Seven structures. Seven different kinds of order. A collection of living forms drawn from geometry, topology and dynamical systems.
Move over a study to turn it. Drag on touchscreens. Use the sliders to change the mathematics.
Real equations. No random particles.
Focus a drawing and use ← ↑ ↓ → to turn it.
A torus in four dimensions, seen through a three-dimensional projection. A close relative of the Hopf fibration, with a woven surface that changes shape as the fourth dimension turns.
Two circles of radius 1/√2 lie in perpendicular planes in ℝ⁴. We rotate the x–w plane, then stereographically project to ℝ³.
(x,y,z,w) = (cos u, sin u, cos v, sin v) / √2
P = (x,y,z) / (1 − w)A familiar ring transformed into an asymmetric, flowing shell. Geometric inversion brings one side close and sends the other away, while preserving circles or turning them into lines.
Start with a ring torus of major radius 1.8 and tube radius 0.65. Invert its points around a centre a inside the hole. The result belongs to the family of Dupin cyclides.
T(u,v) = ((1.8 + 0.65 cos v) cos u,
(1.8 + 0.65 cos v) sin u, 0.65 sin v)
I(T) = (T − a) / ‖T − a‖²A soap-film-like neck opens into a spiral staircase. Move the slider from the catenoid to the helicoid: every intermediate surface in this family has zero mean curvature.
The catenoid C and its conjugate helicoid H form an associate family. This drawing shows a finite parameter patch; its intermediate members can intersect themselves.
C = (cosh v cos u, cosh v sin u, v)
H = (sinh v sin u, −sinh v cos u, u)
X = cos θ C + sin θ H, 0 ≤ θ ≤ π/2A repeating labyrinth of curved passages, with no obvious beginning or end. Cut into the structure to reveal its interior. This example uses the standard trigonometric approximation.
The exact gyroid is a triply periodic minimal surface. Here we triangulate the zero set below: a familiar nodal approximation, not the exact minimal surface. Luminous contours trace the surface. The slider cuts away part of the drawing.
sin x cos y + sin y cos z + sin z cos x = 0Angular waves become a sculptural flower. Lobes emerge and recede as two modes combine; warm and cool threads trace the two directions across the surface.
On the unit sphere, z(x² − y²) and z(x³ − 3xy²) are proportional to real spherical harmonics of degrees 3 and 4. We mix them, then plot absolute amplitude as radius. This is a radial visualisation, not a literal vibrating membrane.
f = (1 − b)·3z(x² − y²) + b·3z(x³ − 3xy²)
r = 0.28 + 1.5|f|, 0 ≤ b ≤ 1A trajectory winds between two butterfly wings, following three deterministic equations. Change ρ to redraw its path. The moving light follows the computed trajectory.
This is the Lorenz system with σ = 10 and β = 8/3. We integrate with fourth-order Runge–Kutta, discard an initial transient, then display a finite trajectory. The light loops over that recorded path; the chaotic solution itself is not periodic.
dx/dt = 10(y − x)
dy/dt = x(ρ − z) − y
dz/dt = xy − (8/3)zCircles inside circles, with smaller structures appearing between every pair. This study uses an Apollonian packing, a special case connected to Kleinian limit sets. A Möbius transformation shifts the view without breaking the tangencies.
Starting with four mutually tangent circles, the Descartes relation gives the next circle in each gap. We draw a finite packing and transform it inside the unit disk. It is one construction from this wider mathematical world, not a rendering of every pattern in Indra’s Pearls.
(k₁ + k₂ + k₃ + k₄)² = 2(k₁² + k₂² + k₃² + k₄²)
M(z) = (z − a) / (1 − āz), |a| < 1Indra’s Pearls · David Mumford ↗
The complex Descartes theorem ↗