A Note on Affinely Regular Polygons

0.19 MB·English

by  Christian Huck

A NOTE ON AFFINELY REGULAR POLYGONS
CHRISTIANHUCK
Abstract. Theaffinelyregularpolygonsincertainplanarsetsarecharacter-
ized. Itisalsoshownthattheobtainedresultsapplytocyclotomicmodelsets
8 and,additionally,haveconsequences inthediscretetomographyofthesesets.
0
0
2

n 1. Introduction
a
J Chrestenson[6] has shown that any (planar) regular polygon whose vertices are
1 contained in d for some d 2 must have 3,4 or 6 vertices. More generally,
Z ≥
2 Gardner and Gritzmann [9] have characterized the numbers of vertices of affinely
regular lattice polygons, i.e., images of non-degenerate regular polygons under a
]
G non-singular affine transformation of the plane whose vertices are contained in the
square lattice 2 or,equivalently, in some arbitraryplanar lattice L. It turned out
M Z
thatthe affinelyregularlattice polygonsareprecisely the affinely regulartriangles,
. parallelograms and hexagons. As a first step beyond the case of planar lattices,
h
t this short text provides a generalization of this result to planar sets Λ that are
a
non-degenerateinsomesenseandsatisfyacertainaffinityconditiononfinitescales
m
(Theorem3.3). Theobtainedcharacterizationcanbeexpressedintermsofasimple
[
inclusion of real field extensions of and particularly applies to algebraic Delone
Q
1 sets,thusincludingcyclotomic model sets. Thesecyclotomicmodelsetsrangefrom
v periodic examples, given by the vertex sets of the square tiling and the triangular
8
tiling, to aperiodic examples like the vertices of the Ammann-Beenker tiling, of
1
the Tu¨bingen triangle tiling and of the shield tiling, respectively. I turns out that,
2
3 for cyclotomic model sets Λ, the numbers of vertices of affinely regular polygons
. in Λ can be characterized by a simple divisibility condition (Corollary 4.1). In
1
particular, the result on affinely regular lattice polygons is contained as a special
0
8 case (Corollary 4.2(a)). Additionally, it is shown that the obtained divisibility
0 condition implies a weak estimate in the discrete tomography of cyclotomic model
: sets (Corollary 5.5).
v
i
X
2. Preliminaries and notation
r
a Natural numbers are always assumed to be positive, i.e., = 1,2,3,... and
N { }
we denote by the set of rational primes. If k,l , then gcd(k,l) and lcm(k,l)
P ∈N
denotetheirgreatestcommondivisorandleastcommonmultiple,respectively. The
groupofunits ofagivenring Ris denotedby R×. As usual, fora complexnumber
z , z denotes the complex absolute value, i.e., z = √zz¯, where ¯. denotes
the∈cComp|le|x conjugation. The unit circle in 2 is den|o|ted by S1, i.e., S1 = x
2 x =1 . Moreover,theelementsofS1 areRalsocalleddirections. Forr >0{an∈d
R || | }
x 2,B (x)denotestheopenballofradiusr aboutx. AsubsetΛoftheplaneis
r
∈R
called uniformly discrete if there is a radius r >0 such that every ball B (x) with
r
x 2 containsatmostonepointofΛ. Further,Λiscalledrelatively denseifthere
∈R
isaradiusR>0suchthateveryballB (x)withx 2containsatleastonepoint
R
∈R
The author was supported by the German Research Council (Deutsche Forschungsgemein-
schaft),withintheCRC701,andbyEPSRCviaGrantEP/D058465/1.
1
2 CHRISTIANHUCK
of Λ. Λ is called a Delone set (or Delaunay set) if it is both uniformly discrete and
relativelydense. Fora subset S ofthe plane, we denote by card(S), (S), conv(S)
F
and the cardinality,set offinite subsets, convexhull andcharacteristicfunction
S
of S,1respectively. A direction u S1 is called an S-direction if it is parallel to a
non-zero element of the difference∈set S S := s s′ s,s′ S of S. Further,
− { − | ∈ }
a finite subset C of S is called a convex subset of S if its convex hull contains no
new points of S, i.e., if C = conv(C) S holds. Moreover, the set of all convex
∩
subsets of S is denoted by (S). Recall that a linear transformation (resp., affine
C
transformation) Ψ: 2 2 of the Euclidean plane is given by z Az (resp.,
R → R 7→
z Az+t), where A is a real 2 2 matrix and t 2. In both cases, Ψ is called
7→ × ∈R
singular when det(A)=0; otherwise, it is non-singular. A homothety h: 2 2
R →R
isgivenbyz λz+t,whereλ ispositiveandt 2. Aconvex polygon isthe
7→ ∈R ∈R
convex hull of a finite set of points in 2. For a subset S 2, a polygon in S is
R ⊂ R
a convex polygon with all vertices in S. A regular polygon is always assumed to be
planar, non-degenerate and convex. An affinely regular polygon is a non-singular
affineimageofaregularpolygon. Inparticular,itmusthaveatleast3vertices. Let
U S1 be a finite set of directions. A non-degenerate convex polygon P is called
⊂
a U-polygon if it has the property that whenever v is a vertex of P and u U, the
∈
lineℓv intheplaneindirectionuwhichpassesthroughv alsomeetsanothervertex
u
v′ of P. For a subset Λ , we denote by the intermediate field of / that
Λ
⊂C K C Q
is given by
:= Λ Λ Λ Λ ,
Λ
K Q − ∪ −
(cid:0)(cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
where Λ Λ denotes the difference set of Λ. Further, we set := , the
Λ Λ
− k K ∩R
maximal real subfield of .
Λ
K
Remark 2.1. Note that U-polygons have an even number of vertices. Moreover,
an affinely regular polygon with an even number of vertices is a U-polygon if and
only if each direction of U is parallel to one of its edges.
For n , we always let ζ := e2πi/n, as a specific choice for a primitive nth
n
∈ N
root of unity in . Let (ζ ) be the corresponding cyclotomic field. It is well
n
knownthat (ζ C+ζ¯ )isQthe maximalrealsubfield of (ζ ); see [17]. Throughout
n n n
Q Q
this text, we shall use the notation
n = (ζn), n = (ζn+ζ¯n), n = [ζn], On = [ζn+ζ¯n].
K Q k Q O Z Z
Exceptfor the one-dimensionalcases = = , is animaginaryextension
1 2 n
K K Q K
of . Further, φ will always denote Euler’s phi-function, i.e.,
Q
φ(n)=card k 1 k n and gcd(k,n)=1 .
∈N| ≤ ≤
(cid:0)(cid:8) (cid:9)(cid:1)
Occasionally,we identify with 2. Primes p for which the number 2p+1 is
C R ∈P
primeaswellarecalledSophieGermainprimenumbers. Wedenoteby theset
SG
P
of Sophie Germain prime numbers. They are the primes p such that the equation
φ(n)=2p has solutions. It is not known whether there are infinitely many Sophie
Germain primes. The first few are
2,3,5,11,23,29,41,53,83,89,113,131,173,
{
179,191,233,239,251,281,293,359,419,... ,
}
see entry A005384 of [16] for further details. We need the following facts from the
theory of cyclotomic fields.
Fact 2.2 (Gauß). [17, Theorem 2.5] [ : ] = φ(n). The field extension /
n n
is a Galois extension with Abelian GKaloisQgroup G( / ) ( /n )×,KwheQre
n
K Q ≃ Z Z
a(modn) corresponds to the automorphism given by ζ ζa.
n 7→ n
A NOTE ON AFFINELY REGULAR POLYGONS 3
Since is the maximal real subfield of the nth cyclotomic field , Fact 2.2
n n
k K
immediately gives the following result.
Corollary 2.3. If n 3, one has [ : ]=2. Thus, a -basis of is given
n n n n
≥ K k k K
by 1,ζ . The field extension / is a Galois extension with Abelian Galois
n n
grou{p G(} / ) ( /n )×/ 1k(moQdn) of order [ : ]=φ(n)/2.
n n
k Q ≃ Z Z {± } k Q
Consider an algebraic number field , i.e., a finite extension of . A full -
K Q Z
module in (i.e.,afree -moduleofrank[ : ])whichcontainsthenumber1
O K Z K Q
andisaringiscalledanorder of . Notethatevery -basisof issimultaneously
K Z O
a -basisof ,whence = inparticular. Itturnsoutthatamongthevarious
Q K QO K
ordersof there isone maximal order whichcontainsallthe otherorders,namely
K
the ring of integers in ; see [5, Chapter 2, Section 2]. For cyclotomic fields,
OK K
one has the following well-known result.
Fact 2.4. [17, Theorem 2.6 and Proposition 2.16] For n , one has:
∈N
(a) is the ring of cyclotomic integers in , and hence its maximal order.
n n
O K
(b) On is the ring of integers in n, and hence its maximal order.
k
Lemma 2.5. If m,n , then = .
m n gcd(m,n)
∈N K ∩K K
Proof. The assertion follows from similar arguments as in the proof of the special
case (m,n) = 1; compare [15, Ch. VI.3, Corollary 3.2]. Here, one has to observe
(ζ ,ζ )= = and then to employ the identity
m n m n lcm(m,n)
Q K K K
(1) φ(m)φ(n)=φ(lcm(m,n))φ(gcd(m,n))
instead of merely using the multiplicativity of the arithmetic function φ. (cid:3)
Lemma 2.6. Let m,n . The following statements are equivalent:
∈N
(i) .
m n
K ⊂K
(ii) mn, or m 2(mod4) and m2n.
| ≡ |
Proof. For direction (ii) (i), the assertion is clear if mn. Further, if m
⇒ | ≡
2 (mod4), say m = 2o for a suitable odd number o, and m2n, then
o n
| K ⊂ K
(due to on). However,Fact 2.2 shows that the inclusion of fields =
o 2o m
| K ⊂K K
cannotbe propersincewehave,bymeansofthe multiplicativity ofφ, theequation
φ(m)=φ(2o)=φ(o). This gives .
m n
K ⊂K
For direction (i) (ii), suppose . Then, Lemma 2.5 implies =
m n m
⇒ K ⊂ K K
, whence
gcd(m,n)
K
(2) φ(m)=φ(gcd(m,n))
byFact2.2again. Usingthemultiplicativityofφtogetherwithφ(pj) = pj−1(p 1)
−
for p and j , we see that, given the case gcd(m,n)<m, Equation (2) can
∈P ∈N
only be fulfilled if m 2 (mod4) and m2n. The remaining case gcd(m,n) = m
is equivalent to the rel≡ation mn. | (cid:3)
|
Corollary 2.7. Let m,n . The following statements are equivalent:
∈N
(i) = .
m n
K K
(ii) m=n, or m is odd and n=2m, or n is odd and m=2n.
Remark 2.8. Corollary 2.7 implies that, for m,n 2 (mod4), one has the
6≡
identity = if and only if m=n.
m n
K K
Lemma 2.9. Let m,n with m,n 3. Then, one has:
∈N ≥
(a) = = or m,n 3,4,6 .
m n m n
k k ⇔ K K ∈{ }
(b) or m 3,4,6 .
m n m n
k ⊂k ⇔ K ⊂K ∈{ }
4 CHRISTIANHUCK
Proof. For claim (a), let us suppose = =: first. Then, Fact 2.2 and
m n
k k k
Corollary2.3imply that[ : ]=[ : ]=2. Note that =
m n m n gcd(m,n)
K k K k K ∩K K
is a cyclotomic field containing . It follows that either = =
m n gcd(m,n)
k K ∩K K
= or = = and hence = = = , since
m n m n gcd(m,n) m n
K K K ∩K K k k k k Q
the latter is the only real cyclotomic field. Now, this implies m,n 3,4,6 ;
∈ { }
see also Lemma 2.10(a) below. The other direction is obvious. Claim (b) follows
immediately from the part (a). (cid:3)
Lemma 2.10. Consider φ on n n 2(mod4) . Then, one has:
{ ∈N| 6≡ }
(a) φ(n)/2=1 if and only if n 3,4 .
∈{ }
(b) φ(n)/2 if and only if n := 8,9,12 2p+1 p .
SG
∈P ∈S { }∪{ | ∈P }
Proof. The equivalences follow from the multiplicativity of φ in conjunction with
the identity φ(pj) = pj−1(p 1) for p and j . (cid:3)
− ∈P ∈N
Remark2.11. Letn 2(mod4). ByCorollary2.3,forn 3,thefieldextension
6≡ ≥
/ is a Galois extension with Abelian Galois group G( / ) of order φ(n)/2.
n n
k Q k Q
UsingLemma2.10,oneseesthatG( / )istrivialifandonlyifn 1,3,4 ,and
n
k Q ∈{ }
simple if and only if n , with as defined in Lemma 2.10(b).
∈S S
3. The characterization
The following notions will be of crucial importance.
Definition 3.1. For a set Λ 2, we define the following properties:
⊂R
(Alg) [ : ]< .
Λ
K Q ∞
(Aff) For all F ( ), there is a non-singularaffine transformationΨ: 2
Λ
∈F K R →
2 such that h(F) Λ.
R ⊂
Moreover,Λ is called degenerate when ; otherwise, Λ is non-degenerate.
Λ
K ⊂R
Remark 3.2. If Λ 2 satisfies property (Alg), then one has [ : ]< , i.e.,
Λ
⊂R k Q ∞
is a real algebraic number field.
Λ
k
Before we turn to examples of planar sets Λ having properties (Alg) and (Aff),
let us prove the central result of this text, where we use arguments similar to the
ones used by Gardner and Gritzmann in the proof of [9, Theorem 4.1].
Theorem 3.3. Let Λ 2 be non-degenerate with property (Aff). Further, let
⊂ R
m with m 3. The following statements are equivalent:
∈N ≥
(i) There is an affinely regular m-gon in Λ.
(ii) .
m Λ
k ⊂k
If Λ additionally fulfils property (Alg), then it only contains affinely regular m-gons
for finitely many values of m.
Proof. For (i) (ii), let P be an affinely regular m-gon in Λ. There is then a
⇒
non-singular affine transformation Ψ : 2 2 with Ψ(R ) = P, where R
m m
is the regular m-gon with vertices givenRin→comRplex form by 1,ζ ,...,ζm−1. If
m m
m 3,4,6 ,condition (ii) holds trivially. Suppose 6=m 5. The pairs 1,ζ ,
m
ζ−∈1,{ζ2 li}e on parallel lines and so do their images u6 nder≥Ψ. Therefore, { }
{ m m}
ζ2 ζ−1 Ψ(ζ2) Ψ(ζ−1)
| m− m | = | m − m |.
ζ 1 Ψ(ζ ) Ψ(1)
m m
| − | | − |
Moreover,since Ψ(ζ2) Ψ(ζ−1)andΨ(ζ ) Ψ(1)areelementsofΛ Λ andsince
m − m m − −
z 2 =zz¯for z , we get the relation
| | ∈C
ζ2 ζ−1 2 Ψ(ζ2) Ψ(ζ−1)2
(1+ζ +ζ¯ )2 =(1+ζ +ζ−1)2 = | m− m | = | m − m | .
m m m m ζ 12 Ψ(ζ ) Ψ(1)2 ∈kΛ
m m
| − | | − |
A NOTE ON AFFINELY REGULAR POLYGONS 5
The pairs ζ−1,ζ , ζ−2,ζ2 also lie on parallel lines. An argument similar to
{ m m} { m m}
that above yields
ζ2 ζ−2 2
(ζ +ζ¯ )2 =(ζ +ζ−1)2 = | m− m | .
m m m m ζ ζ−1 2 ∈kΛ
m m
| − |
By subtracting these equations, one gets the relation
2(ζ +ζ¯ )+1 ,
m m Λ
∈k
whence ζ +ζ¯ , the latter being equivalent to the inclusion of the fields
m m Λ
∈ k
.
m Λ
k ⊂k
For (ii) (i), let R again be the regular m-gon as defined in step (i) (ii).
m
⇒ ⇒
Since m 3, the set 1,ζ is an -basis of . Since Λ is non-degenerate, there
m
≥ { } R C
is an element τ with non-zero imaginary part. Hence, one can define an
Λ
∈ K
-linear map L: 2 2 as the linear extension of 1 1 and ζ τ. Since
m
R R → R 7→ 7→
1,τ is an -basis of as well, this map is non-singular. Since and
m Λ
{ } R C k ⊂ k
since 1,ζ is a -basis of (cf. Corollary 2.3), the vertices of L(R ), i.e.,
m m m m
L(1),L{(ζ ),}...,Lk(ζm−1), lie inK , whence L(R ) is a polygon in . By prop-
m m KΛ m KΛ
erty (Aff), there is a non-singular affine transformation Ψ: 2 2 such that
R → R
Ψ(L(R )) is a polygoninΛ. Since compositionsofnon-singularaffine transforma-
m
tionsarenon-singularaffinetransformationsagain,Ψ(L(R ))isanaffinelyregular
m
m-gon in Λ.
For the additional statement, note that, since Λ has property (Alg), one has
[ : ] < by Remark 3.2. Thus, / has only finitely many intermediate
Λ Λ
k Q ∞ k Q
fields. The assertion now follows immediately from condition (ii) in conjunction
with Corollary 2.7, Remark 2.8 and Lemma 2.9. (cid:3)
Let beanimaginaryalgebraicnumberfieldwith = andlet bethering
L L L OL
ofintegersin . Then,everytranslateΛof or isnon-degenerateandsatisfies
the propertiesL(Alg) and (Aff). To this endL, weOfiLrst show that in both cases one
has = . If Λ is a translate of , this follows immediately from the calculation
Λ
K L L
= = ( )= .
Λ
K KL Q L∪L L
If Λ is a translate of , one has to observe that
OL
Λ = O = ( )= ,
K K L Q OL∪OL L
since = implies = . Inthe firstcase,property(Aff) is evident,whereas,
if ΛisLa traLnslateofOL,prOopLerty(Aff) followsfromthe fact thatthere is alwaysa
OL
-basis of that is simultaneously a -basis of . Thus, if F is a finite set,
Z OL Q L ⊂L
then a suitable translate of aF is contained in Λ, where a is defined as the least
common multiple of the denominators of the -coordinates of the elements of F
Q
with respect to a -basis of that is simultaneously a -basis of . Hence, for
Q L Z OL
these two examples, property (Aff) may be replaced by the stronger property
(Hom) ForallF ( ),thereisahomothetyh: 2 2 suchthath(F) Λ.
Λ
∈F K R →R ⊂
Thus, we have obtained the following consequence of Theorem 3.3.
Corollary 3.4. Let be an imaginary algebraic number field with = and let
L L L
be the ring of integers in . Let Λ be a translate of or a translate of .
OFuLrther, let m with m 3L. Denoting the maximal reaLl subfield of by ,OthLe
∈ N ≥ L l
following statements are equivalent:
(i) There is an affinely regular m-gon in Λ.
(ii) .
m
k ⊂l
Further, Λ only contains affinely regular m-gons for finitely many values of m.
6 CHRISTIANHUCK
Remark 3.5. In particular, Corollary 3.4 applies to translates of imaginary cy-
clotomic fields and their rings of integers, with = for a suitable n 3; cf.
n
l k ≥
Facts 2.2 and 2.4 and also compare the equivalences of Corollary 4.1 below.
4. Application to cyclotomic model sets
Remarkably, there are Delone subsets of the plane satisfying properties (Alg)
and (Hom). These sets were introduced as algebraic Delone sets in [13, Definition
4.2]. Note that algebraic Delone sets are always non-degenerate, since this is true
for all relatively dense subsets of the plane. It was shown in [13, Proposition 4.15]
that the so-called cyclotomic model sets Λ are examples of algebraic Delone sets;
cf. Section 4 of [13] and [13, Definition 4.6] in particular for the definition of
cyclotomic model sets. Any cyclotomic model set Λ is contained in a translate of
,wheren 3,inwhichcasethe -module iscalledtheunderlying -module
n n
O ≥ Z O Z
of Λ. With the exception of the crystallographic cases of translates of the square
lattice and translates of the triangular lattice , cyclotomic model sets are
4 3
O O
aperiodic (i.e., they have no translational symmetries) and have long-range order;
cf.[13,Remarks4.9and4.10]. Well-knownexamplesofcyclotomicmodelsetswith
underlying -module are the vertex sets of aperiodic tilings of the plane like
n
Z O
the Ammann-Beenker tiling [1, 2, 11] (n = 8), the Tu¨bingen triangle tiling [3, 4]
(n = 5) and the shield tiling [11] (n = 12); cf. [13, Example 4.11] for a definition
ofthe vertexsetofthe Ammann-Beenkertiling andsee Figure 1 and[13, Figure1]
for illustrations. For further details and illustrations of the examples of cyclotomic
model sets mentioned above, we refer the reader to [14, Section 1.2.3.2]. Clearly,
any cyclotomic model set Λ with underlying -module satisfies
n
Z O
(3) ( )= ,
Λ n n n
K ⊂Q O ∪O K
whence . It was shown in [13, Lemma 4.14] that cyclotomic model sets Λ
Λ n
k ⊂k
withunderlying -module evensatisfythefollowingstrongerversionofproperty
n
Z O
(Hom) above:
(Hom) ForallF ( ),thereisahomothetyh: 2 2 suchthath(F) Λ.
n
∈F K R →R ⊂
This property enables us to prove the following characterization.
Corollary 4.1. Let m,n with m,n 3. Further, let Λ be a cyclotomic model
∈N ≥
set with underlying -module . The following statements are equivalent:
n
Z O
(i) There is an affinely regular m-gon in Λ.
(ii) .
m n
k ⊂k
(iii) m 3,4,6 , or .
m n
∈{ } K ⊂K
(iv) m 3,4,6 , or mn, or m=2d with d an odd divisor of n.
∈{ } |
(v) m 3,4,6 , or .
m n
∈{ } O ⊂O
(vi) Om On.
⊂
Proof. Direction(i) (ii)isanimmediateconsequenceofTheorem3.3inconjunc-
⇒
tion with Relation (3). For direction (ii) (i), let R againbe the regular m-gon
m
⇒
as defined in step (i) (ii) of Theorem 3.3. Since m,n 3, the sets 1,ζ and
m
⇒ ≥ { }
1,ζ are -bases of . Hence, one can define an -linear map L: 2 2 as
n
{ } R C R R → R
thelinearextensionof1 1andζ ζ . Clearly,thismapisnon-singular. Since
m n
7→ 7→
and since 1,ζ is a -basis of (cf. Corollary 2.3), the vertices
m n m m m
okf L⊂(Rk), i.e., L(1),L{(ζ ),}...,Lk(ζm−1), lie inK , whence L(R ) is a polygon in
m m m Kn m
. Because Λ has property (Hom), there is a homothety h: 2 2 such that
n
K R →R
h(L(R )) is a polygonin Λ. Since homotheties are non-singularaffine transforma-
m
tions, h(L(R )) is an affinely regular m-gon in Λ. As an immediate consequence
m
of Lemma 2.9(b), we get the equivalence (ii) (iii). Conditions (iii) and (iv) are
⇔
A NOTE ON AFFINELY REGULAR POLYGONS 7
Figure 1. A central patch of the eightfold symmetric Ammann-
Beenker tiling of the plane with squares and rhombi, both having
edge length 1. Therein, an affinely regular 6-gon is marked.
equivalentbyLemma2.6. Finally,theequivalences(iii) (v)and(ii) (vi)follow
immediately from Fact 2.4. ⇔ ⇔ (cid:3)
Although the equivalence (i) (iv) in Corollary 4.1 is fully satisfactory, the
⇔
following consequence deals with the two cases where condition (ii) can be used
more effectively.
Corollary 4.2. Let m,n with m,n 3. Further, let Λ be a cyclotomic model
∈N ≥
set with underlying -module . Consider φ on n n 2(mod4) . Then,
n
Z O { ∈N| 6≡ }
one has:
(a) If n 3,4 , there is an affinely regular m-gon in Λ if and only if m
∈ { } ∈
3,4,6 .
{ }
(b) If n , there is an affinely regular m-gon in Λ if and only if
∈S
m 3,4,6,n , if n=8 or n=12,
∈{ }
(cid:26) m 3,4,6,n,2n , otherwise.
∈{ }
Proof. By Lemma 2.10(a), n 3,4 is equivalent to φ(n)/2 = 1, thus condition
∈ { }
(ii)ofCorollary4.1specializesto = ,thelatterbeingequivalenttoφ(m)=2,
m
k Q
which means m 3,4,6 ;cf. Corollary 2.3. This proves the part (a).
∈{ }
By Lemma 2.10(b), n is equivalent to φ(n)/2 . By Corollary 2.3, this
∈ S ∈ P
shows that [ : ] = φ(n)/2 . Hence, by condition (ii) of Corollary 4.1, one
n
k Q ∈ P
either gets = or = . The former case implies m 3,4,6 as in the
m m n
k Q k k ∈ { }
proof of the part (a), while the proof follows from Lemma 2.9(a) in conjunction
with Corollary 2.7 in the latter case. (cid:3)
Example 4.3. As mentioned above, the vertex set Λ of the Ammann-Beenker
AB
tiling is a cyclotomic model set with underlying -module . Since 8 , Corol-
8
Z O ∈S
lary 4.2 now shows that there is an affinely regular m-gon in Λ if and only if
AB
m 3,4,6,8 ; see Figure 1 for an affinely regular 6-gon in Λ . The other so-
AB
∈ { }
lutions are rather obvious, in particular the patch shown also contains the regular
8-gon R , given by the 8th roots of unity.
8
8 CHRISTIANHUCK
Forfurtherillustrationsandexplanationsoftheaboveresults,wereferthereader
to [14, Section 2.3.4.1] or [12, Section 5]. This references also provide a detailed
descriptionof the construction ofaffinely regularm-gons in cyclotomic model sets,
given that they exist.
5. Application to discrete tomography of cyclotomic model sets
Discretetomography isconcernedwiththeinverseproblemofretrievinginforma-
tion about some finite object frominformation about its slices; cf. [8, 9, 12, 13, 14]
andalsoseethe refencestherein. Atypicalexampleisthe reconstruction ofafinite
point setfrom its (discrete parallel) X-raysin a smallnumber of directions. Inthe
following, we restrict ourselves to the planar case.
Definition 5.1. Let F ( 2), let u S1 be a direction and let be the set of
u
∈F R ∈ L
lines in direction u in 2. Then, the (discrete parallel) X-ray of F in direction u is
R
the function X F : := 0 , defined by
u u 0
L →N N∪{ }
X F(ℓ):=card(F ℓ)= (x).
u F
∩ 1
Xx∈ℓ
In[13],westudiedtheproblemofdetermining convexsubsetsofalgebraicDelone
sets Λ by X-rays. Solving this problem amounts to find small sets U of suitably
prescribedΛ-directionswiththepropertythatdifferentconvexsubsetsofΛcannot
havethesameX-raysinthedirectionsofU. Moregenerally,onedefinesasfollows.
Definition 5.2. Let ( 2), and let m . Further, let U S1 be a finite
E ⊂ F R ∈ N ⊂
set of directions. We say that is determined by the X-raysin the directions of U
if, for all F,F′ , one has E
∈E
′ ′
(X F =X F u U) = F =F .
u u
∀ ∈ ⇒
LetΛ 2 beaDelonesetandletU S1 beasetoftwoormorepairwisenon-
⊂R ⊂
parallel Λ-directions. Suppose the existence of a U-polygon P in Λ. Partition the
verticesof P into two disjointsets V,V′, where the elements of these sets alternate
round the boundary of P. Since P is a U-polygon, each line in the plane parallel
tosomeu U thatcontainsapointinV alsocontainsapointinV′. Inparticular,
one sees th∈at card(V)=card(V′). Set
′
C :=(Λ P) (V V )
∩ \ ∪
and, further, F := C V and F′ := C V′. Then, F and F′ are different convex
∪ ∪
subsets of Λ with the same X-rays in the directions of U. We have just proven
direction(i) (ii)ofthefollowingequivalence,whichparticularlyappliestocyclo-
⇒
tomic model sets, since any cyclotomic model set is an algebraicDelone set by [13,
Proposition 4.15].
Theorem 5.3. [13, Theorem6.3] Let Λ be an algebraic Delone set and let U S1
⊂
beasetoftwoormorepairwise non-parallelΛ-directions. Thefollowing statements
are equivalent:
(i) (Λ) is determined by the X-rays in the directions of U.
C
(ii) There is no U-polygon in Λ.
Remark 5.4. Trivially, any affinely regular m-gon P in Λ with m even is a U-
polygon in Λ with respect to any set U S1 of pairwise non-parallel directions
⊂
having the property that each element of U is parallel to one of the edges of P.
ThesetU thenconsistsonlyofΛ-directionsand,moreover,satisfiescard(U) m/2.
≤
BycombiningCorollary4.1,direction(i) (ii)ofTheorem5.3andRemark5.4,
⇒
one immediately obtains the following consequence.
A NOTE ON AFFINELY REGULAR POLYGONS 9
Corollary 5.5. Let n 3 and let Λ be a cyclotomic model set with underlying
≥
-module . Suppose that there exists a natural number k such that, for
n
Z O ∈ N
any set U of k pairwise non-parallel Λ-directions, the set (Λ) is determined by the
C
X-rays in the directions of U. Then, one has
k > max 3,lcm(n,2) .
2
n o
Remark 5.6. In the situation of Corollary 5.5, the question of existence of a
suitable number k is a much more intricate problem. So far, it has only been
∈N
answered affirmatively by Gardner and Gritzmann in the case of translates of the
squarelattice(n=4),whencecorrespondingresultsholdforalltranslatesofplanar
lattices,inparticularfortranslatesofthetriangularlattice(n=3);cf.[9,Theorem
5.7(ii)and (iii)]. Moreprecisely,it is shownthere that, for these cases,the number
k =7isthesmallestamongallpossiblevaluesofk. Itwouldbeinterestingtoknow
if suitable numbers k exist for all cyclotomic model sets.
∈N
LetusfinallynotethefollowingrelationbetweenU-polygonsandaffinelyregular
polygons. The proof uses a beautiful theorem of Darboux [7] on second midpoint
polygons; cf. [10] or [8, Ch. 1].
Proposition 5.7. [9, Proposition 4.2] If U S1 is a finite set of directions, there
⊂
exists a U-polygon if and only if there is an affinely regular polygon such that each
direction of U is parallel to one of its edges.
Remark 5.8. AU-polygonneednotitselfbeanaffinelyregularpolygon,evenifit
isaU-polygoninacyclotomicmodelset; cf.[9,Example4.3]forthe caseofplanar
lattices and [14, Example 2.46] or [12, Example 1] for related examples in the case
of aperiodic cyclotomic model sets.
Acknowledgements
I am indebted to Michael Baake, Richard J. Gardner, Uwe Grimm and Peter
A. B. Pleasants for their cooperation and for useful hints on the manuscript. In-
teresting discussions with Peter Gritzmann and Barbara Langfeld are gratefully
acknowledged.
References
[1] R.Ammann,B.Gru¨nbaumandG.C.Shephard,Aperiodictiles,DiscreteComput. Geom.8
(1992), 1–25.
[2] M. Baake and D. Joseph, Ideal and defective vertex configurations in the planar octagonal
quasilattice,Phys. Rev. B 42(1990), 8091–8102.
[3] M.Baake,P.Kramer,M.SchlottmannandD.Zeidler,Thetrianglepattern–anewquasiperi-
odictilingwithfivefoldsymmetry,Mod. Phys. Lett.B 4(1990), 249–258.
[4] M.Baake,P.Kramer,M.SchlottmannandD.Zeidler,Planarpatternswithfivefoldsymmetry
assectionsofperiodicstructuresin4-space,Int. J. Mod. Phys. B 4(1990), 2217–2268.
[5] Z.I.BorevichandI.R.Shafarevich,Number Theory,AcademicPress,NewYork,1966.
[6] H.E.Chrestenson,Solutiontoproblem5014,Amer. Math. Monthly 70(1963), 447–448.
[7] M.G. Darboux,Surunprobl`emedeg´eom´etrie´el´ementaire, Bull. Sci. Math. 2(1878), 298–
304.
[8] R. J. Gardner, Geometric Tomography, 2nd ed., Cambridge University Press, New York,
2006.
[9] R.J.GardnerandP.Gritzmann,Discretetomography: determinationoffinitesetsbyX-rays,
Trans. Amer. Math. Soc.349(1997), 2271–2295.
[10] R.J.GardnerandP.McMullen,OnHammer’sX-rayproblem,J.London Math.Soc.(2)21
(1980), 171–175.
[11] F.Ga¨hler,Matchingrulesforquasicrystals: thecomposition-decompositionmethod,J.Non-
Cryst. Solids 153-154(1993), 160–164.
[12] C. Huck, Uniqueness in discrete tomography of planar model sets, notes (2007);
arXiv:math/0701141v2 [math.MG]
10 CHRISTIANHUCK
[13] C.Huck,UniquenessindiscretetomographyofDelonesetswithlong-rangeorder,submitted;
arXiv:0711.4525v1 [math.MG]
[14] C.Huck,DiscreteTomography ofDeloneSetswithLong-RangeOrder,Ph.D.thesis(Univer-
sita¨tBielefeld),Logos Verlag,Berlin,2007.
[15] S.Lang,Algebra,3rded.,Addison-Wesley,Reading,MA,1993.
[16] N.J.A.Sloane(ed.),TheOnlineEncyclopediaofIntegerSequences,publishedelectronically
athttp://www.research.att.com/~njas/sequences/
[17] L.C.Washington, Introduction toCyclotomic Fields,2nded.,Springer,NewYork,1997.
DepartmentofMathematicsandStatistics,TheOpenUniversity,WaltonHall,Mil-
ton Keynes,MK7 6AA,United Kingdom
E-mail address: [email protected]