Patterns

Five studies of simple rules, drawn slowly by the page. Give them a minute.

Order without intention phyllotaxis from repeated turns through the golden angle: \(\theta = n \cdot 137.5\ldots^\circ,\; r \propto \sqrt{n}\)

Almost in agreement a damped Lissajous curve traced by two oscillations whose frequencies form a small ratio \(p:q\) and drift slightly out of tune: \(x = e^{-\delta t}\sin(pt + \varphi),\; y = e^{-\varepsilon t}\sin(qt + \psi)\)

Built without a builder a cellular automaton grown for thirteen generations from a single seed; a block appears wherever exactly one of its six neighbours is occupied

The shape of its habits a Clifford attractor formed by iterating a single orbit for a few hundred thousand steps: \(x' = \sin(ay) + c\cos(ax),\; y' = \sin(bx) + d\cos(by)\)

A world in octaves fractal Brownian motion on a rotating sphere, where broad continents emerge first and finer octaves of three-dimensional gradient noise add coastlines and islands: \(h(\mathbf{s}) = \sum_{k=0}^{6} 0.55^k P_k(1.35 \cdot 2^k\mathbf{s}),\; \mathbf{s} \in S^2\)