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.
Correlation dimension, Lyapunov exponent, attractor filling
Interaction strength, spin-glass frustration, criticality
Chaos vs regular dynamics, classification confidence
Skewness, kurtosis, tail asymmetry, non-Gaussianity
Spectrum width, dominant singularity, scaling nonlinearity
Turbulence cascade, intermittency, K41/She-Leveque scaling
heteroskedasticity, regime switches, geometric non-stationarity
Transition predictability, time irreversibility, edge-of-chaos complexity
Conditional entropy, memory depth, sample entropy
Dominant frequency, spectral slope, bandwidth, periodicity
Large-scale curvature, intrinsic dimension, graph expansion
Escape rate, fractal dimension, boundary complexity
Hierarchy depth, branching, boundary clustering
Dynamical trapping, basin structure, Julia dimension
GF(2) linear recurrences, LFSRs, XorShift generators, bit-level linear dependencies
Geometric rigidity, volume invariance, Margulis thickness
Holes, loops, connected components, topological persistence
Diffusion dimensionality, spectral shape, cluster structure
Graph degree distribution, clustering, small-worldness
Polynomial recurrences, algebraic varieties, non-Boolean pattern lattice
p-adic clustering, divisibility depth, modular structure
Gaps, dust fraction, ternary self-similarity
Information gradient, statistical curvature, parameter sensitivity
Directional clustering, angular uniformity
Distribution shape, transport cost, self-similarity
Zipf exponent, vocabulary richness, frequency decay
Zipf exponent, vocabulary richness, frequency decay
D4 lattice structure, 4-byte symmetry constraints
Lattice alignment, algebraic constraint
Hexagonal byte-pair symmetry
Hyperbolic layering, vertical drift
Icosahedral symmetry, 5-fold rotational order
600-cell alignment, non-crystallographic 4D symmetry
Correlation twist, phase coupling, area accumulation
Causal ordering, lightcone structure, timelike fraction
Scale invariance, cross-ratio stability, projective curvature
Spherical layering, radial drift
Shear flow, geodesic divergence, rotation-shear coupling
Hyperbolic splitting, exponential divergence
Phase space area, trajectory stationarity, recurrence
Piecewise-linear regimes, slope diversity, envelope structure
Local roughness, regularity spectrum, singularity strength
Curvature cascade, non-periodic boundary content, monotone runs
Scale coherence, distributional drift, self-similarity
Path roughness, variation index, regularity
Eightfold diffraction symmetry, Bragg peak contrast, cardinal-diagonal anisotropy
Tenfold diffraction symmetry, Golden Mean scaling
Twelvefold diffraction symmetry, square-triangle tiling order
Hat motif correlation, tortuosity self-similarity, chirality
Substitution inflation symmetry, linear complexity, bounded discrepancy
Fivefold diffraction symmetry, Bragg peak contrast, aperiodic order
Sevenfold diffraction symmetry, Danzer tiling order
Exotic geometry guide: Information Theory. Part of the Structure Atlas — 211 data sources analyzed through 200 metrics across 54 exotic geometries.
Exotic geometry guide: Logarithmic Spiral. Part of the Structure Atlas — 211 data sources analyzed through 200 metrics across 54 exotic geometries.
Exotic geometry guide: Recurrence Quantification. Part of the Structure Atlas — 211 data sources analyzed through 200 metrics across 54 exotic geometries.
Exotic geometry guide: Torus T^2. Part of the Structure Atlas — 211 data sources analyzed through 200 metrics across 54 exotic geometries.