Wulff Shape for Nonperiodic Arrangements

Letters in Mathematical Physics45: 81–94, 1998.© 1998Kluwer Academic Publishers. Printed in the Netherlands.

81

Wulff Shape for Nonperiodic Arrangements

KÁROLY BÖRÖCZKY, JR.1 and UWE SCHNELL21Mathematical Institute of the Hungarian Academy of Sciences, H-1364 Budapest Pf. 127,Hungary. e-mail: [email protected] Institut, Universität Siegen, D-57068 Siegen, Germany.e-mail: [email protected]

(Received: 17 February 1998)

Abstract. We define an analogue of the Gibbs–Curie energy for quasi-crystals. We show that thereexists a Wulff-type shape optimizing this energy, which is always a convex polytope. This way wecan model the triacontahedron and the dodecahedron as shapes of quasi-crystals, according to reality.

We also provide a shorter proof for the well-known formula for the number of points of a non-periodic set in a large convex domain, which also yields a nontrivial error term in the cases connectedto real quasi-crystals.

Mathematics Subject Classifications (1991):82D25 (11K38, 52B60, 52C17).

Key words: quasicrystals, Wulff shape, discrepancy, packing.

1. Introduction and Results

It is well known that the underlying structure of quasi-crystals can be modelledby considering the points of a certain higher-dimensional lattice3 filtered bya windowW , and projecting them into an irrational 3-plane5. The filtering isnecessary because the projection of the whole lattice is dense in5. Usually, thewindow is a convex, bounded, open setW ⊂ 5⊥, and the lattice points consideredare the ones which project intoW .

Similarly, as for lattices, one can consider the density of a nonperiodic point set.Theorem 1 is well known, but in Section 4 we provide a rather short proof usinguniform distribution arguments, and can even estimate the error of the approxima-tion. For any convex and bounded setC, we denote by|C| its relative volume in itsaffine hull. Occasionally we useV (·) for the full dimensional volume.

THEOREM 1. For any convex bodyK in 5,

limr→∞

#(3 ∩ rK)|rK| = |W |

detL= δ.

In the case of real quasi-crystals,3 is eitherZ6, or the face centered hypercubiclatticeD6, or the body centered latticeB6. Note thatD6 is the family of integer

82 KAROLY BÖRÖCZKY, JR. AND UWE SCHNELL

vectors whose coordinates sum up to an even number, andB6 is the family ofinteger vectors whose coordinates are either all even or all odd (see, e.g., [14]).Each of them are invariant under the natural action of the icosahedral groupG6.For these lattices

#(3 ∩ rK)|rK| = δ +O

(√log r

r112

)holds. Note that in most applications eitherZ6 orD6 are used.

The two maximal invariant subspaces of the icosahedral group are chosen to be5 and its dual space5⊥; they are the image and the kernel, respectively, of the6× 6-matrix

√5 −1 −1 −1 −1 −1

−1√

5 −1 1 1 −1

−1 −1√

5 −1 1 1

−1 1 −1√

5 −1 1

−1 1 1 −1√

5 −1

−1 −1 1 1 −1√

5

(see, e.g., [22]).5 has an orthogonal basis

{w1 = (τ,0,0,1, τ,1)t , w2 = (1, τ, τ,0,−1,0)t ,

w3 = (0,1,−1,−τ,0, τ )t }and5⊥ has an orthogonal basis

{w4 = (1,0,0,−τ,1,−τ)t , w5 = (−τ,1,1,0, τ,0)t ,w6 = (0,−τ, τ,−1,0,1)t },

whereτ = 1/2(√

5+ 1) is the golden ratio.In this paper the windowW is the interior of the projection of the unit cube

(0,1)6 into5⊥.W is a rhombic triacontahedron with edge length 1/√

2. Its volumeis V (W) = 2

√2 sin(π/5)(τ + 1) = 0.1919. . .. Finally, 3 is the projection of

(W +5) ∩ L into5.In Section 2 we prove that the set of normalsu ∈ 5, for which there are planes

orthogonal tou with positive density of3-points is given by

G = {(R + τRH) · x: x ∈ Z6 primitive} (1)

whereR andH are certain matrices (cf. Section 2).

WULFF SHAPE FOR NONPERIODIC ARRANGEMENTS 83

Foru ∈ G there is a corresponding vectoru′ ∈ 5⊥, such that lin(u, u′)∩L is arational plane, and we define the four-dimensional latticeL4(u) = L∩ lin(u, u′)⊥.We prove that the maximal density of3-points in a plane with normalu ∈ G is

δ2(u) = q(u)

detL4(u)(2)

whereq(u) is given by the 2-volume of the intersection of a corresponding planeE2(u

′) in5⊥ with the windowW (cf. Section 2).A polytopeQ in 5 is called aG-polytope if each facet has a normal inG.

Because of the distribution of the linear densities, it is natural to introduce theGibbs–Curie surface energy forG-polytopes in analogy to periodic crystals (cf. [6,13]).

DEFINITION 1. LetQ be aG-polygon in5 such that the facets ofQ are orthog-onal tou1, . . . , uk ∈ G with 2-volumesf1, . . . , fk. Then the Gibbs–Curie surfaceenergy with parameter% is defined as

E(Q, %) =k∑i=1

(% − δ2(ui)

2δ

)fi.

Remark. E(Q, %) was already established for the classical two-dimensionalPenrose tiling in [4]. It was obtained as a limit of finite packing densities duringthe inflation process. The inflation process for quasicrystals is corresponding to theinteger dilatation in the lattice and the periodic case. In fact the energy has the samestructure as the density deviation, which was introduced in [19, 25, 26] for periodiccrystals.

Now we want to minimizeE(R, %) for fixed%. The candidate for this minimumis the Wulff shape

W(%) ={z ∈ 5:

∣∣∣∣z u

‖u‖∣∣∣∣ 6 % − δ2(u)

2δ, for all uinG

}, (3)

which has nonempty interior for

% > %∗ = sup

{δ2(u)

2δ:u ∈ G

}.

THEOREM 2. If % > %∗ thenW(%) is a polytope.

Note that the condition% > %∗ can be interpreted in the language of theparametric density (see [1, 19, 25, 26]); namely, clusters are%-denser than two-dimensional arrangements.


The first geometric proof of the following theorem appeared in [6]. For the sakeof completeness, in Section 5 we present a short proof based on a classical theoremof Minkowski.

THEOREM 3. Let % > %∗ and letQ be aG-polytope with volumeV (R) =V (W(%)). Then

E(Q, %) > E(W(%), %),

with equality only ifQ is a translate ofW(%).

Finally we prove that for a certain range of% the Wulff shape is a rhombictriacontahedron ifL = Z6 and a dodecahedron ifL = D6.

THEOREM 4. (a).If %∗ < % < 2%∗ and L = Z6 thenW(%) is a rhom-bic triacontahedron.(b) If %∗ < % < 1.21%∗ and L = D6 thenW(%) is adodecahedron.

We remark that both, the rhombic triacontahedron and the dodecahedron, appearas shapes of single grains of certain quasicrystals (AlLiCu and AlFeCu, see [11],p. 62, 179 and p. 86, 198).

2. The Planar Densities

In this section we consider the density of3-points of planes in5. Besides lattices,we also consider translates of lattices, the so-called grids. By a basis of a grid wemean a basis of the original lattice.

Let p ∈ 3 andE = {z ∈ 5:uz = 0} a plane in5 orthogonal tou 6= 0. ByEp = Ep(u) we denote the planep+E. Choose a basis{a1, . . . , ak} of L(p, u) =(Ep +5⊥) ∩ L. If the projection ofL(p, u) has a positive density inEp then twoof theai , saya1 anda2 project to linearly independent vectorsa1 anda2. Since5has a basisw1, w2, w3 using coordinates from the fieldQ[τ ], the same holds fora1

anda2. By construction,u is orthogonal toa1 anda2 and, hence, we may chooseuso that its coordinates lie also inQ[τ ]. Carrying out the calculations above showsthatu = Rx + τRHx for certainx ∈ Z6 whereR andH are the matrices

R =

0 1 1 0 0 0

0 0 0 1 1 0

0 0 0 1 −1 0

1 0 0 0 0 −1

0 1 −1 0 0 0

1 0 0 0 0 1

and H =

0 1 0 0 0 0

1 1 0 0 0 0

0 0 0 1 0 0

0 0 1 1 0 0

0 0 0 0 0 1

0 0 0 0 1 1

.


Observe thatH has the propertyH 2 = H + I .On the other hand, letu = a + τb ∈ 5, with a = Rx, b = RHx andx ∈ Z6.

Note thatA = RH−1R−1 is the symmetric matrix

A = 1

2

−1 1 1 1 1 1

1 −1 1 −1 −1 1

1 1 −1 1 −1 −1

1 −1 1 −1 1 −1

1 −1 −1 1 −1 1

1 1 −1 1 1 −1

and satisfies the propertiesAt = A andA2+A = I .

Foru ∈ G, we define

u′ = b− τa = (RH − τR)x. (4)

Since(RH − τR)t(R + τRH) = 0 and5 = Im(R + τRH) it follows thatu′ ∈ 5.

Now considerE4(u) = (lin(a, b))⊥ andL4(u) = E4(u) ∩ L. The orthogonalprojection ofL4(u) onto5 is contained in the 2-planeE2(u) orthogonal tou, andthe orthogonal projection ofL4(u) onto5⊥ is contained in the 2-planeE2(u

′)orthogonal tou′. Hence, the set of3-points inE2(u) come from the lattice pointsin L4(u) whose projections into5⊥ are contained inW ∩ E2(u

′). We deduce byTheorem 1 that

δ2(E2(u)) = |W ∩ E2(u′)|

detL4(u).

Foru ∈ G let q(u) be the maximal area of an intersection ofW with a 2-planeorthogonal tou′. Denote the projection of somev ∈ L into5 (5⊥) by p (p′), andhence the density ofEp(u) = p + E2(u) is analogously

δ2(Ep(u)) = |W ∩ (p′ + E2(u

′))|detL4(u)

.

If v = αa + βb, then

p′ = β − τατ + 2

u′.

Choosingβ andα so thatβ−τα tends to zero, the density insideEp(u) approachesits supremum among the parallel planes. Therefore the supremumδ2(u) of thesedensities is

δ2(u) = q(u)

detL4(u). (5)


We remark that

detL4(u) = detL · detL2(u),

whereL2(u) = L∗ ∩ lin(a, b) andL∗ is the dual lattice toL (cf. [18]).Next we consider some examples for the planar densities. For this we observe

that the projection of the unit cube into5 is a rhombic icosahedron, which has12 (acute) vertices with valence 5, forming together an icosohedron and 20 ver-tices with valence 3, forming together a dodecahedron. Since these directions arethe facet normals either of the dodecahedron or the icosahedron, we call thesedirections in5 the dodecahedral directionsZD and the icosahedral directionsZI .Further we denote byZT the set of 30 (triacontahedral) directions, which are or-thogonal to the faces of the triacontahedron resulting from projecting(0,1)6 into5. It is easy to check thatZD,ZI , ZT ⊂ G.

EXAMPLES (1) Foru ∈ ZT the vectorsa andb′ = b (or b′ = b − a) are of theform a = 2(±ei,±ej ) andb′ = 2(±ek,±el) with ab′ = 0. The determinant ofL4(u) is 2 if L = Z6 and detL4(u) = 4 if L = D6. The correspondingu′ ∈ 5⊥is also a triacontahedral direction and the maximal intersection is a certain octagonwith areaq(u) = 1/5(6τ + 7). Hence, the densities with respect toZ6 andD6 are

δ2(u) = 1/10(6τ + 7) = 1.67082. . . δ′2(u) = 1/20(6τ + 7) = 0.83541. . . .

(2) Foru ∈ ZD the vectorsa andb are of the forma = 2ei andb = (±1, . . .,±1). The determinant ofL4(u) is

√5 in both casesL = Z6 andL = D6. The cor-

respondingu′ ∈ 5⊥ is also an dodecahedral direction and the maximal intersectionis a regular decagon with areaq(u) = 2 sin(π/5)(τ +1). Hence, the densities withrespect toZ6 andD6 are

δ2(u) = δ′2(u) = 2/5 sin(π/5)(3τ + 1) = 1.37638. . . .

(3) For u ∈ ZI the vectorsa andb are both of the form(±1, . . . ,±1) withab = 0. The determinant ofL4(u) is 3 in both casesL = Z6 andL = D6. The cor-respondingu′ ∈ 5⊥ is also an icosahedral direction and the maximal intersectionis a twelvegon (convex hull of two regular hexagons) with areaq(u) = 3

√3(τ−1).

Hence the densities with respect toZ6 andD6 are

δ2(u) = δ′2(u) =√

3(τ − 1) = 1.070466. . . .

3. The Wulff Shape

Proof of Theorem2. SinceW(%) is compact it suffices to prove that%z 6∈ W(%),for all z ∈ 5 with ‖z‖ = 1.


We considerz as a three-dimensional vector in the basis{w1, w2, w3}. All in-teger linear combinations of this basis are contained inG. By Lemma 2 in [26],obtained by simultanous Diophantine approximation, there is a sequenceun ∈ G,a sequenceαn, and a constantc > 0 such that

‖z− αnun‖ < c

‖un‖3/2 .

If ϕn is the angle betweenz andun this means that sinϕn < c/(‖un‖3/2) and itfollows

(%z)un

‖un‖ = % cosϕn > %(1− sin2ϕn) > % − %c2

‖un‖3 .

If un = an+ τbn, then there are constantsc1, c2 with detL4(u) < c1‖an‖ · ‖bn‖ <c2‖un‖2. Hence

(%z)un

‖un‖ > % − 1

detL4(u)

%c2c2

‖un‖and the assertion follows from‖un‖ →∞. 2

The latticeL2(u) and soL4(u) can be produced by differentx ∈ Z6 in thedefinition ofG.

LEMMA 1. (a) L2(u) is given by one of its elements: Ify ∈ L2(u)\{0} thenL2(u) = L ∩ lin(y,Ay). (b) If L2(u) ∩ L2(v) 6= {0}, for u, v ∈ G, thenu andvare parallel.

Proof. (a) SinceA has only irrational eigenvalues, it follows thaty andAy arelinear independent. Ifu = Rx + τRHx, theny = αRx + βRHx ∈ lin(a, b). Itfollows

Ay = αRH−1x + βRx = αR(Hx − x)+ βRx = (β − α)Rx + αRHx,and so lin(y,Ay) = lin(a, b).

(b) Let u = (R + τRH)x andv = (R + τRH)y. From (a) it follows thatL2(u) = L2(v) and so there areα, β ∈ R with Ry = αRx + βRHx. It followsy = αx + βHx andHy = αHx + βHx + βx = (α + β)Hx + βx, and hence

y +Hy = (α + τβ)x + (β + τ(α + β))Hx = (α + β)(x + τHx).Therefore,v = (α + τβ) · u. 2

To prove Theorem 4 we first need the following upper bound for the planardensities.


LEMMA 2. If u ∈ G\(ZD ∪ ZI ∪ ZT ) then

δ′2(u) 6 δ2(u) 6 π/8(τ + 1) = 1.02809. . . .

Proof. It suffices to considerL = Z6. If D1(L2(u)) is the length of the shortestnonzero vector inL2(u), then we distinguish five cases:

(1) D1(L2(u)) = 1,(2) D1(L2(u)) =

√2,

(3) D1(L2(u)) =√

3,(4) D1(L2(u)) = 2,(5) D1(L2(u)) >

√5.

If (1) holds thenL2(u) contains a unit vectorei. The unit vectors are containedin the lattices of the 12 dodecahedral directions. If (2) holds thenL2(u) contains az = ei ± ej . These 60 vectors are contained in the lattices of the 30 triacontahedraldirections. By Lemma 1,u is parallel to a direction inZD in case (1), and to adirection inZT in case (2).

If (3) holds thenL2(u) contains az = ei ± ej ± ek. 40 of these vectors producelattices with determinant

√41 and the other 40 are contained in the lattices of the

20 icosahedral directions. In the latter case, Lemma 1 yields thatu is parallel to adirection inZI .

In case (4) holds thenL2(u) contains az = ei±ej±ek±el. All of these vectorsproduce lattices with detL2(u) > 4.

It is easy to see that detL2(u) >√

3/2D1(L2(u))2 and so we have detL2(u) >

4 in case (5). It follows foru 6∈ (ZD ∪ ZI ∪ ZT ) that detL4(u) = detL2(u) > 2.Since the window has a circumradius ofτ/

√2, the assertion follows. 2

Proof of Theorem 4. (a) There are 30 directions inG with the maximal densityδ2 = 2δ%∗, namely the triacontahedral ones. The corresponding conditions obvi-ously lead to a rhombic triacontahedron with vertices of the form(%−%∗)/τ(τ,1,0)tor (% − %∗)/τ(τ,0, τ − 1)t or (%− %∗)/τ(1,1,1)t .

It suffices to prove that these vertices satisfy the conditions for all other direc-tionsu ∈ G which have densityδ2 6 α2δ%∗, with α = 0.824.

If we write the vertices as(%−%∗)e, then it suffices to prove that(%−%∗)‖e‖ 6% − α%∗. This is the case for% 6 (‖e‖ − α)/(‖e‖ − 1) · %∗. The assertion followsfrom√

3/τ 6 ‖e‖ 6 1.176.(b) There are 12 directions inG with the maximal densityδ2 = 2δ%∗, namely

the dodecahedral ones. The corresponding conditions obviously lead to a dodeca-hedron with vertices of the form

% − %∗τ

√τ + 2(τ,0, τ − 1)t or

% − %∗τ

√τ + 2(1,1,1)t .

It suffices to prove that these vertices satisfy the conditions for all other direc-tionsu ∈ G which have densityδ2 6 α2δ%∗, with α = 0.778.


If we write the vertices as(%−%∗)e, then it suffices to prove that(%−%∗)‖e‖ 6% − α%∗. This is the case for% 6 (‖e‖ − α)/(‖e‖ − 1) · %∗. The assertion followsfrom ‖e‖ = √3

√τ + 2/τ > 2.036. . .. 2

4. Projected Points in Large Bodies

LEMMA 3. LetL be a lattice inRd+k, and5 be a lineard-space which is notcontained in any lattice(d + k − 1)-plane. Choose a convex bodyW in 5⊥, anddenote by3 the orthogonal projection onto5 of the points ofL whose projectionsonto5⊥ are contained inW . Then for any convex bodyK in 5 and large r,asymptotically

#(rK ∩3) = |W |detL

· |K| · rd + o(rd).

Proof. The main idea is to translate the original problem into the realm ofuniform distribution, and then to estimate the discrepancy of a certain point setwith the help of the multi-dimensional version of the classical estimate of Erdosand Turán.

We are looking for the behavior of #((rK +W) ∩ L). If M is ak-dimensionalplane withM ∩5 = {0} andW ′ is the projection ofW ontoM parallel to5 thenwe have #((rK + W) ∩ L) = #((rK + W ′) ∩ L) + O(rd−1). It follows that theproblem is affine invariant and in the following we assume thatL = Zd+k. Indexthe orthonormal basese1, . . . , ed+k of Zd+k so that5⊥ ∩ lin{e1, . . . , ed} = {0}.

WithM = lin{ed+1, . . . , ed+k} andW ′ as above we have to determine #((rK +W ′) ∩ L).

There exists a basisv1, . . . , vd of 5 such that

vl = el +k∑j=1

αljed+j , l = 1, . . . , d. (6)

Since5 is not contained in any rational(d+k−1)-plane, a nonzero integer vectorcan not be orthogonal to eachv1, . . . , vd . Thus for any(u1, . . . , uk) ∈ Zk\{0} thereexists an 16 l 6 d such that

k∑j=1

αljuj 6∈ Q. (7)

We cut the orthogonal projectionK ′ of K ontoRd into pieces by the coordinatehyperplanes inRd. Further, we cutW ′ into finitely many pieces modZk. It sufficesto prove the statement for a pair of such pieces and by rescalingK we may assumethatK ′ ⊂ [0,1)d andW ′ ⊂ (0,1]k . Note thatx ∈ 3 if and only if x + W ′


contains an integer vector, or in other words, there exists an integer vectoru suchthat x ∈ u − W ′. Our original problem is equivalent to the following one: ForC = (1, . . . ,1) − W ′ ⊂ [0,1)k , determine the number of integer vectorsy =(y1, . . . , yd ) ∈ rK ′ such that({

d∑l=1

αl1yl

}, . . . ,

{d∑l=1

αlkyl

})∈ C, (8)

where{x} denotes the fractional part ofx ∈ R.This problem can be solved by estimating the discrepancyDN of the family�

in Rd+k whoseN = [r]d elements have the form(y1

r, . . . ,

yd

r,

d∑l=1

αl1yl, . . . ,

d∑l=1

αlkyl

)

for 06 |yi | < r and(y1, . . . , yd) ∈ Zd.We quote the notions and statements connected to discrepancy following [12].For positive expressionsf and g, we sayf � g (f � g) if f < c · g

(f > c · g) wherec > 0 depends only onL and5. Denote by�0 the familyof ({ω1}, . . . , {ωd+k}) where(ω1, . . . , ωd+k) ∈ �. We deduce by the definition ofthe isotropic discrepancyJN and its upper bound (see [12], p. 95) that∣∣∣∣#(�0 ∩ (K ′ + C))

N− |K ′ + C|

∣∣∣∣ 6 JN � D1d+kN . (9)

Hence (compare (8)), the Lemma follows if we verify that for anyε > 0 andlarger,DN < ε. We verify this using the theorem of Erdos–Turán–Koksma ([12],p. 116), which yields that for any positive integerm,

DN � 1

m+

∑h∈Zd+k

1>maxj |hj |6m

1∏d+kj=1 max{|hj |,1}

∣∣∣∣∣ 1

N

∑ω∈�

e2πi〈h,ω〉∣∣∣∣∣ . (10)

Choose somem > 2/ε and forh ∈ Zd+k, 1> maxj |hj | 6 m, set

8l(h, r) = 1

[r]dre−1∑y=0

exp

2πi

hlr+

k∑j=1

αljhd+j

y .

With this notation, (10) reads as

DN � 1

m+

∑h∈Zd+k

1>maxj |hj |6m

1∏d+kj=1 max{|hj |,1}

·d∏l=1

|8l(h, r)| . (11)


Note that always|8l(h, r)| 6 1. Thus, it is sufficient to show that ifr is largethen for anyh in (11), there exists anl such that

|8l(h, r)| < 1

(2m+ 1)d+k· ε. (12)

If ‖s‖ denotes the minimal distance ofs to an integer then∣∣∣∣∣∣[r]−1∑y=0

e2πisy

∣∣∣∣∣∣ < 2∣∣1− e2πis∣∣ � 1

‖s‖ . (13)

If there is aj such thathd+j 6= 0, then it follows from (7) that there is anl suchthat

δ =∥∥∥∥∥∥

k∑j=1

αljhd+j

∥∥∥∥∥∥ > 0

and, hence, for larger, the estimate∥∥∥∥∥∥hlr +k∑j=1

αljhd+j

∥∥∥∥∥∥ > 12δ

and (13) yield (11).On the other hand, ifhd+j = 0, j = 1, . . . , k then hl 6= 0 for somel ∈

{1, . . . , d}. For larger it follows that

|8l(h, r)| = 1

[r]

∣∣∣∣∣1− e2πihl[r]r

1− e2πihl/r

∣∣∣∣∣ < 1

r. (14)

Therefore (12) holds, and we conclude the lemma. 2

Most probably, the error term in Lemma 3 cannot be improved much in general.In most application,5 is algebraic over3; namely, the coordinates with respect tosome bases of3 of some bases of5 are contained in an algebraic extension ofQ.In this case a better error term can be proved using Diophantine approximation.

LEMMA 4. In addition to the conditions of Lemma3, assume that5 can bedefined using an algebraic extension of degreeq ofQ over3. Then for any convexbodyK in 5 and larger, asymptotically

#(rK ∩3) = |W |detL

· |K| · rd +O(rd− 1

q(d+k) (log r)1q

).


Remark.Actually, it is sufficient to assume that in (6), for eachl = 1, . . . , d,Q[αl1, . . . , αlk] has degree at mostq overQ.

Proof.We follow the proof of Lemma 3. It is sufficient to prove that

DN � (log r)d+kq

r1q

(15)

for larger.

Setm = r 1q /(logr)

d+kq in (11). First we prove instead of (12) that ifr is large

then for anyh in (11), there exists anl such that

|8l(h, r)| � mq−1

r. (16)

If hd+j = 0, j = 1, . . . , k, then there exists anl such that|8l(h, r)| < 1/r (see(14)).

If not all thehd+j are zero then there exists anl by (7) that∑k

j=1 αljhd+j 6∈ Q.We deduce by Diophantine approximation (see [5], p. 79) that∥∥∥∥∥∥

k∑j=1

αljhd+j

∥∥∥∥∥∥� 1

mq−1.

Since|hl| 6 m and them/r becomes arbitrary small compared to 1/(mq−1), weconclude (16) by (13).

Therefore the estimate∑h∈Zd+k

maxj |hj |6m

1∏d+kj=1 max{|hj |,1}

� (logm)d+k

and (11) yield (15), and the lemma follows by the choice ofm. 2

Remark. In the case of quasi-crystals, everything can be defined inQ[τ ],and henceq = 2. For the Penrose tiling, the error term is O

(r2− 1

8√

log r)

(the

Penrose plane is irrational in a four-dimensional lattice plane), and for the three-

dimensional case, the error term is O(r3− 1

12√

log r).


5. The Optimality of W (%)

For a convex bodyC in R3, denote byhC(·) its support function. IfQ is a convexpolytope with unit facet exterior normalsu1, . . . , uk and corresponding facet areasf1, . . . , fk then the mixed volumeV (Q,Q,C) is

V (Q,Q,C) = 1

3

k∑i=1

hC(ui)fi. (17)

Observe thatV (Q,Q,Q) = V (Q). Remember thatW(%) is a polytope by Theo-rem 2.

Proof of Theorem3. The definition ofW(%) yields for any unit vectoru whichis parallel to some vector inG, we have

hW(%)(u) 6 % − δ2(u)

2δ.

We deduce by (17) that

E(Q, %) > 3V (Q,Q,W(%)),

and equality holds ifQ = W(%). Assume thatQ is not a translate ofW(%). ThenMinkowski’s inequality andV (Q) = V (W(%)) yields that (see [17])

E(Q, %)3 > 33 · V (Q,Q,W(%))3 > 33 · V (Q)2 · V (W(%))= E(W(%), %)3. 2

Note that using surface integrals, it is easy to extend the definition ofE(Q, %)

and the statement of Theorem 3 to any convex bodyQ.

References

1. Arhelger, V., Betke, U. and Böröczky, Jr., K.: Large finite lattice packings, submitted.2. Betke, U., Henk, M. and Wills, J.M.: Finite and infinite packings,J. reine angew. Math.453

(1994), 165–191.3. Betke, U. and Böröczky, Jr. K.: Large lattice packings and crystals, submitted.4. Böröczky, Jr. K. and Schnell, U.: Quasicrystals and the Wulff-shape, to appear inDiscrete

Comput. Geom.5. Cassels, J. W.:Introduction into Diophantine Approximation, Cambridge Univ. Press, Cam-

bridge, 1957.6. Dinghas, A.: Über einen geometrischen Satz von Wulff über die Gleichgewichtsform von

Kristallen,Z. Kristallogr.105(1943), 304–314.7. Dobrushin, R., Kotecky, R. and Shlosman, S.: it Wulff construction, a global shape from local

interaction, Transl. Math. Monogr. 104, Amer. Math. Soc., Providence, RI, (1992).8. Gruber, P. M. and Lekkerkerker, C. G.:Geometry of Numbers, North-Holland, Amsterdam,

1987.


9. Grünbaum, B. and Shephard, G. C.:Tilings and Patterns, Freeman, San Francisco, 1986.10. Henley, C.: Sphere packings and local environments in Penrose tilings,Phys. Rev.34 (1986),

797–816.11. Janot, C.:Quasicrystals, A Primer, Oxford University Press, New York, 1994.12. Kuipers, L. and Niederreiter, H.:Uniform Distribution of Sequences, New York, 1974.13. Laue, M. V.: Der Wulffsche Satz für die Gleichgewichtsform von Kristallen,Z. Kristallogr.105

(1943), 124–133.14. Niizeki, K.: A classification of special points of icosahedral quasilattices,J. Phys. A: Math.

Gen.22 (1989), 4295–4302.15. Olami, Z. and Alexander, S.: Quasiperiodic packing densities,Phys. Rev.37 (1988), 3973–

3978.16. Penrose, R.: The role of aesthetics in pure and applied mathematical research,Bull. Inst. Math.

Appl.10 (1974), 266–271.17. Schneider, R.:Convex Bodies-the Brunn–Minkowski theory, Cambridge Univ. Press, Cam-

bridge, 1993.18. Schnell, U.: Minimal determinants and lattice inequalities,Bull. London Math. Soc.24 (1992),

606–612.19. Schnell, U.: Periodic sphere packings and Wulff-shape, submitted.20. Senechal, M.: Brief history of geometrical crystallography, in: J. Lima-de-Faria (ed.),Histor.

Atlas of Crystalography, Kluwer, Dordrecht, 1990.21. Senechal, M.:Quasicrystals and Geometry, Cambridge Univ. Press, 1995.22. Verger–Gaugry, J. L.: Approximate icosahedral periodic tilings with pseudo-icosahedral

symmetry in reciprocal space,J. Phys. France49 (1988), 1867–1874.23. Verger-Gaugry, J. L.: On a generalization of the Hermite constant,Period. Math. Hungar., to

appear 1997.24. Wills, J. M.: Finite sphere packings and sphere coverings,Rend. Semin. Mat., Messina, Ser. II

2 (1993), 91–97.25. Wills, J. M.: On large lattice packings of spheres,Geom. Dedicata65 (1997), 117–126.26. Wills, J. M.: Lattice packings of spheres and the Wulff-shape,Mathematika86 (1996), 229–

236.27. Wills, J. M.: Parametric density, online packings and crystal growth,Rend. Circ. Mat. Palermo

(2) 50 (1997), 413–424.

Documents

Wulff Shape for Nonperiodic Arrangements