Spherical quadrangles with three equal sides and rational angles

Abstract

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 π 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 or Triangles

“…Therefore the four angles are related by one equation, and we expect any four angles satisfying the equation determine the quadrilateral. The equation was proposed by Ueno and Agaoka [27], and a sharper version was proved for convex almost equilateral quadrilaterals by Coolsaet [12]. In the following lemma, we give the most comprehensive result, in the sense that we do not require the usual restrictions (which are implicitly assumed in all the other lemmas) that angles and edge lengths must be in (0, 2π).…”

“…They obtained a key equality relating the four angles of any almost equilateral quadrilateral. Coolsaet [12] obtained a sharper version of the equality, and used the equality to find all convex almost equilateral quadrilaterals, such that all angles are rational multiples of π. Still a lot more technical preparations are need to bridge the huge gap between these results and the full classification.…”

“…The second major technique is geometrical, and uses straight edges (part of great arcs) and simple boundary. In particular, we give an updated treatment of the equality obtained by Coolsaet [12], but does not require convexity. Moreover, we also shows that the equality implies the existence of the quadrilateral.…”

Rational Angles and Tilings of the Sphere by Congruent Quadrilaterals

“…It matches the trigonometric diophantine equation in [13, (4)]. In Section 3, we generalise Coolsaet's method [10,Theorem 3.1] to determine rational angles. In Section 5, we shall see in Lemma 5.5 that (2.5) serves as one of the criteria for verifying the geometric existence of tiles.…”

“…Myerson [13] effectively has found all rational solutions to the equation. Coolsaet [10] proved this equation for convex almost equilateral quadrilateral and applied Myerson's solutions to determine its rational angles. There are two major advancements in this paper.…”