Index

A

• Abelian group, 198
• Acknowledgements, 487
• Action, 407, 441
• Adjacency matrices, 181
  • and machine learning, 144
• Adjacency tensors
  • for hypernetworks, 181
• ADM formalism, 259
• Age distribution
  • of edges, 181
• Age of universe, 467
• Algebra of operators, 449
• Algebraic surface, 151
• Alternating groups, 198
  • as symmetry of evolution, 361
• Ambiguity
  • of rule updates, 86
• Amplitudes in quantum theory, 407
  • as complex numbers, 441
• Analyticity
  • see also Continuity
• Ancestors
  • in multiway graphs, 262
• Angular momentum
  • intrinsic of particles, 407
• Apollonian packing, 156
• Arbelos, 155
• Archives, 487
• Arrow of time, 413, 429
• Associative calculi, 477
• Assortativity, 183
• Astrophysical processes, 470
• Asymptotic, 93
• Asymptotic growth, 93, 177
• Automated theorem proving, 204, 436
• Automorphism
  • hypergraph, 366

B

• Ball
  • volume of, 163, 174, 193, 413
• Bell numbers, 93, 95
• Bell’s inequalities, 409
• BellB, 93, 95
• Betweenness centrality, 183
• Binary relations
  • rules based on, 99
• Binomial, 77
• Binomial tree, 77
• Black holes, 409
  • and quantum information, 461
  • as analog of quantum measurement, 446
  • galactic, 470
  • qubit as analog of, 458
• Blockchain, 304
• Boolean expression confluence, 205
• Boost, 417
• Branch pairs, 220
  • and commutators, 450
  • in hypergraphs, 313
• Branchial distance, 451
• Branchial graphs, 260, 370, 486
  • and entanglement, 438
  • weights in, 289
• Branchial motion, 454
• Branchial space, 438, 454
• Branchlike connections, 454
• Branchlike hypersurface, 434, 439
• Branchlike separation, 444
• Breadth-first scan, 255
• Buckyball graph
  • directed, 254
• Buckyball graphs, 174, 179, 188
• Buckyball Sierpiński graphs, 197
• Busy beaver, 135

C

• c
  • see Speed of light
• Cactus-like graphs, 81, 100
• Canonical forms
  • see also Confluence
• Canonical systems, 477
• Canonicalization
  • of rules, 91
• Cantor set, 196
• Category theory, 477
• Causal disconnection, 347
• Causal dynamical triangulation, 477
• Causal graphs, 230, 237, 314, 431, 484
  • disconnection in, 457
  • foliations of, 342
  • in spacetime, 414
  • large-scale structure of, 340
  • limits of, 251
  • typical, 245, 325
• Causal invariance, 205, 208, 307
  • adding completions for, 294
  • and commutation relations, 450
  • and Einstein’s equations, 419
  • and quantum mechanics, 433
  • effective, 290
  • frequency of, 226
  • in rule space, 474
  • in trivalent graphs, 392
  • testing for, 219, 311
  • total, 207, 219
• Causal relationships
  • in substitution systems, 228
• Causal set theory, 477
• Cayley graphs, 197, 400
• Cellular automata
  • additive
  • see Additive cellular automata
  • asynchronous
  • see Sequential cellular automata
  • block
  • see Block cellular automata
  • class 3, 138
  • emulation of, 384
  • foliations in, 385
  • invertible
  • see Reversible cellular automata
  • particles in, 190, 425
  • perturbations in, 190
  • reversible
  • see Reversible cellular automata
  • rule labeling in, 92
  • second-order
  • see Reversible cellular automata
  • totalistic
  • see Totalistic cellular automata
  • see also Additive cellular automata
• Cellular automaton
  • on graph, 185
• Centrality, 183
• Charge, 407
• Circle packing, 156
• Circles
  • area of on sphere, 174
• Class 3 behavior, 138
  • see also Rule 30, 90, etc.
• Class 4 behavior
  • see also Rule 110, etc.
• Classical physics, 432
• Classical-like states, 435
• Classification
  • of behavior, 143
• Closed timelike curves, 346, 410, 432
• Clustering coefficient, 183, 184
• CMB, 431
• Coarse-graining, 429
  • and completions, 294
• Coherent structures
  • see also Localized structures
• Collinearity
  • in multiway graphs, 273
• Colored graphs, 392
• Combinatory logic, 399
• Common successors, 260
• Commutation relations, 450
  • and symmetry operations, 359
  • in spacetime, 456
• Company, 487
• Complete graphs, 261
• Completions, 293
  • by observers, 436
• Complex numbers
  • as amplitudes, 441
• Computation
  • see also Programs
• Computation universality
  • and rule space, 474
• Computational exploration, 477
• Computational irreducibility, 403
  • and quantum observers, 437
• Cones
  • rule generating, 149
• Confluence, 208
  • see also Canonical forms
  • see also Causal invariance
  • see also Church–Rosser property
• Conformal field theory, 477
• Connected components, 131
• Connections (on fiber bundles), 463
• Connectivity, 88
• Consciousness
  • see also Intelligence
  • see also Thinking
• Conservation laws, 359
  • for particles, 426
• Conservation of energy, 408, 423
• Continuity
  • on graphs, 195
• Continuity equation
  • on graph, 196
• Continuous space, 148, 411
  • geodesics in, 192
• Continuum limits
  • of networks, 167
• Contravariant vectors, 192
• Coordinates
  • on ellipsoid, 193
• Correlation function, 188
• Cosmic censorship hypothesis, 461
• Cosmic microwave background, 431
• Cosmological arrow of time, 413
• Cosmological constant, 408, 410
• Cosmology, 430
  • see also Universe
• Covariant derivatives, 194
• Crinkling, 157
• Critical pairs, 220
  • see also Branch pairs
• Crossings
  • in graphs, 186
• CTCs
  • see Closed timelike curves
• Cube
  • as initial condition, 137
  • from multiway graph, 213
• Cubic graphs, 390
• Cubic grid, 164
• Curvature
  • and geodesics, 192
  • in fractional-dimensional space, 199
  • in networks, 174
  • in space, 413
  • in spacetime, 418
  • vs. dimension change, 430
• Cusps, 153
• Cycle index polynomials, 95
• Cycles
  • in graphs, 184
• Cyclic behavior, 132, 140
• Cylinders
  • rule generating, 152

D

• d-dimensional grid, 164
• Dangling connections, 119, 390
• Dark energy, 430
• Dark matter, 409, 469
• de Sitter space, 410
• Decoherence, 436, 456
  • and computational irreducibility, 449
• Degree centrality, 183
• Degree distribution, 184
• Depth-first scan, 256
• Description frame
  • in rule space, 475
• Diameter
  • of graphs, 183
• Diamonds
  • in multiway graphs, 206
• Digital physics, 477
• Dimensions
  • local estimates of, 180
  • of networks, 162
  • of space, 163, 411
• Diophantine equations
  • solutions of, 425
• Disconnected behavior, 113, 126, 134
• Disconnected rules, 96, 130
• Disconnection
  • causal, 347
• Distance matrices, 271
• Distances
  • directed, 258
  • in graph, 163
  • in hypergraph, 410
  • spatial, 152
• Divergence
  • of geodesics, 192
  • on graph, 196
• Dodecahedron graph, 197
• Double‐pushout graph grammars, 477
• DPO graph grammars, 477
• Dynamical systems theory, 189

E

• E=mc², 422
• Early universe, 431
  • oligons in, 469
• Edge effects, 165
• Effective causal invariance, 290
• Eigenmodes
  • of graph, 185
• Eigenvector centrality, 183
• Einstein’s equations, 418, 423
  • and units, 465
• Electromagnetic fields
  • and gauge invariance, 463
• Electrons
  • radius of, 468
• Elementary energy, 464, 468
• Elementary length, 464, 468
• Elementary particles, 424
• Elementary time, 464, 468
• Ellipsoid, 193
  • as model for curvature, 199
• Encodings
  • of models, 404
• Energy, 407, 420
  • conservation of, 423
  • elementary, 464, 468
• Energy-momentum tensor, 422
• Ensembles
  • difficulty of using, 188
• Entanglement, 407, 437
• Entanglement cones, 455
• Entanglement horizon, 460
• Entanglement speed (ζ), 456, 469
• Entropic arrow of time, 413
• Entropy
  • of subgraphs, 178
  • see also Second Law of thermodynamics
• Equational logic, 204, 400
• Equations
  • see also Differential equations
  • see also Diffusion equation
  • see also Dirac equation
  • see also Einstein equations
  • see also Klein–Gordon equation
  • see also Laplace equation
  • see also Maxwell’s equations
  • see also Navier–Stokes equations
  • see also Schrödinger equation
• Equilibrium
  • and multiway systems, 429
  • in graph evolution, 183
• Equivalence
  • Principle of Computational
  • see Principle of Computational Equivalence
• Equivalence to substitution systems, 377
• ER=EPR, 461
• Ergodicity
  • for space, 411
• Error bars
  • for dimension estimate, 166
• Escape velocity
  • branchial, 461
• Euclidean distance, 191
• Euclidean space, 148
  • as model space, 196
• Event horizons, 431, 457
  • as analog of quantum measurement, 446
  • causal and branchial, 461
  • branchial, 457
• Events
  • updating, 228, 303
• Excluded subgraphs, 186
• Expansion of universe, 408, 430
  • and energy, 423
  • units and, 465

F

• Factorial (n!), 256
• Fano plane, 272
• FeatureSpacePlot, 144
• Feynman path integral, 441
• Fiber bundles, 463
• Fibonacci sequence, 131, 261, 301
• Fibonacci tree, 201
• Field theory
  • see also Quantum field theory
• Finite groups, 198
• Finite-size effects
  • on dimension, 165
• Finite-size systems, 140
• Flat (associative), 478
• Fluid flow, 441
• Folds
  • in surfaces, 153
• Foliations
  • and branchial graphs, 275
  • and coordinates, 255
  • and fiber bundles, 463
  • and realistic observers, 415
  • causal, 237
  • in cellular automata, 385
  • of causal graphs, 342
• Fractal pattern
  • dimension of, 167
  • rule generating, 110, 160
• Fractals
  • see also Nesting
• Free group, 197
• Freezing time
  • in quantum measurement, 446
• Frilled structure, 103
• Functions on graphs, 195

G

• Gases
  • see also Fluid flow
• Gauge invariance, 408, 462
• Gauss’s law
  • on graph, 196
• Gaussian distributions
  • of weights, 290
• General covariance, 415
• General relativity, 408
  • coordinates in, 259
• Generating function, 164
• Generational evolution, 294
• Generational multiway system, 435, 443
• Generational states, 443
• Generators
  • of groups, 197
• Genus
  • for graph drawing, 186
• Geodesic balls, 163
  • volumes of, 174, 193, 413
• Geodesic sphere, 174
• Geodesics, 191
  • bundle shapes of, 419
  • in multiway space, 439
  • in space, 413
• Geometric group theory, 198
• Geometry
  • emergent, 147
  • of branchial space, 438
  • of causal graphs, 321
• Git, 304
• Global clustering coefficient, 183
• Global counter
  • for elements, 304
• Global symmetries, 359
• Globular behavior, 143
• Globular structures, 115, 138
  • and causal graphs, 330
  • with unordered hypergraphs, 397
• God
  • see also Theology
• Golden ratio, 131
• Gorard, Jonathan (Singapore/UK, 1997– ), 487
• Gradient
  • on graph, 196
• Graph diameter, 183
  • in branchial graph, 261
• Graph grammars, 477
• Graph radius, 183
• Graph rewriting systems, 477
• Graph sizes, 187
• Graphons, 189
• Graphs
  • branchial, 260
  • colored, 392
  • correspondence to hypergraphs, 387
  • functions on, 195
  • infinite, 189
  • non-overlapping, 392
  • ordered, 392
  • rendering of, 163
  • types of, 483
  • undirected, 392
  • vertex transitive, 196
• Gravitational constant (G), 409, 465
• Gravity, 420
• Grids
  • as causal graphs, 252, 331
  • dimension of, 163
  • emergence of, 123
  • from causal graphs, 249
  • rule generating, 148
• Groups
  • Cayley graphs of, 197
  • growth in, 198, 217
  • Lie, 369
  • words in, 400
• Growth
  • based on rules, 129

H

• Halting, 87, 135
• Hamiltonians
  • quantum, 441
• Hardy fields, 177
• Hash, 304
• Hausdorff dimension, 167
• Hawking radiation
  • in branchial space, 458
• Heisenberg group, 198
• Hexagonal faces
  • of buckyball, 174, 179
• Hexagonal grid, 424
  • causal graph as, 249
• Hilbert spaces, 440, 454
  • from branchial graphs, 272
• Holographic principle, 409
• Homogeneity
  • in dimension estimate, 166
  • of universe, 431
• Hubble time, 467
• Hyperbolic space, 155
  • curvature of, 175
  • from causal graphs, 248
• Hyperbolicity
  • failure of, 346
• Hypercubes, 271
  • from multiway graph, 213
• Hypergeometric series, 164
• Hypergraph isomorphism, 91, 306
• Hypergraphs
  • automorphisms of, 366
  • emulating graphs with, 387
  • enumeration of, 95
  • events in, 303
  • representation of, 81
  • unifications of, 311
  • unordered, 396
• Hypersphere, 165

I

• Icosahedral group, 369
• Ideal gas
  • of oligons, 469
• Implementation
  • of models, 478
• Incidence geometry, 272
• IndexGraph, 483
• Inequivalent rules, 92
• Inertial frames, 409, 415
• Infinite graphs, 189
• Inflation
  • in cosmology, 431
• Initial conditions, 136
  • and disconnection, 357
  • for universe, 431
  • possible, 95
• Initial creation event, 357
• Intelligence
  • see also Thinking
• Interference, 437
• Intermediate growth
  • in groups, 199
• Intermediate scales
  • of growth, 177
• Intermediate strings
  • in branch resolution, 223
• Intrinsic curvature, 193
• Invariant measure
  • on graphs, 178, 183
• Invariants
  • in graph evolution, 185
  • see also Conserved quantities
• Involutions, 256
• Irreducibility
  • see Computational irreducibility
• Irreversibility, 429
• Isometry
  • in graphs, 179
• Isomorphism
  • hypergraph, 91
  • of hypergraphs, 306

K

• K_3,3 non-planar graph, 186, 424
• Kaons, 429
• Kepler tiling, 159
• Kinetic energy, 422
• Knitting
  • in behavior of rule, 123
• Knuth–Bendix algorithm, 293
• Kuratowski’s theorem, 186, 424

L

• Lagrangian density, 407, 441
• Lambda calculus, 205, 399
• Lambert W function, 467
• Laplacian
  • on graph, 183, 185
• Lapse function, 259
• Lattice operations
  • for group approximation, 369
• Lattice patterns, 125
• Lattice theory, 273
• Left connectivity, 92
• Lemma
  • added by observers, 436
• Length
  • elementary, 464, 468
• Lettuce leaf, 157
• Lie groups, 462
  • as possible limits, 369
• Light cones, 409
  • analog of in rule space, 474
  • and perturbations, 190
  • branchial analog of, 454
  • in substitution systems, 239
• Lightning bolt, 468
• Limiting shape, 153
• Limits, 406
  • in computing dimension, 167
  • see also Continuum limits
• Line-like behavior, 143
• Lineage
  • for elements, 304
• Linear recurrences, 187
  • for substitution system, 202
• Lobed surface, 151
• Local clustering coefficient, 184, 261
• Local conservation laws
  • see also Conservation laws
• Local gauge invariance, 462
• Local symmetries, 365
• Locality
  • and causal cones, 239
• Localized structures, 425
• Log differences, 165
• Loop quantum gravity, 477
• Loops
  • in causal graphs, 345
• Lorentz invariance, 408
• Lorentz transformations, 417
• Lorentzian manifold, 259
• LQG, 477

M

• Machine learning, 144
• Mandelbrot set
  • rule looking like, 81, 100
• Manhattan distance, 191
• Manifolds, 192, 196
  • from emulating Turing machine, 382
• Many-worlds models, 433
• Mass, 422
• Mass–energy equivalence, 422
• Mathematica
  • patterns
  • see Patterns (in Wolfram Language)
• Matrices
  • adjacency, 181
• Matter, 420
• Maximum scan, 203
• Mean clustering coefficient, 183
• Measurement
  • in quantum mechanics, 442
• Mesh
  • rule generating, 148
• Metric tensors, 192, 419
• Minkowski space, 259
• Minors
  • graph, 186
• Mixed arity, 127
• Mobile automata
  • as analogy, 333
• Model spaces, 196
• Models
  • see also Simulations
  • see also Ultimate theory of physics
• Module, 478
• Molecular dynamics, 441
  • as analogy, 411
• Momentum, 407, 420
• Monoids, 400
• Motion, 415
  • in branchial space, 454
• Multicausal continuum, 456
• Multiedges, 80
• Multigraphs
  • multiway graphs as, 287
• Multiple relations, 84, 110
• Multiple transformation rules, 129
• Multiway causal graphs, 233, 279, 370, 453, 486
  • growth of, 285
• Multiway graphs, 485
  • generational, 295
  • geometry of, 216
  • growth in, 217
  • typical structures of, 212
  • weighted, 287
• Multiway space, 438
• Multiway systems, 305
  • and Cayley graphs, 401
  • and NDTMs, 382
  • and quantum mechanics, 432
  • and reversibility, 429
  • for substitution system, 202
  • history of, 477
  • rule-space, 471

N

• Names
  • of elements, 305
• Natural language
  • picturable nouns in, 144
• NDTMs, 382, 404
• Nearby rules, 145
• Negative energy density, 430
• Negatively curved space, 157, 175
• Neighborhoods
  • in graph, 177
• Nested pattern
  • dimension of, 167
  • rule generating, 110, 160
• Nesting
  • see also Fractals
• Nilpotent groups
  • Cayley graphs of, 198
• Nodes
  • names for, 305
• Non-Abelian groups
  • see also Groups
• Non-commutative geometry, 477
• Non-convex hull, 150
• Non-deterministic Turing machines, 382, 404
• Non-overlapping graphs, 392
• Non-overlapping strings, 219, 226
• Nonlocality
  • in rules, 88
• Normal forms
  • see also Canonical forms
• NP completeness
  • and multiway systems, 383
• NP problems, 382
• Numerological relations, 469

O

• Objective reality
  • in quantum mechanics, 433
• Observers
  • in quantum mechanics, 434, 444
  • in spacetime, 413
• Obstructions
  • in planarity, 186
• Octagon-square tiling, 159
• Oligons, 469
• Operator representation, 399
• Operator-state correspondence, 408
• Operators
  • in quantum mechanics, 449
• Orbifolds, 153
• Ordered graphs, 392
• Orderless, 478
• OrderlessPatternSequence, 481
• Orthogonal components, 192
• Orthogonal directions, 194
• Overlaps
  • in strings, 219, 226

P

• Pairing functions, 92
• Parallel transport, 194
• Partial differential equations (PDEs)
  • on graph, 185
• Partial ordering
  • of events, 255
• Particles, 424
  • mass of, 469
  • perturbations and, 190
  • virtual, 428
  • see also Defects
  • see also Elementary particles
  • see also Localized structures
• Pascal’s triangle, 289
• Path independence, 206
• Path integrals, 409, 441
• Path weights
  • in multiway graphs, 287
• Pattern matching, 478
• Pattern rules, 399
• Pattern variables
  • rules in terms of, 386
• PDEs
  • see Partial differential equations
• Pentagons
  • generated by rule, 154
  • of buckyball, 174, 179, 188
• Permutations
  • as symmetry operation, 359
  • limits of, 369
  • of rule elements, 91
• Perturbations
  • and particles, 428
  • in evolution, 189
• Physical constants, 464
• Pi (π)
  • randomness of digits in, 411
• Picturable nouns, 144
• Piskunov, Max (Russia/USA, 1992– ), 487
• Planarity, 185, 424
• Planck energy, 468
• Planck length, 465
• Planck time, 465
• Planck units, 465
• Planck’s constant, 409, 451, 465
• Plane
  • hyperbolic, 157
• Pointers, 304
• Polygons
  • created from rule, 79
  • rule generating, 147
• Polynomial growth
  • in groups, 198, 217
• Posets, 255
• Post canonical system, 477
• Power series
  • for metric, 192
• Powers of rules, 367
• Pregeometry, 477
• Principle of Computational Equivalence, 143, 402
  • and rulial causal invariance, 474
• Product log function, 467
• Production systems, 477
• Products of rules, 367
• Programs
  • see also Cellular automata, etc.
  • see also Computation
• Projective spaces, 272
• Proofs
  • of causal invariance, 223

Q

• QOF
  • see Quantum observation frame
• Quantified variables
  • rules in terms of, 386
• Quantum computers, 410
• Quantum field theory
  • vacuum in, 428
• Quantum frame, 434, 444
• Quantum holography, 477
• Quantum information paradox, 461
• Quantum measurement, 442
• Quantum mechanics, 407, 432
• Quantum observation frame, 434, 444
• Quantum relativity, 477
• Quantum Zeno effect, 455
• Quasiconformal analog, 197
• Qubits, 447, 458

R

• Radial behavior, 143
• Radius
  • of graphs, 183
• Ramsey theory, 189
• Random rules, 142
• Random sampling
  • of rules, 126
• Random triangulation, 164
• Random updating order, 112
• Randomness
  • generation of, 186
  • generation of in evolution, 411
• Rank-p tensor, 196
• Reconstruction
  • of surface, 150
• Reconvergence, 206
• Recursion
  • see also Fractals
  • see also Nesting
• Recursive construction
  • of patterns, 161
• Reference frames, 415
  • quantum, 433
• References, 488
• Regge calculus, 477
• Registry of Notable Universes, 92, 487
• Relations
  • in groups, 197
• Relativistic invariance, 415
• Relativity, 417
  • in rule space, 475
• Renormalization group, 188
• Repetitive behavior, 140
  • in causal graphs, 337
  • structures with
  • see Localized structures
• Replacements
  • order of, 203
• Resolution
  • of branch pairs, 220
• Rest mass, 422
• Reversibility, 408, 429
  • of cellular automata
  • see Reversible cellular automata
  • see also Irreversibility
• Rewrite systems
  • see also Multiway systems
• Ricci scalar curvature, 174, 192, 193, 413
  • in branchial space, 456
• Ricci tensor, 192, 194
  • spacetime, 418
• Riemann tensor, 194, 413, 419
  • in branchial space, 456
• Riemannian manifold, 192
• Rotation group, 369
• Rotational invariance, 408
• Rule 30, 411
  • emulation of, 384
• Rule 60
  • see also Additive cellular automata
  • see also Rule 90
  • see also Sierpiński pattern
• Rule 90
  • see also Additive cellular automata
  • see also Rule 60
  • see also Sierpiński pattern
• Rule 110, 402
  • particles in, 425
• Rule space, 474
  • see also Rulial space
• Rule-space multiway system, 472
• Rules
  • all possible, 471
  • basic form of, 73
  • canonical form of, 91
  • disconnected, 130
  • multiple transformation, 129
  • nearby, 145
  • number of, 92, 93
  • products of, 367
  • random, 142
  • random sampling of, 126
  • representation of, 81
  • signatures of, 91
• Rulial space, 474

S

• S-matrix, 440
• Samosa, 159
• Scalar curvature
  • see also Ricci scalar curvature
• Scale invariance, 188
• Scales (mathematical)
  • see also Limits
• Schläfli symbol, 157
• Schrödinger equation, 441
• Schwarzschild radius, 460
• Second Law of thermodynamics, 429
  • and quantum observers, 437, 449
• Sectional curvature, 413
• Self-loops, 79, 95
  • as fertile initial conditions, 138
• Self-similarity, 160
  • see also Fractals
  • see also Nesting
• Semi-Thue systems, 477
• Semigroups, 400
• Sensitive dependence
  • see also Chaos theory
• Sequential substitution systems, 203
• Sequential updating, 232
• Shape
  • generated by rule, 153
• Shift vector, 259
• Short codes, 92
• Shortest paths
  • see also Geodesics
• Sierpiński graphs, 197, 315, 389
• Sierpiński pattern
  • dimension of, 167
  • local subgraphs in, 179
  • rule generating, 110, 160, 161
  • see also Rule 60
  • see also Rule 90
• Signatures
  • of rules, 91
  • of substitution systems, 212
• Simplification order, 205
• Simulations
  • see also Models
• Simultaneity, 415
• Single‐pushout graph grammars, 477
• Singularities
  • in branchial space, 461
  • in shape generated, 153
  • quantum coordinate, 446
• Slow growth, 122
• Smooth surface, 150
• Snake states, 298
• SO(3), 369
• Sorting rule, 205, 236, 237
  • and foliations, 255
  • causal graph of, 249
  • in spacetime, 416
  • multiway causal graph for, 283
• Space
  • structure of, 410
  • see also Geometry
• Spacelike connections, 454
• Spacelike hypersurfaces, 410, 416
• Spacetime
  • see also Gravity
  • see also Space
• Spacetime causal graph, 484
• Spatial hypergraph, 414, 483
• Special relativity, 408, 417
• Speed of light (c), 418, 456, 465
  • branchial analog of, 454, 456, 469
• Sphere graphs, 174, 179, 188
  • directed, 254
• Spheres
  • approximation to, 151
  • as model for curvature, 199
  • volumes of, 165
  • volume of in curved space, 174, 193
• Spherical coordinates, 151
• Spin, 407
• Spin networks, 477
• SPO graph grammars, 477
• Square grid, 163
• Standard updating order, 87, 132, 304
  • not done by SubsetReplace, 483
• State transition diagrams, 140
• States graphs, 208, 485
• Statistical averages, 411
• Statistical mechanics, 188
• Stopping, 87, 135
• Strange attractors
  • see also Chaos theory
  • see also Class 3 behavior
• String substitution systems, 201
  • emulation of, 377
  • in quantum mechanics, 435
• String theory, 410, 477
• StringReplace, 203, 248
• Strong hyperbolicity
  • failure of, 346
• Subdivision
  • geometry from, 158
• SubsetReplace, 482
• Substitution systems, 201
  • emulation of, 377
  • number of, 212
  • see also Multiway systems
• Successors
  • common, 260
• Summer School, 487
• Superpositions, 444
• Superrelativity, 477
• Surface
  • 2D, 148
• Surface reconstruction, 150
• Symbolic dynamics, 369
  • on graphs, 178
• Symbolic expressions, 478
  • to represent rules, 399
• Symbolic names
  • for elements, 305
• Symmetric groups, 198, 401
  • as symmetry of evolution, 360, 365
• Symmetries, 359
• Symmetry
  • local gauge, 462
  • of initial conditions, 138
• Synchronization
  • in cellular automata, 384

T

• Tails
  • of branch resolution, 224
• Tangent bundle, 413
• Tangent space, 195
• Taylor series, 177
  • for metric, 192
• Tensor fields, 413
• Tensor function
  • on graph, 196
• Tensor networks, 477
• Term-rewriting systems, 208, 477
• Termination, 87, 119, 130, 135
  • in substitution systems, 206
• Ternary edges, 81
• Ternary operators, 399
• Ternary relations
  • rules based on, 107
• Ternary tree, 104
• Tetrahedron graph, 160
• Theology
  • see also God
• Theorem proving, 204
  • completions in, 294
• Theorem-proving systems, 436
• Thermal equilibrium
  • see also Equilibrium
• Thermalization
  • and computational irreducibility, 449
• Thue–Morse sequence, 201
• Tilings
  • hyperbolic, 157, 248
  • octagon-square, 159
  • see also Penrose tilings
• Time
  • elementary, 464, 468
  • freezing of, 446, 461
  • nature of, 413
• Time dilation, 408, 418
  • quantum analog of, 455
• Time reversal violation, 429
• Time travel, 432
• Topochronology, 477
• Topological structure
  • generated by rule, 102
• Topology
  • generated by rule, 154
• Topos theory, 477
• Torus graph, 165
  • as causal graph, 252
• Total causal invariance, 207, 219, 311
• Transcendental equation, 467
• Transformation rules
  • symbolic, 478
• Transients, 141
• Transition amplitude, 440
• Transition matrix, 202
• Translation cone
  • in rule space, 475
• Tree-like behavior, 143
• Trees
  • dimension of, 168
  • from causal graphs, 246
  • from substitution system, 201
  • generated from rule, 77
  • in hyperbolic space, 156
  • in multiway graphs, 214
  • minimal rules generating, 97
  • reductions on, 399
  • rule generating, 147
  • ternary, 104
• Triangular tiling, 157
• Triangulation, 164
• Trivalent graphs, 390
  • nesting in, 160
• Tubes
  • as causal graphs, 339
  • rule generating, 152
  • volume of, 193
• Tupling function, 92
• Turing machines
  • as analogy, 333
  • emulation of, 379
  • non-deterministic, 382, 404
  • rule labeling in, 92
• Twistor space, 456
• Twistor theory, 477
• Two-point function, 188
• Types
  • for tagging rules, 399

U

• U(1), 369
• Unary edges, 83
• Unary relations, 134
  • rules based on, 96
• Uncertainty principle, 408, 452
• Undecidability, 403
  • in multiway graphs, 204
  • of causal invariance, 224
  • of classification, 143
  • of completions, 294
  • of disconnection, 133
  • of infinite branching, 352
• Undirected graphs, 392
• Unforgeable word constant, 226
• Unifications, 224
  • in hypergraphs, 311
• Units, 464, 468
• Universal algebra, 400
• Universal computation, 402
• Universal Turing machine, 379, 402
• Universe
  • see also Cosmology
• Unordered hypergraphs, 396
• Updating events, 303
  • in substitution systems, 228
• Updating order, 87, 203
  • random, 112
• UUID
  • for elements, 304

V

• Vacuum, 410
  • particles in, 428
• Vector function
  • on graph, 195
• Velocity
  • foliation representing, 416
• Vertex counts, 183, 187
• Vertex degrees
  • in branchial graphs, 262
• Vertex transitive graphs, 196
• Virtual particles, 409, 428
• Visual inspection, 144
• Visualization
  • graph, 186
• Volumes
  • of geodesic ball, 165

W

• Wave-particle duality, 408, 451
• Waves
  • dimension, 430
• Weighted branchial graphs, 439
• Weighted multiway graphs, 287, 439
• Weyl tensor, 419
• Wolfram Function Repository, 487
• Wolfram Language, 478, 487
  • patterns
  • see Patterns (in Wolfram Language)
  • updating order in, 203
• Wolfram Research, 487
• Wolfram Summer School, 487
  • see also Summer School
• wolframphysics.org, 487
• World lines, 422

X-Z

     

• Young tableaux, 256
• Zeno effect, 455
• Zeta
  • maximum entanglement speed, 456, 469