Exactly six years ago I cited Wikipedia’s orientability article as an example of the site’s dense language when it came to mathematical topics:

In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point. A choice of a normal vector allows one to use the right-hand rule to define a “clockwise” direction of loops in the surface, as needed by Stokes’ theorem for instance. More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a “clockwise” orientation for all loops in the manifold.

I understood this, as I said at the time. But it doesn’t flow well, and assumes too much prior knowledge. Forgive the tautololgy, but an introduction is supposed to introduce a topic. This would have been better served being in a “background” section.

I checked out the article again today. It’s still a bit dense, but it’s vastly improved:

In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of “clockwise” and “anticlockwise”. It generalizes the concept of curve orientation, which for a plane simple closed curve is defined based on whether the curve interior is to the left or to the right of the curve. A space is orientable if such a consistent definition exists.

I’ve noticed mathematics articles have got a lot better in the last few years on Wikipedia. If you’ve been involved in this, thank you!