Understanding the Geography of the Gauss map
Gauss map
 

The Gauss map takes a point on a smoothly embedded orientable surface to its (unit) normal. We can think of the normal as being a point on a sphere - this allows us to draw some amusing pictures below. If we draw the normals as spikes then we get something that looks like a cactus.

If you have the Flash plugin here's a day in the life of another brave new world with a different way of seeing what the Gauss map does.

The planet Earth is (approximately) a sphere. What we have done above is to pull back onto the torus a map of the world drawn on the sphere. Here's the Earth for comparison.

One sees:

  • the Gauss map is singular at the north and south poles.
  • off this set the Gauss map is 2-1.

Here's a snapshot where one sees this........
Degree of a map

 
The torus with a pullback of the earth via the Gauss map.
  • The north pole (a point on the sphere) has been stretched (to a circle).
  • The continents appear twice: once on the inside and once on the outside bands of the torus.

We can actually (see and) say a bit more: the continents on the inner band are mirror images of those on the outer band.

Calculating the degree

The degree is the number of times a country appears with the correct orientation minus the number of times it appears as it's mirror image.


In the picture above each country appears exactly exactly twice:

  • once with the correct orientation (+)
  • and once as a mirror image (-)


The degree of the Gauss map is 1-1 = 0

 
 
Here is a surface of genus 2.

Again we calculate the degree but it's a little bit trickier here. We'll do it twice to be sure.

Calculating the degree (I)

There are five copies of the Arabian peninsula (the little boot next to Africa).

  • Two with the correct orientation. (+2)
  • Three are "mirror images " - that is locally the Gauss map reverses the orientation. (-3)

The degree of the Gauss map is 2-3 = -1

Calculating the degree (II)

Note that some countries like Brazil only appear 3 times.


The degree of the Gauss map is 1-2 =-1.

 

Trefoil

Torus knots

Just for fun here's another embedding of the torus.
It's actually a knotted torus - a trefoil.
and here's the pull back of the Earth.
What's the degree of it's Gauss map?