Fundamental Polygon - Standard Fundamental Polygons

Standard Fundamental Polygons

An orientable closed surface of genus n has the following standard fundamental polygon:

This fundamental polygon can be viewed as the result of gluing n tori together, and hence the surface is sometimes called the n-fold torus. ("Gluing" two surfaces means cutting a disk out of each and identifying the circular boundaries of the resulting holes.)

A non-orientable closed surface of (non-orientable) genus n has the following standard fundamental polygon:

Alternately, the non-orientable surfaces can be given in one of two forms, as n Klein bottles glued together (this may be called the n-fold Klein bottle, with non-orientable genus 2n), or as n glued real projective planes (the n-fold crosscap, with non-orientable genus n). The n-fold Klein bottle is given by the 4n-sided polygon

(note the final is missing the superscript −1; this flip, as compared to the orientable case, being the source of the non-orientability). The (2n + 1)-fold crosscap is given by the 4n+2-sided polygon

That these two cases exhaust all the possibilities for a compact non-orientable surface was shown by Henri Poincaré.