Topology Explorer

Mathematical Principles & Theorems

Investigates topological invariants preserved under homeomorphisms (continuous bijective maps with continuous inverses). Euler Characteristic \(\chi = V - E + F = 2 - 2g\) classifies closed orientable surfaces by genus \(g\) (Sphere \(g=0, \chi=2\); Torus \(g=1, \chi=0\); Double Torus \(g=2, \chi=-2\)). Non-orientable surfaces: Möbius Strip (one-sided, single continuous boundary curve, \(\chi=0\)) and Klein Bottle (closed non-orientable surface with no boundary, \(\chi=0\)).

Operating Instructions

• Select topological manifold: Möbius Strip, Torus, Klein Bottle, or Sphere.
• Configure rotation speed, mesh resolution, and twist count using numeric inputs.
• Click and drag on the 3D WebGL stage to inspect surface continuity and parametric coordinates.
• Review topological invariant metrics (Euler characteristic \(\chi\), genus \(g\), orientability, boundary components).