Interactive experiment
Topology puzzle
Two shapes are topologically equivalent if one can be continuously deformed into the other — stretching and bending allowed, cutting and gluing forbidden. The number of holes (genus) is the key invariant. A coffee mug and a donut are the same shape. A sphere and a torus are not.
Are these shapes topologically equivalent?
Sphere
Genus 0
vs
Torus
Genus 1
What is topology? Topology studies properties that survive
continuous deformation. Two shapes are equivalent (homeomorphic) if you can morph one into the other
without cutting or gluing. The genus — the number of holes — is the simplest invariant:
a sphere has genus 0, a torus has genus 1, a double torus genus 2. Orientability matters too:
a Möbius strip has only one side, and a Klein bottle is a closed non-orientable surface.