Interactive geometry laboratory

Gömböc Demo

Explore an analytical Gömböc surface model by Millard L. Sloan.

SurfaceSloan Model 2

β0.030

Radius0.968–1.029

Approx. Kminscanning…

Motionresting

low r

high r

Grab and release to drop · extra-wide rolling floor · drag empty space to orbit · wheel/pinch or View size to zoom

The 3D renderer could not start. Check that WebGL is enabled and that the Three.js CDN is reachable.

Surface controls

All angles below use Sloan's convention: θ is polar angle; φ is azimuth.

Perturbation 0.030

βsphere ⟶ stronger relief

This demo uses Sloan's nonlinear second model.

Gravity lab · no assist running

Gravity9.8 m/s²

Simulation speed1.00×

Gravity Floor friction

Press directly on the Gömböc to grab it. Move while holding, then release to drop or toss it. The solver uses sampled floor contact, collision impulses, friction, approximate spherical inertia, and numerical velocity damping.

Probe point

θ · polarπ/2

φ · azimuthπ

r—

P(θ)—

x—

y—

z—

φ − P—

Rendering

Object view size42%

Surface coloring

Radius color contrast125%

Mesh quality

Actions

OBJ export uses the current β and mesh quality. The HTML itself loads Three.js from a CDN.

Radial field on the sphere

Equirectangular map of r(θ,φ). Click anywhere to move the probe.

φ → · θ ↓

Latitude profile

At the selected θ, compare r over a full azimuthal turn.

θ = π/2

Sloan's second model

r⁴ = 1 + 4β sin(θ) cos[φ − (3π/2)(cosθ − cos³θ/3)]

P(θ) = (3π/2)(cosθ − cos³θ/3) · x = r sinθ cosφ, y = r sinθ sinφ, z = r cosθ

Stable point Sθ = π/2, φ = π; this is the equatorial radial minimum.

Unstable point Uθ = π/2, φ = 0; this is the equatorial radial maximum.

How the phase worksThe red ridge follows φ = P(θ), where the cosine term is +1. The green valley follows φ = P(θ)+π, where it is −1.

Convexity caution. A 2026 reanalysis reports a mono-monostatic and strictly convex sub-regime for sufficiently small β (about β ≲ 0.036 for the analyzed parameterization). This is not a rigorous demonstration; it adopts an approximate finite-difference Gaussian-curvature scan for surface analysis and an approximate rigid-body solver for the gravity simulation.