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 structure-function stats (ESS/K41-SL residual, increment skewness/autocorr) as generic 1D discriminators (cascade not literally validated)
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
LFSR / GF(2)-rank / Walsh bit-algebra + increment smoothness; low-entropy predictability (NOT non-orientability or GF(2)-linearity)
S^3 trajectory occupancy/transition stats vs a 600-cell scaffold + Hopf-phase ACF (NOT Mobius / non-orientable structure; S^3/2I is orientable)
Geometric rigidity, volume invariance, Margulis thickness
Persistent loops (H1) in reconstructed phase space; smooth simple limit cycles
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
Cantor-coordinate gaps (largest = the 1/3 hole), value-cardinality, jump-size concentration
Khintchine-Lévy deviation and Liouville-class partial-quotient outliers on continuous signals
Information gradient, statistical curvature, parameter sensitivity
Shannon entropy, complexity, redundancy
KS distance of the value distribution to RMT Poisson/GOE/GUE surmises; order-invariant marginal-shape test (NOT true level correlations / quantum chaos)
shoulder weight / marginal-distribution shape
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 root-graph, orthogonality, Gram-covariance & edge-adjacency-spectrum stats (NOT triality -- no order-3 automorphism)
E8 root-alignment-magnitude variance + Gram-quantization fidelity (genuine 240-root projection)
G2 root-direction sector occupancy + pair-angle |cos|-alignment kurtosis (NOT hexagonal-symmetry detection)
Hyperbolic layering, vertical drift
Adjacency of consecutive H3-root cells; discrete-symbolic vs smooth (NOT icosahedral / 5-fold 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
(value,increment) phase-portrait occupancy/reflection/shoelace-area/recurrence stats (NOT phase-volume preservation / Hamiltonian isolation)
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
Spectral self-similarity at silver-ratio (~2x) frequency rescalings; spectral coloredness (NOT eightfold diffraction; octagonal embed is dead code)
1D spectral self-similarity at the dodecagonal Pisot ratio 2+sqrt(3) (NOT 12-fold diffraction / Stampfli tiling)
Symbolic-dynamics stats: subword-complexity growth, discrepancy, return-gap & ACF-peak regularity (NOT substitution-inflation symmetry)
Fivefold diffraction symmetry, Bragg peak contrast, aperiodic order
1D spectral self-similarity at the heptagonal ratio rho~2.247 & conjugates (NOT sevenfold diffraction / Danzer tiling)
AR-process structure, linear short-memory dependence, white-noise residuality
heteroskedasticity, regime switches, geometric non-stationarity
Conditional entropy, memory depth, sample entropy
Time-reversal symmetry breaking; conservative-vs-dissipative dynamics; nonlinear/non-Gaussian asymmetry
Anisochronous oscillation — amplitude-dependent frequency (Van der Pol, Duffing, Stuart-Landau shear, Ruelle-Takens cascade); also separates AM (high amp spread, zero shear) from FM (zero amp spread, time-varying frequency)
Within-cycle positional asymmetry of argmax/argmin. On smooth waveforms reads rise/fall time (sawtooth, ECG QRS, relaxation oscillators). On binary/sparse-spike signals reads duty-cycle skew (skewed PWM, sparse spike trains, Prime Indicator / von Mangoldt). Insensitive to anisochronicity, amplitude variation, and harmonic-symmetric distortion (square, clipped sine, triangle).