Non-side-to-side tilings of the sphere by congruent triangles with any irrational angle

TL;DR

This paper classifies non-side-to-side tilings of the sphere by congruent triangles, especially focusing on cases with irrational angles, revealing families of tilings and unique configurations, and outlining a scheme for rational angles.

Contribution

It introduces new tools for classifying such tilings and provides a comprehensive classification for triangles with irrational angles, including families and unique tilings.

Findings

Existence of 1-parameter families of tilings with many 2-layer earth map configurations

Identification of a unique 8-tile tiling for a specific triangle

Discovery of a sporadic 16-tile tiling for another specific triangle

Abstract

We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families of triangles each admitting many 2-layer earth map tilings with $2n$($n\geq3$) tiles, together with rotational modifications for even $n$; a 1-parameter family of triangles each admitting a unique tiling with $8$ tiles; and a sporadic triangle admitting a unique tiling with $16$ tiles. Then a scheme is outlined to classify the case with all angles being rational in degree, justified by some known and new examples.