Tuesday, April 10, 2007

Your Favorite Lenny? Some alternate choices

Excellent alternate choices for Your Favorite Lenny:

Top to bottom, left to right: Bruce, Bernstein, Briscoe, daVinci, Smalls, Nimoy, Sheldon (last name Leonard)

Any of these or others could be Your Favorite Lenny; but My Favorite Lenny is still Leonhard Euler.

Let's start with the Euler characteristic, sometimes known as the Euler-Poincaré characteristic. (They didn't work together; Henri Poincaré shows up more than a century later and extends the work Euler did on this.)

Think of a cube. It has 6 faces, 8 corners (we will call them vertices) and 12 edges.

Consider a pyramid. It has 5 faces (including the base), 5 vertices and 8 edges.

The Euler characteristic says F + V = E + 2. This is true for any 3-d geometric shape with flat surfaces for faces, as long as there is no hole that goes through the shape; the technical term is polyhedron. For surfaces with holes (think of a donut, or drilling a rectangular tunnel through a cube that extends from one face to another), the formula changes to F + V = E + 2 + 2G, where G is the number of such holes, known as the genus of the solid.

You might say, "Hey, Matty Boy! What about a sphere? It has one face, no vertices and no edges. 1 + 0 = 0 + 2 is wrong! Ha, ha, ha!"

You're right, but the shape hasn't been properly Eulerized; we need at least 1 vertex, and if there are at least 2 vertices, then we need edges as well to create faces. Let's pick a point on the sphere and call that a vertex. 1 + 1 = 0 + 2.

Let's draw a Great Circle that goes through that point. (A Great Circle is the largest circle that can be drawn on a sphere, like the equator on Earth.) We now have 1 vertex and 1 edge, and the sphere now has 2 faces, call them north and south. 2 faces, 1 vertex, 1 edge. 2 + 1 = 1 + 2.

My Favorite Lenny.


Tara Mobley said...

Interesting. I'll need to read up more on this Lenny. Must play more with math.

Padre Mickey said...

I vote for Lenny of Lenny and Squigy fame.

Matty Boy said...

Many are the Lennys one could chose.

I gots Euler with nothin' to lose.

Some people say "Euler" is the one word math test. If you see the name and pronounce it U-ler, you didn't learn any math. If you know it's Oy-ler, you learned some math. (Or you're German.)

Karla said...

Lenny, King of Sparta!
Lenny Brezhnev!
Lenny Di Caprio
Lenny McCoy!

joshsonofmichael said...

leonard cohen. he liked girls that were all edge and no face.

luna_teak said...

gotta be daVince...it's daHair