Fact: a map of the Earth can either
- accurately depict areas, or
- accurately depict angles.
But not both. Thanks for the info, The Borsuk-Ulam Theorem. (one of its corollaries is that subsets of Rⁿ are not homeomorphic to Sⁿ)
Gerardus Mercator’s projection preserves angles and local shape relationships. So it’s good for sailing. The mathematical term is a conformal mapping from the Earth S² to the page R² [0,1].
(see what I did there? North ain’t up. Away from Earth is up.)
DETERMINANT = 1
Mercator’s projection makes Africa look small and Greenland look big, because of their distances from the equator. But try navigating with this puppy.
In matrix terms, equal area projections are mapped from S² to R² by a matrix with determinant 1. (If you had to use a bunch of different matrices on a manifold and stitch them together, some sum of determinants would have to be 1.)
Area and angle aren’t the only two qualities of a map, of course. That’s why there are so many alternative projections. For example, the Robinson projection we used in my elementary school is neither conformal nor area-preserving.
You can spend all week reading about them on W***pedia or Mathworld. Gall-Peters, Lambert, Aitoff, Hammer, Goode, Boggs, van der Grinten, Tobler, Mollweide, Gnomonic, Eckart, Collignon, Kavrayskiy, pseudocylindrical sinusoidal projections, and Tissot’s indicatrix of deformation are all cool.
Also, thinking about maps is a good way to start visualizing mathematical functions as transformations of an entire space rather than thinking of the f(a), f(b), f(c)'s individually.
There is a related result to the Borsuk-Ulam theorem, called — literally — the Hairy Ball Theorem. I am not making this up.