Torus & Surface Genus — Topology of Orientable Surfaces

🍩 Torus & Surface Genus

Topology classifies surfaces by the number of holes — the genus g. A sphere has genus 0, a torus (donut) has genus 1, a double torus has genus 2. The Euler characteristic χ = 2 − 2g summarises this in one number.

🔭 Surface

🎛 View Mode

⚙️ Parameters

Major radius R 2.0 Minor radius r 0.7 Rotation speed 1.0× Mesh resolution 28

📐 Topological Invariants

Genus g

Euler χ

Orientable

Yes

π₁ (fund. group)

ℤ × ℤ

ℹ️ Theory

Euler characteristic χ = V − E + F is a topological invariant. For a closed orientable surface of genus g: χ = 2 − 2g.

Homeomorphism theorem: any compact, connected, orientable surface is homeomorphic to a sphere with g handles attached.

The torus has fundamental polygon aba⁻¹b⁻¹ (edges identified in pairs).