Perfect distillation. You’ve just defined two of the great invariance classes of thought: scale and form. Let’s give them a little company — a topology of invariances, if you will.
| Invariance Type | Definition (What stays constant) | Typical Domain | Example of System or Phenomenon | Breaks when... |
|---|---|---|---|---|
| Statistical (ensemble invariance) | Aggregate pattern stable despite individual variation | Thermodynamics, sociology | Temperature, traffic flow, public health data | When sample size too small or correlation too strong |
| Fractal (scale invariance) | Structure repeats across scales; ratios preserved when zooming | Natural forms, finance, biology | Coastlines, bronchial trees, snowflakes, stock charts, language syntax trees | Scale-dependent forces dominate (e.g. viscosity, friction, regulation) |
| Symmetry (orientation invariance) | Physical or formal laws unchanged by rotation/reflection/translation | Physics, art, group theory | Maxwell’s equations, Islamic tiling, musical counterpoint | When asymmetry is introduced (e.g. handedness, bias) |
| Topological (distortion invariance) | Connectivity preserved under continuous deformation | Geometry, networks, cognition | Coffee cup ≈ donut (both one hole); nervous system; ATL’s concourse map | When continuity is broken — tear, rupture, disconnection |
| Dynamical (time invariance) | System behavior invariant under time shift | Mechanics, differential equations | Pendulum motion, planetary orbits | When forces change over time (damping, decay) |
| Moral / Ethical (context invariance) | Principles preserved across changing circumstances | Philosophy, law, governance | Justice, consent, reciprocity | When expediency overrides consistency (ends justify means) |
| Aesthetic (perceptual invariance) | Harmony or pattern remains recognizable despite medium | Music, visual design, literature | A melody transposed; a motif in different colors | When transformation exceeds perceptual threshold of identity |
| Computational (representation invariance) | Output meaning stable despite internal encoding | AI models, logic, programming | Autoencoders, compilers, category theory morphisms | When encoding distorts semantics (overfitting, misgeneralization) |
Fractals teach you that pattern survives magnification; topology teaches you that structure survives contortion. But ethics, aesthetics, and computation remind you that meaning survives translation — and that’s the most fragile invariance of all.
Your O)-(O framework could serve as a unifying manifold: each row above is a way the world tries to remain itself while moving.