Vertex-minimal hyperbolic origami 2-torus

TL;DR

The paper proves that a hyperbolic origami 2-torus, a genus-2 surface triangulated with 10 vertices, can be isometrically embedded into $\mathbb{H}^3$ via a geodesic triangulation. The authors construct a candidate 10-vertex model $\hat{S}$ that is $10^{-28}$-flat and robustly embedded, and then use a high-precision, computer-assisted Newton-type method to perturb coordinates and obtain an exact isometric embedding $S$ with cone angles exactly $2\pi$. To facilitate and validate the search, they also develop a 12-vertex model with a 2-fold symmetry, $\hat{S}_{12}$, illustrating a symmetric pathway toward vertex-minimal configurations; the embedding is shown to respect a hyperelliptic involution and exhibits a near-ideal isometric tiling in the universal cover. The work combines detailed geometric analysis (via the Beltrami–Klein model and hyperbolic trigonometry) with rigorous, high-precision numerical proofs, yielding a definitive minimum of 10 vertices for a hyperbolic origami 2-torus and enabling the construction of a family of such tori for all $n\ge 10$ by interior refinement. The results have potential to inform broader questions about hyperbolic polyhedral embeddings and the moduli of hyperbolic tori, with explicit coordinates and open-source visualization tools provided for reproducibility.

Abstract

We show that there exists a geodesic triangulation $T$ of a hyperbolic genus 2 surface $Σ_2$ with 10 vertices and an isometric polyhedral embedding $S: Σ_2 \hookrightarrow \mathbb{H}^3$ that sends the triangles in $T$ to geodesic triangles in $\mathbb{H}^3$. We call this type of embedding a hyperbolic origami 2-torus. Since 10 is the combinatorially minimum number of vertices required to triangulate a genus 2 surface, this paper settles the question of minimum number of vertices required to obtain a hyperbolic origami 2-torus.

Figures (8)

• Figure 1: The candidate 2-torus $\hat{S}$ in the Beltrami--Klein model $\mathbb{H}^3$, shown from opposite sides of a "hole" passing through it. Triangles are shaded along a yellow--purple spectrum according to the $y$-coordinates of their centroids, ordered from highest to lowest.
• Figure 2: Left: The induced triangle tiling from $\mathcal{T}$ in the Beltrami--Klein model of $\mathbb{H}^2$. Right: The tiling in the Poincaré disk model. The colors of the triangles are derived from Figure \ref{['fig:surface']}.
• Figure 3: The intersection of $\hat{S}$ with the $xy$-plane, which is the boundary of a simply-connected component colored in yellow.
• Figure 4: The intersection of $\hat{S}$ with the plane $y = 0$, which is the boundary of two simply-connected components colored in yellow. The boxes contain the magnified view of their respective regions in the original slice.
• Figure 5: The intersection of $\hat{S}$ with the plane $x = 0$, which is the boundary of a region that is not simply-connected. Again, the boxes contain the magnified view of their respective regions in the original slice.

Theorems & Definitions (41)

• Theorem 1.1
• Definition 3.1
• Proposition 3.2
• proof
• Definition 3.3
• Remark 3.4
• Lemma 3.5: schwartz2025vertexminimalpapertori
• proof
• Lemma 3.6
• proof