Genus, orientability, and the Euler characteristic
A surface is a two-dimensional manifold - intuitively, something that locally looks like a flat plane, but globally can be curved and have interesting topology. Surfaces are classified by a few key properties: their genus (number of holes), whether they're orientable, and their Euler characteristic.
One of the most beautiful results in topology is the Classification of Surfaces Theorem, which states that every compact surface can be completely classified by these properties. In this module, you'll explore these concepts through interactive 3D visualizations and calculators.
Explore different topological surfaces in 3D. Each surface has unique properties like genus (number of holes) and orientability.
The simplest closed surface with no holes.
The Euler characteristic (χ) is a topological invariant defined as χ = V - E + F, where V = vertices, E = edges, and F = faces. For any polyhedron homeomorphic to a sphere, χ = 2!
A regular hexahedron with 6 square faces
The Euler characteristic is a topological invariant! No matter how you subdivide a surface (adding vertices, edges, and faces), the value of χ = V - E + F stays the same. All five Platonic solids have χ = 2 because they're all homeomorphic to a sphere.
The genus is the number of "holes" or "handles" in a surface. Add or remove handles to see how the genus changes the surface's topology and Euler characteristic.
Every closed, orientable surface can be classified by its genus! The Classification of Surfaces Theorem states that any such surface is homeomorphic to a sphere with g handles attached. The genus completely determines the topology.
The Classification of Surfaces Theorem states that every compact surface can be classified by just three properties: whether it's closed, whether it's orientable, and its Euler characteristic. Answer the questions below to classify your surface!
A closed surface has no edges or boundaries, like a sphere or torus. An open surface has at least one boundary, like a disk or Möbius strip.
An orientable surface has two distinct sides (like a sphere). A non-orientable surface only has one side (like a Möbius strip or Klein bottle).
The Euler characteristic is calculated as χ = V - E + F (vertices - edges + faces). Common values: sphere (2), torus (0), double torus (-2).
Every compact surface is homeomorphic to either: (1) a sphere with g handles (orientable), (2) a sphere with k cross-caps (non-orientable), or (3) a surface with boundary. The Euler characteristic completely determines which surface you have!
1. Surface Gallery: We explored standard surfaces in 3D - the sphere, torus, double torus, Möbius strip, and Klein bottle. Each has unique topological properties.
2. Euler Characteristic: The formula χ = V - E + F is a powerful topological invariant. All five Platonic solids have χ = 2 because they're homeomorphic to a sphere!
3. Genus and Handles: Adding handles to a sphere increases its genus. The genus completely determines the topology of closed orientable surfaces: χ = 2 - 2g.
4. Classification: Every compact surface can be classified by checking if it's closed, if it's orientable, and computing its Euler characteristic. These three properties tell you everything!
Next Up: Now that you understand surfaces, you're ready to dive into knot theory! You'll learn how to visualize knots in 3D, understand knot diagrams, and explore Reidemeister moves - the fundamental operations that preserve knot equivalence.