Spherical tilings by congruent quadrangles over pseudo-double wheels (III) - the essential uniqueness in case of convex tiles

TL;DR

This paper investigates conditions under which spherical monohedral quadrangular tilings of certain topologies are isohedral, establishing that specific face counts and tile shapes guarantee this property, thus advancing classification of such tilings.

Contribution

It proves that spherical monohedral quadrangular tilings with certain face counts and convex tiles are necessarily isohedral, addressing an inverse problem in spherical tiling classification.

Findings

Tilings with 6 faces of the specified topology are isohedral.

Tilings with 8 faces with kite, dart, or rhombus tiles are isohedral.

Convex tile tilings of the specified topology are isohedral.

Abstract

In [B.Gruenbaum, G.C. Shephard, Spherical tilings with transitivity properties, in: The geometric vein, Springer, New York, 1981, pp. 65-98], they proved "for every spherical normal tiling by congruent tiles, if it is isohedral, then the graph is a Platonic solid, an Archimedean dual, an n-gonal bipyramid (n>2), or an n-gonal trapezohedron (i.e., the pseudo-double wheel of 2n faces)". In the classification of spherical monohedral tilings, one naturally asks an "inverse problem" of their result: For a spherical monohedral tiling of the above mentioned topologies, when is the tiling isohedral? We prove that for any spherical monohedral quadrangular tiling being topologically a trapezohedron, if the number of faces is 6, or 8, if the tile is a kite, a dart or a rhombi, or if the tile is convex, then the tiling is isohedral.