and arithmetic of crystallographic packings

Alex Kontorovicha,b,1 and Kei Nakamuraa

aDepartment of Mathematics, Rutgers University, New Brunswick, NJ 08854; and bSchool of Mathematics, Institute for Advanced Study, Princeton, NJ 08540

Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved November 21, 2018 (received for review December 12, 2017)

We introduce the notion of a “crystallographic sphere packing,” argument leading to Theorem 3 comes from constructing circle defined to be one whose limit set is that of a geometrically packings “modeled on” combinatorial types of convex polyhedra, finite hyperbolic reflection group in one higher dimension. We as follows. exhibit an infinite family of conformally inequivalent crystallo- graphic packings with all radii being reciprocals of integers. We (♥): Polyhedral Packings then prove a result in the opposite direction: the “superintegral” Let Π be the combinatorial type of a convex . Equiv- ones exist only in finitely many “commensurability classes,” all in, alently, Π is a 3-connected‡ planar graph. A version of the at most, 20 dimensions. Koebe–Andreev–Thurston Theorem§ says that there exists a 3 geometrization of Π (that is, a realization of its vertices in R with sphere packings | crystallographic | arithmetic | polyhedra | straight lines as edges and faces contained in Euclidean planes) Coxeter diagrams having a midsphere (meaning, a sphere tangent to all edges). This midsphere is then also simultaneously a midsphere for the he goal of this program is to understand the basic “nature” of Πb. Fig. 2A shows the case of a cuboctahedron Tthe classical Apollonian gasket. Why does its integral struc- and its dual, the . ture exist? (Of course, it follows here from Descartes’ Kissing 2 Stereographically projecting to Rc, we obtain a cluster (just Circles Theorem, but is there a more fundamental, intrinsic meaning, a finite collection) C of circles, whose nerve (that is, explanation?) Are there more like it? (Around a half-dozen tangency graph) is isomorphic to Π, and a cocluster, C, with similarly integral circle and sphere packings were previously b known, each given by an ad hoc description.) If so, how many nerve Πb, which meets C orthogonally. Again, the example of the more? Can they be classified? We develop a basic unified frame- cuboctahedron is shown in Fig. 2B. Definition 4: The orbit P = P(Π) = Γ ·C of the cluster C work for addressing these questions, and find two surprising D E phenomena: under the group Γ = Cb generated by reflections through the (♥) there is indeed a whole infinite zoo of integral sphere cocluster Cbis said to be modeled on the polyhedron Π. packings, and (♠) up to “commensurability,” there are only finitely many Lemma 5. An orbit modeled on a polyhedron is a crystallographic Apollonian-like objects, over all dimensions. packing. n−1 See Fig. 3 for a packing modeled on the cuboctahedron. Such Definition 1: By an S -packing (or just packing) P of packings are unique up to conformal/anticonformal maps by n Rcn := R ∪ {∞}, we mean an infinite collection of oriented Mostow rigidity, but Mobius¨ transformations do not generally (n − 1)- (or co-dim-1 planes) so that: (i) The interiors preserve arithmetic. of spheres are disjoint, and (ii) The union of the interiors of the Definition 6: We call a polyhedron Π integral if there exists spheres is dense in space. The bend of a sphere is the recipro- some packing modeled on Π, which is integral. ∗ cal of its (signed) radius. To be dense but disjoint, the spheres It is not hard to see that the cuboctahedron is indeed inte- in the packing P must have arbitrarily small radii, so arbitrarily gral, as is the , which corresponds to the classical large bends. If every sphere in P has integer bend, then we call the packing integral. Without more structure, one can make completely arbitrary Significance constructions of integral packings. A key property enjoyed by the classical Apollonian circle packing and connecting it to the This paper studies generalizations of the classical Apollonian theory of “thin groups” (see refs. 1–3) is that it arises as the limit circle packing, a beautiful geometric fractal that has a surpris- set of a geometrically finite reflection group in hyperbolic space ing underlying integral structure. On the one hand, infinitely of one higher dimension. many such generalized objects exist, but on the other, they Definition 2: We call a packing P crystallographic if its limit may, in principle, be completely classified, as they fall into, set is that of some geometrically finite reflection group Γ < only finitely, many “families,” all in bounded dimensions. n+1 Isom(H ). This definition is sufficiently general to encompass all previ- Author contributions: A.K. and K.N. wrote the paper.y ously proposed generalizations of Apollonian gaskets found in The authors declare no conflict of interest.y the literature, including refs. 4–10. With these two basic and This article is a PNAS Direct Submission.y general definitions in place, we may already state our first main Published under the PNAS license.y result, confirming (♥). 1 To whom correspondence should be addressed. Email: [email protected] Published online December 26, 2018. Theorem 3. There exist infinitely many conformally inequivalent *In dimensions n = 2, that is, for circle packings, the bend is just the curvature. integral crystallographic packings. However, in higher dimensions n ≥ 3, the various “curvatures” of an (n − 1)-sphere are We show in Fig. 1 but one illustrative example, whose only proportional to 1/radius2, not 1/radius; so, we instead use the term “bend.” “obvious” is a vertical mirror image. It turns out (but ‡Recall that a graph is k-connected if it remains connected whenever fewer than k may be hard to tell just from the picture) that this packing does vertices are removed. indeed arise as the limit set of a Kleinian reflection group. The §See, e.g., ref. 11 for an exposition of a proof.

436–441 | PNAS | January 8, 2019 | vol. 116 | no. 2 www.pnas.org/cgi/doi/10.1073/pnas.1721104116 Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 4 4 ssml h ru eeae yrflcin nbt h cluster the both in reflections cocluster, by and generated group the packing polyhedral simply a is of case the In steshr neir r olne disjoint. longer no are interiors sphere the as group as this in write spheres may all through reflections plus its define otrvc n Nakamura and Kontorovich group that recall notation. more some need we proof, the imply itself by first not the does so, packing; crystallographic of same part the poly- to many rise infinitely give that out hedra turns it because here polyhedral inequivalence integral inequivalent conformally many packings. infinitely to 7. 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Fig. 8 1 9 9 12 12 2 0 ¶ 1 0 4 58 2 9 256 256 290 140 140 199 199 126 292 292 8 8 71 71 138 188 188 58 239 239 284 187 186 193 193 172 176 56 60 60 56 176 172 215 215 oko rhme l 1) e lotevepito Shitarneet”i the in arrangements” “Schmidt of viewpoint (13). the Sheydvasser and also (9) see Stange (12); of al. work et Graham of work 58 202 eae oino uepcigfrtecasclAolna aktaoearayin already arose gasket Apollonian classical the for superpacking of notion related A 181 154 200 200 65 65 126 209 152 152 209 292 292 290 256 256 202 229 172 194 78 258 167 85 260 278 214 121 121 47 47 242 120 116 32 170 162 162 158 211 28 10 268 268 250 96 124 124 167 167 131 260 260 260 260 220 106 256 256 42 248 248 290 251 268 70 200 98 154 98 196 122 115 256 238 294 118 262 241 241 219 184 261 254 156 156 142 34 124 124 152 152 284 110 155 186 148 148 72 58 272 216 161 161 254 212 188 188 288 288 181 102 102 208 184 184 246 187 266 88 88 125 289 255 255 289 166 248 220 220 268 268 234 234 64 244 40 40 206 ento 9: Definition oeta h uepcigi o akn yordefinition our by packing a not is superpacking the that Note 8: Definition eunn otegnrlstigo rsalgahcpackings, crystallographic of setting general the to Returning implies course of This following the show to able are we difficulties, these Despite problem. this with difficulties basic some out point us Let 23 212 212 148 280 158 60 60 50 218 171 134 272 100 100 110 268 190 103 103 210 90 166 189 109 249 249 92 29 230 299 98 278 268 140 140 248 248 32 32 264 68 68 178 292 282 61 22 274 183 183 128 128 154 96 96 288 288 41 41 178 114 200 232 190 228 66 287 287 176 176 220 188 242 248 97 97 229 172 172 59 282 112 112 134 40 256 256 168 196 196 168 124 177 177 280 280 242 170 9 18 9 288 226 232 238 281 281 160 160 268 268 218 101 181 240 240 92 116 230 285 285 294 174 70 260 260 260 260 192 244 262 164 265 265 268 112 274 36 280 280 292 188 45 264 160 63 54 63 184 254 269 284 72 108 126 72 280 99 180 198 189 252 110 152 140 140 211 80 202 232 232 169 169 127 127 124 44 44 118 269 168 215 215 132 132 132 132 194 194 224 224 94 82 268 268 179 173 291 208 164 155 131 64 64 210 58 228 266 134 122 14 119 286 160 192 192 52 52 206 288 46 130 244 244 270 136 136 208 208 275 196 286 286 216 280 280 239 239 262 152 172 296 80 120 100 100 120 80 224 224 209 209 206 206 284 284 298 220 220 200 180 180 200 128 286 134 222 162 68 282 213 296 296 62 204 250 218 298 178 144 42 121 121 220 220 229 56 56 166 296 248 248 205 205 84 84 256 236 196 98 98 140 233 233 230 230 294 212 243 75 125 268 192 86 259 130 269 94 141 113 132 8 16 16 292 242 274 142 57 57 46 155 24 13 32 32 232 232 234 234 56 112 112 238 144 144 61 77 240 240 226 212 34 292 116 100 230 250 68 276 276 128 252 204 133 280 78 169 156 156 91 169 70 286 296 296 124 163 194 254 298 238 256 196 134 152 162 162 286 82 296 244 244 38 20 20 212 212 84 84 248 248 253 276 43 214 208 208 185 185 30 30 182 159 159 258 189 265 265 176 176 50 259 269 120 120 100 208 226 226 220 180 212 212 17 152 214 52 52 216 296 126 278 238 238 280 280 106 222 299 44 96 214 199 199 229 110 132 292 292 233 56 122 198 294 294 284 51 242 124 124 254 168 102 268 224 187 211 104 289 238 238 289 272 272 Γ ecl h atra latter the call we Γ; 123 276 181 176 94 14 256 256 246 166 72 72 260 272 262 38 62 74 220 184 237 280 136 93 227 88 21 150 250 190 190 197 78 142 249 230 28 244 198 35 160 49 49 16 275 256 84 246 182 245 172 nitga rsalgahcpcig Tecrlsaelbldwith labeled are circles (The packing. crystallographic integral An 202 53 206 155 264 264 134 124 276 188 232 128 112 64 64 128 130 240 240 125 262 246 68 134 290 292 292 140 210 (♥). 162 162 230 164 164 220 76 168 22 2 174 174 100 100 200 200 202 182 104 206 212 htifiieymn oyer r integral, are polyhedra many infinitely that 7, Theorem 121 121 112 280 99 125 242 158 235 107 72 254 216 144 144 262 288 288 148 148 290 294 206 80 80 182 156 267 205 97 86 196 101 246 172 299 236 214 225 225 128 272 220 229 194 237 288 256 256 224 260 70 204 268 193 214 216 steobtof orbit the is 146 76 269 285 217 4 69 241 245 77 12 6 262 292 221 197 292 262 166 160 supergroup, 150 142 230 224 208 200 76 165 146 70 216 161 214 204 293 128 289 289 268 194 145 260 256 256 208 129 236 172 200 196 196 173 169 169 156 206 148 148 144 144 294 290 288 288 254 53 248 244 244 242 212 P 49 206 202 200 200 76 85 220 81 81 210 140 68 118 256 48 134 209 130 173 128 64 64 128 246 262 278 208 110 110 240 124 188 179 28 230 299 276 253 104 228 228 nntl ayplhdaaeitga n ierise give and integral are polyhedra many Infinitely 249 249 226 206 190 100 100 264 264 202 248 172 260 84 70 184 40 196 98 98 298 131 296 294 236 236 246 232 160 172 172 78 151 170 161 272 62 190 230 36 36 150 74 29 264 264 5 250 229 220 277 72 72 206 280 173 235 166 262 94 152 260 174 256 256 189 276 44 117 272 272 132 110 224 268 198 20 20 103 242 184 284 294 140 140 25 25 106 214 207 207 126 224 30 109 136 136 203 267 154 50 152 105 105 70 120 120 254 180 220 195 150 38 176 176 116 263 296 208 208 46 107 196 196 64 276 162 162 124 34 286 93 70 16 16 194 83 161 298 156 182 182 156 296 296 286 219 68 141 204 157 235 24 280 100 189 241 176 205 116 142 142 252 32 32 137 137 250 190 40 224 185 185 212 230 254 112 120 112 120 292 226 144 168 144 264 240 240 238 257 257 46 116 116 234 234 14 162 130 130 232 232 276 252 252 252 252 293 147 52 52 250 280 191 191 289 289 156 156 170 121 132 110 132 121 242 130 292 50 152 94 240 86 = Γ 296 299 42 192 56 56 e 20 20 212 84 84 148 148 248 248 262 179 268 196 98 140 98 118 294 244 244 166 79 79 275 197 sasmdt rs stelmtsto discrete a of set limit the as arise to assumed is 236 284 284 220 220 116 116 175 175 178 68 85 248 248 218 221 204 289 289 162 298 80 120 120 80 134 211 296 296 217 217 282 200 180 180 200 86 69 115 152 218 52 224 224 130 196 256 262 208 208 242 271 271 254 286 64 64 206 261 192 192 160 270 244 18 140 82 184 56 228 266 228 65 65 277 224 224 283 142 254 104 110 36 36 119 119 232 232 262 72 126 108 72 275 299 180 90 252 252 270 133 208 253 136 88 85 284 188 254 91 70 164 55 55 116 116 298 292 280 280 174 268 28 28 244 238 129 129 281 281 272 92 22 221 218 236 236 230 160 200 160 62 240 240 124 170 158 217 217 238 112 84 84 112 149 226 168 196 196 168 242 236 134 184 280 280 150 296 296 10 256 256 66 188 250 107 182 182 104 260 260 260 260 220 176 176 226 32 32 242 114 133 139 290 259 259 ie packing a Given 48 70 92 92 146 247 228 247 26 260 128 112 128 80 80 223 223 176 96 96 144 122 103 103 196 196 68 68 155 178 284 214 156 195 143 182 195 143 169 169 282 266 166 224 108 108 118 264 128 248 248 281 281 268 278 92 40 40 269 158 214 76 76 168 168 86 283 106 244 60 60 60 50 238 288 100 206 110 90 190 210 287 287 254 218 268 158 148 202 192 252 252 252 252 88 88 269 149 166 58 109 21 234 234 184 184 89 89 212 28 124 165 233 233 288 288 152 152 142 49 49 122 254 245 268 248 248 162 := Γ 116 170 e 198 242 286 221 250 78 248 278 160 201 201 260 172 The 258 225 225 256 256 290 140 140 152 152 126 8 8 138 188 188 58 200 200 284 186 117 172 56 56 186 138 284 277 289 289 256 256 250 D 242 81 81 220 106 293 277 196 98 156 156 124 186 148 148 58 272 288 288 231 231 184 184 266 88 88 185 185 166 248 234 234 192 202 158 60 50 237 252 252 252 252 267 100 254 26 106 151 151 76 76 251 213 238 168 128 214 158 244 278 156 68 68 178 224 271 271 53 282 284 274 196 196 232 232 128 128 277 70 224 224 225 225 146 C 104 260 261 121 121 284 182 188 62 184 157 158 236 136 136 18 238 b 116 128 128 244 164 36 37 140 205 206 169 169 254 86 218 256 152 212 200 200 176 176 116 116 298 , 118 208 208 196 98 50 148 148 295 295 ob h ru eeae by generated group the be to Γ, 248 101 130 e 132 292 242 274 P 225 225 250 32 32 hΓ, 232 232 superpacking, 289 289 C 296 296 194 196 162 82 244 244 165 212 212 214 1 182 50 100 101 oepantemi da in ideas main the explain To 3. Theorem 222 292 292 122 294 242 245 72 72 170 E 196 98 197 226 128 128 290 200 ne t uegop htis, that supergroup, its under 288 P . P esrs h conformal the stress We 3. Theorem f ymtygroup symmetry 288 i := 200 162 290 128 128 226 197 196 98 170 72 72 122 245 242 294 292 292 222 < 101 50 100 182 214 212 212 82 165 162 244 244 196 194 296 296 289 289 32 32 232 232 Γ 250 e 225 225 132 274 242 130 101 292 50 148 148 248 98 196 118 295 295 208 208 116 116 176 176 298 200 200 86 212 152 218 256 169 169 254 206 205 18 140 37 36 P 164 128 128 116 244 62 136 136 158 238 Isom( 157 236 184 121 188 121 182 104 284 70 261 146 · 26 260 128 128 53 225 225 224 224 277 196 196 68 68 232 232 274 178 284 156 282 271 271 224 128 278 158 214 76 76 168 106 213 244 50 60 238 151 151 100 251 254 158 202 192 267 252 252 252 252 237 88 88 166 58 185 185 234 234 184 184 248 266 124 231 231 288 288 148 148 186 272 156 156 196 98 P 106 277 220 293 81 81 242 250 256 256 289 289 8 8 138 277 284 186 56 56 P 117 172 58 186 200 200 188 188 138 126 284 ihsmer group symmetry with 152 152 140 140 290 256 256 172 225 225 78 258 201 201 160 260 278 248 221 198 250 242 286 116 170 21 162 28 10 49 49 248 248 268 122 245 . 254 P 142 124 152 152 165 58 89 89 212 109 288 288 233 233 184 184 88 88 166 149 234 234 40 40 192 202 148 158 60 60 60 50 218 269 252 252 252 252 100 110 268 210 90 190 254 287 287 26 106 206 86 76 76 288 238 168 168 128 214 158 118 244 92 108 108 283 f 269 278 268 281 281 166 248 248 32 32 195 182 195 169 156 143 143 169 264 68 68 214 178 224 266 282 48 155 103 103 22 122 284 196 196 128 80 112 80 128 176 96 96 144 70 114 133 223 223 146 92 92 28 28 104 260 139 247 228 247 107 66 259 259 176 176 220 290 182 182 188 242 226 150 260 260 260 260 84 112 112 84 62 184 134 250 256 256 196 196 168 168 124 296 296 149 280 280 217 217 158 236 242 170 18 P 226 238 129 129 160 160 200 55 55 218 221 240 240 92 116 116 230 236 236 174 70 272 281 281 85 88 238 244 164 268 36 36 280 280 91 292 188 298 133 136 254 208 284 = 72 126 108 72 253 180 90 252 270 252 119 119 110 299 275 104 262 232 232 56 65 65 142 254 283 224 224 277 82 20 20 184 64 64 140 228 266 228 14 192 192 160 52 206 130 244 270 261 208 208 H 69 115 196 286 254 271 271 86 242 262 256 218 152 80 120 120 80 224 224 200 180 180 200 217 217 211 134 162 68 85 282 296 296 116 116 221 204 218 298 79 79 289 289 178 42 175 175 220 220 118 56 56 197 166 248 248 84 84 284 284 236 196 98 140 98 275 244 244 179 50 148 148 294 212 268 192 86 262 248 248 130 of , 94 152 299 296 110 121 132 132 121 16 16 52 52 240 292 242 170 156 156 191 191 46 147 289 289 24 130 130 280 250 293 116 116 252 252 252 252 32 32 162 276 40 232 232 234 234 112 120 112 120 257 257 238 144 168 144 264 240 240 226 212 34 292 116 185 185 254 190 100 230 137 137 142 142 224 205 5 176 189 250 68 252 157 204 241 141 280 219 235 83 156 182 182 156 70 161 286 296 296 124 194 298 93 46 162 162 286 64 38 20 20 107 276 196 196 25 25 208 208 30 116 296 P bsn oain we notation, Abusing . 263 176 176 50 n 70 105 120 120 105 180 220 195 150 152 254 154 109 126 203 136 136 106 267 207 207 44 214 103 140 140 224 184 29 132 110 198 294 117 284 242 268 224 189 272 272 36 36 174 276 152 94 256 256 166 72 72 173 260 262 62 206 74 235 220 280 277 150 229 250 190 40 264 264 78 161 hsmeans This algebraic. 230 28 170 272 151 172 172 160 131 84 70 232 246 98 98 196 236 236 100 100 184 294 296 172 298 202 +1 190 104 260 206 248 264 264 124 48 226 249 249 228 228 110 110 299 179 253 230 276 188 64 128 64 128 130 208 240 173 118 278 262 246 209 68 134 256 140 210 81 81 220 49 85 76 2 200 200 202 53 206 212 242 244 244 248 254 144 144 288 288 148 148 290 294 206 169 156 169 173 129 196 196 200 172 of 208 236 128 h supergroup the (Π), 145 194 256 256 260 70 204 268 289 289 293 214 161 216 146 76 165 4 200 208 224 142 230 150 160 166 12 6 262 292 197 221 ¶ ). P 292 262 77 69 245 241 76 217 285 146 70 216 214 193 269 204 128 268 194 260 220 256 288 256 224 237 214 101 229 86 272 225 225 236 172 196 97 299 80 80 205 246 22 182 156 206 267 148 148 72 P 107 216 144 144 294 290 288 288 262 125 112 254 158 121 121 99 235 242 104 212 280 206 100 100 182 202 200 200 16 76 174 174 168 220 164 164 162 162 140 210 230 14 68 125 292 292 290 134 130 128 64 64 112 128 246 262 21 240 240 124 188 53 28 232 276 134 35 155 206 49 49 264 264 202 172 84 182 245 246 160 38 256 275 198 78 142 17 88 197 62 244 190 190 230 249 150 74 93 136 250 184 220 72 72 227 280 237 166 123 262 94 176 272 104 260 181 51 246 256 256 102 56 211 124 124 276 44 187 Π 122 289 238 238 272 272 289 168 132 110 224 268 198 20 20 254 242 284 294 294 199 199 233 106 292 292 214 96 52 52 126 222 229 30 30 299 216 280 280 238 238 50 278 214 152 296 120 120 100 212 212 180 220 13 208 38 43 176 176 226 226 189 269 259 159 159 182 265 265 185 185 258 208 208 84 84 214 212 212 82 253 276 248 248 162 162 244 244 152 124 134 196 ihsymmetry with 296 34 286 . 70 8 16 16 194 163 256 78 238 298 254 169 156 156 91 169 133 296 296 128 286 68 204 24 280 100 116 252 32 32 276 276 250 61 77 212 230 56 112 112 292 226 144 144 240 240 238 46 234 234 14 57 57 232 232 155 142 132 274 113 242 130 292 94 141 86 rigorously 125 75 42 269 192 56 56 212 84 84 259 268 196 140 98 98 243 62 121 121 294 205 205 230 230 166 233 233 236 220 220 256 144 178 68 248 248 296 218 204 229 128 213 162 298 222 250 120 80 120 100 100 80 46 134 296 296 282 200 220 180 180 200 220 206 206 209 209 172 152 286 52 52 298 284 284 224 224 216 130 296 196 58 262 239 239 208 208 136 136 119 280 280 122 286 286 64 64 206 131 275 44 44 192 160 192 270 244 244 134 9 18 9 210 288 82 155 132 132 132 132 286 164 208 228 266 173 179 94 194 194 118 124 127 127 215 215 168 291 268 268 80 224 224 110 269 169 169 36 202 211 45 140 140 152 54 63 63 232 232 ssufficiently is 72 72 108 126 180 198 99 189 252 184 112 160 280 284 269 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286 115 296 296 124 212 itself, 298 256 196 205 274 196 81 81 134 152 82 64 297 110 155 299 290 245 245 38 20 11 157 194 84 84 112 84 248 122 214 196 278 298 280 217 244 181 30 284 87 116 87 182 145 194 288 204 299 290 176 50 294 188 100 100 80 208 170 298 212 212 280 91 133 152 193 52 52 184 184 216 217 53 219 74 280 106 221 222 what 44 96 214 140 140 97 121 110 77 121 56 122 228 294 294 284 231 242 124 124 268 248 127 127 272 272 238 238 36 36 276 276 118 152 266 14 37 131 72 72 260 272 109 275 62 74 220 184 136 259 150 164 190 40 78 254 169 91 169 170 247 260 172 172 277 175 175 16 240 149 242 179 207 207 84 84 232 196 98 98 211 236 236 100 100 86 184 294 296 172 298 202 ver- 104 260 206 248 193 264 264 197 225 225 48 226 92 109 228 228 110 110 276 231 231 64 128 64 128 112 235 130 208 240 240 215 262 246 68 134 284 221 153 136 253 289 289 290 290 256 257 292 292 210 140 295 295 299 162 162 216 218 164 164 220 58 76 275 168 298 2 60 174 174 285 100 100 291 200 140 104 212 218 112 220 280 264 136 244 138 71 230 212 73 280 236 75 288 148 290 92 29 130 80 208 56 83 68 86 196 196 200 4 1 145 258 208 214 220 0 101 101 223 256 256 260 0 204 204 268 1 216 216 4 146 146 676 76 4 9 199 199 0 203 203 2 224 224 4 142 142 2 227 227 3 230 230 4 245 245 6 160 160 6 12 6 262 262 9 292 292 8 285 285 268 260 220 256 256 223 214 208 5 2 8 1 1 2 4 0 9 6 8 5 9 9 rhgnlcutradccutrpi ihnerves with pair cocluster and cluster orthogonal dual with generally packings these resulting while that, the observe on and havoc face, a wreak seed or the vertex doubling a including along “growths,” call we which polyhedra, also See pyramid. hexagonal 21. the is polyhedron integral superintegral. not but integral are that packings crystallographic 13. stronger strictly Lemma a is latter the that condition. superintegral discover we inequivalent superintegral, conformally many infinitely packings. crystallographic to rise 12. Theorem statement. stronger following the prove actually we 7, superintegral of choices all for same the universal, are supergroups commensurate) (but different 8.) section 12, ref. in defined is “superintegrality” of notion superpacking its in i.2. Fig. B A oprove To 14: Remark also was packing integral known previously every Although is polyhedron a that say we packings, polyhedral to Returning 11. Remark 10: Definition emtiainadcutrccutrpi for pair cluster/cocluster and Geometrization Π= b hr xs nntl aycnomlyinequivalent conformally many infinitely exist There hmi oeaern (A) dodecahedron. rhombic edfiecranoeain n“seed” on operations certain define we 12, Theorem nntl ayplhdaaesprnerladgive and superintegral are polyhedra many Infinitely fsm akn oee ni s oprove To is. it on modeled packing some if utoeeapeo nitga u o super- not but integral an of example one Just hl ifrn ymtygroups symmetry different While ecl packing a call We PNAS P f sa nee.(oeta nunrelated an that (Note integer. an is | aur ,2019 8, January P Π, An (B) midsphere. their and Π, b superintegral Γ. P | Π so , h superpackings the Γ, e o.116 vol. and Π= P Π. b (growth cuboctahedron feeybend every if | Γ o 2 no. Theorem edto lead Remark ) | and 437

MATHEMATICS 25 24 27 27 24 25 27 25 27 24 24 27 25 27 252427 27 24 25 27 25 27 24 24 27

28 28 28 28 28 28 28 28 Theorem 19. 8 26 26 8 8826 26 If P is a superintegral crystallographic packing, then 29 29 14 6 6614 29 29 14 7 17 15 15 17 17 15 12 3 3 12 18 18 1233 12 18 # 10 10 10 its supergroup Γ is arithmetic. 18 18 18 18 e 22 2 22 222 22 20 20 20 20 4 6 24 24 6 4 446624 24 27 27 27 27 In fact, to conclude arithmeticity, it is sufficient that the orbit 25 29 29 25 25 29 29 25

27 5 5 27 27 5527

6 24 24 6 6624 24 20 20 20 20 4 22 22 4 4422 22 18 18 1118 18 under the supergroup Γe of a single sphere S ∈ P has all integer 10 10 10 12 12 18 18 12 12 18 15 15 15 7 2 17 17 2 17 3 3 14 14 3314 bends. Let us sketch a proof. To a (positively oriented) sphere 29 29 29 29

26 26 6 6626 26 8 28 28 8 28 28 8828 28 28 28

25 24 27 27 24 25 27 25 27 24 24 27 25 27 252427 27 24 25 27 25 27 24 24 27 S of center z = (z1, ... zn ) and radius r, we attach the “inversive coordinates” Fig. 3. A packing modeled on the cuboctahedron, shown with cocluster.   vS : = b, b, bz .

Here, b = 1/r is the bend, and b = 1/r is the cobend, that is, the P(seed) are usually conformally inequivalent, the superpackings b b reciprocal of the coradius, the latter defined as the radius of the Pfare essentially preserved, in fact sphere after inversion through the unit sphere; see the discus- sion in, e.g., refs. 17 and 18. The vector vS lies on a one-sheeted Pf(growth) ⊂ Pf(seed). hyperboloid Q = −1, where Q is the (universal) “discriminant” form, In particular, if a polyhedron is superintegral, then all of its  1  2 growths are also superintegral, and hence integral. This proves Q =  1 . Theorem 12 Theorem 3 2 and hence . −In−1

(♠): Classifying Superintegral Crystallographic Packings In these coordinates, Toward the opposite general problem of classifying integral Γe < OQ (R) [1] and superintegral crystallographic packings, we make two basic observations. The first, having nothing to do with integrality, is a right action by Mobius¨ transformations on the row vector vS . shows that the entire theory of crystallographic packings is Since Γe is a lattice, it is essentially (up to finite index components) “low”-dimensional. Zariski dense in OQ ; hence, the orbit O = vS · Γe of S is essentially Zariski dense in the quadric Q = −1. There is then a choice of Theorem 15. Crystallographic packings can only exist in dimensions cluster CS ⊂ O of n + 2 spheres whose matrix V of inversive coor- n < 996. dinates has (full) rank n + 2. Make such a choice arbitrarily. This To prove this, we need the following. cluster V has a Gram matrix of inversive products,

Lemma 16. The supergroup Γe of a crystallographic packing P with † n+1 G := V· Q ·V , [2] symmetry group Γ is a lattice, that is, it acts on H with finite covolume. which is invertible (also has rank n + 2). Let We first sketch a proof of this lemma. Let Γ be a symme- try group for P; then, it is assumed to be geometrically finite −1 (recall that this means some uniform thickening of the convex F := G core of Γ has finite volume). Since Γ is a reflection group, it has n+1 an essentially unique fundamental polyhedron F := Γ\H . be its inverse, which also induces a quadratic form having The domain of discontinuity Ω of Γ (that is, the complement in signature (1, n + 1). Then Γe is conjugate to a “bends” group, n+1 dH of its limit set ΛΓ) is the union of disjoint open geometric balls, since the limit set ΛΓ is assumed to coincide with the set of −1 Ae := V· Γe ·V < OF (R), limit points of P. The quotient Ω/Γ is then a disjoint union of finitely many open ends. For each end, we develop the domain which now acts on the left on the (second) column vector of under the Γ-action and fill an open ball, the boundary of which is bends b = V· (0, 1, 0, ... , 0)† in V; this vector b lies on the cone then an (unoriented) sphere in P. A geodesic hemisphere above such a ball is a frontier of the flare, cutting the walls it meets F = 0, and Ae is a lattice in OF (R). Though a priori real valued, of F either tangentially or at right angles (for, otherwise, the we claim that F is in fact rational. Indeed, by assumption, the spheres in P would overlap). Hence, when we form the super- Ae-orbit group Γe by adjoining to Γ reflections through all of the spheres B = A·e b in P, we obtain a discrete action, and moreover the original n+2 lies in Z ∩ {F = 0} and is Zariski dense in the cone. However, domain of discontinuity Ω has been entirely cut out, rendering Γe a quadratic form having a Zariski dense set of integer points B on a lattice. the cone F = 0 is easily seen to be rational, as claimed. Next, Returning to Theorem 15, Vinberg (14) and Prokhorov (15) A showed that hyperbolic reflection lattices can only exist in dimen- we observe that, since e is a linear action, it in fact preserves a Λ sions n < 996, and hence crystallographic packings are similarly full-rank Z-lattice . However, the group bounded in dimension, proving the theorem. (The number 996 is Λ not expected to be sharp.) OF = {g ∈ OF (R): gΛ = Λ} Next, we show that not only is the dimension bounded, but if we assume superintegrality, then (up to commensurability) there is easily seen to be congruence and contains Ae. Hence, Ae is are only finitely many Apollonian-like objects. arithmetic, as is its conjugate Γe. This proves Theorem 19. Definition 17: Two crystallographic packings are said to be commensurate if their supergroups are.

# Theorem 18. There are only finitely many commensurability classes Recall that a real hyperbolic lattice is arithmetic (of the simplest type, as all reflec- tion lattices are) if it is commensurate with the automorphism group of a hyperbolic of superintegral crystallographic packings, all of dimension n ≤ 20. quadratic form over the ring of integers of a totally real number field (see, e.g., To prove this theorem, we show the following. ref. 16).

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Belolipetsky Bianchi by group reflective classification of complete (25) the on sub- reflective decomposition suitable a a find with Eq. to exist) form effort, not the does of some it after prove able, (or group yet not are we Eq. in as exist. decomposition answering a Hence, ing ideas. similar 23 uses Question direction forward the and group, this packing. under crystallographic orbit a cluster the Then cocluster. the that so ie ftewlsitretorthogonally. intersect walls the if line, (ii disjoint; (i (iii by: are connected interiors are vertices walls’ two and walls, reflecting subgroups, reflection for algorithm Vinberg’s it from (26); of comes issue Shaiheev execution The of since. the paper ever minor literature early the a an in has to propagated literature has traced the be in can in out, diagram that turns Coxeter mistake It the case unity). that of only case, root Eisenstein cube this The the the on adjoining 25. group is, Bianchi (that ref. the integers is in data Eq. 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C ete(hn ru eeae yrflcin through reflections by generated group (thin) the be C u ro fti eutrelies result this of proof our 24, Theorem −6]) sete ijit agn,o rhgnlt any to orthogonal or tangent, disjoint, either is = PNAS {1}, C scmesrt oamxmlreflec- maximal a to commensurate is sete ijito agn,and tangent, or disjoint either is | hc ie ftewlsme tacusp; a at meet walls the if line, thick a ) aur ,2019 8, January hoe 28 Theorem C b h eutn akn ssonin shown is packing resulting The . C b = rpoigta n cannot one that proving or 3, 3 ,3, {2, sntsrihfradfrom straightforward not is . . . n ec ie ieto rise gives hence and 2 (Z[ | 6}, , o.116 vol. √ .Vrie denote Vertices −6]). otdln,i the if line, dotted a ) iv ( or m; π/ P C | em 16, Lemma eak26, Remark o 2 no. ne the under =Γ := C · | no ) [4] 439 , is

MATHEMATICS D E A C, Cb can be computed to have (symmetric) Gram matrix (see 2 5 Eq. )  −11111110 0 0 0 0 0 √2  √ √ √ √ 3 6 · −1 1 5 7 5 1 0 2 3 4 3 4 3 2 3 0 0  √ √ √ √   · · −1 1 5 7 5 0 0 2 3 4 3 4 3 2 3 0   √ √ √ √   · · · −1 1 5 7 2√3√ 0 0 2 3 4√3 4√3 0   · · · · −1 1 5 4√3 2√3√ 0 0 2 3 4√3 0   · · · · · −1 1 4 3 4 3 2 3 0 0 2 3 0  G =  √ √ √ √ . 2  · · · · · · −1 2 3 4 3 4 3 2 3 0 0 0   · · · · · · · −1 1 5 7 5 1 1   · · · · · · · · −1 1 5 7 5 1   · · · · · · · · · −1 1 5 7 1   ········ · ·−1 1 5 1   ········ · · ·−1 1 1  ············−1 1 1 ·············−1 4 3 [5] Vinberg’s Arithmeticity Criterion (28) (see also ref. 29, Theo- rem 3.1) says in this context that Γe is arithmetic if and only if B cyclic products of 2G are always integers. This is almost the case for Eq. 5, except for the entry √2 in the top right; hence Γe is 25 27 32 25 24 27 27 24 25 32 27 25322724 24 27 32 25 27 3 28 26 26 28 28 28 34 34 8 30 30 8 8 6 14 29 29 1466 14 nonarithmetic (see Lemma 20). However, it is nearly so; indeed, 15 17 1715 15 17 30 18 1830 30 18 34 12 3 3 12 34 34 1 10 10 10 Γe, viewed as a subgroup of OQ (R) (see Eq. 1), can be conju- 18 18 22 22 1 36 2 36 20 20 gated to lie in OQ ( [ ]) with unbounded denominators in its 4 6 24 24 6 4 4 Z 3 27 27 entries. The latter group is a perfectly nice S-arithmetic lattice 25 29 29 25 27 5 5 27

24 24 6 20 20 6 in the product OQ (R) × OQ (Q3), but Γe is already a lattice on 4 22 36 36 22 4 4 18 18 1 O ( ) Γ 10 10 10 projection to the first factor, Q R . This too implies that e is 34 12 12 34 34 1 30 18 1830 30 18 15 17 2 1715 15 17 nonarithmetic and, in this sense, is reminiscent of constructions 14 3 3 14 14 30 29 29 30 34 34 6 8 28 26 26 28 8 6628 28 8 of nonarithmetic groups by Deligne and Mostow (30). It is inter- 25 27 32 25 24 27 27 24 25 32 27 25322724 24 27 32 25 27 esting to understand whether all integral but nonsuperintegral packings arise this way.

Local–Global Principles

Fig. 5. (A) The reflecting walls in Fig. 4. (B) The packing resulting from the We conclude with a discussion of whether local–global princi- orbit generated by Cb on the cluster C in the decomposition Eq. 4. ples hold for bends of crystallographic circle (n = 2) packings. (For higher-dimensional sphere packings, this problem becomes easier; see, e.g., ref. 31.) As explained in ref. 1 for the case of the which, for the Eisenstein integers, has extra stabilizers due to the classical Apollonian packing, the “asymptotic” local–global prin- larger group of units. The true diagram is ciple is proved in ref. 32. This method was extended in the thesis of Zhang (33) to show the same statement for packings modeled on the octahedron. Most recently, Fuchs, Stage, and Zhang (34) showed that the Bourgain–Kontorovich method extends to the and the Eisenstein Bianchi group has a subgroup with Coxeter following context: diagram Theorem 31. Let P be a packing with symmetry group Γ and let C ∈ P. Assume that there is a circle C 0 ∈ P tangent to C so that This last diagram supports a decomposition as in Eq. 3 by tak- the stabilizer of C 0 in Γ is a congruence (Fuchsian) group. Then the ing either C = {1} or C = {3}. We are thus finished sketching the orbit Γ · C satisfies an asymptotic local–global principle. only nonimmediate case of Theorem 24. The assumption of the existence of such a companion cir- Remark 30: In fact, it turns out that all previously known inte- cle C 0 is a generalization of Sarnak’s observation (35) in the gral circle packings (and many new ones) arise in this way as limit classical Apollonian case that such leads to certain shifted k sets of thin subgroups of reflective Bianchi groups. binary quadratic forms representing bends in the orbit. We To prove Theorem 25, we apply the Structure Theorem show that this condition is both satisfied and not satisfied infi- to (manipulations of) certain other Coxeter diagrams, e.g., nitely often. Vinberg’s diagrams (27) in dimensions n + 1 ≤ 14 for the reflec- tive subgroup of the integer orthogonal group preserving the Theorem 32. The assumptions (and hence conclusions) of Theorem 2 2 2 form −2x0 + x1 + ··· + xn+1. In dimensions n + 1 = 15, 16, 17 31 are satisfied for infinitely many conformally inequivalent super- and 21, arithmetic reflection groups are known, but currently not integral crystallographic packings. The same statement holds with crystallographic packings. “are satisfied” replaced by “are not satisfied.” Thus, even the asymptotic local–global problem remains open Integral but Nonsuperintegral Packings in this generality. Lacking evidence against the local–global prin- Let us say more about what happens in Remarks 14 and ciple in all of these examples, we conjecture that it does indeed 21. When Π is the hexagonal pyramid, its supergroup Γe = hold.

ACKNOWLEDGMENTS. The authors benefitted tremendously from numer- ous enlightening conversations about this work with Arthur Baragar, Misha kNote that Stange (9) defines what she calls “K-Apollonian circle packings,” which are Belolipetsky, Elena Fuchs, Jeremy Kahn, Jeff Lagarias, Alan Reid, Igor Rivin, not required to fill the (and hence are not “packings” by our definition); these Peter Sarnak, Kate Stange, Akshay Venkatesh, Alex Wright, and Xin Zhang. exist for every Bianchi group. We also thank the referee for comments and suggestions. A.K. is partially

440 | www.pnas.org/cgi/doi/10.1073/pnas.1721104116 Kontorovich and Nakamura Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 7 otrvc 21)Lte oBl ue vial thttps://math.rutgers.edu at Available Duke. Bill to Letter (2017) A Kontorovich groups. 17. reflection hyperbolic Arithmetic (2016) M Belolipetsky noncompact a 16. with reflections of groups discrete of Absence (1986) MN Prokhorov arXiv:1708.03405. 15. superpackings. and orders Vinberg Quaternion 14. (2017) A Sheydvasser pack- 13. circle Apollonian (2006) CH Yan AR, Wilks CL, Mallows JC, Lagarias RL, Graham 12. and constructions Extremal polytopes: Convex (2007) GM Ziegler 11. revisited. packings, Apollonian dimensional Higher (2017) A Baragar 10. rn M-434,a salBntoa cec onaingat Simons a grant, (FRG) Foundation Science Group Binational of Research Israel Division an Focused DMS-1463940, Grant Grant NSF CAREER (DMS)-1455705, (NSF) Sciences Foundation Mathematical Science National by supported otrvc n Nakamura and Kontorovich .MxelG(92 peepcig n yeblcrflcingroups. reflection hyperbolic and packings Sphere packings. sphere (1982) infinite G of class Maxwell new A 5. (1974) DW Boyd 4. groups. matrix thin on Notes (2014) P Sarnak sieve. 3. affine the and distribution of Levels (2014) A Kontorovich 2. .Sag E(08 h ploinsrcueo inh groups. Bianchi of structure Apollonian configura- The Apollonian (2018) Irreducible KE (2010) Stange C circles. 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