Dihedral Tilings of the Sphere by Regular Polygons and Quadrilaterals II: Regular Polygons with High Gonality and Rhombi

Abstract

We classify the dihedral edge-to-edge tilings of the sphere by regular polygons with gonality at least 5 and rhombi.

Figures (21)

• Figure 1: Rhombus and regular polygons
• Figure 2: The earth map type, an infinite family of tilings with $\beta^2\gamma, \alpha\beta\gamma^c$
• Figure 3: The earth map type: prisms consist of two regular $m$-gons and $m$ rhombi, $m\ge5$ and $\diamond=\beta$
• Figure 4: The four Archimedean type tilings, by regular pentagons and rhombi, $\diamond=\beta$
• Figure 5: Three triangular fusions of the snub dodecahedron

Theorems & Definitions (24)

• Theorem
• Lemma 2.1
• Lemma 2.2: Counting Lemma, luk2
• Lemma 2.3
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof