Positive and negative Euler characteristic (Geometry and the Imagination)
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Conway, Doyle, Gilman, Thurston; The Geometry Center  
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 k, then the Euler characteristic of the quotient orbifold is 2/k, since the Euler characteristic of the sphere is 2. However, the converse is not true...  


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Math Topics:  NonEuclidean Geometry, Symmetry/Tessellations 
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