FAMOUS EQUATIONS

The mathematics behind the art. Each piece in our gallery emerges from equations that have shaped our understanding of nature, chaos, and beauty.

Where Mathematics Becomes Art

Every artwork in our gallery is generated from mathematical equations—formulas discovered by scientists, mathematicians, and pioneers of chaos theory. Here are some of the most significant.

The Lorenz System

The butterfly that started chaos theory
1963
Edward Lorenz
$$ \frac{dx}{dt} = \sigma(y - x) \frac{dy}{dt} = x(\rho - z) - y \frac{dz}{dt} = xy - \beta z$$

The Discovery

In 1963, MIT meteorologist Edward Lorenz was running weather simulations when he made a fateful shortcut—rounding a number from 0.506127 to 0.506. The result was completely different weather. This tiny change led to the discovery of deterministic chaos and the famous "butterfly effect."

The Story

The butterfly-shaped attractor that emerged from his equations became the icon of chaos theory. It shows that even simple, deterministic systems can be fundamentally unpredictable. The shape you see is actually a single continuous line that never crosses itself and never repeats.

σ = 10, ρ = 28, β = 8/3 (classic parameters)

Available as Prints

lorenz 01 lorenz 02

The Rössler System

The simplest chaos
1976
Otto Rössler
$$ \frac{dx}{dt} = -y - z \frac{dy}{dt} = x + ay \frac{dz}{dt} = b + z(x - c)$$

The Discovery

Otto Rössler set out to find the simplest possible system that could produce chaos. He succeeded—these three equations with just one nonlinear term (xz) create a beautiful folding attractor.

The Story

The Rössler attractor demonstrates that chaos doesn't require complexity. A single nonlinear term is enough. The system spirals outward on a plane, then folds back through a third dimension, never quite repeating. It's chaos distilled to its essence.

a = 0.2, b = 0.2, c = 5.7

Available as Prints

rossler 01 rossler 02

Spherical Harmonics

The shapes of atomic orbitals
1782
Laplace & Legendre
$$Y_l^m(\theta, \phi) = \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}} P_l^m(\cos\theta) e^{im\phi}$$

The Discovery

These functions describe how things vary on the surface of a sphere. In quantum mechanics, they define the angular shapes of electron orbitals—the fundamental architecture of all atoms.

The Story

When Schrödinger solved his equation for the hydrogen atom in 1926, spherical harmonics emerged naturally. The s, p, d, f orbitals you learn in chemistry are spherical harmonics with different quantum numbers. Every element in the periodic table owes its chemistry to these shapes.

l = degree (0,1,2,...), m = order (-l to +l)

Available as Prints

spherical harmonic 01 spherical harmonic 02 spherical harmonic 03

The Golden Spiral

φ = 1.618033988749...
Ancient
Nature
$$ \phi = \frac{1 + \sqrt{5}}{2} \theta = \frac{2\pi}{\phi^2} \approx 137.5°$$

The Discovery

The golden angle (137.5°) appears throughout nature because it's the most irrational angle—the one that most evenly distributes points around a circle without creating obvious spokes.

The Story

Sunflowers, pinecones, pineapples, and cacti all arrange their seeds and scales using the golden angle. It's not mysticism—it's optimization. This angle maximizes packing efficiency and sunlight exposure. Nature discovered calculus before we did.

Divergence angle ≈ 137.507764°

Available as Prints

phyllotaxis 01 phyllotaxis 02 phyllotaxis 03

Reaction-Diffusion

How the leopard got its spots
1952
Alan Turing
$$ \frac{\partial u}{\partial t} = D_u \nabla^2 u + f(u,v) \frac{\partial v}{\partial t} = D_v \nabla^2 v + g(u,v)$$

The Discovery

In his final major work, Alan Turing proposed that animal patterns emerge from two chemicals that react with each other while diffusing at different rates.

The Story

Turing's 1952 paper "The Chemical Basis of Morphogenesis" explained zebra stripes, leopard spots, and fingerprint whorls using mathematics. Two morphogens—an activator and an inhibitor—create stable patterns spontaneously. This was Turing's gift to biology: the mathematics of life's patterns.

Du/Dv ratio and reaction kinetics determine pattern type

Available as Prints

reaction diffusion 01 reaction diffusion 02 reaction diffusion 03

The Three-Body Problem

The unsolvable dance
1687+
Newton / Poincaré
$$ \vec{F}_{ij} = G\frac{m_i m_j}{|\vec{r}_j - \vec{r}_i|^2}\hat{r}_{ij} m_i\frac{d^2\vec{r}_i}{dt^2} = \sum_{j \neq i} \vec{F}_{ij}$$

The Discovery

Newton solved the two-body problem exactly. But add a third mass, and the equations become unsolvable in closed form—one of the oldest open problems in physics.

The Story

Henri Poincaré proved in 1890 that most three-body configurations are chaotic—no formula can predict their long-term behavior. This discovery founded the field of dynamical systems. Every solar system with three or more bodies lives in this chaotic regime.

G = gravitational constant, masses m₁, m₂, m₃

Available as Prints

three body 01 three body 02 three body 03

Hilbert Curve

A line that fills a plane
1891
David Hilbert
$$ \lim_{n \to \infty} H_n : [0,1] \to [0,1]^2 \text{surjective and continuous}$$

The Discovery

A one-dimensional line that passes through every point in a two-dimensional square. This seemingly impossible object challenged 19th-century notions of dimension.

The Story

Hilbert constructed his curve as a limit of increasingly complex recursive patterns. Each iteration doubles the detail while maintaining continuity. In the limit, the curve visits every point in the unit square—a line becomes a plane. Today, Hilbert curves are used in database indexing and image processing.

Iteration depth n → ∞

Available as Prints

space filling 01 space filling 02 space filling 03

License the Mathematics

Every equation above is available as a licensed vector file. Bring the beauty of mathematical discovery into your space — printed at any size you need.

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