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Characterization of reduced polygons in the hyperbolic plane

Characterize the class of reduced polygons in the hyperbolic plane H^2 by providing necessary and sufficient conditions for a hyperbolic polygon to be reduced.

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Background

In Euclidean and spherical geometry, reduced polygons are fully characterized and are all odd-gons. In contrast, hyperbolic geometry exhibits qualitatively different behavior, including the existence of reduced rhombi.

Because of these differences, a complete characterization in H2 is not presently known; this paper thus focuses on the subclass of ordinary reduced polygons for which certain structural properties are available.

References

Interestingly enough, the characterization of hyperbolic reduced polygons is still unclear, but clearly it must be different from the Euclidean and spherical ones; there exist reduced rhombi on the hyperbolic plane, while Euclidean and spherical reduced polygons are all odd-gons (see Lassak ).

— On the area of ordinary hyperbolic reduced polygons (2403.11360 - Sagmeister, 17 Mar 2024) in Section 1 (Introduction)