Reckoning / Shape / Topology lens
Shape · the lens
Stretch, don’t tear
Geometry measures; topology asks what survives when measuring is forbidden. If a shape may be bent and stretched but not torn or glued, a cube becomes a sphere and a pipe becomes a doughnut, but the doughnut can never become the ball: the hole cannot be removed without tearing.
The Euler characteristic
Draw any network of vertices, edges and faces on a surface and count: V − E + F. For any surface that can be stretched onto a sphere the answer is always 2; for a torus it is always 0, however the network is drawn. Euler noticed the rule for polyhedra in 1758, and it became one of the first invariants of topology: a number that stretching cannot change.
A cube (8 − 12 + 6), a pyramid (5 − 8 + 5) and a wedge (6 − 9 + 5) all give 2. Triangulated meshes of a sphere and a torus generated by the build script give 2 and 0. A hollow sphere has two separate boundary spheres: 2 + 2 = 4.
Every shape on this site
Plane figures: the characteristic of the region (disc 1, ring 0). Solids: the characteristic of the boundary surface.