Discrete Tomography of Icosahedral Model Sets
The discrete tomography of B-type and F-type icosahedral model sets is investigated, with an emphasis on reconstruction and uniqueness problems. These are motivated by the request of materials science for the unique reconstruction of quasicrystalline…
Authors: Christian Huck
DISCRETE TOMOGRAPHY OF ICOSAHEDRAL MODEL SETS CHRISTIAN HUCK Abstra t. The disrete tomograph y of B-t yp e and F-t yp e iosahedral mo del sets is in v es- tigated, with an emphasis on reonstrution and uniqueness problems. These are motiv ated b y the request of materials siene for the unique reonstrution of quasirystalline stru- tures from a small n um b er of images pro dued b y quan titativ e high resolution transmission eletron mirosop y . 1. Intr odution Disr ete tomo gr aphy (the w ord tomograph y is deriv ed from the Greek τ oµoσ , meaning a slie) is onerned with the in v erse problem of retrieving information ab out some nite ob jet from (generally noisy) information ab out its slies. A t ypial example is the r e onstrution of a nite p oin t set in Eulidean 3 -spae from its line sums in a small n um b er of diretions. More preisely , a ( disr ete p ar al lel ) X-r ay of a nite subset of Eulidean d -spae R d in diretion u giv es the n um b er of p oin ts of the set on ea h line in R d parallel to u . This onept should not b e onfused with X -ra ys in diration theory , whi h pro vide rather dieren t information on the underlying struture that is based on statistial pair orrelations; ompare [10℄, [12 ℄ and [19 ℄. In the lassial setting, motiv ated b y rystals, the p ositions to b e determined form a subset of a ommon translate of the ubi lattie Z 3 or, more generally , of an arbitrary lattie L in R 3 . In fat, man y of the problems in disrete tomograph y ha v e b een studied on Z 2 , the lassial planar setting of disrete tomograph y; see [ 21 ℄, [17 ℄ and [16 ℄. Bey ond the ase of p erfet rystals, one has to tak e in to aoun t wider lasses of sets, or at least signian t deviations from the lattie struture. As an in termediate step b et w een p erio di and random (or amorphous) Delone sets, w e onsider systems of ap erio di or der , more preisely , so-alled mo del sets (or mathemati al quasirystals ), whi h are ommonly regarded as go o d mathematial mo dels for quasirystalline strutures in nature [38 ℄. Our in terest in the disrete tomograph y of mo del sets is mainly motiv ated b y the task of struture determination of quasirystals, a new t yp e of solids diso v ered 25 y ears ago; see [ 33 ℄ for the pioneering pap er and [37 , 25 , 11℄ for ba kground and appliations. More preisely , w e address the problem of uniquely reonstruting three-dimensional quasirystals from their images under quan titativ e high r esolution tr ansmission ele tr on mir os opy (HR TEM) in a small n um b er of diretions. In fat, in [26℄ and [36℄ a te hnique is desrib ed, based on HR TEM, whi h an eetiv ely measure the n um b er of atoms lying on lines parallel to ertain diretions; it is alled QUANTITEM ( QU an titativ e AN alysis of T he I nformation from T ransmission E letron M irosop y). A t presen t, the measuremen t of the n um b er of atoms lying on a line an only b e appro ximately a hiev ed for some rystals; f. [26 , 36℄. Ho w ev er, it is reasonable to exp et that future dev elopmen ts in te hnology will impro v e this situation. 1 2 C. HUCK In this text, w e onsider b oth B-typ e and F-typ e i osahe dr al mo del sets Λ in 3 -spae whi h an b e desrib ed in algebrai terms b y using the i osian ring ; f. [8 ℄, [27℄ and [29 ℄. Note that the terminology originates from the fat that the underlying Z -mo dules (to b e explained in Setion 3) of B-t yp e and F-t yp e iosahedral mo del sets an b e obtained as pro jetions of b o dy-en tred and fae-en tred h yp erubi latties in 6 -spae, resp etiv ely . The F-t yp e iosa- hedral phase is the most ommon among the iosahedral quasirystals. Belo w, w e nev ertheless dev elop the theory for b oth the B-t yp e (also alled I-t yp e) and the F-t yp e phase. W ell kno wn examples of iosahedral quasirystals inlude the aluminium allo ys AlMn and AlCuF e; f. [22 ℄ for further examples. In pratie, only X -ra ys in Λ -diretions, i.e. , diretions parallel to non-zero elemen ts of the dierene set Λ − Λ of Λ ( i.e. , the set of in terp oin t v etors of Λ ) are reasonable. This is due to the fat that X -ra ys in non- Λ -diretions are meaningless sine the resolution oming from su h X -ra ys w ould not b e go o d enough to allo w a quan titativ e analysis neigh b ouring lines are not suien tly separated. In fat, in order to obtain appliable results, one ev en has to nd Λ -diretions that guaran tee HR TEM images of high resolution, i.e. , yield dense lines in the orresp onding quasirystal Λ . An y lattie L in R d an b e slied in to latties of dimension d − 1 . More generally , mo del sets ha v e a dimensional hierar h y , i.e. , an y mo del set in d dimensions an b e slied in to mo del sets of dimension d − 1 . In Prop osition 3.16, it is sho wn that generi (to b e explained in Setion 3) B-t yp e and F-t yp e iosahedral mo del sets an b e slied in to (planar) ylotomi mo del sets , whose disrete tomograph y w e ha v e studied earlier; f. [4 , 24 ℄ and [23℄. The latter observ ation will b e ruial, sine it enables us to use the results on the disrete tomograph y of ylotomi mo del sets, slie b y slie. Using the sliing of generi iosahedral mo del sets in to ylotomi mo del sets and the results from [4℄, it w as sho wn in [24℄ that the algorithmi problem of r e onstruting nite subsets of a large lass of generi iosahedral mo del sets Λ ( i.e. , those with p olyhedral windo ws) giv en X -ra ys in two Λ -diretions an b e solv ed in p olynomial time in the real RAM-mo del of omputation (Theorem 4.3 ). Sine this r e onstrution pr oblem an p ossess rather dieren t solutions, one is led to the in v estigation of the orresp onding uniqueness pr oblem , i.e. , the (unique) determination of nite subsets of a xed iosahedral mo del set Λ b y X -ra ys in a small n um b er of suitably presrib ed Λ -diretions. Here, a subset E of the set of all nite subsets of a xed iosahedral mo del set Λ is said to b e determine d b y the X -ra ys in a nite set U of diretions if dieren t sets F and F ′ in E annot ha v e the same X -ra ys in the diretions of U . Sine, as demonstrated in Prop osition 5.1, an y xed n um b er of X -ra ys in Λ -diretions is insuien t to determine the en tire lass of nite subsets of a xed iosahedral mo del set Λ , it is neessary to imp ose some restrition in order to obtain p ositiv e uniqueness results. In Prop osition 5.3 , it is sho wn that the nite subsets F of ardinalit y less than or equal to some k ∈ N of a xed iosahedral mo del set Λ are determined b y an y set of k + 1 X -ra ys in pairwise non-parallel Λ -diretions. Prop osition 5.6 then sho ws that, for ev ery R > 0 and an y xed iosahedral mo del set Λ , there are t w o non-parallel Λ -diretions su h that the set of b ounded subsets of Λ with diameter less than R is determined b y the X -ra ys in these diretions. F or our main result, w e restrit the set of nite subsets of a xed iosahedral mo del set Λ b y onsidering the lass of onvex subsets of Λ . They are nite sets C ⊂ Λ whose on v ex h ulls on tain no new p oin ts of Λ , i.e. , nite sets C ⊂ Λ with C = con v ( C ) ∩ Λ . By using the DISCRETE TOMOGRAPHY OF ICOSAHEDRAL MODEL SETS 3 sliing of generi iosahedral mo del sets in to ylotomi mo del sets again, it is sho wn that there are four pairwise non-parallel Λ -diretions su h that the set of on v ex subsets of an y iosahedral mo del set Λ are determined b y their X -ra ys in these diretions (Theorem 5.12 ). In fat, it turns out that one an ho ose four Λ -diretions whi h pro vide uniqueness and yield dense lines in iosahedral mo del sets, the latter making this result lo ok promising in view of real appliations (Example 5.14 and Remark 5.15). Finally , w e demonstrate that, in an appro ximativ e sense, this result holds in a far more general (and relev an t) situation, where one deals with a whole family of generi iosahedral mo del sets at the same time, rather than dealing with a single xed iosahedral mo del set. 2. Preliminaries and not a tion Natural n um b ers are alw a ys assumed to b e p ositiv e, i.e. , N = { 1 , 2 , 3 , . . . } . Throughout the text, w e use the on v en tion that the sym b ol ⊂ inludes equalit y . W e denote the norm in Eulidean d -spae R d b y k · k . The unit sphere in R d is denoted b y S d − 1 , i.e. , S d − 1 = { x ∈ R d | k x k = 1 } . Moreo v er, the elemen ts of S d − 1 are also alled dir e tions . Reall that a homothety h : R d → R d is giv en b y x 7→ λx + t , where λ ∈ R is p ositiv e and t ∈ R d . W e all a homothet y exp ansive if λ > 1 . If x ∈ R , then ⌊ x ⌋ denotes the greatest in teger less than or equal to x . F or r > 0 and x ∈ R d , B r ( x ) is the op en ball of radius r ab out x . F or a subset S ⊂ R d , k ∈ N and R > 0 , w e denote b y card( S ) , F ( S ) , F ≤ k ( S ) , D 0 su h that ev ery ball B r ( x ) with x ∈ R d on tains at most one p oin t of Λ . F urther, Λ is alled r elatively dense if there is a radius R > 0 su h that ev ery ball B R ( x ) with x ∈ R d on tains at least one p oin t of Λ . Remark 3.6. Let Λ b e an iosahedral mo del set with windo w W . Then, Λ is a Delone set in R 3 ( i.e. , Λ is b oth uniformly disrete and relativ ely dense) and is of nite lo al omplexity ( i.e. , Λ − Λ is losed and disrete). Note that Λ is of nite lo al omplexit y if and only if for ev ery r > 0 there are, up to translation, only nitely man y p oin t sets (alled p athes of diameter r ) of the form Λ ∩ B r ( x ) , where x ∈ R 3 ; f. [35, Prop osition 2.3℄. In fat, Λ is ev en a Meyer set , i.e. , Λ is a Delone set and Λ − Λ is uniformly disrete; ompare [ 27℄. F urther, Λ is an ap erio di mo del set, i.e. , Λ has no translational symmetries. Moreo v er, if Λ is r e gular , Λ is pur e p oint dir ative , i.e. , the F ourier transform of the auto orrelation densit y that arises b y plaing a delta p eak (p oin t mass) on ea h p oin t of Λ lo oks purely p oin t-lik e; f. [35 ℄. If Λ is generi, Λ is r ep etitive , i.e. , giv en an y pat h of radius r , there is a radius R > 0 su h that an y ball of radius R on tains at least one translate of this pat h; f. [35 ℄. If Λ is regular, the frequeny of rep etition of nite pat hes is w ell dened, i.e. , for an y pat h of radius r , the n um b er of o urrenes of translates of this pat h p er unit v olume in the ball B r (0) of radius r > 0 ab out the origin 0 approa hes a non-negativ e limit as r → ∞ ; f. [34 ℄. Moreo v er, if Λ is b oth generi 8 C. HUCK Figure 1. A few slies of a pat h of the iosahedral mo del set Λ B ico (left) and their . ⋆ -images inside the iosahedral windo w in the in ternal spae (righ t), b oth seen from the p ositiv e x -axis. and regular, and, if a suitable translate of the windo w W has full iosahedral symmetry ( i.e. , if a suitable translate of the windo w W is in v arian t under the ation of the group Y ⋆ h of order 120 , where Y ⋆ h := Y ⋆ ∪ ( − Y ⋆ ) and Y ⋆ is the group of rotations of order 60 generated b y the t w o matries that arise from the t w o matries in ( 3 ) b y applying the onjugation . ′ to ea h en try), then Λ has full iosahedral symmetry Y h := Y ∪ ( − Y ) in the sense of symmetries of LI-lasses, meaning that a disrete struture has a ertain symmetry if the original and the transformed struture are lo ally indistinguishable (LI) ( i.e. , up to translation, ev ery nite pat h in Λ also app ears in an y of the other elemen ts of its LI-lass and vi e versa ); see [3℄ for details. T ypial examples are balls and suitably orien ted v ersions of the iosahedron, the do deahedron, the rhom bi triaon tahedron (the latter also kno wn as Kepler's b o dy) and its dual, the iosido deahedron. Example 3.7. F or a generi regular iosahedral mo del set with full iosahedral symmetry Y h , onsider Λ B ico := Λ B ico (0 , s + W ) , where s := 10 − 3 (1 , 1 , 1) t and W is the regular iosahedron with v ertex set Y ⋆ h ( τ ′ , 0 , 1) t ; see Figure 1 for an illustration. 3.2. Cylotomi mo del sets as planar setions of iosahedral mo del sets. In this setion, w e shall demonstrate that b oth B-t yp e and F-t yp e iosahedral mo del sets Λ an b e niely slied in to ylotomi mo del sets with underlying Z -mo dule Z [ ζ 5 ] , where the slies are in tersetions of Λ with translates of the h yp erplane H ( τ , 0 , 1) in R 3 orthogonal to ( τ , 0 , 1) t . DISCRETE TOMOGRAPHY OF ICOSAHEDRAL MODEL SETS 9 F rom no w on, w e alw a ys let ζ 5 := e 2 π i/ 5 , as a sp ei hoie of a primitiv e 5 th ro ot of unit y in C . Oasionally , w e iden tify C with R 2 . Remark 3.8. It is w ell kno wn that the 5 th ylotomi eld Q ( ζ 5 ) is an algebrai n um b er eld of degree 4 o v er Q . Moreo v er, the eld extension Q ( ζ 5 ) / Q is a Galois extension with Ab elian Galois group G ( Q ( ζ 5 ) / Q ) ≃ ( Z / 5 Z ) × , where a ( mo d 5) orresp onds to the automorphism giv en b y ζ 5 7→ ζ a 5 ; f. [39, Theorem 2.5℄. Note that, restrited to the quadrati eld Q ( τ ) , b oth the Galois automorphism of Q ( ζ 5 ) / Q that is giv en b y ζ 5 7→ ζ 3 5 and its omplex onjugate automorphism ( i.e. , the automorphism giv en b y ζ 5 7→ ζ 2 5 ) indue the unique non-trivial Galois automorphism . ′ of Q ( τ ) / Q (determined b y τ 7→ 1 − τ ). F urther, Z [ ζ 5 ] is the ring of in tegers in Q ( ζ 5 ) ; f. [39, Theorem 2.6℄. The ring Z [ ζ 5 ] also is a Z [ τ ] -mo dule of rank t w o. More preisely , one has the equalit y Z [ ζ 5 ] = Z [ τ ] ⊕ Z [ τ ] ζ 5 ; f. [4 , Lemma 1(a)℄. Sine ζ 3 5 is also a primitiv e 5 th ro ot of unit y in C , one further has the equalit y Z [ ζ 5 ] = Z [ ζ 3 5 ] = Z [ τ ] ⊕ Z [ τ ] ζ 3 5 . Denition 3.9. Cylotomi mo del sets with underlying Z -mo dule Z [ ζ 5 ] Λ cyc ( t, W ) arise from the ut and pro jet s heme (4 ) b y setting d := m := 2 , L := Z [ ζ 5 ] and letting the star map . ⋆ 5 : L → R 2 b e either giv en b y the non-trivial Galois automorphism of Q ( ζ 5 ) / Q , dened b y ζ 5 7→ ζ 3 5 , or its omplex onjugate automorphism. Remark 3.10. The star map . ⋆ 5 as dened in Denition 3.9 is a monomorphism of Ab elian groups. F urther, the image of the map ˜ . 5 : L → R 2 × R 2 , dened b y α 7→ ( α, α ⋆ 5 ) , is indeed a lattie in R 2 × R 2 . Finally , one an v erify that the image L ⋆ 5 is indeed a dense subset of R 2 . F or the general setting, w e refer the reader to [4, 24 , 23 ℄. By [24 , Lemma 1.84(a)℄ (see also [23 , Lemma 25(a)℄), for all ylotomi mo del sets Λ with underlying Z -mo dule Z [ ζ 5 ] , the set of Λ -diretions is preisely the set of Z [ ζ 5 ] -diretions. Example 3.11. F or illustrations of ylotomi mo del sets with underlying Z -mo dule Z [ ζ 5 ] , see Figure 2 on the left and Figure 3; f. Prop osition 3.16 and Example 3.17 b elo w. Lemma 3.12. F or L ∈ { Im( I ) , I 0 } , the fol lowing e quations hold: (a) L ∩ H ( τ , 0 , 1) = Z [ τ ](0 , 1 , 0) t ⊕ Z [ τ ] 1 2 ( − 1 , − τ ′ , τ ) t . (b) ( L ∩ H ( τ , 0 , 1) ) ⋆ = L ⋆ ∩ H ( τ ′ , 0 , 1) . Pr o of. P art (a) follo ws from Equations (1) and (2) together with the relations Im( I ) = 1 2 M B and I 0 = 1 2 M F . P art (b) follo ws from the iden tit y (( τ , 0 , 1) t ) ⋆ = ( τ ′ , 0 , 1) t . Denition 3.13. W e denote b y Φ the R -linear isomorphism Φ : H ( τ , 0 , 1) → C , determined b y (0 , 1 , 0) t 7→ 1 and 1 2 ( − 1 , − τ ′ , τ ) t 7→ ζ 5 . F urther, Φ ⋆ will denote the R -linear isomorphism Φ ⋆ : H ( τ ′ , 0 , 1) → C , determined b y (0 , 1 , 0) t 7→ 1 and 1 2 ( − 1 , − τ , τ ′ ) t 7→ ζ 3 5 . Lemma 3.14. The maps Φ and Φ ⋆ ar e isometries of Eulide an ve tor sp a es, wher e H ( τ , 0 , 1) , H ( τ ′ , 0 , 1) and C ar e r e gar de d as two-dimensional Eulide an ve tor sp a es in the anoni al way. Mor e over, identifying C with the xy -plane in R 3 , Φ and Φ ⋆ extend uniquely to dir e t rigid motions of R 3 , i.e., elements of the gr oup SO(3 , R ) . Pr o of. The rst assertion follo ws from the follo wing iden tities: w w r (0 , 1 , 0) t + s 1 2 ( − 1 , − τ ′ , τ ) t w w = | r + s ζ 5 | = p r 2 + s 2 − r sτ ′ , w w r (0 , 1 , 0) t + s 1 2 ( − 1 , − τ , τ ′ ) t w w = | r + s ζ 3 5 | = p r 2 + s 2 − r sτ . 10 C. HUCK Figure 2. The en tral slie of the pat h of Λ B ico from Figure 1 (left) and its . ⋆ -image inside the (mark ed) deagon ( s + W ) ∩ H ( τ ′ , 0 , 1) (righ t), b oth seen from p erp endiular viewp oin ts. The additional statemen t is immediate. Lemma 3.15. L et L ∈ { Im( I ) , I 0 } . Via r estrition, the maps Φ and Φ ⋆ indu e isomorphisms of r ank two Z [ τ ] -mo dules: L ∩ H ( τ , 0 , 1) Φ − → Z [ ζ 5 ] , L ⋆ ∩ H ( τ ′ , 0 , 1) Φ ⋆ − → Z [ ζ 5 ] . Pr o of. This follo ws immediately from the denition of Φ and Φ ⋆ together with Lemma 3.12 and Remark 3.8. Prop osition 3.16. L et Λ b e a generi i osahe dr al mo del set with underlying Z -mo dule L , say Λ = Λ ico ( t, W ) . Then, for every λ ∈ Λ , one has the identity Φ ( Λ ∩ ( λ + H ( τ , 0 , 1) )) − λ = z ∈ Z [ ζ 5 ] z ⋆ 5 ∈ W λ , wher e . ⋆ 5 is the Galois automorphism of Q ( ζ 5 ) / Q , dene d by ζ 5 7→ ζ 3 5 and W λ := Φ ⋆ ( W ∩ (( λ − t ) ⋆ + H ( τ ′ , 0 , 1) )) − ( λ − t ) ⋆ . Thus, the sets of the form Φ ( Λ ∩ ( λ + H ( τ , 0 , 1) )) − λ , (5) wher e λ ∈ Λ , ar e ylotomi mo del sets with underlying Z -mo dule Z [ ζ 5 ] . Pr o of. First, onsider Φ( µ ) , where µ ∈ ( Λ ∩ ( λ + H ( τ , 0 , 1) )) − λ . It follo ws that µ ∈ L ∩ H ( τ , 0 , 1) and ( µ + ( λ − t )) ⋆ = µ ⋆ + ( λ − t ) ⋆ ∈ W . Lemma 3.15 implies that Φ( µ ) ∈ Z [ ζ 5 ] , sa y Φ( µ ) = α + β ζ 5 for suitable α, β ∈ Z [ τ ] . One has Φ( µ ) ⋆ 5 = α ′ + β ′ ζ 3 5 = Φ ⋆ ( µ ⋆ ) ∈ W λ . DISCRETE TOMOGRAPHY OF ICOSAHEDRAL MODEL SETS 11 Figure 3. Another t w o slies of the pat h of Λ B ico from Figure 1. Con v ersely , supp ose that z ∈ Z [ ζ 5 ] satises z ⋆ 5 ∈ W λ . Then, there are suitable α, β ∈ Z [ τ ] su h that z = α + β ζ 5 and, onsequen tly , z ⋆ 5 = α ′ + β ′ ζ 3 5 ∈ W λ . By denition of W λ , one has z ⋆ 5 = Φ ⋆ ( µ ) , where µ ∈ H ( τ ′ , 0 , 1) satises µ + ( λ − t ) ⋆ ∈ W . Clearly , there exist r , s ∈ R su h that µ = r (0 , 1 , 0) t + s 1 2 ( − 1 , − τ , τ ′ ) t , whene Φ ⋆ ( µ ) = r + sζ 3 5 . The linear indep endene of 1 and ζ 3 5 o v er R no w implies that r = α and s = β , so that µ ∈ L ⋆ . Moreo v er, one an v erify that one has µ − ⋆ ∈ ( Λ ∩ ( λ + H ( τ , 0 , 1) )) − λ and Φ( µ − ⋆ ) = α + β ζ 5 = z . This pro v es the laimed iden tit y . The assertion is no w immediate. Example 3.17. F or an illustration of the on ten t of Prop osition 3.16 in ase of the iosahedral mo del set Λ B ico from Example 3.7 , see Figures 2 and 3. 3.3. The translation mo dule of iosahedral mo del sets. In order to shed some ligh t on the set of Λ -diretions of an iosahedral mo del set Λ with underlying Z -mo dule L , w e rst ha v e to establish a relation b et w een iosahedral mo del sets and their underlying Z -mo dules. W e denote b y m τ the Z [ τ ] -mo dule endomorphism of Q ( τ ) 3 , giv en b y m ultipliation b y τ , i.e. , α 7→ τ α . F urthermore, w e denote b y m τ ⋆ the Z [ τ ] -mo dule endomorphism of ( Q ( τ ) 3 ) ⋆ , giv en b y α ⋆ 7→ ( τ α ) ⋆ . Lemma 3.18. The map m τ ⋆ is ontr ative with ontr ation onstant 1 /τ ∈ (0 , 1) , i.e. , the e quality k m τ ⋆ ( α ⋆ ) k = (1 /τ ) k α ⋆ k holds for al l α ∈ Q ( τ ) 3 . Pr o of. F or α ∈ Q ( τ ) 3 , observ e that k m τ ⋆ ( α ⋆ ) k = k ( τ α ) ⋆ k = k τ ′ α ⋆ k = (1 /τ ) k α ⋆ k . Lemma 3.19. L et Λ b e an i osahe dr al mo del set with underlying Z -mo dule L , say Λ = Λ ico ( t, W ) . Then, for any F ∈ F ( L ) , ther e is an exp ansive homothety h : R 3 → R 3 suh that h ( F ) ⊂ Λ . Pr o of. F rom in t( W ) 6 = ∅ and the denseness of L ⋆ in R 3 , one gets the existene of a suitable α 0 ∈ L with α 0 ⋆ ∈ in t ( W ) . Consider the op en neigh b ourho o d V := in t( W ) − α 0 ⋆ of 0 in 12 C. HUCK R 3 . Sine the map m τ ⋆ is on trativ e b y Lemma 3.18 (in the sense whi h w as made preise in that lemma), the existene of a suitable k ∈ N is implied su h that ( m τ ⋆ ) k ( F ⋆ ) ⊂ V . Hene, one has { ( τ k α + α 0 ) ⋆ | α ∈ F } ⊂ in t( W ) ⊂ W and, further, h ( F ) ⊂ Λ , where h : R 3 → R 3 is the expansiv e homothet y giv en b y x 7→ τ k x + ( α 0 + t ) . As an easy appliation of Lemma 3.19 , one obtains the follo wing result on the set of Λ - diretions for iosahedral mo del sets Λ . Prop osition 3.20. L et Λ b e an i osahe dr al mo del set with underlying Z -mo dule L . Then, the set of Λ -dir e tions is pr e isely the set of L -dir e tions. Pr o of. Sine one has Λ − Λ ⊂ L , ev ery Λ -diretion is an L -diretion. F or the on v erse, let u ∈ S 2 b e an L -diretion, sa y parallel to α ∈ L \ { 0 } . By Lemma 3.19, there is a homothet y h : R 3 → R 3 su h that h ( { 0 , α } ) ⊂ Λ . It follo ws that h ( α ) − h (0) ∈ ( Λ − Λ ) \ { 0 } . Sine h ( α ) − h (0) is parallel to α , the assertion follo ws. 4. Complexity In the pratie of quan titativ e HR TEM, the determination of the rotational orien tation of a quasirystalline prob e in an eletron mirosop e an rather easily b e a hiev ed in the diration mo de. This is due to the iosahedral symmetry of gen uine iosahedral quasirystals. Ho w ev er, the X -ra y images tak en in the high-resolution mo de do not allo w us to lo ate the examined sets. Therefore, as already p oin ted out in [4℄, in order to pro v e pratially relev an t and rigorous results, one has to deal with the non-anhor e d ase of the whole lo al indistinguishability lass (or LI-lass, for short) LI( Λ ) of a regular, generi iosahedral mo del set Λ , rather than dealing with the anhor e d ase of a single xed iosahedral mo del set Λ ; reall Remark 3.6 for the equiv alene relation giv en b y lo al indistinguishabilit y and ompare also [18℄. Remark 4.1. In the rystallographi ase of a lattie L in R 3 , the LI-lass of L onsists of all translates of L in R 3 , i.e. , one has LI( L ) = { t + L | t ∈ R 3 } . In partiular, LI( L ) simply onsists of one translation lass. The en tire LI-lass LI( Λ ico ( t, W )) of a regular, generi iosahedral mo del set Λ ico ( t, W ) an b e sho wn to onsist of all generi iosahedral mo del sets of the form Λ ico ( t, s + W ) and all patterns obtained as limits of sequenes of generi iosahedral mo del sets of the form Λ ico ( t, s + W ) in the lo al top ology (L T). Here, t w o patterns are ε -lose if, after a translation b y a distane of at most ε , they agree on a ball of radius 1 /ε around the origin; see [3, 35 ℄. Ea h su h limit is then a subset of some Λ ico ( t, s + W ) , but s migh t not b e in a generi p osition. Note that the LI-lass LI( Λ ) of an iosahedral mo del set Λ on tains un ountably many (more preisely , 2 ℵ 0 ) translation lasses; f. [3℄ and referenes therein. In view of the ompliation desrib ed ab o v e, w e m ust mak e sure that w e deal with nite subsets of generi iosahedral mo del sets of the form Λ ico ( t, s + W ) , i.e. , subsets whose . ⋆ -image lies in the interior of the windo w. This restrition to the generi ase is the prop er analogue of the restrition to p erfe t latties and their translates in the rystallographi ase. Analogous to the lattie ase [15 , 16℄ and the ase of ylotomi mo del sets [ 4℄, the main algorithmi problems of the disrete tomograph y of iosahedral mo del sets lo ok as follo ws. DISCRETE TOMOGRAPHY OF ICOSAHEDRAL MODEL SETS 13 Denition 4.2 (Consisteny , Reonstrution, and Uniqueness Problem) . Let L = Im( I ) (resp., L = I 0 ), let W ⊂ R 3 b e a windo w and let u 1 , . . . , u m ∈ S 2 b e m ≥ 2 pairwise non- parallel L -diretions. The orresp onding onsisteny , reonstrution and uniqueness problems are dened as follo ws. Consisteny . Giv en funtions p u j : L 3 u j → N 0 , j ∈ { 1 , . . . , m } , whose supp orts are nite and satisfy supp( p u j ) ⊂ L L u j , deide whether there is a nite set F whi h is on tained in an elemen t of I B g ( W ) (resp., I F g ( W ) ) and satises X u j F = p u j , j ∈ { 1 , . . . , m } . Reonstr ution . Giv en funtions p u j : L 3 u j → N 0 , j ∈ { 1 , . . . , m } , whose supp orts are nite and satisfy supp( p u j ) ⊂ L L u j , deide whether there exists a nite subset F of an elemen t of I B g ( W ) (resp., I F g ( W ) ) that satises X u j F = p u j , j ∈ { 1 , . . . , m } , and, if so, onstrut one su h F . Uniqueness . Giv en a nite subset F of an elemen t of I B g ( W ) (resp., I F g ( W ) ), deide whether there is a dieren t nite set F ′ that is also a subset of an elemen t of I B g ( W ) (resp., I F g ( W ) ) and satises X u j F = X u j F ′ , j ∈ { 1 , . . . , m } . One has the follo wing tratabilit y result, whi h w as pro v ed for the ase of B-t yp e iosahedral mo del sets b y om bining the results from Setion 3.2 with those presen ted in [4℄; f. [24 , Theorem 3.33℄ for the details. The pro of for the F-t yp e ase is similar and w e prefer to omit the straigh tforw ard details here. Belo w, for L ∈ { Im( I ) , I 0 } , the L -diretions in S 2 ∩ H ( τ , 0 , 1) will b e alled L ( τ , 0 , 1) -dir e tions . By Lemma 3.12 (a), the set of Im( I ) ( τ , 0 , 1) -diretions and the set of I ( τ , 0 , 1) 0 -diretions oinide. Theorem 4.3. L et L ∈ { Im( I ) , I 0 } . When r estrite d to two L ( τ , 0 , 1) -dir e tions and p olyhe- dr al windows, the pr oblems Consisteny , Reonstr ution and Uniqueness as dene d in Denition 4.2 an b e solve d in p olynomial time in the r e al RAM-mo del of omputation. Remark 4.4. F or a detailed analysis of the omplexities of the ab o v e algorithmi problems in the B-t yp e ase, w e refer the reader to [ 24 , Chapter 3℄. Note that ev en in the an hored planar lattie ase Z 2 the orresp onding problems Consisteny , Reonstr ution and Uniqueness are NP -hard for three or more Z 2 -diretions; f. [15 , 16℄. 5. Uniqueness 5.1. Simple results on determination of nite subsets of iosahedral mo del sets. In this setion, w e presen t some uniqueness results whi h only deal with the anhor e d ase of determining nite subsets of a xed iosahedral mo del set Λ b y X -ra ys in arbitr ary Λ - diretions; f. Prop osition 3.20. As already explained in Setion 1, X -ra ys in non- Λ -diretions are meaningless in pratie. Without the restrition to Λ -diretions, the nite subsets of a xed iosahedral mo del set Λ an b e determined b y one X -ra y . In fat, an y X -ra y in a non- Λ -diretion is suitable for this purp ose, sine an y line in 3 -spae in a non- Λ -diretion passes through at most one p oin t of Λ . The next result represen ts a fundamen tal soure of diulties 14 C. HUCK in disrete tomograph y . There exist sev eral v ersions; ompare [21, Theorem 4.3.1℄, [13, Lemma 2.3.2℄, [5 , Prop osition 4.3℄, [24 , Prop osition 2.3 and Remark 2.4℄ and [23 , Prop osition 8℄. Prop osition 5.1. L et Λ b e an i osahe dr al mo del set with underlying Z -mo dule L , say Λ = Λ ico ( t, W ) . F urther, let U ⊂ S 2 b e an arbitr ary, but xe d nite set of p airwise non-p ar al lel L -dir e tions. Then, F ( Λ ) is not determine d by the X -r ays in the dir e tions of U . Pr o of. W e argue b y indution on card( U ) . The ase card( U ) = 0 means U = ∅ and is ob vious. Supp ose the assertion to b e true whenev er card( U ) = k ∈ N 0 and let card( U ) = k + 1 . By indution h yp othesis, there are dieren t elemen ts F and F ′ of F ( Λ ) with the same X -ra ys in the diretions of U ′ , where U ′ ⊂ U satises card( U ′ ) = k . Let u b e the remaining diretion of U . Cho ose a non-zero elemen t α ∈ L parallel to u su h that α + ( F ∪ F ′ ) and F ∪ F ′ are disjoin t. Then, F ′′ := ( F ∪ ( α + F ′ )) − t and F ′′′ := ( F ′ ∪ ( α + F )) − t are dieren t elemen ts of F ( L ) with the same X -ra ys in the diretions of U . By Lemma 3.19 , there is a homothet y h : R 3 → R 3 su h that h ( F ′′ ∪ F ′′′ ) = h ( F ′′ ) ∪ h ( F ′′′ ) ⊂ Λ . It follo ws that h ( F ′′ ) and h ( F ′′′ ) are dieren t elemen ts of F ( Λ ) with the same X -ra ys in the diretions of U ; f. Lemma 2.6 . Remark 5.2. An analysis of the pro of of Prop osition 5.1 sho ws that, for an y nite set U ⊂ S 2 of k pairwise non-parallel L -diretions, there are disjoin t elemen ts F and F ′ of F ( Λ ) with card( F ) = card( F ′ ) = 2 ( k − 1) and with the same X -ra ys in the diretions of U . Consider an y on v ex subset C of R 3 whi h on tains F and F ′ from ab o v e. Then, the subsets F 1 := ( C ∩ Λ ) \ F and F 2 := ( C ∩ Λ ) \ F ′ of F ( Λ ) also ha v e the same X -ra ys in the diretions of U . Whereas the p oin ts in F and F ′ are widely disp ersed o v er a region, those in F 1 and F 2 are on tiguous in a w a y similar to atoms in a quasirystal; ompare [ 15 , Remark 4.3.2℄ and [23 , Remark 2.4 and Figure 2.1℄ (see also [23 , Remark 32 and Figure 5℄). Originally , the pro of of the follo wing result is due to Rén yi; f. [32 ℄ and ompare [21 , Theorem 4.3.3℄. Prop osition 5.3. L et Λ b e an i osahe dr al mo del set with underlying Z -mo dule L . F urther, let U ⊂ S 2 b e any set of k + 1 p airwise non-p ar al lel L -dir e tions, wher e k ∈ N 0 . Then, F ≤ k ( Λ ) is determine d by the X -r ays in the dir e tions of U . Mor e over, for al l F ∈ F ≤ k ( Λ ) , one has G F U = F . Pr o of. Let F , F ′ ∈ F ≤ k ( Λ ) ha v e the same X -ra ys in the diretions of U . Then, one has card( F ) = card( F ′ ) b y Lemma 2.2 (a) and F , F ′ ⊂ G U F b y Lemma 2.4. But w e ha v e G U F = F sine the existene of a p oin t in G U F \ F implies the existene of at least card( U ) ≥ k + 1 p oin ts in F , a on tradition. It follo ws that F = F ′ . Remark 5.4. In partiular, the additional statemen t of Prop osition 5.3 demonstrates that, for a xed iosahedral mo del set Λ with underlying Z -mo dule L , the unique reonstrution of sets F ∈ F ≤ k ( Λ ) from their X -ra ys in arbitrary sets of k + 1 pairwise non-parallel L -diretions U ⊂ S 2 merely amoun ts to ompute the grids G U F . Let Λ b e an iosahedral mo del set with underlying Z -mo dule L . Remark 5.2 and Prop osition 5.3 sho w that F ≤ k ( Λ ) an b e determined b y the X -ra ys in an y set of k + 1 pairwise non-parallel L -diretions but not b y 1 + ⌊ log 2 k ⌋ pairwise non-parallel X -ra ys in L -diretions. Ho w ev er, in pratie, one is in terested in the determination of nite sets b y X -ra ys in a small n um b er of diretions sine after ab out 3 to 5 images tak en b y HR TEM, the ob jet ma y b e damaged or ev en destro y ed b y the radiation DISCRETE TOMOGRAPHY OF ICOSAHEDRAL MODEL SETS 15 energy . Observing that the t ypial atomi strutures to b e determined omprise ab out 10 6 to 10 9 atoms, one realizes that the last result is not pratial at all. The follo wing result w as pro v ed in [ 24, Theorem 2.8(a)℄; see also [ 23 , Theorem 13(a)℄. Prop osition 5.5. L et d ≥ 2 , let R > 0 , and let Λ ⊂ R d b e a Delone set of nite lo al omplexity. Then, the set D 0 . Then, the set D 0 (whi h is rather natural in pratie), then Theo- rem 5.16 allo ws us to write DISCRETE TOMOGRAPHY OF ICOSAHEDRAL MODEL SETS 19 1 v ol( W ) Z s + W y d λ ( y ) ≈ 1 card ( F − t ) X x ∈ F − t x ⋆ = 1 card ( F ′ − t ) X x ∈ F ′ − t x ⋆ ≈ 1 v ol( W ) Z ( s ′ +( t ′ − t ) ⋆ )+ W y d λ ( y ) . Consequen tly , s + Z W y d λ ( y ) ≈ ( s ′ + ( t ′ − t ) ⋆ ) + Z W y d λ ( y ) , and hene s ≈ s ′ + ( t ′ − t ) ⋆ . The latter means that, appro ximately , b oth F − t and F ′ − t are elemen ts of the set C ( Λ B ico (0 , s + W )) . No w, it follo ws in this appro ximativ e sense from prop ert y (C) and Theorem 5.12 that F − t ≈ F ′ − t , and, nally , F ≈ F ′ . Remark 5.19. The ab o v e analysis suggests that, for all xed windo ws W ⊂ R 3 whose b oundary b d( W ) has Leb esgue measure 0 in R 3 , the sets of the form ∪ Λ ∈I B g ( W ) C ( Λ ) (resp., ∪ Λ ∈I F g ( W ) C ( Λ ) ) are appro ximately determined b y the X -ra ys in the four presrib ed L ( τ , 0 , 1) - diretions of U ico , where L = Im( I ) (resp., L = I 0 ); f. Examples 6.12 and 6.15. A dditionally , in the pratie of quan titativ e HR TEM, the resolution oming from the diretions of U ico is lik ely to b e rather high, whi h mak es this appro ximativ e result lo ok ev en more promising in view of real appliations; f. Remark 5.15 . 6. Outlook F or a more extensiv e aoun t of b oth uniqueness and omputational omplexit y results in the disrete tomograph y of Delone sets with long-range order, w e refer the reader to [ 24 ℄. This referene also on tains results on the in terativ e onept of su essive determination of nite sets b y X -ra ys and further extensions of settings and results that are b ey ond our sop e here; ompare also [23℄. Although the results of this text and of [24℄ giv e satisfying answ ers to the basi problems of disrete tomograph y of iosahedral mo del sets, there is still a lot to do to reate a to ol that is as satisfatory for the appliation in materials siene as is omputerized tomograph y in its medial or other appliations. First, w e b eliev e that it is an in teresting problem to haraterize the sets of Λ -diretions in gener al p osition ha ving the prop ert y that, for all iosahedral mo del sets Λ , the set of on v ex subsets of Λ is determined b y the X -ra ys in these diretions; ompare [13, Problems 2.1 and 2.3℄. Seondly , it w ould b e in teresting to ha v e exp erimen tal tests in order to see ho w w ell the ab o v e results w ork in pratie. Sine there is alw a ys some noise in v olv ed when ph ysial measuremen ts are tak en, the latter also requires the abilit y to w ork with impreise data. F or this, it is neessary to study stabilit y and instabilit y results in the disrete tomograph y of iosahedral mo del sets in the future; f. [ 1℄. A kno wledgements It is m y pleasure to thank Mi hael Baak e, Uw e Grimm, P eter Gritzmann, Barbara Langfeld and Reinhard Lü k for v aluable disussions and suggestions. This w ork w as supp orted b y 20 C. 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