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Geometry and arithmetic of crystallographic sphere 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 polyhedron. Equiv- ones exist only in finitely many “commensurability classes,” all in, alently, Π is a 3-connectedz 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 j crystallographic j arithmetic j polyhedra j 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 dual polyhedron Πb. Fig. 2A shows the case of a cuboctahedron Tthe classical Apollonian gasket. Why does its integral struc- and its dual, the rhombic dodecahedron. 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 MATHEMATICS 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 [ f1g, we mean an infinite collection of oriented Mostow rigidity, but Mobius¨ transformations do not generally (n − 1)-spheres (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 tetrahedron, 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] 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” symmetry 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. www.pnas.org/cgi/doi/10.1073/pnas.1721104116 PNAS Latest Articles j 1 of 6 Downloaded by guest on September 29, 2021 9 162 169 162 169 162 169 162 169 162 169 162 169 162 169 162169 162 169 169 162 196 288 289 242 200 242289 200 288 225 289 288 200 196 242 289200 242 288 225 289 242 288 200 196 242 225 289 289 225242 196 200 288 242 289 225 288 242 200 289 242 196 200 288 289 225 288 200 289 242 200 242 289 288 196 242 288 289 200 242 225 289 121 128 98 274 100 292 128 290 121 81 128 121 98 218 128 81 98 100 178128 121 298 292 98 128 128 98 292 298 121128178 100 98 81 128 218 98 121 128 81 121 290128 292 100 274 98 128 121 98 128 81 121 98 128 260 148 212 72 164 244 226 170 72 260 146 250 148 194 212 72 72 212 194 148 250 146 260 72 170 226 244 164 72 212 148 260 170 260 260 72 164 268 292 296 296 278 278 296 296 292 268 268 292 50 244 222 49 228 254 196 50 49 248 196 244 50 50 244 196 248 49 50 196 254 228 49 222 244 50 49 50 228 292 182 172 292 182 293 168 295 294 294 295 168 293 182 292 172 182 292 294 172 295 168 106156 269 152 122 262 262 148 152 156 158 271 148130 130148 271 158 156 152 148 262 262 122 152 269 156 106 130 299 220 100 220 116 36 236 223 220 231 116 214 245 245 214 116 231 220 223 236 36 116 220 100 220 299 220 237 245214 100 231 299 199 206 196 206 199 82 259 197 173 175 259 196 259 175 173 197 259 82 196 175 174 166 159 158 166 165 251 165 165 251 165 166 158 159 166 74 235 272 32 272 150 150 32 211 151 32 32 151 211 32 150 150 272 32 272 149 32 134 124 256 140 248 248 140 256 124 248 124 140 25 104 68 68 68 68 104 25 25 179 104 240 296 296 58 296 296 58 296 296 296 86 277 192 192 277 86 298 84 52 298 184 132 132 184 298 52 84 298 132 84 52 184 124 125 256 256 139 124 124 139 256 256 125 124 127 50 253 118 118 253 50 118 281 78 253 232 232 160 110 116 76 76 232 232 76 76 116 110 160 232 232 232 110 152 257 103 282 224 18 101 250 250 18 224 282 103 257 152 224 18 152 250 100 100 101 100 100 234 234 234 234 233 232 233 234 234 234 234 232 131 286 258 93 252 252 252 232225 225 252 252 252 93 258 286 225 252 197 184 208 208 184 184 208 208 184 184 208 208 184 184 208 87 204 176 176 204 115 287 202 284 176 176 284 202 287 115 204 176 176 204 202 284 176 287 84 256 276 276 254 79 136 62 277 277 62 136 79 254 276 276 256 136 77 185 56 185 128 128 185 56 185 77 77 210 56 253 254 254 253 254 224120 224 224 120 224 149 52 91 237 214 214 237 91 52 52 34 91 70 70 70 70 70 70 70 231 212 212 156 156 212 212 156 153 205 205 205 144 144162 162 144 144 176 96 62 60 284 284 60 60 284 191 188 128 128 188 191 166262 188 128 266 26 26 294 276 137 137 276 55 53 136 136 53 55 136 55 276 280 280 165 80 280 280 248 112 112 244 244 112 112 244 159 38 109 158 158 158264 138 9 252 252 252 252 252 9 252 252 234 234 113 234 59 20 143 238 106 106 238 143 20 106 20 238 44 8 88 88 8 88 88 8 88 88 8 88 264 Definition 10: 108 108 108 64 superintegral 184 273 204 We call a packing if every bend 28 266 28 268 288 28 186 90 90 266 90 90 186 266 89 89 170217 P 268 266 266 268 268 190 299 266 299 278 77 188 188 188 284 248 184 119 248 248 119 184 248 279 119 180 180 180 244 271 283 283 271 241 208 268 268 268 268 208 241 254 90 90 254 261 110 242 110 109 109 110 242 110 261 162 246 228 236 168 168 236 228 246 236 148 224 224 148 148 224 224 148 166 24 218 217 218 218 217 218 192 280 280 206 206 280 280 238 281 238 48 283 283 48 238 281 238 72 64 64 72 296 296 72 64 64 72 31 30 150 86 201 201 86 150 256 256 29 221 256 256 256 256 221 29 256 256 202 195 195 202 196 284 193 258 92 92 259 258
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