📐 Torus & Genus Explorer
Compare closed orientable surfaces with genus g = 0–3. Rotate the 3D model, toggle wireframe, and watch the Euler characteristic χ = 2−2g update. Drag to spin — scroll to zoom.
Surface (genus g)
Display
Gauss-Bonnet
For sphere: K=1/R², ∫KdA=4π=2π·2
2-cell disc presentation: trivial.
closed orientable surface:
χ = 2 − 2g
g=# handles / holes
Topology of Surfaces
Euler Characteristic
For any simplicial complex: χ = V − E + F. For a closed orientable surface of genus g: χ = 2 − 2g. The sphere has χ = 2 (no handles), the torus χ = 0 (one handle), the double torus χ = −2, and so on. The Euler characteristic is a topological invariant — it does not change under continuous deformations (homeomorphisms) of the surface.
Gauss-Bonnet Theorem
The Gauss-Bonnet theorem connects geometry to topology: ∬_M K dA + ∮_∂M κ_g ds = 2πχ(M). For a closed surface (no boundary): ∬ K dA = 2πχ. For a sphere of radius R, K = 1/R² everywhere, and ∬ K dA = 4π = 2π · 2 = 2πχ. For a torus there are regions of positive and negative Gaussian curvature that integrate to exactly zero.
Fundamental Group π₁
The fundamental group classifies loops in a space up to continuous deformation. For the sphere, π₁(S²) is trivial (every loop contracts). For the torus T², π₁ = ℤ × ℤ, generated by a meridian and a longitude. For genus-g surfaces, π₁ is generated by 2g loops with the single relation a₁b₁a₁⁻¹b₁⁻¹···a_g b_g a_g⁻¹b_g⁻¹ = 1 (surface group).
Classification Theorem
Every closed orientable surface is homeomorphic to either a sphere or a connected sum of tori T² # T² # ··· (g copies). This completely classifies them by genus. Adding a handle increases genus by 1 (χ decreases by 2). Non-orientable surfaces (RP², Klein bottle) are classified separately. The Riemann surfaces in complex analysis are precisely these smooth orientable surfaces equipped with a complex structure.