Dihedral f-Tilings of the Sphere Induced by the Möbius Triangle $(2,3,4)$

Abstract

We classify the special families of dihedral folding tilings of the sphere derived from the Möbius triangle $(2,3,4)$. Our study emerges from the study of isometric foldings in the Riemann sphere and meets at the juncture of the triangle group $Δ(2,3,4)$. The juxtaposition enables us to apply the classification theorem of edge-to-edge tilings of the sphere by congruent triangles and introduces a group theoretical method. The two prototiles of each family consist of the Möbius triangle and another polygon induced by a reflection of the triangle group acting on the Möbius triangle. To enumerate the tilings, we give two solutions to solve the associated constraint satisfaction problem. The methods are not exclusive to this problem and therefore applicable to similar problems of more general settings.

Figures (14)

• Figure 1: The Möbius triangle $\triangle abc$, the kite $\square a^2b^2$, and $\alpha=\tfrac{1}{4}\pi$, $\beta=\tfrac{1}{3}\pi$ and $\gamma=\tfrac{1}{2}\pi$ and the isosceles triangles $\triangle \bar{a}c^2$ and $\triangle\bar{b}c^2$
• Figure 2: The plane drawings of the barycentric subdivision of the octahedron $B\mathcal{O}$ and its flip modification $FB\mathcal{O}$; the arrows in each picture converge to a single vertex
• Figure 3: Locations of $c$-assignments in $B\mathcal{O}$ and $FB\mathcal{O}$
• Figure 4: The even $c$-assignments at a vertex
• Figure 5: The spherical deltoidal icositetrahedron and the spherical pseudo-deltoidal icositetrahedron

Theorems & Definitions (15)

• Theorem
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 4.1