Tessellation of the Hyperbolic Plane

Tessellation of the hyperbolic plane by octagons.

Jordan Curve Theorem.

Jordan Curve Theorem states: Any simple closed curve c on the 2-sphere separates it into two components U and V. Each component has boundary c. Thinking of the 2-sphere as the compactification of the plane, the theorem seems intuitively plausible. However, the proof is nontrivial because of theorem's generality. From the right picture, it is clear that the presented curve is both simple and closed. However, the inside and outside are not readily obvious in the left picture.

Sierpinski's Carpet

Sierpinski's carpet is a fractal. It is obtained as follows: Start with a square S . Divide S into 9 identical squares via vertical and horizontal line segments. Remove the middle square. Now divide each of the remaining squares into 9 squares like before, and remove each middle square. Repeat the process.

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

Klein Bottle

Leaf of a foliation on the torus

Hyperbolic Plane

Lines in the Poincare disk model of the hyperbolic plane