Here’s another example of what mathematicians mean by an “ugly discontinuity”.
The Torus is the Cartesian product of circles ◯×◯. I.e. an abstract geometry in which concrete angular measurement pairs (or triples or quadruples or quintuples or …) are realised.
The Sphere is … not that.
It’s nontrivial to recognise that ◯×◯≠sphere. For example the people who wrote the Starfox battle mode drew the screen as a sphere but programmed the battle mode on a torus.
By the Hairy Ball Theorem we know that spheres are different to independent pairs of circles. Specifically: one circle “vanishes” at the top and bottom of the other, to make a sphere. Changing your latitude coordinate at the North Pole leaves you in the same place. In other words “two” collapses to “one” at the poles which also implies that, for consistency, latitude needs to be close to collapse around 89°N—not at all like ◯×◯. where the two capstans spin freely independent of one another.
(This is half of the “joke” … or, “prank”, or “not-funny joke” in my twitter location. I designate myself at (−90,45) so you can imagine a person spinning around uselessly as they try to “walk in a circle” on the South Pole. OK … it’s only slightly funny even to me.)
This is like how globes can represent the Earth much better than maps on a flat sheet of paper. Since it’s impossible to map R² onto S², flat maps can never be perfect. (The fact that the difference is merely a point—that is R² does map onto S²{0}—is a distraction from how distorted real maps get. Look how different Greenland looks from the North versus the European view
Furthermore the torus can’t be deformed into a sphere, and it’s difficult for mathematicians to see the relationships between high-dimensional and low-dimensional spheres. (And this has something to do with the story of what Grigory Perelman achieved in solving that Clay Prize.)
The savvy way of talking about this is to say that the sphere has ugly symmetries. How can I say that when the Sphere is a Platonically perfect elementary shape?! The Sphere is so perfect that mass in outer space likes to form itself into that most balanced of balanced shapes.
Basically because when you hold the globe with two fingers and your friend spins it, the antipodes where your two fingers are holding it don’t move. (Yes, neighbourhoods around them move—but “points” in the infinitely-deep-down-continuum-set-of-measure-zero sense are singularities (erm, singularities in the 1/z sense, not in the “black hole” sense).)
Tomorrow: a post on a statistical application of the humble circle.
isomorphismes 