Dihedral Tilings of the Sphere by Regular Polygons and Quadrilaterals I: Quadrilaterals with Equal Opposite Edges

Abstract

We classify the dihedral edge-to-edge tilings of the sphere by regular polygons and quadrilaterals with equal opposite edges (edge configuration xyxy).

Figures (13)

• Figure 1: The quadrilateral with edges $x,y$ and angles $\beta,\gamma$; and the regular polygons with edges $x$ and angles $\alpha$
• Figure 2: The infinite family of tilings of prism type, $\diamond=\beta$
• Figure 3: The sporadic tilings, $\diamond=\beta$
• Figure 4: The arrangements of $\alpha \vert \gamma$ and $\gamma\vert\gamma$ and $\beta\vert\beta$ and $\alpha^3, \alpha\beta\gamma$
• Figure 5: The tiling with $\alpha^3$ and the tiling with $\alpha\beta\gamma$

Theorems & Definitions (16)

• Theorem
• Lemma 2.1: Parity Lemma
• proof
• Lemma 2.2: Counting Lemma
• Lemma 2.3
• Lemma 2.4
• Proposition 3.1
• proof
• Proposition 3.2
• proof