Holey Hyperbolic Polyforms

Abstract

A polyform is a planar figure formed by gluing congruent regular polygons along entire edges. We study polyforms in hyperbolic ${p,q}$-tessellations and the extremal problem of minimizing the number of tiles needed to realize exactly $h$ holes. Denoting this minimum by $g_{p,q}(h)$, we establish general lower and upper bounds, compute exact values in several small cases, and give a sufficient structural condition for a polyform to have $h$ holes and $g_{p,q}(h)$ tiles.

Figures (5)

• Figure 1: Examples of hyperbolic polyforms with multiple holes. Left: a $\{5,4\}$–polyform with six holes. Right: a $\{7,3\}$–polyform with three holes. Holes are defined as bounded connected components of the complement.
• Figure 2: Polyforms realizing the values in Table \ref{['table:g(h)vals']}, arranged by tessellation (rows) and number of holes (columns).
• Figure 3: Examples of the polyforms $B_{p,q}(h)$ used in the proof of Theorem 1.6. The highlighted edges are ultraparallel and serve as reflection axes in the construction of polyforms with an increasing number of holes.
• Figure 4: The puncturing process. Top left. The $\{2k,k\}$-tessellation. Top right. For each tile in the $\{2k,k\}$-tessellation, its center is added as a vertex, and edges are added between this center and the vertices of the tile, resulting in the $\{3, 2k\}$-tessellation. Bottom left. The vertices of the $\{3, 2k\}$-tessellation are colored gray and white. The subgraph of gray vertices is the original $\{2k, k\}$-tessellation; the white vertices are the added centers. Bottom right. The dual of $\{3, 2k\}$ is the $\{2k,3\}$-tessellation with an associated coloring of tiles.
• Figure 5: Left. An extremal polyform with 6 tiles in the $\{2k,k\}$-tessellation when $k=4$. Right. The corresponding punctured $\{2k,3\}$-polyform with 6 holes.

Theorems & Definitions (42)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Conjecture
• Lemma 2.1