Tessellation of the Hyperbolic Plane

Tessellation of the hyperbolic plane by octagons.

Train Tracks

Let c be a simple closed curve around the middle and left hole. Let f = gh, where h interchanges the right and middle holes counterclockwise, and g interchanges the left and middle holes clockwise. The picture on the left shows the action of f^2 on c. The one on the right reveals its train track.

Trefoil Knot

The trefoil knot is the simplest knot there is. To get it, start with a rope, tie up the rope, then glue its endpoints.

Likorish Generators

Dehn twists about 11 curves generate the mapping class group Mod(S) of the given surface. For a closed surface S of genus g, Dehn twists about 3g-1 curves generate Mod(S).

Genus 5 Surface

Mobius Band

Leaf of a foliation on the torus