Dihedral Tilings of the Sphere by Regular Polygons and Quadrilaterals I: Squares and Rhombi

Abstract

We classify the dihedral edge-to-edge tilings of the sphere by squares and rhombi.

Figures (21)

• Figure 1: The prototiles: square and rhombus
• Figure 2: The earth map type tilings: $\ast = \gamma^c$ and $\star = \gamma^{c-1}$ for $c\ge2$
• Figure 3: The four Platonic and Archimedean type tilings, $\diamond=\beta$
• Figure 4: The two sporadic tilings, $\diamond=\beta$
• Figure 5: Triangular fusions of the snub cube

Theorems & Definitions (18)

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