Spherical quadrangles with three equal sides and rational angles

When the condition of having three equal sides is imposed upon a (convex) spherical quadrangle, the four angles of that quadrangle cannot longer be freely chosen but must satisfy an identity. We derive two simple identities of this kind, one involving ratios of sines, and one involving ratios of tangents, and improve upon an earlier identity by Ueno and Agaoka. The simple form of these identities enable us to further investigate the case in which all of the angles are rational multiples of pi and produce a full classification, consisting of 7 infinite classes and 29 sporadic examples. Apart from being interesting in its own right, these quadrangles play an important role in the study of spherical tilings by congruent quadrangles

Tilings of the sphere by congruent quadrilaterals II: edge combination a3ba^3 b with rational angles

Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of $a^3b$-quadrilaterals with all angles being rational degrees. There are $12$ sporadic and $3$ infinite sequences of quadrilaterals admitting the $2$-layer earth map tilings together with their modifications, and $3$ sporadic quadrilaterals admitting $4$ exceptional tilings. Among them only $3$ quadrilaterals are convex. New interesting non-edge-to-edge triangular tilings are obtained as a byproduct.Comment: 36 pages, 36 figures, 10 table

On the Goncharov depth conjecture and a formula for volumes of orthoschemes

We prove a conjecture of Goncharov, which says that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. We give an explicit formula for this presentation, involving a summation over trees that correspond to decompositions of a polygon into quadrangles. Our second result is a formula for volume of hyperbolic orthoschemes, generalizing the formula of Lobachevsky in dimension $3$ to an arbitrary dimension. We show a surprising relation between two results, which comes from the fact that hyperbolic orthoschemes are parametrized by configurations of points on $\mathbb{P}^1.$ In particular, we derive both formulas from their common generalization.Comment: 49 pages, 7 figure