A Class of Models with the Potential to Represent Fundamental Physics
  1. Introduction
  2. Basic Form of Models
  3. Typical Behaviors
  4. Limiting Behavior and Emergent Geometry
  5. The Updating Process for String Substitution Systems
  6. The Updating Process in Our Models
  7. Equivalence and Computation in Our Models
  8. Potential Relation to Physics
  9. Additional Material
  10. References
  11. Index

4.2 Hyperbolic Space

Among all possible rules, the formation of geometrical shapes of the kind we have just been discussing is very rare. Slightly more common is the type of behavior that we see in a rule like:

{{x, y}, {y, z}} -> {{w, x}, {w, y}, {x, y}, {y, z}}
RulePlot[ResourceFunction[ "WolframModel"][{{x, y}, {y, z}} -> {{w, x}, {w, y}, {x, y}, {y, z}}]]
Graph[Rule @@@ #, GraphLayout -> "SpringElectricalEmbedding", ImageSize -> Tiny, VertexStyle -> ResourceFunction["WolframPhysicsProjectStyleData"]["SpatialGraph", "VertexStyle"], EdgeStyle -> ResourceFunction["WolframPhysicsProjectStyleData"]["SpatialGraph", "EdgeLineStyle"]] & /@ ResourceFunction[ "WolframModel"][{{x, y}, {y, z}} -> {{w, x}, {w, y}, {x, y}, {y, z}}, {{0, 0}, {0, 0}}, 9, "StatesList"]

Essentially the same behavior also occurs in a mixed-arity rule with a single relation on the left-hand side:

{{x, y}} -> {{z, y, x}, {y, z}, {z, x}}
RulePlot[ResourceFunction[ "WolframModel"][{{x, y}} -> {{z, y, x}, {y, z}, {z, x}}]]
Flatten[Append[ ResourceFunction["WolframModelPlot"][#, "MaxImageSize" -> 100] & /@ ResourceFunction[ "WolframModel"][{{x, y}} -> {{z, y, x}, {y, z}, {z, x}}, {{0, 0}}, 7, "StatesList"], {ResourceFunction["WolframModelPlot"][ ResourceFunction[ "WolframModel"][{{x, y}} -> {{z, y, x}, {y, z}, {z, x}}, {{0, 0}}, 8, "FinalState"], "MaxImageSize" -> 100, VertexCoordinateRules -> { 1 -> {10.17470974389601, 8.49613742700209}, 3 -> {5.522696341420668, 7.338440718767156}, 6 -> {6.3409702377458155`, 4.311759255597379}, 7 -> {5.496888459165139, 10.472976361208069`}, 12 -> {3.615129292794065, 4.450896697724719}, 13 -> {8.4834898391255, 3.766154816205485}, 14 -> {7.394559375500452, 11.633466188026262`}, 15 -> {2.8984489031323792`, 9.605475891687881}, 24 -> {3.895172802770809, 2.840468525286524}, 25 -> {2.985887559656875, 5.67951058539453}, 26 -> {9.705650118588965, 4.6138773593047375`}, 27 -> {6.971951283425833, 2.168266710532894}, 28 -> {5.483577197699321, 12.73655210751767}, 29 -> {8.7719756599412, 11.234319688281573`}, 30 -> {2.6065366385527637`, 8.230190879914598}, 31 -> {2.7362421862854522`, 11.234118436779987`}, 48 -> {2.6104867160506307`, 2.770610986610823}, 49 -> {4.743472706572571, 2.6262659671477575`}, 50 -> {3.540819922379263, 6.463609329320128}, 51 -> {1.9400317339712698`, 4.5219696852351845`}, 52 -> {9.359034857699994, 2.9573980903550328`}, 53 -> {9.7744564327544, 5.7134912930396275`}, 54 -> {6.054569140344993, 2.2329413652144137`}, 55 -> {7.810359332633349, 1.7201595953950017`}, 56 -> {6.157928916138276, 13.43101395008935}, 57 -> {4.63310743523114, 12.40996821196174}, 58 -> {9.292251306406886, 10.29264323347189}, 59 -> {7.975647412991421, 12.786275585073462`}, 60 -> {1.298412074095909, 9.080242385549548}, 61 -> {3.3459073735497626`, 7.591555166611972}, 62 -> {3.4989145850458265`, 11.642225441828133`}, 63 -> {1.4776734738176203`, 10.967150300145452`}, 96 -> {2.6265786460091918`, 1.9877656342767989`}, 97 -> {2.3999069988493984`, 3.2813803053477923`}, 98 -> {5.249912896997562, 3.1018501323600587`}, 99 -> {3.7828523421324425`, 1.8184148918204628`}, 100 -> {2.3236435514081957`, 5.985684236440766}, 101 -> {4.226436985543674, 6.845361219968333}, 102 -> {2.0760875908713823`, 3.9765707592100723`}, 103 -> {1.527821895377047, 4.9860118385133925`}, 104 -> {9.84876882196794, 3.159437438000381}, 105 -> {8.870775373288968, 2.4930122847313956`}, 106 -> {9.713215771465247, 6.4273452299596885`}, 107 -> {9.564443374464055, 4.695650922862196}, 108 -> {6.189775047562619, 1.2005698384463175`}, 109 -> {5.870942968708712, 2.856650647566711}, 110 -> {8.203304931214394, 2.1457231834793618`}, 111 -> {7.235115999016531, 0.9031052519385188}, 112 -> {5.341861835395375, 14.01802403630179}, 113 -> {6.676217641866845, 13.150808370635533`}, 114 -> {4.639102506185069, 11.759231589519993`}, 115 -> {4.46964750087078, 13.450833824145157`}, 116 -> {9.016294317888168, 11.491306163738942`}, 117 -> {9.324712327813536, 9.483695256436661}, 118 -> {7.387402164569019, 13.041722819752499`}, 119 -> {8.480836067034538, 12.790499516136181`}, 120 -> {1.0175063412620267`, 8.517774213006785}, 121 -> {1.2842755208751715`, 9.648945645876593}, 122 -> {4.106322307324142, 7.622664064007106}, 123 -> {2.041436015749269, 7.71235477691487}, 124 -> {2.360864094882011, 12.182473720501463`}, 125 -> {4.136196607784628, 11.344981944377079`}, 126 -> {1.4160857670602205`, 10.415930600982302`}, 127 -> {1.2835747076180641`, 11.725459301967838`}, 192 -> {2.0605343341705344`, 1.855543156917466}, 193 -> {2.951337055647875, 1.9326988633882491`}, 194 -> {2.6503322363490667`, 3.634247736365396}, 195 -> {1.8831410865730231`, 2.592977474834938}, 196 -> {4.7117874795572, 2.258614373212065}, 197 -> {5.507986998304639, 3.5727581484903213`}, 198 -> {3.3928863219780556`, 1.7956816594501495`}, 199 -> {4.030541053506976, 1.5965062639952539`}, 200 -> {2.4126927621381484`, 6.239169757610148}, 201 -> {1.993384811031858, 5.6902173679425525`}, 202 -> {4.663096900739548, 6.990433952542747}, 203 -> {3.3935442478727955`, 6.565465859574131}, 204 -> {1.324961996319851, 3.9187174952680772`}, 205 -> {2.4714018034136513`, 3.964098169729651}, 206 -> {1.7583015325366371`, 5.256304300292178}, 207 -> {1.0023278842983847`, 4.530770171567826}, 208 -> {9.786668835484088, 2.432248416002552}, 209 -> {9.948506215284393, 3.5282724205299636`}, 210 -> {8.56337519631959, 2.703035079087936}, 211 -> {9.195870791017624, 2.043545277484997}, 212 -> {9.453227824296258, 5.862678627849127}, 213 -> {9.760124390150489, 6.818075025592382}, 214 -> {9.530018631915722, 4.168840988006668}, 215 -> {9.467628925052926, 4.9701284089048015`}, 216 -> {5.798428706239622, 1.0842086338706933`}, 217 -> {6.482929233335493, 1.052079096074601}, 218 -> {5.907983446844438, 3.347020887766741}, 219 -> {5.66721804704877, 1.9770175999694504`}, 220 -> {8.036121999275705, 1.2383188948813042`}, 221 -> {8.206638817742807, 2.613926297780714}, 222 -> {6.913404064236986, 0.9420147932776375}, 223 -> {7.533885422605626, 0.5855590352420348}, 224 -> {5.53534194758248, 14.41945841669519}, 225 -> {5.043689800326046, 13.89330569355554}, 226 -> {6.813267843508938, 12.697925406148947`}, 227 -> {6.269316939980895, 13.989899808151852`}, 228 -> {4.205252245004947, 12.555596907314735`}, 229 -> {4.8038092945374515`, 11.302690474098675`}, 230 -> {4.720908934682256, 13.678544129069051`}, 231 -> {4.0629689798184, 13.450733774753438`}, 232 -> {9.14889398710512, 11.239376726671393`}, 233 -> {8.731406979969686, 11.931213888824908`}, 234 -> {9.42538430770116, 8.98925299915938}, 235 -> {8.93212552219056, 10.112468313284895`}, 236 -> {7.574149461906626, 13.618221989401357`}, 237 -> {7.168391526666329, 12.720759256428687`}, 238 -> {8.59323574079489, 12.437982496182078`}, 239 -> {8.2248800916467, 13.506578453642224`}, 240 -> {0.3954149682270211, 8.820653643679009}, 241 -> {1.3020785200396694`, 8.30520045815128}, 242 -> {1.6621713724958307`, 9.766291503487286}, 243 -> {0.5434946366412812, 9.510233170570245}, 244 -> {3.215283797799815, 7.743105658801523}, 245 -> {4.617948919083308, 7.612715441795215}, 246 -> {1.6465575462112751`, 7.914964805554587}, 247 -> {2.1983844797056356`, 7.475321580418564}, 248 -> {2.549407325777497, 12.451426854459541`}, 249 -> {1.9759388269268425`, 12.108507647160456`}, 250 -> {4.506638457006838, 10.97099084325866}, 251 -> {3.3193991054790533`, 11.896335192925319`}, 252 -> {0.7309589207606464, 10.943348448769393`}, 253 -> {1.7476992060481733`, 10.13382023278189}, 254 -> {1.5808897877956716`, 11.865005944651491`}, 255 -> {0.7001102298755331, 11.707638271311527`}, 256 -> {18.7387554885444, 1.2794820788594188`}, 257 -> {18.35944764914279, 1.5153297101981624`}, 258 -> {18.1875670272798, 2.9271234608181245`}, 259 -> {18.95542130386331, 1.9546039496195315`}, 260 -> {17.77052414654499, 2.1687257867118612`}, 261 -> {17.12593520460679, 3.3327645987765284`}, 262 -> {18.43645318704022, 1.779902210261322}, 263 -> {17.989469069153383`, 1.6192032438309973`}, 264 -> {17.33259820889451, 4.298582276697555}, 265 -> {17.89625529395596, 3.9812410552277204`}, 266 -> {15.321927825221675`, 5.006886584367881}, 267 -> {16.458370019541235`, 4.327965533714803}, 268 -> {19.094072030197395`, 2.9932258699664103`}, 269 -> {18.284852453759715`, 3.1299338130061827`}, 270 -> {18.18574910654754, 4.049135442605229}, 271 -> {19.074995841266393`, 3.388018275880864}, 272 -> {15.264242533317995`, 3.4213201492840946`}, 273 -> {15.235103821735498`, 4.477256500634358}, 274 -> {15.933846463544734`, 3.5035360164876947`}, 275 -> {15.466933425514359`, 3.0372004829578576`}, 276 -> {15.897312991700451`, 6.833357088045831}, 277 -> {15.304936289492687`, 7.448722613446059}, 278 -> {15.70919565235456, 5.026894175294209}, 279 -> {15.88735197238305, 6.041477702547438}, 280 -> {17.217420411598578`, 1.496341178371262}, 281 -> {16.765202089269362`, 1.6384124581973951`}, 282 -> {16.892212746029493`, 3.2244525561387083`}, 283 -> {17.29516690078701, 2.044128794703713}, 284 -> {15.97535388084484, 2.1218293085057347`}, 285 -> {15.927415221330929`, 3.316406954035385}, 286 -> {16.54654914376718, 1.6398436559238343`}, 287 -> {16.136292231344925`, 1.60695815838103}, 288 -> {13.082446073904803`, 6.546066190050237}, 289 -> {13.297140307435107`, 6.058367416563898}, 290 -> {11.915836598094568`, 6.792745142797633}, 291 -> {12.510641292578205`, 6.72845138654665}, 292 -> {13.586218302675796`, 5.1840106001373805`}, 293 -> {13.780069887886054`, 6.118003064509507}, 294 -> {13.146065806332398`, 5.574211782157779}, 295 -> {13.188812971106891`, 4.953200167563724}, 296 -> {11.03580978992025, 8.830213270907297}, 297 -> {11.514939294629686`, 8.658904181360896}, 298 -> {9.754976559119925, 8.220436807048149}, 299 -> {10.555214050671585`, 8.359681570361321}, 300 -> {12.625203240946634`, 8.051435222113296}, 301 -> {12.459880745450832`, 7.408725429647387}, 302 -> {11.812348321328578`, 8.517193220724256}, 303 -> {12.424714442563698`, 8.45841413314397}, 304 -> {18.617335853202977`, 5.319590581266202}, 305 -> {18.08324496734576, 5.506321311857555}, 306 -> {17.04882498891599, 6.061441286534572}, 307 -> {18.318274597032758`, 5.673357176255218}, 308 -> {16.79486091600649, 5.42206314312673}, 309 -> {15.54711549559176, 5.307846895296838}, 310 -> {17.961914344911794`, 5.685968517659683}, 311 -> {17.61961673066874, 5.883892173363255}, 312 -> {15.491013509908575`, 6.654111703766665}, 313 -> {16.127015405388743`, 6.3603665778927665`}, 314 -> {14.018570948449465`, 6.636645936233051}, 315 -> {14.84016561137184, 6.741332864172207}, 316 -> {17.471880513979425`, 5.259741053661548}, 317 -> {16.85272216050957, 5.316440582391797}, 318 -> {16.23895649148957, 5.861391787176203}, 319 -> {17.124340469747555`, 5.231500767751133}, 320 -> {14.379410361745288`, 14.65196150826595}, 321 -> {13.809692957063266`, 14.02701574963982}, 322 -> {14.600631607529392`, 13.371663473845777`}, 323 -> {14.718536928888367`, 14.428515916945718`}, 324 -> {13.20935613449849, 12.9627595321291}, 325 -> {13.053318680802665`, 11.681725473586834`}, 326 -> {13.599665186723309`, 13.913411478922221`}, 327 -> {13.206638648919307`, 13.611568363715937`}, 328 -> {15.964025084302046`, 12.437763763290203`}, 329 -> {15.879660184723864`, 12.975757709099902`}, 330 -> {15.063345419548842`, 10.884947492263498`}, 331 -> {15.685740948673963`, 11.87605782708162}, 332 -> {15.473903683401963`, 14.131522176201079`}, 333 -> {14.824854128177078`, 13.277775910668964`}, 334 -> {15.843899555241505`, 13.222751891117822`}, 335 -> {15.866748963100388`, 14.064873621759856`}, 336 -> {10.539515601522258`, 10.736083576325125`}, 337 -> {10.69217712744977, 9.870853269221179}, 338 -> {11.353558163516919`, 10.84749310974494}, 339 -> {10.807868574044901`, 11.115608281300398`}, 340 -> {9.910791185906254, 8.471554371360828}, 341 -> {9.81674485140098, 7.458338716272725}, 342 -> {10.343937905749836`, 9.636778001527034}, 343 -> {10.381868998990665`, 9.050237577124888}, 344 -> {12.41949553553432, 13.217166381858704`}, 345 -> {11.98858619482119, 12.867976102044347`}, 346 -> {12.793102515365659`, 11.688616633903356`}, 347 -> {12.717644117531934`, 12.80171383221538}, 348 -> {11.293886970987833`, 12.154629568258134`}, 349 -> {11.490934832783095`, 11.025021876302393`}, 350 -> {11.761067418947992`, 12.79926724681619}, 351 -> {11.349045510671221`, 12.687467132591909`}, 352 -> {20.271333966192564`, 8.965054985781368}, 353 -> {20.017994266889904`, 9.431542268718125}, 354 -> {18.58007983465817, 8.669735024015566}, 355 -> {19.831648086830224`, 8.688309199914528}, 356 -> {19.371181719352613`, 10.226716206373217`}, 357 -> {18.15666848731374, 9.952781985561844}, 358 -> {19.95955227844548, 9.678947900571814}, 359 -> {19.93963187238929, 10.165168746557603`}, 360 -> {16.9754496564222, 8.128175808716808}, 361 -> {17.690540805624703`, 7.938331109066248}, 362 -> {15.294013315469536`, 7.8064804765721}, 363 -> {16.351913384908848`, 8.142568913644304}, 364 -> {19.233838065215764`, 8.069650939471318}, 365 -> {18.52744106181223, 8.45711406832441}, 366 -> {17.94706599559249, 7.716174422101309}, 367 -> {19.01076359679024, 7.762793335484749}, 368 -> {18.012014634123624`, 13.071090308299082`}, 369 -> {17.178834911068172`, 12.517927971159649`}, 370 -> {18.040920471452665`, 12.026593061522496`}, 371 -> {18.356564488537824`, 12.893818139690165`}, 372 -> {16.22909963576365, 11.410420500846277`}, 373 -> {15.31146758508006, 10.665204724732103`}, 374 -> {17.079303375541397`, 12.286693759604645`}, 375 -> {16.52596719073692, 12.062912132605911`}, 376 -> {19.711373014297575`, 11.095217353036368`}, 377 -> {19.616045810902985`, 11.553419604924184`}, 378 -> {18.064123413287525`, 10.211892524095667`}, 379 -> {19.21908532404918, 10.76468980039723}, 380 -> {19.189773610057667`, 12.54685054799994}, 381 -> {18.20169216629831, 11.874156157100648`}, 382 -> {19.56353841558557, 11.801197463442174`}, 383 -> {19.661503798202045`, 12.442760194265137`}, 384 -> {1.8691816327553177`, 1.3485549659186007`}, 385 -> {1.8350504470156195`, 2.007617492175618}, 386 -> {3.1638170078886945`, 2.2015828502111323`}, 387 -> {2.3452420935715534`, 1.3448388126387132`}, 388 -> {1.9415843651588613`, 3.1090105298952655`}, 389 -> {2.9697760677783567`, 3.8540918530132515`}, 390 -> {1.7290466326210419`, 2.2383649855449206`}, 391 -> {1.5395697079182664`, 2.6745554088481938`}, 392 -> {4.6533852262381945`, 2.421444941760983}, 393 -> {4.428294464229643, 1.8757900213418517`}, 394 -> {5.726862116828631, 3.895085909758719}, 395 -> {4.9434616156505795`, 3.0652596700205716`}, 396 -> {3.18696760672239, 1.1672353668289361`}, 397 -> {3.3599230900922397`, 2.072708582869418}, 398 -> {4.24744629741511, 1.727889870583832}, 399 -> {3.5515313773913215`, 1.042667295929156}, 400 -> {1.6742264099775213`, 6.073368558721967}, 401 -> {2.7043168989694757`, 6.274588824348263}, 402 -> {2.151039350286135, 5.489258663763903}, 403 -> {1.4504495442771432`, 5.7714864204415015`}, 404 -> {4.044579800391335, 6.780480198052568}, 405 -> {4.979067829480787, 7.068945248071374}, 406 -> {2.9760627945127127`, 6.334292931424521}, 407 -> {3.571153530749396, 6.734180538539314}, 408 -> {1.1366453803236276`, 3.5426435743933498`}, 409 -> {1.0166299143230686`, 4.0198404277362325`}, 410 -> {2.826646089938228, 4.092006347730651}, 411 -> {1.6870973179124706`, 3.608121573846643}, 412 -> {0.9205205258976232, 5.017318238906172}, 413 -> {2.1274697748558054`, 5.2702113806623485`}, 414 -> {0.9051824247985466, 4.24666287049311}, 415 -> {0.5611423499014171, 4.646921276784331}, 416 -> {10.099865524673385`, 2.2794275943074784`}, 417 -> {9.673358595581615, 2.104422614791816}, 418 -> {9.96549298576773, 3.844051424303155}, 419 -> {10.127874814555595`, 2.8084443519564966`}, 420 -> {8.698222676229491, 1.9583179480369077`}, 421 -> {8.39229031741017, 3.0219895183195127`}, 422 -> {9.45318019395323, 1.9777415640788696`}, 423 -> {9.07132019146989, 1.5852969953910043`}, 424 -> {9.235183543941597, 6.053713506908798}, 425 -> {9.337603025859355, 5.478313744978044}, 426 -> {9.93661677764339, 7.014022968836638}, 427 -> {9.349693369734098, 6.517788760961554}, 428 -> {9.713077143496962, 4.0504811728926065`}, 429 -> {9.525709013595574, 3.890021534231551}, 430 -> {9.698322748832087, 5.143992500616164}, 431 -> {9.205878149911355, 4.4755001420855125`}, 432 -> {5.7359413247867375`, 0.4343489838434571}, 433 -> {5.654166733806448, 1.290219667502146}, 434 -> {6.638401968898531, 1.311090600168602}, 435 -> {6.129740655501775, 0.3960966802625739}, 436 -> {5.589631998536661, 2.6741081865753795`}, 437 -> {6.004581742175376, 3.7041826753091947`}, 438 -> {5.592851331266756, 1.530294261691186}, 439 -> {5.438575772376027, 2.067001155507924}, 440 -> {8.284188595347134, 1.1601256004389766`}, 441 -> {7.951074572156767, 0.812831057324171}, 442 -> {8.120557268536858, 3.022254705062173}, 443 -> {8.156124093334014, 1.7848372437696274`}, 444 -> {6.9371707767110165`, 0.1829491044719979}, 445 -> {6.879215881658812, 1.2425427816655097`}, 446 -> {7.766258569438685, 0.6617753107831408}, 447 -> 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1.021199394878174}, 289 -> {12.589543784148182`, 1.0382085494996716`}, 290 -> {12.129826801928141`, 2.7176947901732085`}, 291 -> {12.022272855246753`, 1.5575767689348474`}, 292 -> {13.534283743382279`, 1.3094982920993665`}, 293 -> {13.522834031277895`, 2.4642996673857738`}, 294 -> {12.83136087041154, 1.0214530784657754`}, 295 -> {13.30576104231454, 0.8092620330934874}, 296 -> {11.854049306972723`, 4.955983846314114}, 297 -> {11.84216570532573, 4.289753369120032}, 298 -> {11.100729013838018`, 6.154788281194646}, 299 -> {11.678007766705763`, 5.515505814284875}, 300 -> {11.680707203722088`, 2.518298526582603}, 301 -> {12.217895260429884`, 2.9464154647960275`}, 302 -> {11.55566302857546, 3.882559207844009}, 303 -> {11.49157191786629, 2.8935245354897887`}, 304 -> {16.79983415120599, 0.7545012348998767}, 305 -> {16.63614321885152, 1.6001439367777888`}, 306 -> {15.521715176949325`, 1.262337197772565}, 307 -> {16.42129192675526, 0.603225710279899}, 308 -> {16.318686948383043`, 2.9397660612280667`}, 309 -> {15.609920258901852`, 3.830758172745522}, 310 -> {16.619000525174844`, 1.8547677856831353`}, 311 -> {16.59076237525316, 2.400247742676207}, 312 -> {14.12549703801689, 0.669659953024965}, 313 -> {14.563975527020105`, 0.4430725467808596}, 314 -> {13.776371810405557`, 2.5434701051645465`}, 315 -> {14.071553590997246`, 1.3079095337399398`}, 316 -> {15.778178990962797`, 0.1882781188228453}, 317 -> {15.65543073344583, 1.2588942761353934`}, 318 -> {14.80763968483301, 0.3669615688046175}, 319 -> {15.568704076832809`, -0.01848276421305961}, 320 -> {13.303813910872893`, 14.81601058816587}, 321 -> {12.851140442015963`, 14.154235277382728`}, 322 -> {13.738183129795836`, 13.57346780650036}, 323 -> {13.68022823474363, 14.633061483693872`}, 324 -> {12.461274918120633`, 13.031173344396244`}, 325 -> {12.49684174291779, 11.793755883103698`}, 326 -> {12.666324439297881`, 14.003179530841699`}, 327 -> {12.333210416107514`, 13.655884987726893`}, 328 -> {15.178823239078621`, 12.749009432657946`}, 329 -> {15.024547680187892`, 13.285716326474684`}, 330 -> {14.612817269279272`, 11.111827912856675`}, 331 -> {15.027767012917987`, 12.14190240159049}, 332 -> {14.487658355952874`, 14.419913907903297`}, 333 -> {13.978997042556117`, 13.504919987997269`}, 334 -> {14.9632322776482, 13.525790920663724`}, 335 -> {14.881457686667911`, 14.381661604322414`}, 336 -> {11.411520861543293`, 10.340510446080359`}, 337 -> {10.91907626262256, 9.672018087549706}, 338 -> {11.091689997859074`, 10.92598905393432}, 339 -> {10.904321867957686`, 10.765529415273264`}, 340 -> {11.26770564172055, 8.298221827204316}, 341 -> {10.680782233811257`, 7.801987619329232}, 342 -> {11.279795985595293`, 9.337696843187826}, 343 -> {11.382215467513051`, 8.762297081257072}, 344 -> {11.546078819984757`, 13.230713592774865`}, 345 -> {11.164218817501418`, 12.838269024087001`}, 346 -> {12.225108694044478`, 11.794021069846357`}, 347 -> {11.919176335225156`, 12.857692640128963`}, 348 -> {10.489524196899053`, 12.007566236209374`}, 349 -> {10.651906025686918`, 10.971959163862715`}, 350 -> {10.944040415873033`, 12.711587973374055`}, 351 -> {10.517533486781263`, 12.536582993858392`}, 352 -> {20.05625666155323, 10.169089311381539`}, 353 -> {19.7122165866561, 10.569347717672759`}, 354 -> {18.48992923659884, 9.54579920750352}, 355 -> {19.696878485557022`, 9.7986923492597}, 356 -> {18.930301693542177`, 11.207889014319228`}, 357 -> {17.79075292151642, 10.72400424043522}, 358 -> {19.60076909713158, 10.796170160429638`}, 359 -> {19.48075363113102, 11.273367013772521`}, 360 -> {17.046245480705252`, 8.081830049626557}, 361 -> {17.641336216941937`, 8.481717656741349}, 362 -> {15.63833118197386, 7.747065340094496}, 363 -> {16.57281921106331, 8.035530390113301}, 364 -> {19.166949467177503`, 9.044524167724369}, 365 -> {18.46635966116851, 9.326751924401968}, 366 -> {17.91308211248517, 8.541421763817606}, 367 -> {18.943172601477126`, 8.742642029443903}, 368 -> {17.065867634063327`, 13.773343292236714`}, 369 -> {16.369952714039538`, 13.088120717582038`}, 370 -> {17.257475921362406`, 12.743302005296453`}, 371 -> {17.430431404732257`, 13.648775221336933`}, 372 -> {15.673937395804067`, 11.750750918145298`}, 373 -> {14.890536894626017`, 10.92092467840715}, 374 -> {16.189104547225003`, 12.940220566824019`}, 375 -> {15.964013785216453`, 12.394565646404887`}, 376 -> {19.077829303536383`, 12.141455179317676`}, 377 -> {18.888352378833606`, 12.577645602620949`}, 378 -> {17.64762294367629, 10.961918735152619`}, 379 -> {18.675814646295784`, 11.707000058270605`}, 380 -> {18.272156917883095`, 13.471171775527157`}, 381 -> {17.453582003565955`, 12.614427737954738`}, 382 -> {18.782348564439026`, 12.808393095990253`}, 383 -> {18.74821737869933, 13.46745562224727}}]}]]

We can think of the structure that is produced as being like a binary tree of triangles:

CloudGet["https://wolfr.am/LDlGWNOC"]; Table[Show[hgraph[k], Overlay3CycleTree[hgraph[k]], ImageSize -> 100], {k, 1, 6}]

The same structure can be produced from an Apollonian circle packing (e.g. [18][1: p985]):

CloudGet["https://wolfr.am/LDlGWNOC"]; Show[#, ImageSize -> 120] & /@ (Append[#, Show[#]] &[ CirclePack[2000]])

If each triangle is required to have the same area, the structure can be rendered in 2D as:

pts = {{0, -1.`20.}, {-1.`20., 0}, {0, 1.`20.}, {1.`20., 0}, { 1.5`20., -1.5`20.}, { 1.5`20., 1.5`20.}, {-1.5`20., 1.5`20.}, {-1.5`20., -1.5`20.}, { 1.12132034355964257320253308631454711785`20., \ -2.70710678118654752440084436210484903928`20.}, { 2.70710678118654752440084436210484903928`20., \ -1.12132034355964257320253308631454711785`20.}, { 2.70710678118654752440084436210484903928`20., 1.12132034355964257320253308631454711785`20.}, { 1.12132034355964257320253308631454711785`20., 2.70710678118654752440084436210484903928`20.}, \ {-1.12132034355964257320253308631454711785`20., 2.70710678118654752440084436210484903928`20.}, \ {-2.70710678118654752440084436210484903928`20., 1.12132034355964257320253308631454711785`20.}, \ {-2.70710678118654752440084436210484903928`20., \ -1.12132034355964257320253308631454711785`20.}, \ {-1.12132034355964257320253308631454711785`20., \ -2.70710678118654752440084436210484903928`20.}, { 0.79419852642987161560934934273958486775`20., \ -3.99270561650904772080251126955401660287`20.}, { 2.46847367459241271032466037638313767368`20., \ -3.69433192638305998443269247571841841951`20.}, { 3.69433192638305998443269247571860510585`20., \ -2.46847367459241271032466037638312913734`20.}, { 3.99270561650904772080251126955401660287`20., \ -0.79419852642987161560934934273960840279`20.}, { 3.99270561650904772080251126955401660287`20., 0.79419852642987161560934934273958486775`20.}, { 3.69433192638305998443269247571860510585`20., 2.46847367459241271032466037638313767368`20.}, { 2.46847367459241271032466037638313767368`20., 3.69433192638305998443269247571860510585`20.}, { 0.79419852642987161560934934273958486775`20., 3.99270561650904772080251126955401660287`20.}, \ {-0.79419852642987161560934934273960840279`20., 3.99270561650904772080251126955401660287`20.}, \ {-2.46847367459241271032466037638312913734`20., 3.69433192638305998443269247571860510585`20.}, \ {-3.69433192638305998443269247571841841951`20., 2.46847367459241271032466037638313767368`20.}, \ {-3.99270561650904772080251126955401660287`20., 0.79419852642987161560934934273958486775`20.}, \ {-3.99270561650904772080251126955401660287`20., \ -0.79419852642987161560934934273960840279`20.}, \ {-3.69433192638305998443269247571841841951`20., \ -2.46847367459241271032466037638312913734`20.}, \ {-2.46847367459241271032466037638312913734`20., \ -3.69433192638305998443269247571841841951`20.}, \ {-0.79419852642987161560934934273960840279`20., \ -3.99270561650904772080251126955401660287`20.}, { 0.55102318312238618372497126620459485886`20., \ -5.59463226576418582728196711338111292191`20.}, { 1.83044317910220187218586706530083656464`20., \ -6.03416248806451987781867073030208119309`20.}, { 2.96120847205624827553663714154394936184`20., \ -5.54003139109315793301630224365458535486`20.}, { 3.78487807448266062413424710532448130705`20., \ -4.61188727767775329761713898646067309991`20.}, { 4.61188727767775329761713899489091891892`20., \ -3.78487807448266062413424710556894732372`20.}, { 5.54003139109315793301630224594187326079`20., \ -2.9612084720562482755366371415424867543`20.}, { 6.03416248806451987781867073030011109744`20., \ -1.83044317910220187218586706530128789947`20.}, { 5.59463226576418582728196711338111292191`20., \ -0.55102318312238618372497126620460635533`20.}, { 5.59463226576418582728196711338111292191`20., 0.55102318312238618372497126620459485886`20.}, { 6.03416248806451987781867073030011109744`20., 1.83044317910220187218586706530083656464`20.}, { 5.54003139109315793301630224594187326079`20., 2.96120847205624827553663714154394936184`20.}, { 4.61188727767775329761713899489091891892`20., 3.78487807448266062413424710532448130705`20.}, { 3.78487807448266062413424710532448130705`20., 4.61188727767775329761713899489091891892`20.}, { 2.96120847205624827553663714154394936184`20., 5.54003139109315793301630224594187326079`20.}, { 1.83044317910220187218586706530083656464`20., 6.03416248806451987781867073030011109744`20.}, { 0.55102318312238618372497126620459485886`20., 5.59463226576418582728196711338111292191`20.}, \ {-0.55102318312238618372497126620460635533`20., 5.59463226576418582728196711338111292191`20.}, \ {-1.83044317910220187218586706530128789947`20., 6.03416248806451987781867073030011109744`20.}, \ {-2.9612084720562482755366371415424867543`20., 5.54003139109315793301630224594187326079`20.}, \ {-3.78487807448266062413424710556894732372`20., 4.61188727767775329761713899489091891892`20.}, \ {-4.61188727767775329761713898646067309991`20., 3.78487807448266062413424710532448130705`20.}, \ {-5.54003139109315793301630224365458535486`20., 2.96120847205624827553663714154394936184`20.}, \ {-6.03416248806451987781867073030208119309`20., 1.83044317910220187218586706530083656464`20.}, \ {-5.59463226576418582728196711338111292191`20., 0.55102318312238618372497126620459485886`20.}, \ {-5.59463226576418582728196711338111292191`20., \ -0.55102318312238618372497126620460635533`20.}, \ {-6.03416248806451987781867073030208119309`20., \ -1.83044317910220187218586706530128789947`20.}, \ {-5.54003139109315793301630224365458535486`20., \ -2.9612084720562482755366371415424867543`20.}, \ {-4.61188727767775329761713898646067309991`20., \ -3.78487807448266062413424710556894732372`20.}, \ {-3.78487807448266062413424710556894732372`20., \ -4.61188727767775329761713898646067309991`20.}, \ {-2.9612084720562482755366371415424867543`20., \ -5.54003139109315793301630224365458535486`20.}, \ {-1.83044317910220187218586706530128789947`20., \ -6.03416248806451987781867073030208119309`20.}, \ {-0.55102318312238618372497126620460635533`20., \ -5.59463226576418582728196711338111292191`20.}, \ {-8.82526348925847426852402385155172376707`17.034535949251175, \ -1.30910417496527758625161316043272474563`17.033641220531187}, \ {-8.60395399378296035339567691607117741313`16.77516083395495, \ -2.15517828152853602905144313192084376115`16.77470680701365}, \ {-7.91791301234602057822105557681832979014`16.781946434870076, \ -2.83307457668812429555399420546063180445`16.781542187064247}, \ {-7.60809348503042668672579049508962667741`16.69360412228167, \ -3.59836023010762696428258572794985579709`16.69333338670229}, \ {-7.71876993056995754333201385207778349273`16.485338827866748, \ -4.6264526527812567562991769980891887432`16.48521426714185}, \ {-7.23113854984243469857080969083717431409`16.47264180519843, \ -5.36297785565941535606411801962901422502`16.472567306844535}, \ {-6.05056443863198370609762787035059609581`16.606920188461768, \ -5.48391194992202377784551399353748118763`16.606876042578037}, \ {-5.48391194992202377784551399353748118763`16.606876042578037, \ -6.05056443863198370609762787035059609581`16.606920188461768}, \ {-5.36297785565941535606411801962901422502`16.472567306844535, \ -7.23113854984243469857080969083717431409`16.47264180519843}, \ {-4.6264526527812567562991769980891887432`16.48521426714185, \ -7.71876993056995754333201385207778349273`16.485338827866748}, \ {-3.59836023010762696428258572794985579709`16.69333338670229, \ -7.60809348503042668672579049508962667741`16.69360412228167}, \ {-2.83307457668812429555399420546063180445`16.781542187064247, \ -7.91791301234602057822105557681832979014`16.781946434870076}, \ {-2.15517828152853602905144313192084376115`16.77470680701365, \ -8.60395399378296035339567691607117741313`16.77516083395495}, \ {-1.30910417496527758625161316043272474563`17.033641220531187, \ -8.82526348925847426852402385155172376707`17.034535949251175}, \ {-0.38525186847282515601338759957429252803`17.65204355589572, \ -7.84198193616441487727518548839545152795`17.655921511225028}, { 0.38525186847282515601338759957428454815`17.65204355589572, \ -7.84198193616441487727518548839545191999`17.655921511225028}, { 1.30910417496527758625161316043269074486`17.03364122053118, \ 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If we tried to render this with every triangle roughly the same size, then even in 3D the best we could do would be to have something that crinkles up more and more at the edge, like an idealized lettuce leaf:

CloudGet["https://wolfr.am/LDlGWNOC"]; ResourceFunction["GraphReconstructedSurface"][ List @@@ EdgeList[hgraph[10]], 2, EdgeStyle -> Directive[AbsoluteThickness[1.3], Hue[0.61, 0.3, 0.85]], VertexSize -> 2, VertexStyle -> Directive[Hue[0.65, 0.64, 0.68], Specularity[Hue[0.71, 0.6, 0.64], 10]]]

But just as we can think of the grids we discussed before as being regularly laid out in ordinary 2D or 3D space, so now we can think of the object we have here as being regularly laid out in a hyperbolic space [19][20] of constant negative curvature.

In particular, the object corresponds to an infinite-order triangular tiling of the hyperbolic plane (with Schläfli symbol {3,}). There are a variety of ways to visualize the hyperbolic plane. One example is the Poincaré disk model in which hyperbolic-space straight lines are rendered as arcs of circles orthogonal to the boundary:

CloudGet["https://wolfr.am/LDlGWNOC"]; Table[Graphics[{{LightGray, Circle[]}, Style[{Line[{{0, -1}, {0, 1}}], Table[ PLine[#[[1]], #[[2]]] & /@ N[Partition[Table[2 k Pi/(2^n), {k, 0, 2^n - 1}], 2, 1, 1]], {n, 1, step + 1}]}, ResourceFunction["WolframPhysicsProjectStyleData"][ "SpatialGraph"]["EdgeLineStyle"]]}, ImageSize -> 130, PlotRange -> {{-1, 1}, {-1, 1}}], {step, 5}]

(The particular graph here happens to be the Farey graph [21].)