Decomposing Centrally Symmetric Convex Polyhedral Surfaces into Parallelograms

Abstract

Let $\mathcal{M}_{2N}(δ_1, δ_2,\dots, δ_N)$ be the moduli space of centrally symmetric convex polyhedral surfaces with $2N$ labeled vertices and prescribed cone-deficits $δ_1$, $δ_2$, $\dots$, $δ_N$. We show that $\mathcal{M}_{2N}(δ_1, δ_2,\dots, δ_N)$ has the structure of a real hyperbolic manifold of dimension $2N-3$. When $N=4$ and $5$, we show that every surface in $\mathcal{M}_{2N}(δ_1, δ_2,\dots, δ_N)$ can be decomposed into at most $2\binom{2N-2}{2}$ parallelograms, and the decomposition is invariant under the antipodal map. Using the edge-lengths of these parallelograms as coordinates, we show that the moduli space of centrally symmetric polyhedral surfaces with $8$ unlabeled vertices and cone-deficits $\fracπ{2}$ is isometric to the quotient of a real hyperbolic regular ideal $5$-simplex by the dihedral group $D_6$.

Figures (30)

• Figure 1: Two directed cycles on the surface of a cube. The second one is strictly shorter.
• Figure 2: The surface $\mathscr{S}^+$ is cut open and unfolded to a planar polygon $P_{\mathscr{S}^+}$.
• Figure 3: Directed edges on the surface of a cube that give rise to different local frames.
• Figure 4: Vertices $1^+$ and $2^+$ "collide" to $v_{12}$, resulting in a surface with one fewer vertex.
• Figure 5: A loop arrangement with $6$ loops (red) on a sphere with $8$ labeled vertices.

Theorems & Definitions (48)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof