A Note on Affinely Regular Polygons

A NOTE ON AFFINEL Y REGULAR POL YGONS CHRISTIAN HUCK Abstract. The affinely regular polygons in certain planar sets ar e characte r- ized. It is also sho wn that the obta ined results apply to cyclotomic model s ets and, additiona lly , ha v e conseq uences in the discrete tomograph y of these sets. 1. Introduction Chrestenson [6] has shown tha t any (planar ) reg ular po lygon whose vertices are contained in Z d for s ome d ≥ 2 must hav e 3 , 4 o r 6 v e r tices. More generally , Gardner and Gritzmann [9] hav e c haracterized the num b er s of vertices of affinely regular lattic e p olygo ns , i.e., images of non-degenera te regula r p olygons under a non-singular affine tra nsformation of the plane whose vertices are co ntained in the square lattice Z 2 or, equiv alently , in some arbitr a ry planar lattice L . It turned out that the affinely reg ular lattice po lygons are pr ecisely the affinely regular triang les, parallelog rams and hexa gons. As a first step beyond the case of planar lattices, this sho rt text provides a genera lization of this r esult to planar se ts Λ that ar e non-degenera te in some sense and satisfy a certain affinit y condition on finite scales (Theorem 3 .3). The obtained characteriza tion can be expressed in terms of a simple inclusion o f real field extensions o f Q and par ticularly applies to algebr aic Delone sets , thus including cyclo tomic mo del sets . These cyclotomic mo del sets rang e from per io dic examples, given by the vertex sets of the squa re tiling and the triang ula r tiling, to ap erio dic examples like the vertices o f the Ammann-Beenker tiling, of the T ¨ ubingen tr ia ngle tiling and of the shield tiling, resp ectively . I turns out that, for cyclotomic mo del sets Λ , the n umber s of v er tice s of affinely r egular polygons in Λ can b e characterized by a simple divisibility condition (Corolla ry 4.1). In particular, the result on affinely reg ular lattice p oly gons is co nt ained as a s pe cial case (Corollary 4.2(a)). Additionally , it is shown that the obta ined divisibilit y condition implies a weak estimate in the discr ete tomo gr aphy of c y clotomic mo del sets (Coro llary 5.5). 2. Preliminaries and not a tion Natural num bers ar e always assumed to be p o sitive, i.e., N = { 1 , 2 , 3 , . . . } and we denote by P the set of rationa l primes. If k , l ∈ N , then gcd( k , l ) and lcm( k , l ) denote their grea test common divisor and least common m ultiple, resp ectively . The group of units of a given ring R is denoted by R × . As usua l, for a complex num b er z ∈ C , | z | denotes the co mplex absolute v alue, i.e ., | z | = √ z ¯ z , where ¯ . denotes the complex co njugation. The unit cir cle in R 2 is denoted by S 1 , i.e., S 1 = { x ∈ R 2 | | x | = 1 } . Moreov er, the e lement s of S 1 are also called dir e ct ions . F o r r > 0 and x ∈ R 2 , B r ( x ) denotes the op en ball of radius r a b out x . A subset Λ of the plane is called uniformly discr et e if there is a r adius r > 0 such that every ball B r ( x ) with x ∈ R 2 contains at mo st o ne point of Λ . F urther, Λ is c a lled r elatively dense if there is a radius R > 0 such that every ball B R ( x ) with x ∈ R 2 contains at least one point The author w as supp orted b y the German Research Council (Deutsc he F orsc h ungsgemein- sc haft), within the CR C 701, and b y EPSRC via Gran t EP/D058465/1 . 1 2 CHRISTIAN HUCK of Λ . Λ is calle d a Delone set (or Delaunay set ) if it is b oth uniformly discr ete and relatively dense. F or a subs e t S of the plane, we denote by card( S ), F ( S ), conv( S ) and 1 S the ca rdinality , set of finite subsets, co nvex h ull and characteristic function of S , r esp ectively . A dir ection u ∈ S 1 is called an S -dir e ct ion if it is parallel to a non-zero element of t he difference set S − S := { s − s ′ | s, s ′ ∈ S } of S . F urther , a finite subset C of S is called a c onvex su bset of S if its co nv ex hull contains no new p oints of S , i.e., if C = conv( C ) ∩ S holds. Moreov er, the set of a ll c onv e x subsets of S is denoted by C ( S ). Reca ll that a line ar tr ansformatio n (res p., affine tr ansformation ) Ψ : R 2 → R 2 of the Euclidea n plane is given by z 7→ Az (resp., z 7→ Az + t ), where A is a real 2 × 2 matrix and t ∈ R 2 . In bo th cas es, Ψ is called singular when det( A ) = 0; otherwise, it is non-s ingular. A homothety h : R 2 → R 2 is given by z 7→ λz + t , where λ ∈ R is p ositive and t ∈ R 2 . A c onvex p olygo n is the conv ex hull of a finite set of p oints in R 2 . F or a subse t S ⊂ R 2 , a p olygon in S is a conv ex p olygon with a ll vertices in S . A r e gular p olygon is a lwa y s as s umed to b e planar, non-de g enerate and co nv ex . An affinely r e gular p olygon is a non-singular affine image of a regula r p oly g on. In particular, it m ust hav e at least 3 v e rtices. Le t U ⊂ S 1 be a finite set o f directions. A non-degener ate c onv ex p olyg on P is called a U -p olygon if it has the prop erty that whene ver v is a vertex of P and u ∈ U , the line ℓ v u in the plane in direction u which pa sses through v also meets ano ther vertex v ′ of P . F or a subset Λ ⊂ C , we denote by K Λ the intermediate field o f C / Q that is given by K Λ := Q  Λ − Λ  ∪  Λ − Λ  , where Λ − Λ denotes the difference set of Λ . F urther, we se t k Λ := K Λ ∩ R , the maximal re a l subfield of K Λ . Remark 2.1 . Note tha t U -p olyg ons hav e an even num b er of vertices. Mor eov er, an affinely regular p olyg on with a n even num ber of vertices is a U -p olygo n if and only if each dir ection of U is para llel to o ne of its edges. F or n ∈ N , we always let ζ n := e 2 π i/n , a s a sp ecific choice for a primitive n th ro ot of unity in C . Let Q ( ζ n ) b e the corres po nding cyclo tomic field. It is well known that Q ( ζ n + ¯ ζ n ) is the max imal real subfield of Q ( ζ n ); s ee [17]. T hr oughout this text, we s hall use the no tation K n = Q ( ζ n ) , k n = Q ( ζ n + ¯ ζ n ) , O n = Z [ ζ n ] , O n = Z [ ζ n + ¯ ζ n ] . Except for the one-dimensiona l cas e s K 1 = K 2 = Q , K n is an imag ina ry extensio n of Q . F ur ther, φ will always denote E uler’s phi-function, i.e., φ ( n ) = ca rd  k ∈ N | 1 ≤ k ≤ n a nd gcd( k, n ) = 1  . Occasiona lly , we identif y C with R 2 . P rimes p ∈ P for which the num b er 2 p + 1 is prime as well ar e called Sophie Germain prime num b ers. W e denote b y P SG the set of Sophie Ger main pr ime num bers . They are the primes p such that the equation φ ( n ) = 2 p has solutio ns . It is not known whether there are infinitely many Sophie Germain primes. The fir st 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 e nt ry A005 384 of [1 6] for further details. W e need the following facts fr om the theory o f cyclotomic fields. F act 2. 2 (Gauß ) . [17, Theorem 2.5 ] [ K n : Q ] = φ ( n ) . The field extension K n / Q is a Galo is extension with Ab elian Galo is gr oup G ( K n / Q ) ≃ ( Z /n Z ) × , wher e a (mo d n ) c orr esp onds t o the aut omorphism given by ζ n 7→ ζ a n . A NOTE ON AFFINEL Y REGULAR POL YGONS 3 Since k n is the maxima l rea l subfield of the n th cyclo to mic field K n , F act 2 .2 immediately g ives the following result. Corollary 2. 3 . If n ≥ 3 , one has [ K n : k n ] = 2 . Thus, a k n -b asis of K n is given by { 1 , ζ n } . The field ext ension k n / Q is a Galois extension with A b elian Galois gr oup G ( k n / Q ) ≃ ( Z /n Z ) × / {± 1 (mo d n ) } of or der [ k n : Q ] = φ ( n ) / 2 . Consider an algebra ic num b er field K , i.e., a finite e x tension o f Q . A full Z - mo dule O in K (i.e., a free Z -mo dule of rank [ K : Q ]) which con tains the n um ber 1 and is a ring is calle d an or der o f K . No te that every Z -basis of O is sim ultaneously a Q -basis of K , whence Q O = K in particular. It turns out that among the v ar ious orders of K there is o ne maximal or der which c ontains all the other orders, na mely the r ing of int egers O K in K ; see [5, Chapter 2, Section 2]. F o r cycloto mic fields, one has the following well-known result. F act 2. 4. [17, Theorem 2.6 and Prop os itio n 2.16] F or n ∈ N , one has: (a) O n is the ring of cyclotomic inte gers in K n , and henc e its maximal or der. (b) O n is the ring of inte gers in k n , and henc e its maximal or der. Lemma 2.5. If m , n ∈ N , then K m ∩ K n = K gcd( m,n ) . Pr o of. The ass e rtion follows fro m s imilar ar guments as in the pr o of of the sp ecial case ( m, n ) = 1; co mpare [15, Ch. VI.3, Corolla r y 3.2]. Here, one has to obser ve Q ( ζ m , ζ n ) = K m K n = K lcm( m,n ) and then to employ the ide ntit y (1) φ ( m ) φ ( n ) = φ (lcm( m, n )) φ (gcd( m, n )) instead of merely using the multiplicativit y o f the a r ithmetic function φ .  Lemma 2.6. L et m, n ∈ N . The fol lowing st atements ar e e qu ivalent: (i) K m ⊂ K n . (ii) m | n , or m ≡ 2 (mo d 4) and m | 2 n . Pr o of. F or directio n (ii) ⇒ ( i), the a ssertion is clea r if m | n . F urther, if m ≡ 2 (mo d 4), say m = 2 o for a suitable o dd num b er o , and m | 2 n , then K o ⊂ K n (due to o | n ). How ever, F act 2.2 shows that the inclusion of fields K o ⊂ K 2 o = K m cannot b e pr op er since we ha v e, by means of the multip licativity of φ , the eq uation φ ( m ) = φ (2 o ) = φ ( o ). This g ives K m ⊂ K n . F or dire ction (i) ⇒ (ii), supp ose K m ⊂ K n . Then, Lemma 2 .5 implies K m = K gcd( m,n ) , whence (2) φ ( m ) = φ (gcd( m, n )) by F a ct 2.2 aga in. Using the multiplicativit y of φ tog ether with φ ( p j ) = p j − 1 ( p − 1 ) for p ∈ P and j ∈ N , we see that, given the case gcd( m, n ) < m , Equation (2 ) can only b e fulfilled if m ≡ 2 (mo d 4) and m | 2 n . The remaining case gcd( m, n ) = m is equiv a le nt to the r e lation m | n .  Corollary 2.7. L et m, n ∈ N . The fol lowing statements ar e e qu ivalent: (i) K m = K n . (ii) m = n , or m is o dd and n = 2 m , or n is o dd and m = 2 n . Remark 2.8. Coro llary 2.7 implies that, for m, n 6≡ 2 (mod 4), one has the ident it y K m = K n if and only if m = n . Lemma 2.9. L et m, n ∈ N with m, n ≥ 3 . Then, one has: (a) k m = k n ⇔ K m = K n or m, n ∈ { 3 , 4 , 6 } . (b) k m ⊂ k n ⇔ K m ⊂ K n or m ∈ { 3 , 4 , 6 } . 4 CHRISTIAN HUCK Pr o of. F or claim (a), let us supp os e k m = k n =: k first. Then, F act 2.2 and Corollar y 2.3 imply that [ K m : k ] = [ K n : k ] = 2. Note that K m ∩ K n = K gcd( m,n ) is a c y clotomic field containing k . It follows that either K m ∩ K n = K gcd( m,n ) = K m = K n or K m ∩ K n = K gcd( m,n ) = k and hence k m = k n = k = Q , since the la tter is the only rea l cyclo tomic field. Now, this implies m, n ∈ { 3 , 4 , 6 } ; see also L e mma 2.10(a) b elow. The other direction is obvious. Claim (b) follows immediately fr o m the pa rt (a).  Lemma 2.10. Consider φ on { n ∈ N | n 6≡ 2 (mod 4) } . Then, one has: (a) φ ( n ) / 2 = 1 if and only if n ∈ { 3 , 4 } . (b) φ ( n ) / 2 ∈ P if and only if n ∈ S := { 8 , 9 , 1 2 } ∪ { 2 p + 1 | p ∈ P SG } . Pr o of. The equiv a lences follow fro m the multiplicativity of φ in conjunction with the identit y φ ( p j ) = p j − 1 ( p − 1 ) for p ∈ P and j ∈ N .  Remark 2.11. Let n 6≡ 2 (mod 4). By Corolla ry 2.3, for n ≥ 3, the field extensio n k n / Q is a Galois extensio n with Ab elian Galois group G ( k n / Q ) of or der φ ( n ) / 2. Using Lemma 2.10, one sees that G ( k n / Q ) is trivial if and only if n ∈ { 1 , 3 , 4 } , and simple if and only if n ∈ S , with S as defined in Lemma 2.10(b). 3. The characteriza tion The following notions will b e of crucial impo rtance. Definition 3. 1. F or a set Λ ⊂ R 2 , we define the following pr op erties: (Alg) [ K Λ : Q ] < ∞ . (Aff ) F or all F ∈ F ( K Λ ), there is a non-singular affine transformatio n Ψ : R 2 → R 2 such that h ( F ) ⊂ Λ . Moreov er, Λ is called de gener ate when K Λ ⊂ R ; other wise, Λ is non-degener ate. Remark 3.2 . If Λ ⊂ R 2 satisfies pr op erty (Alg), then o ne has [ k Λ : Q ] < ∞ , i.e., k Λ is a real algebraic num ber field. Before we turn to examples of planar sets Λ having pr op erties (Alg) and (Aff ), let us prov e the central result of this text, where we us e arguments similar to the ones used b y Gar dner and Gritzmann in the pro of of [9, Theore m 4.1]. Theorem 3. 3. L et Λ ⊂ R 2 b e non- de gener ate with pr op erty (Aff ). F urther, let m ∈ N with m ≥ 3 . The fol lowing st atements ar e e qu ivalent: (i) Ther e is an affinely r e gular m -gon in Λ . (ii) k m ⊂ k Λ . If Λ additiona l ly fulfils pr op erty (Alg), t hen it only c ontains affinely r e gular m -gons for finit ely many values of m . Pr o of. F or (i) ⇒ (ii), le t P be an affinely regular m - gon in Λ . There is then a non-singular affine trans fo rmation Ψ : R 2 → R 2 with Ψ( R m ) = P , where R m is the r e g ular m -gon with vertices g iven in c o mplex form by 1 , ζ m , . . . , ζ m − 1 m . If m ∈ { 3 , 4 , 6 } , condition (ii) holds trivially . Supp ose 6 6 = m ≥ 5. The pairs { 1 , ζ m } , { ζ − 1 m , ζ 2 m } lie on par allel lines a nd so do their images under Ψ. Therefore, | ζ 2 m − ζ − 1 m | | ζ m − 1 | = | Ψ( ζ 2 m ) − Ψ ( ζ − 1 m ) | | Ψ( ζ m ) − Ψ (1) | . Moreov er, since Ψ( ζ 2 m ) − Ψ( ζ − 1 m ) and Ψ( ζ m ) − Ψ(1) are elemen ts of Λ − Λ and since | z | 2 = z ¯ z for z ∈ C , we get the rela tion (1 + ζ m + ¯ ζ m ) 2 = (1 + ζ m + ζ − 1 m ) 2 = | ζ 2 m − ζ − 1 m | 2 | ζ m − 1 | 2 = | Ψ( ζ 2 m ) − Ψ ( ζ − 1 m ) | 2 | Ψ( ζ m ) − Ψ (1) | 2 ∈ k Λ . A NOTE ON AFFINEL Y REGULAR POL YGONS 5 The pairs { ζ − 1 m , ζ m } , { ζ − 2 m , ζ 2 m } als o lie on par a llel lines. An ar g ument similar to that ab ove yields ( ζ m + ¯ ζ m ) 2 = ( ζ m + ζ − 1 m ) 2 = | ζ 2 m − ζ − 2 m | 2 | ζ m − ζ − 1 m | 2 ∈ k Λ . By subtracting these equations, o ne gets the relation 2( ζ m + ¯ ζ m ) + 1 ∈ k Λ , whence ζ m + ¯ ζ m ∈ k Λ , the latter b eing equiv alent to the inclusion of the fields k m ⊂ k Λ . F or (ii) ⇒ (i), let R m again b e the regular m - gon as defined in step (i) ⇒ (ii). Since m ≥ 3, the set { 1 , ζ m } is an R -basis of C . Since Λ is no n-degenera te, there is an elemen t τ ∈ K Λ with non- zero imagina ry part. Hence, one can define an R -linear map L : R 2 → R 2 as the linear ex tension o f 1 7→ 1 and ζ m 7→ τ . Since { 1 , τ } is an R -basis of C as well, this map is non-singular . Since k m ⊂ k Λ and since { 1 , ζ m } is a k m -basis of K m (cf. Coro llary 2.3), the vertices of L ( R m ), i.e., L (1) , L ( ζ m ) , . . . , L ( ζ m − 1 m ), lie in K Λ , whence L ( R m ) is a p olygon in K Λ . By prop- erty (Aff ), there is a non-singular affine transfor mation Ψ : R 2 → R 2 such that Ψ( L ( R m )) is a p o lygon in Λ . Since comp ositions o f non-sing ular affine transforma - tions are non-singular affine transformatio ns again, Ψ( L ( R m )) is an affinely regular m -gon in Λ . F or the additional sta tement, no te that, since Λ has prop er ty (Alg), one has [ k Λ : Q ] < ∞ by Remar k 3.2. Th us, k Λ / Q has only finitely many intermediate fields. The asser tion now follows immediately fr o m condition (ii) in conjunction with Cor ollary 2.7, Rema rk 2.8 a nd Lemma 2.9.  Let L be an imaginary algebraic num ber field with L = L and let O L be the ring of integers in L . Then, every translate Λ of L or O L is non-degener ate and satisfies the prop er ties (Alg) and (Aff ). T o this end, we first show that in b oth cases one has K Λ = L . If Λ is a trans late of L , this follows immediately fro m the calc ula tion K Λ = K L = Q ( L ∪ L ) = L . If Λ is a translate of O L , one has to obser ve that K Λ = K O L = Q ( O L ∪ O L ) = L , since L = L implies O L = O L . In the first case, prop erty (Aff ) is evident, whereas, if Λ is a translate of O L , prop er ty (A ff ) follows from the fact that there is always a Z -basis of O L that is simultaneously a Q -basis of L . Thus, if F ⊂ L is a finite set, then a suitable translate of aF is co ntained in Λ , where a is defined a s the leas t common multiple of the denomina tors of the Q -co ordina tes of the elements o f F with resp ect to a Q -basis of L that is simultaneously a Z -basis of O L . Hence, for these tw o examples, pro pe rty (Aff ) may b e replace d by the stronger prop erty (Hom) F or all F ∈ F ( K Λ ), there is a ho mothety h : R 2 → R 2 such that h ( F ) ⊂ Λ . Thu s, we hav e obtained the following cons e quence of Theorem 3.3. Corollary 3. 4 . L et L b e an imaginary algebr aic n u mb er field with L = L and let O L b e t he ring of inte gers in L . L et Λ b e a t r anslate of L or a tr anslate of O L . F urther, let m ∈ N with m ≥ 3 . Denoting the maximal r e al su bfield of L by l , t he fol lowing statements ar e e quivalent: (i) Ther e is an affinely r e gular m -gon in Λ . (ii) k m ⊂ l . F urther, Λ only c ontains affinely re gular m -gons for fi nitely many values of m . 6 CHRISTIAN HUCK Remark 3.5. In particular , Corollar y 3 .4 applies to trans lates of imaginar y cy- clotomic fields and their rings of integers, with l = k n for a suitable n ≥ 3 ; cf. F ac ts 2.2 a nd 2 .4 and a lso compare the equiv alences of Cor ollary 4.1 below. 4. Applica tion to cyclotomic mo del sets Remark ably , there are Delone subs ets of the plane satisfying pr op erties (Alg) and (Hom). These sets were introduced as algebr aic Delone sets in [1 3, Definition 4.2]. Note that alg ebraic Delone sets are alwa ys non-degener ate, since this is tr ue for all relatively dense subsets o f the plane. It was shown in [13, Pr op osition 4.15 ] that the s o-called cyclotomic m o del s et s Λ a re examples o f alg e br aic Delone sets; cf. Section 4 of [13] and [13, Definition 4.6] in particular for the definition of cyclotomic mo del se ts . An y cyclo tomic mo del s et Λ is contained in a tra nslate o f O n , where n ≥ 3, in whic h case the Z -mo dule O n is called the underlying Z -mo dule of Λ . With the exception o f the crys tallogra phic ca ses o f tr anslates of the s q uare lattice O 4 and transla tes of the triangula r lattice O 3 , cyclotomic mo del sets are ap erio dic (i.e., they have no tra nslational symmetries) a nd hav e lo ng -range order; cf. [13, Remark s 4.9 and 4 .10]. W ell-known examples of cyclo tomic model sets with underlying Z -mo dule O n are the vertex sets o f a p er io dic tilings of the pla ne like the Ammann-B eenker tiling [1, 2 , 11] ( n = 8), the T ¨ ubingen triang le tiling [3, 4] ( n = 5) and the shield tiling [11] ( n = 12); cf. [13, Example 4.11 ] for a definition of the vertex set of the Ammann-Be enker tiling and se e Figure 1 and [13, Figure 1 ] for illustrations. F or further details and illustrations of the examples of cyclotomic mo del sets mentioned ab ov e, we refer the reader to [14, Sectio n 1.2.3 .2]. Clearly , any cy c lotomic mo del s et Λ with underlying Z -mo dule O n satisfies (3) K Λ ⊂ Q ( O n ∪ O n ) = K n , whence k Λ ⊂ k n . It was shown in [13, Lemma 4.14] that c y clotomic mo del sets Λ with under lying Z -mo dule O n even s atisfy the following s tronger version o f prop erty (Hom) ab ove: (Hom ) F or all F ∈ F ( K n ), there is a homothety h : R 2 → R 2 such that h ( F ) ⊂ Λ . This prop er ty enables us to prov e the following characteriza tion. Corollary 4. 1. Le t m, n ∈ N with m , n ≥ 3 . F urther, let Λ b e a cyclotomic mo del set with underlying Z -mo dule O n . The fol lowing st atements ar e e qu ivalent: (i) Ther e is an affinely r e gular m -gon in Λ . (ii) k m ⊂ k n . (iii) m ∈ { 3 , 4 , 6 } , or K m ⊂ K n . (iv) m ∈ { 3 , 4 , 6 } , or m | n , or m = 2 d with d an o dd divisor of n . (v) m ∈ { 3 , 4 , 6 } , or O m ⊂ O n . (vi) O m ⊂ O n . Pr o of. Dire ction (i) ⇒ (ii) is an immediate consequence o f Theorem 3.3 in conjunc- tion with Relation (3). F or direction (ii) ⇒ (i), let R m again be the regular m -go n as defined in step (i) ⇒ (ii) o f Theor e m 3.3. Since m, n ≥ 3, the sets { 1 , ζ m } and { 1 , ζ n } are R -bases o f C . Hence, one can define an R -linear ma p L : R 2 → R 2 as the linear ex tension of 1 7→ 1 and ζ m 7→ ζ n . Clearly , this map is non-singular . Since k m ⊂ k n and since { 1 , ζ m } is a k m -basis of K m (cf. Cor ollary 2 .3), the vertices of L ( R m ), i.e., L (1) , L ( ζ m ) , . . . , L ( ζ m − 1 m ), lie in K n , whence L ( R m ) is a po ly gon in K n . Because Λ has pro p e r ty (Hom ), there is a homothety h : R 2 → R 2 such that h ( L ( R m )) is a p olygon in Λ . Since homotheties ar e non- singular affine transfor ma- tions, h ( L ( R m )) is an affinely reg ular m -g on in Λ . As an immediate conseque nc e of Lemma 2 .9(b), we get the equiv alence (ii) ⇔ (iii). Conditions (iii) a nd (iv) are A NOTE ON AFFINEL Y REGULAR POL YGONS 7 Figure 1. A central patch of the eightfold s ymmetric Ammann- Beenker tiling of the plane with squares and rhombi, b oth having edge length 1. Ther ein, an affinely r e gular 6-g on is marked. equiv alent by Lemma 2 .6. Finally , the equiv alences (iii) ⇔ (v) and (ii) ⇔ (vi) fo llow immediately fr o m F act 2.4.  Although the equiv alence (i) ⇔ (iv) in Coro llary 4.1 is fully satisfactor y , the following consequence deals with the tw o cases where condition (ii) c an b e used more effectively . Corollary 4. 2. Le t m, n ∈ N with m , n ≥ 3 . F urther, let Λ b e a cyclotomic mo del set with un derlying Z -mo dule O n . Consider φ on { n ∈ N | n 6≡ 2 (mod 4) } . Then, one has: (a) If n ∈ { 3 , 4 } , ther e is an affinely r e gular m -gon in Λ if and only if m ∈ { 3 , 4 , 6 } . (b) If n ∈ S , ther e is an affinely r e gular m -gon in Λ if and only if  m ∈ { 3 , 4 , 6 , n } , if n = 8 or n = 12 , m ∈ { 3 , 4 , 6 , n, 2 n } , otherwise. Pr o of. By Lemma 2 .10(a), n ∈ { 3 , 4 } is equiv alent to φ ( n ) / 2 = 1 , thus condition (ii) of Corollar y 4.1 sp ecia lizes to k m = Q , the latter b eing equiv alen t to φ ( m ) = 2, which means m ∈ { 3 , 4 , 6 } ; cf. Co rollary 2.3. This pr ov e s the part (a). By Lemma 2.10(b), n ∈ S is equiv alent to φ ( n ) / 2 ∈ P . By Co r ollary 2.3, this shows that [ k n : Q ] = φ ( n ) / 2 ∈ P . Hence, by condition (ii) of Co rollar y 4 .1, one either gets k m = Q or k m = k n . The former case implies m ∈ { 3 , 4 , 6 } a s in the pro of o f the par t (a ), while the pro o f follows from Lemma 2 .9(a) in conjunction with Cor ollary 2.7 in the latter cas e.  Example 4.3. As mentioned ab ov e, the vertex s e t Λ AB of the Ammann-Beenker tiling is a cyclo tomic mo del set with under lying Z -mo dule O 8 . Since 8 ∈ S , Corol- lary 4.2 now shows that there is an a ffinely re g ular m -gon in Λ AB if and o nly if m ∈ { 3 , 4 , 6 , 8 } ; see Fig ure 1 for an affinely reg ula r 6-g on in Λ AB . The other so- lutions are rather obvious, in particular the patch shown also contains the re g ular 8-gon R 8 , given by the 8th ro ots of unity . 8 CHRISTIAN HUCK F or further illustrations a nd explanatio ns of the abov e res ults, we refer the rea der to [14, Section 2.3 .4.1] or [12, Section 5]. This refer ences also provide a detailed description of the cons tr uction of affinely reg ular m -g ons in cyclo tomic mo del sets, given tha t they exis t. 5. Applica tion to discrete tomo graphy o f cyclotomic model sets Discr ete tomo gr aphy is concer ned with the in verse pro blem of r etrieving informa - tion ab o ut some finite ob ject from infor mation ab out its slices; cf. [8, 9, 12, 13, 14] and also see the refences therein. A t ypical example is the re c onstruction of a finite po int set from its ( discr ete p ar al lel ) X -r ays in a small num ber of directions. In the following, w e restrict ourselves to the pla nar case. Definition 5. 1. Let F ∈ F ( R 2 ), let u ∈ S 1 be a dire c tion and let L u be the s et o f lines in dir ection u in R 2 . Then, the ( discr et e p ar al lel ) X-r ay of F in dir e ction u is the function X u F : L u → N 0 := N ∪ { 0 } , defined by X u F ( ℓ ) := card( F ∩ ℓ ) = X x ∈ ℓ 1 F ( x ) . In [13], we s tudied the problem of determining conv ex subsets of algebr aic Delone sets Λ by X -r ays. So lving this pr oblem a mo unt s to find small s ets U o f suitably prescrib ed Λ -directio ns with the prop erty that different co n vex subsets of Λ c annot hav e the same X -r ays in the directions of U . More genera lly , one defines as follows. Definition 5.2. Let E ⊂ F ( R 2 ), and let m ∈ N . F urther, le t U ⊂ S 1 be a finite set of directions. W e s ay that E is determine d by the X -rays in the direc tio ns of U if, for all F , F ′ ∈ E , o ne has ( X u F = X u F ′ ∀ u ∈ U ) = ⇒ F = F ′ . Let Λ ⊂ R 2 be a Delo ne set and let U ⊂ S 1 be a s et of tw o or more pair wise non- parallel Λ -dire c tio ns. Suppose the existence of a U -p olyg on P in Λ . Partition the vertices of P into tw o disjoint sets V , V ′ , where the elements of these sets alter na te round the b o unda ry of P . Since P is a U -p olyg on, each line in the pla ne par allel to some u ∈ U that cont ains a p oint in V also contains a point in V ′ . In particular , one sees that ca rd( V ) = card( V ′ ). Set C := ( Λ ∩ P ) \ ( V ∪ V ′ ) and, further, F := C ∪ V and F ′ := C ∪ V ′ . Then, F and F ′ are different conv ex subsets o f Λ with the same X -r ays in the dir ections o f U . W e have just prov en direction (i) ⇒ (ii) of the following equiv ale nc e , which par ticularly applies to cyclo- tomic mo del sets, since any cyclo tomic mo del set is an algebr aic Delone set by [13, Prop ositio n 4.15]. Theorem 5. 3. [13, Theor em 6.3 ] L et Λ b e an algebr aic Delone set and let U ⊂ S 1 b e a set of two or mor e p airwise non-p ar al lel Λ - dir e ct ions. The fol lowing statements ar e e quivalent: (i) C ( Λ ) is determine d by t he X -r ays in the dir e ctions of U . (ii) Ther e is no U - p olygon in Λ . Remark 5.4. T rivially , any affinely regular m -gon P in Λ with m even is a U - po lygon in Λ with res pe c t to any set U ⊂ S 1 of pairwis e non-pa rallel directio ns having the pr op erty that each element of U is para llel to one o f the edges of P . The set U then co nsists only of Λ -directions a nd, moreover, s atisfies card( U ) ≤ m / 2. By com bining Corollary 4.1, direction (i) ⇒ (ii) of Theorem 5.3 and Remar k 5.4, one immediately obtains the following consequence. A NOTE ON AFFINEL Y REGULAR POL YGONS 9 Corollary 5.5. L et n ≥ 3 and let Λ b e a cyclotomic m o del set with underlying Z -mo dule O n . Supp ose that ther e exists a natur al numb er k ∈ N such t hat, for any set U of k p airwi se non-p ar al lel Λ - dir e ctions, the set C ( Λ ) is determine d by the X -r ays in the dir e ctions of U . Then, one has k > max n 3 , lcm( n, 2) 2 o . Remark 5. 6. In the situation of Corollary 5.5, the question of ex istence of a suitable num ber k ∈ N is a muc h mo re intricate pro blem. So far, it has only b een answered affirma tively b y Gardner and Gritzmann in the cas e of tra nslates of the square lattice ( n = 4), whence corresp onding r e s ults hold for all translates of planar lattices, in particular for trans lates of the triangular lattice ( n = 3); cf. [9, Theorem 5.7(ii) a nd (iii)]. More prec is ely , it is shown there that, for these cases, the n um ber k = 7 is the smalles t among all p ossible v alues of k . It would be interesting to know if suitable num b er s k ∈ N exist fo r all cy c lotomic mo del sets. Let us finally note the following relatio n betw een U -p olyg ons and affinely regula r po lygons. The pro of uses a b eautiful theorem of Dar b o ux [7] on second midp oint po lygons; cf. [10] o r [8, Ch. 1]. Prop ositi o n 5. 7 . [9, Pro po sition 4.2] If U ⊂ S 1 is a fin ite set of dir e ctions, ther e exists a U -p olygon if and only if t her e is an affinely r e gular p olygon such that e ach dir e ction of U is p ar al lel to one of its e dges. Remark 5.8. A U -p olygon need not itself be an affinely regular polyg on, even if it is a U -p olyg on in a cyclotomic mo del s e t; cf. [9, Example 4.3 ] for the case of planar lattices and [1 4, Exa mple 2.46] or [12, Example 1] for rela ted examples in the case of ap erio dic cyclotomic mo del sets. Ackno wledgements I am indebted to Michael Baa ke, Richard J. Ga rdner, Uwe Grimm and Peter A. B. Pleasants for their co op eratio n and for useful hint s on the manuscript. In- teresting discussions with Peter Gritzmann and Ba rbara La ngfeld ar e gr atefully ackno wledged. References [1] R. Ammann, B. Gr¨ un baum and G. C. Shephard, Aperi odic tiles, Discr ete Comput. Ge om. 8 (1992), 1–25. [2] M. Baake and D. Joseph, Ideal and defective verte x configurations in the planar o ctagonal quasilattice, Phys. R ev. B 42 (1990), 8091–8102 . [3] M. Baak e, P . Kramer, M. Sch lottmann and D. Zeidler, The triangle pattern – a new quasip eri- odic tili ng with fiv efold s ym m etry , Mo d. Phys. Lett. B 4 (1990), 249–258. [4] M. Baake, P . K ramer, M. Schlottmann and D. Zeidler, Planar patterns with fivefold symmetry as sections of perio dic structures in 4-space, Int. J. Mo d. Phys. B 4 (1990) , 2217–2268. [5] Z. I. Borevic h and I. R. Shafarevic h, Numb er The ory , Academic Press, New Y ork, 1966. [6] H. E. Chrestenson, Solution to problem 5014 , Amer. Math. M onthly 70 (1963), 447–448. [7] M. G. Darb oux, Sur un probl ` eme de g ´ eo m´ etrie ´ el ´ emen taire, Bul l. Sci. Math. 2 (1878), 298– 304. [8] R. J. Gardner, Ge ometric T omo gr aphy , 2nd ed., Cambridge Universit y Pr ess, New Y ork, 2006. [9] R. J. Gardner and P . Gritzmann, Discr ete tomography : determination of finite s ets by X-rays, T r ans. Amer. Math. So c. 349 (1997), 2271–229 5. [10] R. J. Gardner and P . McMullen, On Hammer’s X-ra y problem, J. L ondon Math. So c . (2) 21 (1980), 171–175 . [11] F. G¨ ahler, Matc hing r ules for quasicrystals: the composition-decomp osition met hod, J. Non- Cryst. Solids 1 53-154 (1993), 160–164 . [12] C. Huck, Uniqueness i n discrete tomograph y of planar model sets, notes (2007); arXiv:ma th/07011 41v2 [m ath.MG] 10 CHRISTIAN HUCK [13] C. Huck, Uniqueness in discrete tomograph y of Delone sets with long-range order, submitte d; arXiv:07 11.4525v 1 [math.MG] [14] C. Huck , Discr ete T omo gr aphy of Delone Sets with L ong-R ange Or der , Ph.D. thesis (Univer- sit¨ at Bielefeld), Logos V erlag, Berlin, 2007. [15] S. Lang, A lgebr a , 3rd ed., Addison-W esley , Reading, MA, 199 3. [16] N. J. A. Sloane (ed.), The Online Encyclop e dia of Inte ger Se quenc es , published electronically at http:/ /www.res earch.at t.com/~njas/sequences/ [17] L. C. W ashington, Intr o duction to Cyclotomic Fields , 2nd ed., Springer, New Y ork, 1997. Dep ar tment of Ma themat ics and S t a tistics, The Open University, W al ton Hall, Mil- ton Keynes, MK7 6AA, United King dom E-mail addr ess : c.huck@ope n.ac.uk