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🍩 Torus & Surface Genus

About this simulation

This simulation explores the topology of closed orientable surfaces: the sphere, the torus (donut) and the double torus. Topology classifies such surfaces by their genus — the number of holes — which makes a coffee mug and a donut topologically identical. These ideas underpin everything from the shape of the universe in cosmology to the connectivity of computer networks and the structure of DNA.

How it works

• A parametric mesh is generated for the chosen surface (sphere, torus or double torus).
• Each vertex is rotated in 3D and projected onto the 2D canvas with perspective.
• Wireframe, shaded (painter's algorithm) and polygon-net views render the same geometry differently.
• The topological invariants — genus, Euler characteristic and fundamental group — update live.

Key equations

χ = V − E + F = 2 − 2g — χ is the Euler characteristic, V/E/F are the vertices, edges and faces of any triangulation, and g is the genus (number of holes).

Controls

• Surface — switch between torus (g=1), double torus (g=2) and sphere (g=0).
• View mode — wireframe, shaded or fundamental polygon net.
• Major / minor radius — reshape the torus tube and ring.
• Rotation speed & mesh resolution — adjust auto-spin and detail.
• Drag the canvas to rotate the surface manually.

Did you know?

A torus and a sphere can never be smoothly deformed into one another — you cannot remove a hole without cutting. This is captured by the fundamental group: the sphere's is trivial, while the torus's is ℤ × ℤ, reflecting its two independent loops.