[Paper Review] Tilings of the Sphere by Geometrically Congruent Pentagons I

Ka Yue Cheuk, Ho Man Cheung+1 more|arXiv (Cornell University)|Oct 8, 2013

Quasicrystal Structures and Properties8 references3 citations

TL;DR

This paper proves that the only edge-to-edge tiling of the sphere by congruent pentagons—where one tile has all vertices of degree 3 and exactly three of five possible edge length combinations—is the minimal dodecahedron tiling. The result establishes a uniqueness constraint on geometrically congruent pentagonal tilings under specific topological and metric conditions.

ABSTRACT

We show that there are no edge-to-edge tilings of the sphere by congruent pentagons beyond the minimal dodecahedron tiling, such that there is a tile with all vertices having degree 3 and the edge length combinations are three of the five possibilities.

Motivation & Objective

• To determine whether non-dodecahedral edge-to-edge tilings of the sphere by congruent pentagons exist under specific geometric and topological constraints.
• To analyze the structural limitations imposed by requiring all vertices of one tile to have degree 3.
• To classify possible edge length combinations and eliminate configurations beyond three of five possible types.
• To establish that the dodecahedron is the only such tiling under the given constraints.
• To contribute to the broader classification of spherical tilings by convex pentagons with geometric congruence.

Proposed method

• Applied combinatorial and topological constraints to analyze vertex and edge configurations in spherical pentagonal tilings.
• Used the condition that one tile must have all vertices of degree 3 to restrict possible tiling structures.
• Evaluated all possible edge length combinations, focusing on exactly three of the five theoretical possibilities.
• Employed symmetry and geometric congruence to eliminate non-dodecahedral configurations.
• Conducted exhaustive case analysis on vertex cycles and edge adjacency patterns.
• Leveraged known results on spherical polyhedra and Euler's formula to validate structural consistency.

Experimental results

Research questions

• RQ1 Are there any edge-to-edge spherical tilings by congruent pentagons beyond the dodecahedron that satisfy the degree-3 vertex condition on one tile?
• RQ2 Can such tilings exist when only three of the five possible edge length combinations are used?
• RQ3 What structural constraints arise from requiring geometric congruence and edge-to-edge adjacency in spherical pentagonal tilings?
• RQ4 Is the dodecahedron the unique solution under these constraints?
• RQ5 What role do vertex degrees and edge length combinations play in limiting the existence of such tilings?

Key findings

• No edge-to-edge tiling of the sphere by congruent pentagons exists beyond the dodecahedron under the specified conditions.
• The dodecahedron is the unique tiling where one tile has all vertices of degree 3 and exactly three of the five edge length combinations are used.
• All other configurations satisfying the degree-3 and three-edge-length constraints are geometrically impossible on the sphere.
• The combination of geometric congruence, edge-to-edge adjacency, and vertex degree constraints severely limits tiling possibilities.
• The result confirms a strong uniqueness property for spherical pentagonal tilings under these specific constraints.
• The analysis confirms that no new families of such tilings can be constructed beyond the known dodecahedral case.