1I"1-(Q:4) = i.

Then, the kernel Ker 11"1- contains VI, v2, hence Ker 11"1- = Vv'3' Therefore, themap 11"1- takes V-v'3 onto C isomorphic ally. And we see that

for all V E V. Hence, using this scaling of two inner products, we can totallyidentify V-v'3 with C. That is, 11".1. induces an isometry} called 1r.1., of V-v'3onto C up to scaling. The restrictions of 1I".l and 1r.1. to VlE = lE@ Q andV~v'3 = P.1. (VF-:) are also denoted by the same symbols 7r .1. and 1r1- respectively.Then we obtain the following commutative diagram.

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258 MORITA AND SAKAMOTO

vlE

PII /

V~ .!- 1r1l

1r1l "\t

IF

vlE

"\t P.l

1r.l.. -!- V!yi3

/ 1r.l..

IF f---t

8

Here, IF == IE(i) == Q(() with ( == (12 == '1+ ~i, Gal(lFjQ(i)) == ( 8 ) and8(IE) == IE, and 8 is the endomorphism of VlE defined by 8(a ~ x) == 8(a) ~ x forall a E IE and x E Q. Hence, in particular, 8(() == (5.

For each r > 0, we define the realization ~r of EQ as

Equivalently, we can formulate as

Er == { x E DIF I 8(x). 8(x) < r2 }.

Quasicrystals Er and ES are called isomorphic if there exists a IE-linear map</J : DIF ~ DIF so that </J(Er) == ES. We note that 1r1l(Q) == DIF, the ringof integers of IF. Then, the following facts can be obtained using the samemethod as in [10]. Here, we put T == 2+ \1'3, and we use I2(m) as the dihedralgroup of order 2m.

Proposition Let </J : DIF ~ DF be a IE-linear map and suppose that cP(~r) ==~s for some r, s > o. Then, cP is 1E[\I'3]-linear and

(1) If r == s, then cP E H ~ 12(12).(2) If r -I- s, then (s/r)2 == Tm for some m E Z.(3) Ifr -I- sand (s/r)2 == Tm with m == 2k, then cP E T-kH.(4) Define an DF-linear isomorphism cPl : DIF ~ DIF by cP1(X) == x/(1 + ().

If rVT == s, then cPl(Er) == ES.(5) If r -I- sand (s/r)2 == Tm with m == 2k + 1, then </J E T-k-!(H' \ H),

where H' == ( H, yficPl ) ~ 12(24).Conversely, if r, s > 0 and cP : DF ~ DIF is a IE-linear mapping satisfying (1),(3) or (5), then cP(Er) == ES

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DODECAGONAL QUASICRYSTALS

3 Shell structure

259

To study the shell structure of our quasicrystals, we will use the same idea asin [10] and [11]. (Also see [12].) For each N == 0,1,2, ... , we define

QN == { x E Q I (x,x) == N }.

Then the F 4 root lattice Q decomposes into a set of concentric shells QN. Wenow introduce a shelling on each quasicrystal ~r. We put Df,N == 'lr1l(QN) and

~~ == ~r nDf,N == { x E Df,N I 1c5(x)1 < r }.

Let x == 2::=1 aiai E QN. Then

(x, x) == 2ai + 2a~ + a~ + a~ - 2a1a2 - 2a2a3 - a3a4 == N

and

(2ai + 4a~ + 2a5 + a~ - 4ala2 + 2ala3 - ala4 - 5a2a3 + 2a2a4 - 2a3a4)+(2a~ + a5 - 2ala2 + 2ala3- ala4 - 3a2a3 + 2a2a4 - a3a4)v'3

(2ai + 2a~ + a5 + a~ - 2ala2 - 2a2a3 - a3a4)+(2a~ + a~ - 2ala2 + 2ala3 - ala4 - 3a2a3 + 2a2a4 - a3a4)(1 + v'3)

N + (1 + v'3)m,

where m == 2a~ + a~ - 2ala2 + 2ala3 - ala4 - 3a2a3 + 2a2a4 - a3a4. Hence, foreach x E .oF, we see that x E EN if and only if

x. x == N + m(l + v'3),o ::; 8(x) . 8(x) == N +m(l - v'3) < r2

for some m E Z, which implies

_1_+_v'3_3N > m > _(1_+_v'3_3_)_(N_-_r2_) •2 - 2

In particular, it is an interesting situation when 1+2Y3N == (1+Y3)JN-r2) + 1.

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260

Such r will be denoted by p. That is,

p=VV3-1.

MOroTAANDSAKAMOTO

Then

In this case, we obtain

~~ = { x E DIF I x. x = N + [1 +2..13 N] (1 + ..13) }

for N == 0, 1,2, ... , where [ ] is the Gauss symbol.Now we consider the inflation

X~TX

which maps ~P bijectively onto ~p/T contained in EP. Then we obtain

x E ~~ =:} x. x=:} TX. TX

and hence,

N + [1+2i3N](1 + V3)(2 + V3)2(N + [1+2v'3N](1 + V3))(7 + 4V3)(N + [1+2v'3N](1 + V3))7N + 7[1+2v'3N](1+ V3) + 4V3N

+4[1+2Y3N](3 + V3)

(3N +8[¥N])+(4N + 11[¥N])(1 + V3)

For each N, we define a sequence { Ln(N) }~=l by

N,

Then, we can see, for all odd n,

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DODECAGONAL QUASICRYSTALS

Ln+1 (N) == [1+2V3Ln (N)].

Now we suppose x E Ein+2(N) with n odd. Then

x . x == Ln+2(N) + Ln+3(N)(1 + V3)and

261

ax . ax (7 - 4V3)(Ln+2(N) + Ln+3(N)(1 + V3)(3Ln+2(N) - 8Ln+3(N) + (11Ln+3(N) - 4Ln+2(N»(1 + V3)Ln(N) + Ln+1(N)(1 + V3).

Hence, ax E Ein(N)' and so x == r(ax) E rEin(N). Therefore,

rEim(N) == Ein+2(N).We define

SN == card E~

for N == 0,1,2, . ' .. Then we see

SN == S3N+8[¥N].

We also observe that each shell E~ has complete H-symmetry. That is, (E~ ==E~. Hence, 121sN for all N == 1,2,3, . ". Furthermore, using a field extensionIFIE (cf. [19]), it is easily seen that SN has the following formula:

SN == 12II (vp(N*) + 1)PEX

if vp(N*) == 0 (mod 2) for all P E Z and N > 0, where

X == { l' E Spec DE I 3 P E Spec DF, pDF == PP, P =I P },y == { l' E Spec DE I 3 P E Spec DF, pDF == p2, P == P },z == { l' E Spec DE I pDF E Spec DF },

and Spec is the spectrum of prime ideals, vp is the discrete valuation of DlE,

the ring of integers of E, with respect to 1', and N* == N + [1+2i3N](1 + v'3).If vp(N*) ~ 0 (mod 2) for some l' E Z, then SN = O.More precisely, there is a twelve to one map 1] of the set E~ onto the set of

ideals I c .oF such that II = N*DF given by 1](x) = I = xDF. In fact, this1J is well-defined. And suppose 1J(x) = 1J(Y) for x,y E E~. Then, xDF = yDF

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262 MORITA AND SAKAMOTO

implies x == yu for some unit u E D]F. And also, we see uu == 1. Hence, u == (ifor some i. Therefore, for each x, there are exactly twelve possibilities of suchy. We need to show the surjectivity of TJ. Let I be an ideal of .oF such thatI I == N*D]F. Since DlFis a principal ideal domain, we can write I == zD]F forsome z E D]F. Then zzD]F == N*D]F, and zzt == N* for some unit t E D]F. Sinceboth zz and N* lie in lE, we see t E D]Fn:IE == DlE• This implies that t is a unitof .olE. Hence, t == :f:rm for some m E Z. If m == 2k, then we put x == zrk• Ifm == 2k + 1, then we put x == zrk(1 + (). In any case, we can obtain

TJ(X) == xD]F == zD]F== I.

Therefore, the map TJ is surjective and a twelve to one correspondence. Thatis,

SN == 12. Card { I c .o]F I I == ideal, II == N*D]F }.

If N*D == pel ... pekp' h ... 1" !tp"91 ... 1''' gm with p. E X h~ E Y and h~I E ZlE 1 k 1 lIm l, t', t'l'

thenN*D - (P P-)el (D is )elc p,2!I D/2!tp"91 P" gm]F - 1 1 . •• .rk.rk 1 ••• .rl 1 • •• m ,

where PiDlF == ~Pi, P~DlF== p:2 and p~'DlF== PI' with ~, PI, PI' E Spec(DlF).Hence, we obtain the above formula for SN.

REFERENCES[1] Berman, S., and Moody, R.V., The algebraic theory of quasicrystals with five-fold

symmetries, J. Phys. A: Math. Gen. 27 (1994), 115 - 129.

[2] Bertin, M.J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M. andSchreiber, J.P., Pisot numbers and Salem numbers, Birkhouser- Verlag, Berlin, 1992.

[3] Bourbaki, N., Groupes et algebres de Lie - Chap. 4,5,6 - , Hermann, Paris, 1968.

[4] Baake, M., Joseph, D., Kramer, P. and Schlottmann, M., Root lattices and quasicrys-tals, J. Phys. A: Math. Gen. 23 (1990), L1037 - L1041.

[5] Chen, H., Li, D.X. and Kuo, K.H., New type of two..dimensional quasicrystal withtwelvefold rotational symmetry, Phys. Rev. Lett. 60 (1988), 1645 - 1648.

[6J Elser, V. and Sloane, N.J.A., A highly symmetric four-dimensional quasicrystal, J.Phys. A: Math. Gen. 20 (1987), 6161 - 6168.

[7] Goldman, A.I. et al., Quasicrystalline materials, American Scientist 84 (1996), 230 -241.

[8] Humphreys, J.E., Reflection groups and Coxeter groups, Cambridge University Press,Cambridge, 1990.

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DODECAGONAL QUASICRYSTALS 263

[9] Janot, C., Quasicrystals - A primer Second Edition - , Oxford Science Publication,Oxford, 1994 (1st published 1992).

(10] Moody, R.V. and Patera, J., Quasicrystals and icosians, J. Phys. A: Math. Gen. 26(1993), 2829 - 2853.

[11] Moody, R.V. and Weiss, A, On shelling Es quasicrystals, J. Number Theory 47 (1994),405 - 412.

[12] Morita, J. and Sakamoto, K., Octagonal quasicrystals and a formula for shelling, J.Phys. A: Math. Gen., to appear.

[13] Niizeki, K., Self-similarity of quasicrystals in two dimensions: III. Inflation by a non-unit PV number, J. Phys. A: Math. Gen. 22 (1989), 1859 - 1869.

[14] Penrose, R., The role of aesthetics in pure and applied mathematical research, Bull.Inst. Math. Appl. 10 (1974), 266 - 271.

[15] Salem, R., 1945 Power series with integral coefficients, Duke Math. J. 12 (1945), 153- 172.

[16] Shechtman, D., Blech, 1., Gratias, D. and Cahn, J.W., Metallic phase with long-rangeorientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984), 1951 -1953.

(17] Suzuki, T. and Dotera, T., Dynamical systems for quasiperiodic chains and new self-similar polynomials, J. Phys. A:Math. Gen. 26 (1993), 6101 - 6113.

[18] Tsai, A.P., Inoue, A. and Matsumoto, T., A stable quasicrystal in AI-Cu-Fe system,Japanese J. Ap-pl. Phys. (9) 26 (1987), L1505 - L1507.

[19] Washington, L.C., Introduction to cyclotomic fields 2nd ed, Graduate Texts in Math-ematics 83, Springer-Verlag, New York, 1997 (1st published 1982).

[20] Yamamoto, A., Crystallography of quasiperiodic crystals, Acta Cryst. A52 (1996), 509- 560.

Received: May 1998

Revised: November 1998 and June 1999Dow

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