Each geometry is a question asked of data. The answers are gradients from low to high, with specific atlas sources as landmarks. These pages explain what each geometry detects through contrast and atlas examples, not mathematical definitions.
Interaction strength, spin-glass frustration, criticality
Local roughness, regularity spectrum, singularity strength
Recurrence rate, determinism, laminarity, trapping time
Transition predictability, time irreversibility, edge-of-chaos complexity
Correlation twist, phase coupling, area accumulation
Dominant frequency, spectral slope, bandwidth, periodicity
Triality symmetry, 4-byte structural constraint
Substitution inflation symmetry, linear complexity, bounded discrepancy
Spectrum width, dominant singularity, scaling nonlinearity
Conditional entropy, memory depth, sample entropy
Shannon entropy, complexity, redundancy
Large-scale curvature, intrinsic dimension, graph expansion
Piecewise-linear regimes, slope diversity, envelope structure
Polynomial recurrences, algebraic varieties, non-Boolean pattern lattice
Dynamical trapping, basin structure, Julia dimension
Causal ordering, lightcone structure, timelike fraction
Periodicity, cyclic coverage, uniformity
Directional clustering, angular uniformity
Gaps, dust fraction, ternary self-similarity
p-adic clustering, divisibility depth, modular structure
Distribution shape, transport cost, self-similarity
Information gradient, statistical curvature, parameter sensitivity
Zipf exponent, vocabulary richness, frequency decay
Zipf exponent, vocabulary richness, frequency decay
Hierarchy depth, branching, boundary clustering
Geometric rigidity, volume invariance, Margulis thickness
Diffusion dimensionality, spectral shape, cluster structure
Holes, loops, connected components, topological persistence
Escape rate, fractal dimension, boundary complexity
Graph degree distribution, clustering, small-worldness
GF(2) linear recurrences, LFSRs, XorShift generators, bit-level linear dependencies
Lattice alignment, algebraic constraint
Hexagonal byte-pair symmetry
Icosahedral symmetry, 5-fold rotational order
600-cell alignment, non-crystallographic 4D symmetry
Hyperbolic splitting, exponential divergence
Spherical layering, radial drift
Hyperbolic layering, vertical drift
Shear flow, geodesic divergence, rotation-shear coupling
Scale invariance, cross-ratio stability, projective curvature
Phase space area, trajectory stationarity, recurrence
Skewness, kurtosis, tail asymmetry, non-Gaussianity
Correlation dimension, Lyapunov exponent, attractor filling
Chaos vs regular dynamics, classification confidence
heteroskedasticity, regime switches, geometric non-stationarity
Growth rate, spiral tightness, radial regularity
Path roughness, variation index, regularity
Scale coherence, distributional drift, self-similarity
Fivefold diffraction symmetry, Bragg peak contrast, aperiodic order
Eightfold diffraction symmetry, Bragg peak contrast, cardinal-diagonal anisotropy
Hat motif correlation, tortuosity self-similarity, chirality
Twelvefold diffraction symmetry, square-triangle tiling order
Tenfold diffraction symmetry, Golden Mean scaling
Sevenfold diffraction symmetry, Danzer tiling order