Spherical Tiling by 12 Congruent Pentagons

Abstract

The tilings of the 2-dimensional sphere by congruent triangles have been extensively studied, and the edge-to-edge tilings have been completely classified. However, not much is known about the tilings by other congruent polygons. In this paper, we classify the simplest case, which is the edge-to-edge tilings of the 2-dimensional sphere by 12 congruent pentagons. We find one major class allowing two independent continuous parameters and four classes of isolated examples. The classification is done by first separately classifying the combinatorial, edge length, and angle aspects, and then combining the respective classifications together.

53 pages, 40 figures, spherical geometry

References in corpus (3)

• Tilings of the Sphere by Edge Congruent Pentagons
• Combinatorial Space Tiling
• Combinatorial Tilings of the Sphere by Pentagons

Cited by in corpus (9)

• On Deformed Dodecahedron Tiling
• Tilings of the Sphere by Edge Congruent Pentagons
• Spherical tilings by congruent quadrangles over pseudo-double wheels (III) - the essential uniqueness in case of convex tiles
• Edge-to-edge Tilings of the Sphere by Angle Congruent Pentagons
• Tilings of the Sphere by Congruent Pentagons I
• Tilings of the Sphere by Congruent Pentagons I: Edge Combinations $a^2{b}^{2}c$ and $a^{3}bc$
• Areas of spherical polyhedral surfaces with regular faces
• Moduli Spaces of Pentagonal Subdivision Tilings
• Tilings of the Sphere by Congruent Quadrilaterals with Exactly Two Equal Edges