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Non-Euclidean Dissections

On sphere and hyperbolic plane dissection theory must be adapted to the geometry of the respective space. There is considerably less work done on non-Euclidean dissections than on Euclidean dissections. [1] Nevertheless, we can formulate results equivalent to some of those enunciated for the Euclidean plane, after modifying for the new context.

On spheres and hyperbolic planes we do not have rectangles and parallelograms but we can work with similar basic polygons. They were defined almost a millennium ago by the Persian poet and mathematician Omar Khayyam.

A Khayyam quadrilateral on a sphere or on a hyperbolic plane is a quadrilateral with two opposite sides equal and two right angles formed by the opposite equal sides on a third side. Similarly, Khayyam parallelograms are quadrilaterals with two opposite sides parallel transports of each other along a third side (called base).

With these definitions, and assuming the Archimedean Axiom (A) mentioned in the Euclidean polygonal dissections subsection, it can be shown that the following results hold true:

a) Every hyperbolic triangle (and every small spherical triangle) is equivalent by dissection to a Khayyam parallelogram with the same base as the triangle.

b) Khayyam parallelogram is equivalent by dissection to a Khayyam quadrilateral with the same base.

c) Every small polygon on a sphere is equivalent by dissection to a lune.

A lune is obtained by the intersection of two big circles on the sphere, as in this figure:

To prove the affirmation c) one should first show that for all spherical triangles with the same base and area, the midpoints of the two non-base edges lie on a great circle and the geometric locus of the vertex opposite the base is a curve equidistant from that great circle (see figure below).

d) On a hyperbolic plane, two simple polygons (the sides intersect only at vertices) with the same area, are equivalent by dissection.