### Regular Irregular

In the book The Eightfold Way (nothing to do with Gell-Mann) there is an article by Thurston on the beauty of the Klein quartic curve, which the book is about.

The Klein quartic is tiled regularly by irregular heptagons. Topologically it is a three holed (genus 3) oriented surface, and hence it has a hyperbolic geometry. It has interesting symmetries. Tiling the Poincare disc with the heptagons we see a central heptagon with seven surrounding ones. By following the tiling outwards we generate the sequence 7,7,14,21,35,56,91,147,238, ... which is precisely seven times the Fibonacci sequence! If the three holed surface is squished about it can be made to look like a tetrahedron with tubes for edges. Have fun playing.

The Klein quartic is tiled regularly by irregular heptagons. Topologically it is a three holed (genus 3) oriented surface, and hence it has a hyperbolic geometry. It has interesting symmetries. Tiling the Poincare disc with the heptagons we see a central heptagon with seven surrounding ones. By following the tiling outwards we generate the sequence 7,7,14,21,35,56,91,147,238, ... which is precisely seven times the Fibonacci sequence! If the three holed surface is squished about it can be made to look like a tetrahedron with tubes for edges. Have fun playing.

## 1 Comments:

Cool! It looks like a conformal representation of the hyperbolic plane, or one of the mc escher drawings Penrose uses in his book.

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