funny picture is about:- Imagine you have a square sheet of flexible paper, the topological variant, colored like shown.
- Now bend the top side all around the back, so that a horizontal tube is formed. The top left yellow square get connected to the bottom left yellow square and the same happens to the yellow squares on the right hand side. The magenta figure at the top gets connected to the red figure on the bottom.
- Bend this tube around forming a torus (doughnut), so that the two 1x2 yellow rectangles form a 2x2 yellow square. The blue rectangle on the left gets connected to the cyan one on the right.
- Inspect the coloring of the resulting torus, and you will note that you have 7 "countries", where each one shares a part of its border with the 6 remaining "countries".
This proves, that you require at least 7 colors to color a map on a torus. Using the same simple arguments, like the ones used to prove the 5 color theorem in the plane, you can easily show that 7 colors are sufficient to color any map on a torus.
