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
Phase coupling between frequencies; quadratic nonlinearity; non-Gaussian higher-order structure; harmonic vs cross-mode interaction
Interaction strength, spin-glass frustration, criticality
Nodal regularity, zero-crossing clustering, multi-scale modal complexity
Chaos vs regular dynamics, classification confidence
Skewness, kurtosis, tail asymmetry, non-Gaussianity
Discrete cyclic transport, modular winding bias, recurrent backbone strength on integer-quantized sequences
Spectrum width, dominant singularity, scaling nonlinearity
Turbulence cascade, intermittency, K41/She-Leveque scaling
Transition predictability, time irreversibility, edge-of-chaos complexity
Recurrence rate, determinism, laminarity, trapping time
Dominant frequency, spectral slope, bandwidth, periodicity
Frequency ratio arithmetic, nested periodicities, spirograph closure
Large-scale curvature, intrinsic dimension, graph expansion
Escape rate, fractal dimension, boundary complexity
Compressible/solenoidal energy split, harmonic 1-forms, curl/div ratio, Laplacian energy, source/sink density
Hierarchy depth, branching, boundary clustering
Sequence-dependent dynamical trapping, basin structure, Julia boundary sensitivity
GF(2) linear recurrences, LFSRs, XorShift generators, bit-level linear dependencies
Topological sector transitions on S³/2I
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
Khintchine-Lévy deviation and Liouville-class partial-quotient outliers on continuous signals
Information gradient, statistical curvature, parameter sensitivity
Shannon entropy, complexity, redundancy
Wigner-Dyson vs Poisson level statistics; integrable vs quantum-chaotic spectral signatures
Directional clustering, angular uniformity
Periodicity, cyclic coverage, 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
deep diverse edge walks on the 600-cell with diff-vector closure on H4 roots; gated against degenerate-plateau and pingpong saturators
Correlation twist, phase coupling, area accumulation
Causal ordering, lightcone structure, timelike fraction
Piecewise-linear regimes, slope diversity, envelope structure
Scale invariance, cross-ratio stability, projective curvature
Spherical layering, radial drift
Shear flow, geodesic divergence, rotation-shear coupling
Hyperbolic splitting, exponential divergence
Phase space filling, reflection symmetry, trajectory stationarity, orbital coherence, recurrence
Quadratic-polynomial alignment of marked positions on a 2D lattice
Quadratic-polynomial alignment of marked positions on a 2D lattice
Local roughness, regularity spectrum, singularity strength
Curvature cascade, non-periodic boundary content, monotone runs
Growth rate, spiral tightness, radial regularity
Scale coherence, distributional drift, self-similarity
Path roughness, variation index, regularity
Cross-scale energy cascades, intermittency, self-organized criticality
Eightfold diffraction symmetry, Bragg peak contrast, cardinal-diagonal anisotropy
Twelvefold diffraction symmetry, square-triangle tiling order
Substitution inflation symmetry, linear complexity, bounded discrepancy
Fivefold diffraction symmetry, Bragg peak contrast, aperiodic order
Sevenfold diffraction symmetry, Danzer tiling order