Tilings of the Sphere with Isosceles Triangles

Abstract

The spherical triangles which tile the sphere in an edge-to-edge fashion have been known for some time. However, if we relax the requirement that the triangles must meet edge-to-edge, other tilings are possible. This paper begins the classification of these tilings by characterizing all isosceles triangles that tile the sphere. One infinite family and three sporadic tiles that tile only edge-to-edge are exhibited.

An isosceles triangle that tiles the sphere in exactly three ways

“…1(c) can exist in a tiling, we can show that there is no tiling using only that (3,2) configuration. If we attempt such a tiling, that configuration is forced at every vertex containing a large angle.…”

“…In a recent paper [2] Dawson showed that, among the isosceles triangles, there are three sporadic triangles and one infinite family that tile the sphere but do not do so in an edge-to-edge fashion. In two of the sporadic cases, (150 • , 60 • , 60 • ) and (100 • , 60 • , 60 • ), it is very easy to show that the triangle can tile in one way only (up to reflection) [2].…”

“…In two of the sporadic cases, (150 • , 60 • , 60 • ) and (100 • , 60 • , 60 • ), it is very easy to show that the triangle can tile in one way only (up to reflection) [2]. The tiles in the infinite family {(180 • − 360 • /n, 360 • /n, 360 • /n), n odd} each tile in a large number of different ways, which can be enumerated using Burnside's theorem.…”

Spherical f-Tilings by (Equilateral and Isosceles) Triangles and Isosceles Trapezoids

“…Spherical f-tilings with prototiles a spherical triangle and a r-sided regular polygon (r ≥ 5) was recently obtained in [1]. Dawson and Doyle have also been interested in special classes of spherical tilings, see [6][7][8] for instance.…”

Spherical folding tessellations by kites and isosceles triangles IV

“…Dawson has also been interested in special classes of spherical tilings, see [10], [11] and [12], for instance.…”