A Museum of Mathematical Things

A cabinet of curiosities for the mathematically inclined. Each exhibit presents a named object — a dynamical system, a geometric construction, or an algorithmic process — with its mathematics, its history, and what makes it notable in the literature. Use the dropdown or arrow keys to move between exhibits; the controls beneath each sketch allow you to replay the animation or explore a different instantiation of the same concept.

Exhibit I

Rule 110 Elementary Cellular Automaton

Rule 110 is one of 256 elementary one-dimensional cellular automata, classified by Stephen Wolfram in 1983. The state of each cell in the next generation is determined by the triple formed by the cell and its two immediate neighbours; these eight possible triples are mapped to 0 or 1 according to the binary representation of the rule number (01101110₂ = 110₁₀). What distinguishes Rule 110 is its proven computational universality: in 2004, Matthew Cook demonstrated that it can simulate a universal Turing machine, a result Wolfram had conjectured but whose publication he legally delayed for several years. The animation reveals each generation from top to bottom, beginning from a single live cell. New seed begins instead from a randomly populated first row, producing different but equally deterministic evolution under the identical rule.

Exhibit II

Lissajous Figure at Irrational Frequency Ratio

A Lissajous figure traces the path of a point whose x and y coordinates are independently sinusoidal: x = sin(at + δ), y = sin(bt). When a/b is rational, the curve closes after a finite number of oscillations and repeats exactly. When the ratio is irrational, the path never revisits any point; it fills the bounding square asymptotically, though never completely. The figures were studied independently by Nathaniel Bowditch around 1815 and by Jules Antoine Lissajous, who demonstrated them with tuning forks and a mirror in 1857. The animation traces the curve incrementally, making the non-repeating accumulation visible in real time. New seed selects a different irrational ratio from a curated set (√2, √3, φ, π/e, √5, √7) and randomises the phase offset δ, demonstrating that non-closure is a consequence of irrationality rather than any particular ratio.

Exhibit III

Perlin Noise Landscape Coloured by Altitude

Coherent noise — noise that is continuous and locally correlated rather than independently random at each point — was formalised by Ken Perlin at New York University in 1982. The gradient noise function he produced, which interpolates smoothly between random gradient vectors at integer lattice points, earned Perlin a Technical Academy Award in 1997 for its role in TRON (1982). The landscape above uses a single octave of one-dimensional Perlin noise to determine terrain elevation at each horizontal position; natural terrain typically combines several octaves at diminishing amplitude. The altitude-dependent colouring follows digital elevation model convention: standing water, lower slope, grassland, bare rock above the treeline, snowfield. The sketch scans left to right, building the landscape column by column. New seed changes the noise seed, producing an entirely different terrain under the same colouring regime.

Exhibit IV

The Lorenz Attractor

Edward Lorenz, a meteorologist at MIT, derived this system of three coupled differential equations in 1963 while studying simplified atmospheric convection. Its defining property — that two trajectories beginning from arbitrarily close initial conditions diverge exponentially — is what Lorenz dramatised in his 1972 talk “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” The canonical parameters σ = 10, ρ = 28, β = 8/3 give the attractor a fractal dimension of approximately 2.06. The projection is onto the xz-plane; lines brighten as they are drawn later, so the most recently traced path is the most visible. New seed randomises the starting point, demonstrating sensitive dependence on initial conditions: a different initial position traces a qualitatively different path across the same underlying attractor geometry.

Exhibit V

Langton’s Ant

Christopher Langton introduced this two-dimensional Turing machine in 1986 as part of his work on artificial life. The rules are minimal: the ant turns right on a white square, left on a black one, flips the square’s colour, and moves forward one step. For the first few hundred steps the pattern appears disorganised. Between roughly 500 and 10,000 steps it produces apparently chaotic behaviour. Then, without warning, the ant begins constructing a rectilinear “highway” — a periodic diagonal corridor it extends indefinitely. A proof that this highway always emerges regardless of initial configuration has not been found; it remains one of the more disarming examples of emergent order in discrete systems. Watch the step counter: the transition is abrupt. New seed pre-flips one or two cells near the origin, dramatically altering the early chaotic phase while preserving the eventual emergence of structure.

Exhibit VI

Harmonograph

The harmonograph was a popular Victorian scientific instrument: two coupled pendulums controlled the motion of a pen across paper, tracing figures that were simultaneously mechanical, mathematical, and decorative. The governing equations are sums of damped sinusoids; each pendulum contributes a sinusoidal component whose amplitude decays exponentially as energy dissipates through friction. When the frequency ratio of the two pendulums is a simple rational number (2:3, 3:4) the figure resembles a Lissajous curve; the slight detuning from the integer ratio — a better model of physical pendulums — produces the characteristic inward spiral that made the instrument so prized as a parlour demonstration of the transition between order and equilibrium. The animation traces the pen’s path from first contact to rest. New seed randomises the frequency ratio and phase values, generating a different but equally physical figure from the same class of equations.

Exhibit VII

De Jong Attractor

The de Jong attractor is defined by a discrete iterated map: x_n+1 = sin(a y_n) − cos(b x_n), y_n+1 = sin(c x_n) − cos(d y_n). Peter de Jong, an American programmer and visual artist, introduced these maps in the 1980s as a vehicle for generative art. Unlike the Lorenz or Rössler systems, no differential equations are integrated: each step is computed exactly, and because sin and cos are bounded, every orbit is confined to [−2, 2] × [−2, 2]. The mathematical interest lies in how sensitively the attractor’s character depends on the four parameters: small changes shift the figure from loose filaments to dense rosettes to near-symmetric shells. The sketch accumulates half a million points at low opacity; the attractor’s structure emerges statistically as high-density regions build up by repeated deposition rather than by connected line-drawing. New seed randomises all four parameters, demonstrating how radically the attractor’s geometry changes across a small region of parameter space.

Exhibit VIII

Bifurcation Diagram of the Logistic Map

The bifurcation diagram plots the asymptotic behaviour of x_n+1 = rx_n(1 − x_n) as the growth parameter r increases from 2.5 to 4. The map was studied as a population model by Pierre François Verhulst in 1838, but its dynamical complexity was not catalogued until Robert May’s 1976 paper in Nature, which argued that simple nonlinear recurrences were worthy of serious mathematical attention. For r < 3 the orbit settles to a fixed point. Near r ≈ 3 it bifurcates to a period-2 cycle, then period-4 near r ≈ 3.449, in a cascade reaching chaos at r ≈ 3.570. Within the chaotic region, windows of periodic behaviour persist — most conspicuously a period-3 window near r ≈ 3.830, whose existence is guaranteed by the Li–Yorke theorem. The ratio of successive bifurcation widths converges to the Feigenbaum constant δ ≈ 4.6692, universal across all smooth maps with a single quadratic maximum. The sketch builds one vertical column per r value, drawing the cascade from left to right. New seed zooms into a sub-interval of the parameter axis, exposing the diagram’s fractal self-similarity.

Exhibit IX

Newton Fractal

Newton’s method for finding roots of a complex polynomial f(z) defines a dynamical system: iterate z → z − f(z)/f′(z) from each starting point and record which root the sequence converges to. The resulting basins of attraction — the sets of starting points converging to each root — share a common boundary that is a fractal, a result established by Pierre Fatou and Gaston Julia between 1917 and 1919 and formalised using the theory that now bears their names. For the canonical polynomial z³ − 1 = 0, whose three roots lie equally spaced on the unit circle, the three basins interleave in a self-similar tangle: any point on the boundary between two basins is simultaneously on the boundary of the third. Each pixel is coloured by which root its starting value converges to, shaded by iteration count — brighter pixels converge quickly; dark regions lie near the fractal boundary where convergence is slow. New seed changes the degree of the polynomial from 3 to 4 or 5, producing more roots and correspondingly more intricate basin geometry.

Exhibit X

The Rössler Attractor

Otto Rössler, a German biochemist at the University of Tübingen, published this three-dimensional chaotic system in 1976 with an explicit goal: to construct the simplest possible autonomous continuous-time system capable of chaos. Where the Lorenz system requires three nonlinear terms, Rössler’s requires only one — the product zx in the third equation — making it the minimal known example of its kind at the time. The equations are ẋ = −y − z, ẏ = x + ay, ż = b + z(x − c), with canonical parameters a = b = 0.2, c = 5.7. The xy-projection reveals the attractor’s characteristic single-lobe spiral: the trajectory winds outward in a near-planar orbit until the z-equation injects a sudden vertical excursion that resets it close to the origin. This return map is topologically equivalent to a folded strip, and Rössler proposed it explicitly as a chemical analogue of Lorenz’s atmospheric model. The sketch integrates 8,000 steps and draws each segment at increasing opacity; the earliest orbit is dark, the most recent path is brightest. New seed varies the parameter c across the range 4.0–8.0, altering the depth of the chaotic excursion and the spread of the attractor.

#rossler

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