Positive and negative Euler characteristic



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Positive and negative Euler characteristic

A symmetry pattern on the sphere always gives rise to a quotient orbifold with positive Euler characteristic. In fact, if the order of symmetry is , then the Euler characteristic of the quotient orbifold is , since the Euler characteristic of the sphere is 2.

However, the converse is not true. Not every collection of parts costing less than can be put together to make a viable pattern for symmetry on the sphere. Fortunately, the experts at The Orbifold Shop know the four bad configurations which are too skimpy to be viable:

All other configurations are good. If they form an orbifold with positive orbifold Euler characteristic, they come from a pattern of symmetry on the sphere.

The situation for negative orbifold Euler characteristic is straightforward, but we will not prove it:

Theorem. Every orbifold with negative orbifold Euler characteristic comes from a pattern of symmetry in the hyperbolic plane with bounded fundamental domain. Every pattern of symmetry in the hyperbolic plane with compact fundamental domain gives rise to a quotient orbifold with negative orbifold Euler characteristic.

Since you can spend as much as you want, there are an infinite number of these.



Next: Hyperbolic Geometry Up: Geometry and the Imagination Previous: The Euler characteristic



Peter Doyle