Side-to-side Tiling of the Sphere by Congruent Curvilinear Triangles

TL;DR

This work advances the understanding of sphere tilings by congruent curvilinear triangles, where edges need not be geodesics, by developing a four-type edge framework ($g$, $h$, $r$, $a$) and proving there are exactly sixteen curvilinear triangle prototiles suitable for side-to-side tilings. The authors show that all such tilings are modifications of the classical straight-triangle tilings, and they enumerate the resulting families, including tetrahedral, Platonic-based subdivisions, cube subdivisions, and earth-map tilings, along with their rotation and flip variants. They also identify several straight-tiling families that have no curvilinear analogs, solidifying that curvature and edge-curvilinearity do not produce new tiling types beyond known classifications. The results provide a complete, geometry-grounded classification that links curvilinear tilings to well-studied straight-tiling families, with implications for geometric design and modeling on spherical surfaces.

Abstract

The edge-to-edge tilings of the sphere by congruent polygons, where all edges are straight, have been completely classified. We classify the curvilinear version of the similar triangular tilings, where the edges may not be straight, and find that these are the modifications of the straight triangular tilings.

Figures (39)

• Figure 1: A tiling of the sphere by a bird-like curvilinear triangle.
• Figure 2: Curvilinear side.
• Figure 3: A general curvilinear edge and its isometric transformations.
• Figure 4: Four types of edges and their schematic drawings.
• Figure 5: Add a circle to emphasise $r^{-1}$.

Theorems & Definitions (16)

• Theorem
• Lemma 1
• Lemma 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof