Tessellations of Compact Surfaces
Poincaré’s theorem for a compact polygon II satisfying the side and angle conditions tells us that the identification space SII, obtained by identifying sides of II according to the side pairing, is also an orbit space S/Γ. Here S = SII is S2, ℝ2, or ℍ2—the surface from which II originates—and Γ is the group generated by the side-pairing transformations of II. Because of its interpretation as an orbit space, SII is also called an orbifold.
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