Today I Learned

Some of the things I've learned every day since Oct 10, 2016

Category Archives: topology

99: Non-Orientable Surfaces and the Möbius Strip

The Möbius strip is a canonical and minimal example of a non-orientable surface, in the sense that

  1. a surface is non-orientable if and only if it has the Möbius strip as a topological subspace, and
  2. the Möbius strip is the only surface with this property.

61: Coloring Problem on the Torus

Given any division of a (2-)torus into regions, it is possible to color the regions with 7 distinct colors so that no two adjacent regions have the same color.

This contrasts with the more well-known fact that the plane is 4-colorable.