stylistic:style_transfer
Orientation by triangulation

Any surface has a triangulation: a decomposition into triangles such that each edge on a triangle is glued to at most one other edge. Each triangle is oriented by choosing a direction around the perimeter of the triangle, associating a direction to each edge of the triangle. If this is done in such a way that, when glued together, neighboring edges are pointing in the opposite direction, then this determines an orientation of the surface. Such a choice is only possible if the surface is orientable, and in this case there are exactly two different orientations.

If the figure can be consistently positioned at all points of the surface without turning into its mirror image, then this will induce an orientation in the above sense on each of the triangles of the triangulation by selecting the direction of each of the triangles based on the order red-green-blue of colors of any of the figures in the interior of the triangle.

This approach generalizes to any n -manifold having a triangulation. However, some 4-manifolds do not have a triangulation, and in general for n > 4 some n -manifolds have triangulations that are inequivalent.