Topology can define direction {orientation, topology}. To turn bounded surfaces inside out, project at right angles through plane defined by surface boundary line, to make projected-figure points same distance from plane as original bounded surface {inside out}. Turning figures with surface arrows inside out reverses arrow direction.
After one loop, vector direction is either same or opposite {orientability}|. Orientability is a global property.
On triangulated surfaces {orientable surface}, triangles can orient so sides common to two triangles orient oppositely on two triangles. Spheres are orientable surfaces. Projective planes are not orientable surfaces. If and only if two orientable closed surfaces have same genus, surfaces are homeomorphic.
If arrow goes around Klein bottle or Möbius strip once, arrow reverses direction {version, topology}|, the same as turning figure inside out.
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225