Laminating lattices with symmetrical glue

Laminatin g lattices with symmetrical glue V eit Elser and Simon Gra vel Laboratory of Atomic and Solid State Physics Cornell Unive rsity , Ithaca, NY 14853-2501 Abstract W e use the automorp hism group Aut ( H ) , of holes in the lattice L 8 = A 2 ⊕ A 2 ⊕ D 4 , as the starting point in the constructio n o f sphe re packings in 10 and 12 dimen sions. A seco nd lattice, L 4 = A 2 ⊕ A 2 , enters the con - struction because a subgro up of Aut ( L 4 ) is isom orphic to Aut ( H ) . The lattices L 8 and L 4 , when glued tog ether thro ugh this relationsh ip, provide an alternative construction of th e laminated lattice in twelve dimen sions with kissing number 648. More interestingly , the action of Aut ( H ) on L 4 defines a pair of in variant planes through which dense, non-lattice packings in 10 di- mensions can be constructed. The most symmetric of these is aperiodic with center density 1 / 32 . These constructio ns were pro mpted by an unexpected arrange ment of 378 kissing spheres discovered by a search algorithm. Intr oduction One of the simplest geometrical constr aint problems w ith strong ties to the design of codes is the problem of kissing spher es [1]. The objec t is to pack equal spheres — as many as possible — so that each sphere touches a gi ven sphe re of the same size. The best lo wer bounds on the m aximum number of kiss ing spheres, in all dimensio ns where this probl em has be en studied, are deri ved from integ ral lattices or error correcting codes. Excep tionally symmetric lattices, such as E 8 and the Leech lattice, accou nt for the relativ ely few dimensio ns where the m aximum kiss- ing number has been esta blished . Con verse ly , searche s for goo d solu tions to the kissin g sphere s problem by an unbiased alg orithm test s the scope of the kno w n schemes for designi ng good codes. 1 372 spheres 0.0 0.1 0.2 0.3 0.4 0.5 cos H Θ L 374 spheres 0.0 0.1 0.2 0.3 0.4 0.5 cos H Θ L Figure 1: Res ults of a numerical search for kissing sphere s in 10 dimensions. Sho wn is the distrib ution of cosines of the angl es sub tended by pairs of sphe res. The distrib ution for 37 4 spher es has a pair of sharp peaks at cos ( θ ) = (3 ± √ 3) / 12 . This paper comes in response to some une xpecte d results in the search for kissin g sphere s in 10 dimension s [2]. The algorit hm used in the search worked directl y with the two kin ds of distance constraint s on the sphere centers, x i : k x i k = 2 , k x i − x j k ≥ 2 , i 6 = j . The only assumption that restricted the search was in version symmetry (the oper - ation x i → − x i merely interchange s pairs of spheres) . A rough charac terizati on of a solution is giv en by the distri b ution of the cosin es of the angles subtend ed by pairs of spher es, 4 cos ( θ ij ) = x i · x j . Because of in version symmetry the cosine distrib ution is symmetric about zero. For small nu mbers of spheres there is consid erable f reedom of mov ement i n the soluti on and the cosine distrib ution looks quasi-con tinuou s. Figure 1 (left) shows a typical outpu t produced by the algorithm for 372 spher es. The number 372 is of interes t becaus e this is the maximum kissing number of the densest kn o wn p acking in te n dimen sions, P 10 c [1]. The d ensest k no wn lat tice pack ing has k issing n umber 336. Because there e xists a non -lattice pack ing with kissing number 500 ( P 10 b ), the quasi-con tinuou s distrib ution found for 372 sphere s is not in itself surprising. What came as a surpris e was the distrib ution obt ained for 374 spheres . There the appear ance of sharp peaks , Fig. 1 (right), shows that the sphere arrang ement has tak en a ver y ordered form. Indeed, with the exceptio n of a very small number of sphere s, the cosine s found by the algorithm are (within numerical precision) all contai ned in the set C = { 0 , 1 / 2 , (3 ± √ 3) / 12 } . The irrational v alues in this set are at odds with the simple ratio nal valu es generated by integra l lattice s and error correc ting codes. 2 A foren sic examin ation of the sphere centers obtai ned by the alg orithm has re vea led that there is a maximal set of 378 spheres consistent with the cosine s C . This wa s acco mplished by first findi ng a su itable coord inate system that re produc ed the cosines C when the numerical coordin ates were appro ximated by numbers of the f orm a + b √ 3 , a, b ∈ Q . A basis reduction algorithm was then used to establish that the 378 sphere center s form a Z -module of rank 12. The main out come of the forensic examinati on, and the theme of this paper , is a motiv e for the arr angement of the 378 kissing spheres. In rou gh out line, we first observ e that a dense 12 dimens ional lattic e L 12 can be construct ed by glu- ing togeth er a pair of not excep tionall y dense lattices L 8 and L 4 in eight and four dimensio ns. T his gluing is the k ey to the constructi on of a dense non-lattice pack- ing Q 10 in ten dimensions . The reduc tion in dimens ion occurs entir ely within L 4 , which possess es a symmetr y that decompos es the four dimensional space into a pair of in var iant planes . Moreov er , as this symmetry is non-crys tallogr aphic in two dimensio ns, these pla nes are totally irrationa l subspac es with respect to L 4 . The packin g Q 10 results from the projecti on of L 12 into the orthogon al complement of one o f t hese irra tional p lanes an d e xplain s the ra nk-12 Z -mod ule di scov ered by the algori thm. What emerges as a reasonabl e moti ve is that the constr uction of Q 10 , from the dense L 12 , is such that it is able to preserve many of the sphere contacts of the lattice packi ng. W e will follo w the standard terminology of lattices w hene ver possible [1 ]. In particu lar , the norm of a v ector x is its Euclidea n inner product x · x . The automor - phism gr oup o f a la ttice, A ut (Λ) , is t he group o f unimodular transformati ons of the lattice generators that pre serv es the Euclidea n inn er pro duct. Equi valent ly , Aut (Λ) is the group of real orthogo nal transfor mations on the lattice vect ors that permutes the elements of Λ . W e use the term glue gr oup for any subgroup of Λ ∗ / Λ , where Λ ∗ is the dual of Λ . The depth of a general poin t x , in a lattice Λ , is the minimum norm in the set { x − y : y ∈ Λ , y 6 = x } , and is denote d ∆( x ) . The number of minimum norm e lements is written τ ( x ) and equa ls the kissin g number of Λ when x ∈ Λ . Holes of the lattice A 2 ⊕ A 2 ⊕ D 4 Glue groups of part icular in terest are tho se ge nerated b y the v ertices of the V orono i reg ion, the holes of the lattice. Our constructio n begins with the lattice L 8 = A 2 ⊕ A 2 ⊕ D 4 . 3 L 8 has kissing number 6 + 6 + 24 = 36 and for our choi ce of scale, minimum norm 4 and center density δ 8 = (1 / 2 √ 3) 2 (1 / 8) = 1 / 96 . Each A 2 has two holes and the glue gro up Z 3 has a single generator; D 4 has three holes and glue group Z 2 × Z 2 with two generators. The glue group H generated by the ho les of L 8 thus has four generat ors which, for later con ven ience, we repr esent as vectors under additi on modulo 1: h 1 =  1 3 1 3 1 3 1 3  h 2 =  1 3 1 3 2 3 2 3  (1) h 3 =  1 2 0 1 2 0  h 4 =  0 1 2 0 1 2  The generator h 1 repres ents a hole in on e A 2 compone nt while h 2 repres ents a hole in the other . All four elements, ± h 1 and ± h 2 , thus hav e depth 4 / 3 in L 8 . Another four elements, ± h 1 ± h 2 , represent holes in both A 2 compone nts and hav e depth 8 / 3 . The v ectors h 3 , h 4 , and h 3 + h 4 repres ent th e t hree holes in t he D 4 compone nt of L 8 with depth 2. The 36 elemen ts of H fall into conjugac y classe s with respect to Aut ( L 8 ) . The quotie nt of Aut ( L 8 ) , with respec t to the normal subgrou p that acts trivia lly on H , defines Aut ( H ) , the automorp hism group of the holes. Aut ( H ) is generated by (i) e xcha nge of the t wo h oles within either A 2 compone nt, (ii) e xchang e of the tw o A 2 compone nts, and (ii i) an y permutation of the thre e hole s of D 4 . The 2 2 × 2 × 3! elements of Aut ( H ) ha ve an interestin g representatio n in terms of 4 × 4 matrices acting on the 4-compone nt glue elemen ts by righ t multiplicatio n. The matrices σ =     1 0 0 0 0 1 0 0 0 0 − 1 0 0 0 0 − 1     ρ 4 =     0 0 − 1 0 0 0 0 − 1 1 0 0 0 0 1 0 0     togeth er implement the exchang e of h 1 and h 2 , as w ell as sendin g these to their in vers es (without aff ecting h 3 and h 4 ). T wo other m atrices generate the 6 permu- tation s of the three holes of D 4 : ρ 2 =     0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0     ρ 3 =     0 − 1 0 0 1 − 1 0 0 0 0 0 − 1 0 0 1 − 1     . Matrix ρ 3 cyc lically permutes h 3 , h 4 and h 3 + h 4 (fixing h 1 and h 2 ), while ρ 2 exc hanges h 3 and h 4 (again with no effec t on the A 2 holes whe n c ombined with an approp riate combina tion of σ and ρ 4 ). 4 h orbit size ∆ 8 ( h ) τ 8 ( h ) 0 0 0 0 1 4 36 1 2 0 1 2 0 3 2 8 1 3 1 3 1 3 1 3 4 4/3 3 1 3 1 3 0 0 4 8/3 9 5 6 1 3 5 6 1 3 12 10/3 24 5 6 1 3 1 2 0 12 14/3 72 T able 1: Properties of the symmetry orb its of glu e in L 8 . The orbit represen tati ves h are sums (mod 1) of the g enerat ors (1); ∆ 8 ( h ) is the depth a nd τ 8 ( h ) the number of points in L 8 at squared distanc e ∆ 8 ( h ) . For future calculation s we tabu late data on the 6 conjugac y classes of the L 8 glue element s in T able 1. As an e xample, the e ntry h =  5 6 1 3 5 6 1 3  = h 1 + h 3 is the combin ation of a hole in one A 2 compone nt and a hole in D 4 . There are τ 8 ( h ) = 3 × 8 points in L 8 at the minimum squared distance ∆ 8 ( h ) = 4 / 3 + 2 from h . The symmetry orbit of h has size 12 and is generated by ρ 3 ρ 4 :  5 6 1 3 5 6 1 3   1 3 5 6 2 3 1 6   1 6 1 6 1 6 1 6   1 6 2 3 5 6 1 3   1 3 5 6 1 3 5 6   5 6 5 6 1 6 1 6   1 6 2 3 1 6 2 3   2 3 1 6 1 3 5 6   5 6 5 6 5 6 5 6   5 6 1 3 1 6 2 3   2 3 1 6 2 3 1 6   1 6 1 6 5 6 5 6  Symmetric glue for th e lattice A 2 ⊕ A 2 W e no w interpret the 36 glue element s, when expre ssed as 4-compone nt vec tors, as glue elements of a four dimensional lattice L 4 . Since we want the glue in L 4 to includ e a ll the automorp hisms th e corres pondin g elements ha ve in L 8 , we constr uct L 4 so the 4 × 4 matrices ρ 2 , ρ 3 , ρ 4 , an d σ are elements of Aut ( L 4 ) . These matr ices act on the four generato rs of L 4 by left multipl ication . W e e xhib it L 4 in terms of the projec tions of i ts gen erator s u 1 , u 2 , u 3 , u 4 in tw o orthog onal planes . These a re sho wn in Figure 2; compone nts in the t wo plan es are 5 u 1 ýý u 2 ýý u 3 ýý u 4 ýý u 1 ¦ u 2 ¦ u 3 ¦ u 4 ¦ Figure 2: Projectio ns of the gene rators of L 4 into two ortho gonal planes. distin guished by supersc ripts || and ⊥ . The pair u || 1 , u || 2 genera tes an A 2 lattice , as does the pair u || 3 , u || 4 . The same holds for the projectio n into the other plane . T his much is consisten t with the automorphis m gener ated by ρ 3 , which acts as a ro tation by 120 ◦ in the two planes. The pair u || 3 , u || 4 is rotated by +90 ◦ relati ve to u || 1 , u || 2 , and by − 90 ◦ in the othe r plane. If we fix the coordinates so that u || 1 = u ⊥ 1 and u || 2 = u ⊥ 2 , then this implies u || 3 = − u ⊥ 3 and u || 4 = − u ⊥ 4 . From this we ha ve u i · u j = ( u || i + u ⊥ i ) · ( u || j + u ⊥ j ) = 0 i ∈ { 1 , 2 } , j ∈ { 3 , 4 } thus identifyi ng L 4 as A 2 ⊕ A 2 . W e choose the scale so that L 4 has minimal norm 4 and center density δ 4 = 1 / 12 . The actions of ρ 2 and ρ 4 on the project ed lattice gene rators sho w that these als o generat e isomet ries of the two planes: ρ 2 reflects across a line and ρ 4 rotates by 90 ◦ . Altogether , ρ 2 , ρ 3 , ρ 4 genera te the two-d imensiona l non-crysta llogra phic reflection group G 0 of order 24. The final genera tor of Aut ( H ) , the in v olutio n σ , re verses the signs of u 3 and u 4 relati ve to u 1 and u 2 , and is there fore con jugate to an ex chang e of the t wo planes. W e observ e that the only point of L 4 contai ned in eith er plane is the origin. The data for the glue elements, as for L 8 , again condenses into a table of the symmetry orbits. As a guide for the compu tations in T able 2, we sho w in Figure 3 the compone nts of the glue in th e A 2 plane span ned by u 1 and u 2 (the diagra m for 6 u 1 u 2 A A A A B B B B B C C D D D D D D Figure 3: The four types of glue in one A 2 compone nt of the lattic e L 4 . the other A 2 compone nt is identic al). There are 12 distinct glue elements in this compone nt, and these fa ll into four families: A : [0 0] B :  1 2 0   0 1 2   1 2 1 2  C :  1 3 1 3   2 3 2 3  D :  1 6 1 6   1 6 2 3   2 3 1 6   5 6 1 3   1 3 5 6   5 6 5 6  Glue in the family B has depth 1 from 2 poin ts of A 2 , famil y C has depth 4 / 3 from 3 poin ts, and family D has d epth 1 / 3 from only one point of A 2 . T aking into accoun t both A 2 compone nts we obtai n the info rmation in T able 2. T welve dimensional lattice packing For each elemen t h of the glue gro up H we hav e a glue elemen t x 8 ( h ) for the lattice L 8 and a correspondi ng glue elemen t x 4 ( h ) for the lattice L 4 . The elemen ts x 8 ( h ) + x 4 ( h ) , h ∈ H , form a glue grou p for L 8 ⊕ L 4 and, toge ther with L 8 ⊕ L 4 , form a lattice L 12 . From the data in T ables 1 and 2 we can che ck that L 12 has minimal norm 4 and compute the kissing number . Let x and y be distinct points of L 12 . The vect or x − y belon gs to one of the 36 conjugac y classes with respect to L 8 ⊕ L 4 . A non-zero vector in the class [0 0 0 0] has minimal norm 4 because it must be a minimal vec tor of either L 8 or 7 h orbit size ∆ 4 ( h ) τ 4 ( h ) 0 0 0 0 1 4 12 1 2 0 1 2 0 3 2 4 1 3 1 3 1 3 1 3 4 8/3 9 1 3 1 3 0 0 4 4/3 3 5 6 1 3 5 6 1 3 12 2/3 1 5 6 1 3 1 2 0 12 4/3 2 T able 2: Propert ies of the symmetry orbits of glue in L 4 . ∆ 4 ( h ) is the depth and τ 4 ( h ) the number of points in L 4 at squar ed distance ∆ 4 ( h ) . L 4 ; the number of such ve ctors is 36 + 12 , the sum of the kissing numbers of L 8 and L 4 . If the conjugac y class h of x − y is non-tr iv ial, then its minimal norm is ∆ 8 ( h ) + ∆ 4 ( h ) and the number of such vecto rs is τ 8 ( h ) τ 4 ( h ) . Computati ons for all the co njugac y classes are giv en in T able 3 and verify that L 12 has minimal no rm 4 and kissing number 48 + 96 + 108 + 108 + 288 = 648 . The center density of this latti ce is δ 12 = | H | δ 8 δ 4 = 1 / 32 . These prope rties match those of the 12- dimensio nal laminated lattic e ha ving the highest kissi ng number , Λ max 12 [1]. The latter is one of three, equal density lattices prod uced by laminating one dimensi on at a time, starti ng with Λ 1 = 2 Z and always maximizing the density . L 12 and Λ max 12 were sho wn to be isomorph ic by comparin g their Gram matrices [3]. T en dimensional non-lattice packings Our non -stand ard cons tructio n of Λ max 12 ∼ = L 12 leads rathe r directl y to a no n-latti ce packin g that exp lains the numerical kissi ng number resu lts in 10 dimension s. The constr uction of the non-la ttice packing is closely related to quasicrys tal patterns, of which the best kno w n e xample is the Penrose tiling . T he foc us no w shift s to the Euclidean space X 4 that contain s L 4 . The in var iant plan es of the the group G 0 pro vide a natural ortho gonal decomposi tion X 4 = X || ⊕ X ⊥ . A general poin t x ∈ L 12 has a uniqu e e xpress ion of the form x = x || + x ⊥ + x 8 , w here x 8 is in the space contain ing L 8 . The only role of x 8 in the follo w ing constru ction is throug h its depth in L 8 , which we denote by ∆ 8 ( x ) ; the depth depends only on the 8 h orbit size ∆ 8 ( h ) + ∆ 4 ( h ) τ 8 ( h ) τ 4 ( h ) number 1 2 0 1 2 0 3 4 32 96 1 3 1 3 1 3 1 3 4 4 27 108 1 3 1 3 0 0 4 4 27 108 5 6 1 3 5 6 1 3 12 4 24 288 5 6 1 3 1 2 0 12 6 144 17 28 T able 3: The short vec tors of L 12 in the non-tri vial con jugac y classes of the glue group . associ ated glue element. The non-la ttice packing is then ob tained as follows: Construction Q . Define the subs et S 10 = { x ∈ L 12 : x ⊥ ∈ V ⊥ } , where V ⊥ ⊂ X ⊥ is a bounded domain and specified in detail belo w . From S 10 , a packing in 10 dimensio ns is gi ven by th e set Q 10 = { √ 2 x || + x 8 : x ∈ S 10 } . W e will see that if there are kissin g spheres at x, y ∈ L 12 , that is, z = x − y is a vector of norm 4, then v ery often z || · z || = z ⊥ · z ⊥ . Since this implies that z || + z ⊥ + z 8 and √ 2 z || + z 8 ha ve the same norm, spheres that are kiss ing in L 12 will often ha ve counterparts in Q 10 that are kiss ing as well. Note that if z is in the conjug acy c lass  5 6 1 3 1 2 0  , then ( √ 2 z || + z 8 ) · ( √ 2 z || + z 8 ) ≥ ∆ 8 ( z ) = 14 / 3 > 4 , and the packing constra int is satisfied for any z || . W e deri ve below a domain V ⊥ for which we can prov e that Q 10 is a pack ing with minimal dista nce 2. As a prelimina ry step we identify and characte rize the minimal vecto rs of L 12 that arise in constr uction Q: Lemma. A non-zer o x ∈ L 12 that satisfi es the inequ alities ∆ 8 ( x ) + 2 x || · x || < 4 (2) x ⊥ · x ⊥ ≤ 8 / 3 (3) is a minimal vector of L 12 . Pr oof. Combining (2) and (3 ) we obtain x · x < 14 / 3 + 1 2 ∆ 8 ( x ) . T able 1 , tog ether with ∆ 8 ( x ) < 4 as implied by (2), limits the possibl e v alues of ∆ 8 in the bound 9 repres entati ve x || · x || x ⊥ · x ⊥ 1 3 1 3 1 3 1 3 4 3 4 3 1 3 1 3 − 2 3 1 3 4 3 − 2 √ 3 4 3 + 2 √ 3 1 3 1 3 1 3 − 2 3 4 3 + 2 √ 3 4 3 − 2 √ 3 T able 4: Data on t he three o rbits w ith respect to the reflection group G 0 of the 4 × 9 minimal vect ors of L 12 with ∆ 4 = 8 / 3 . Orbit repre sentati ves are specified in the same 4-compon ent basis used to define the glue group. on x · x . Recallin g that the possible norms of the integral lattice L 12 ∼ = Λ max 12 are 4 , 6 , . . . , we see that only the case ∆ 8 ( x ) = 10 / 3 is unresolv ed. In this case (2) and (3) imply x || · x || < 1 / 3 and x 4 · x 4 < 3 , where x 4 = x || + x ⊥ . W ithout loss of gene rality it is suf fi cient to e xamine just one ele ment of the symmetry orb it of the glue, say h =  1 6 1 6 1 6 1 6  . There are just five solutio ns to the inequality x 4 · x 4 = k P 4 i =1  n i + 1 6  u i k 2 < 3 for integer s n i . These are where all integers are zero, for which x 4 · x 4 = 2 / 3 and x is minimal, or only one of the integer s is nonze ro and has the val ue − 1 . For the latter one obtains x || · x || = (4 ± √ 3) / 3 , both of which are inconsi stent with x || · x || < 1 / 3 . The domain V ⊥ is const rained by the symmetry orbit s of the L 12 minimal v ec- tors, specifically their projectio ns into X ⊥ . Since G 0 acts as the reflection group of order 24 on the set of projec ted minimal ve ctors, the size of the orbits is al ways a m ultiple of 12 (since x ⊥ is non-zero for a minimal vec tor). Consul ting T able 2 we see, for example , that there are 3 × 4 minimal vec tors with x 4 · x 4 = 2 . This single orbit with resp ect to G 0 is also an orbit of Aut ( H ) and is therefore fixed by the i n vol ution σ that exchange s X || and X ⊥ . W e conclude that for minimal v ectors x in thi s conjugac y class we hav e x || · x || = x ⊥ · x ⊥ . The equality of proj ected minimal vect or norms in X || and X ⊥ holds for al l the classes exce pt the class with ∆ 8 = 14 / 3 , where this p ropert y will not b e rel ev ant, and the c lass with ∆ 8 = 4 / 3 . The latter contains 4 × 9 vect ors which form thre e orbits with resp ect to G 0 ; data for these are gi ve n in T able 4. Only the second orbit in T able 4 requires attention, as it violates the prope rty k √ 2 x || k ≥ k x || + x ⊥ k . A v oiding such minimal vec tors is one of the roles of the domain V ⊥ . More specifically , we constru ct V ⊥ such that if y ⊥ , z ⊥ ∈ V ⊥ , then x ⊥ = y ⊥ − z ⊥ is nev er one of the 12 forb idden ve ctors in the orb it of ( u ⊥ 1 + 10 Figure 4: The domain V ⊥ , a regula r dodeca gon, used in construct ion Q. T rans- lation s associated with pairs of opposite edges (arro ws) are the forbidd en vectors of the cons tructio n. V ⊥ includ es half the boundar y (dar k), and in part icular , six dodec agon ver tices. u ⊥ 2 − 2 u ⊥ 3 + u ⊥ 4 ) / 3 . Up t o t ranslat ion in X ⊥ , th ere is a unique dod ecagon with the proper ty that its oppos ite edges are translat ed by v ectors i n t his se t. It is possibl e to includ e half the bounda ry of the dodec agon in the definition of V ⊥ , and we make the definite choice shown in Figure 4 in what follows. V ⊥ has diameter p 8 / 3 , area | V ⊥ | = 2 , and happens to coinci de with the shado w of the V oronoi domain of L 4 = A 2 ⊕ A 2 , althoug h the sign ificance of this is not clea r . Theor em. Construction Q, wit h V ⊥ as specified abo ve, pr oduces a spher e packi ng Q 10 with minimum dista nce 2. Pr oof. Conside r x, y ∈ S 10 and let z = x − y . W e wish to verify that k √ 2 z || + z 8 k ≥ 2 . From the construct ion of S 10 we kno w that z ⊥ · z ⊥ ≤ 8 / 3 (the diamete r of V ⊥ ). The theorem is pro ved if w e can sho w that k √ 2 z || + z 8 k < 2 leads to a contra diction . B ut with this statemen t both hypothes es of the lemma are satisfied, thereb y establi shing that z is a minimal vect or . From o ur exhausti ve anal ysis of the minimal v ectors we know (aga in omitting t he irrele van t class with ∆ 8 = 14 / 3 > 4 ) that k √ 2 z || k ≥ k z || + z ⊥ k with o ne ex ceptio n: the min imal vec tors whose pro jec- tions into X ⊥ coinci de with one of the forb idden vectors . But V ⊥ was des igned so this is impossi ble, hence k √ 2 z || k ≥ k z || + z ⊥ k . Moreo ve r , since z is minimal, this sho w s k √ 2 z || + z 8 k ≥ k z || + z ⊥ + z 8 k = 2 , a contr adictio n. W e know that the packing Q 10 is aperiodic becau se its projection into X || is 11 Figure 5: Sphere ce nters (dots) o f the pa cking Q 10 projec ted into the plane X || form an aperiodic pattern , a reg ion of which is sho wn here. Pairs of kissing spheres in the 10 dimension al packin g may ha ve any of the fiv e projectio ns sho wn. a quasicrys tal. A represen tati ve reg ion of the projection is sho wn in Figure 5, where each point repre sents an L 8 lattice translated by one of 36 glue elements . T o set the scale, the projecti ons of v arious pairs of unit- radius spheres are also sho w n. Even though the sphere projection s over lap, glue translation s of the L 8 lattice s guaran tee that the sp heres for m a packing . There is a smalle st separa tion of projec ted sphere cente rs, and sphere s in this relationshi p amply s atisfy the packing constr aint because the associated conjugac y class of the gluing has ∆ 8 = 14 / 3 . Many sph eres are actually kissing; this is the case for those depicted in the figure. The center density of Q 10 can be computed using a formula dev eloped in the study of quasicr ystals [4, 5]. R estrict ing to just one of the conjugac y class es of glue, and omitting the dilation by √ 2 , the density of cente rs in X || is equal to the 12 Figure 6: The proje cted sphere center s in X || form a quasicrysta l tiling comprising triang les, square s, and 30 ◦ rhombi. produ ct of the center density of L 4 and the area of the domain V ⊥ whose shado w selects the subset of L 4 used in Q 10 : δ || = | V ⊥ | δ 4 . Includi ng all the conjug acy classe s multipli es this by | H | = 36 and the dilati on by √ 2 dimin ishes the density by 2. T o get the density in 10 dimensio ns this density in X || is multiplied by the center density of L 8 : δ 10 = 1 2 | V ⊥ | | H | δ 4 δ 8 = 1 2 | V ⊥ | δ 12 = 1 32 . Because Q 10 is a non-lattice packing , th e sphe re centers fal l into d istinct clas ses. This is e vident in the quasicrysta l patte rn of projected centers in X || , where there is a special cl ass of centers that is ne ver in the short est-sep aration rela tionshi p w ith other centers. When cente rs in this spec ial class are connected by edges, the re- sult, sho w n in Figure 6, is a tiling of triangles, squares and 30 ◦ rhombi. Alternate 13 tilings formed by these three tiles, utilizing the same gluing scheme for the L 8 lat- tices, also corres pond to valid packings (details omitted). The densest of these is obtain ed by the tiling that only uses the triangle, the densest tile of the three . The impro vemen t of the density , to the v alue 7 / (96 + 64 √ 3) ≈ 0 . 0338 , comes at the exp ense of symmetry . Whereas the quasicryst al pattern has point symmetry group G 0 of ord er 24, the point grou p of the triangular tiling is only of order 12. The denses t kno w n packing in 10 dimensio ns has cente r dens ity 5 / 128 ≈ 0 . 03906 [1]. W e concl ude by ret urning to the results of the numerical exp eriment that pro mpted this in vest igation . The arra ngement of 378 kiss ing spheres discov ered by the search algori thm [2] co incide s with the arrangement obt ained from the pack ing Q 10 when the dodeca gonal domain V ⊥ is gi ven a singular centering (translatio n) in X ⊥ . At a singul ar centering one L 8 lattice p rojects t o the exac t c enter of V ⊥ , thereby making the six ver tices of V ⊥ a v ailable to the packing (see Fig. 4). Relativ e to the do- decago n center , the six vert ices hav e glue in the orbi t of  1 3 1 3 0 0  with ∆ 8 = 8 / 3 and ∆ 4 = 4 / 3 . F rom T able 1 we see that τ 8 = 9 sph eres, in each of the L 8 lattice s that project to these vertic es of V ⊥ , make contact with the central sphe re. There is also a set of 12 L 8 lattice s that projec t to a re gular dodec agon within V ⊥ . These are in the orbit of  5 6 1 3 5 6 1 3  with ∆ 8 = 10 / 3 , ∆ 4 = 2 / 3 and τ 8 = 24 . The X || counte rpart of this re lations hip are the man y exa mples o f c omplete dodecagon s en- circlin g tile vertices in Fig. 6. Finally , the L 8 lattice that pro jects to the dode cagon center makes 36 contacts with the central sphere (the kissin g number of L 8 ). The net kissing number is thus 6 × 9 + 12 × 24 + 36 = 37 8 . Refer ences [1] J. H. Conway & N. J. A. Sloane, Sp her e P acki ngs, L attices and Gr oups , (Springe r , 1993). [2] S. Grav el & V . Elser , Di vide and concu r: A general approach to constra int satisf actio n, unpublish ed (2008). [3] N. J. A. Sloane, priv ate communication. [4] V . Elser , The dif fracti on pattern of pro jected structu res, Acta Cryst. A 42 , 36-43 (1986) . [5] R. V . Moody & J. Pater a, Densit ies, minimal dist ances, and cover ings of qu a- sicrys tals, Comm. Math. Phys. 195 , 613-626 (1998). 14