Tilings of the Sphere by Congruent Quadrilaterals or Triangles

Abstract

We completely classify edge-to-edge tilings of the sphere by congruent quadrilaterals. As part of the classification, we also present a modern version of the classification of edge-to-edge tilings of the sphere by congruent triangles. Together with our series of papers that classifies edge-to-edge tilings of the sphere by congruent pentagons, we complete the classification of edge-to-edge tilings of the sphere by congruent polygons.

Figures (74)

• Figure 1: All possible triangles, with distinct $a,b,c$.
• Figure 2: Quadrilaterals suitable for tiling, with distinct $a,b,c$.
• Figure 3: Angle indications when not determined by edges.
• Figure 4: Platonic solids.
• Figure 5: Triangular and quadrilateral earth map tilings.

Theorems & Definitions (91)

• Theorem
• Theorem
• Lemma 1
• proof
• Lemma 2: Parity Lemma
• proof
• Lemma 3
• Lemma 4: Counting Lemma
• proof
• Lemma 5: Balance Lemma