Click to zoom into the Mandelbrot set — the boundary between order and chaos. The set has infinite complexity at every scale. Its boundary has Hausdorff dimension 2 (Shishikura 1998). Each point c is colored by how quickly |z| → ∞ under z ← z² + c.
The Lorenz attractor — a strange attractor from convection equations. Two nearby trajectories diverge exponentially (sensitive dependence). The "butterfly effect": tiny perturbations grow to global consequences. Hausdorff dimension ≈ 2.06.
The 2D Ising model — exactly solved by Onsager (1944). At T_c = 2J/ln(1+√2) ≈ 2.27J, a phase transition separates ordered ferromagnet from disordered paramagnet. Correlation length ξ → ∞ at criticality. Critical exponents: β=1/8, γ=7/4, ν=1.
Quantum double-slit experiment — wave-particle duality. Each particle passes through both slits simultaneously (in superposition), creating interference. The fringe spacing Δy = λL/d. Measuring which slit destroys the interference pattern.
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From quantum mechanics to ecology, from chaos theory to collective intelligence — 10,000 interactive explorations of the mathematical structures underlying reality.