## Genus, Euler Characteristic, Boundary Components

Here’s another GEB triplet for Hofstadter. Compact, orientable surfaces are classified up to homeomorphism by their genus and number of boundary components. Consequentially, the Euler characteristic of such surface may be determined from these two. This diagram shows the relationship of these three quantities.

[…] The Euler characteristic, orientability, and number of boundary pieces uniquely determine what surface your planet is. If you get 2, you’re definitely on a sphere. Adding one boundary piece decreases the Euler characteristic by 1, so if you get a 1, you are on a sphere that got punctured, which is topologically equivalent to a polygon in the plane. If you get a 0, you could be living on a torus, or you could be living on a washer. The list goes on. […]

An Impractical, Ahistorical, Mathematically Elegant Way to Figure out the Earth Is a Sphere – WeeklyTimesNews.com said this on January 28, 2016 at 11:28 pm |