The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this "isoperimetric" problem differs for every given area, as solutions do not scale. Cox conjectured that a regular $k$-gonal tile with 120-degree angles is isoperimetric. For area $π/3$, the regular heptagon has 120-degree angles and therefore tiles many hyperbolic surfaces. For other areas, we show the existence of many tiles but provide no conjectured optima.
On closed hyperbolic surfaces, we verify via a reduction argument using cutting and pasting transformations and convex hulls that the regular $7$-gon is the optimal $n$-gonal tile of area $π/3$ for $3\leq n \leq 10$. However, for $n>10$, it is difficult to rule out non-convex $n$-gons that tile irregularly.
In this paper we give a complete description about normal monohedral tilings of a convex disc with smooth boundary where we have at most three topological discs as tiles. This result is a far-reaching generalization of the results of Kurusa, Lángi and Vígh \cite{KLV2020}. Some further partial results are proved for non-normal tilings.
A convex pentagonal tile is a convex pentagon that admits a monohedral tiling. We show that a convex pentagonal tile that admits a periodic tiling has a property in which the sum of three internal angles of the pentagon is equal to 360°.
In this note we prove that any monohedral tiling of the closed circular unit disc with $k \leq 3$ topological discs as tiles has a $k$-fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain the center, and the first step towards answering a question of Stein appearing in the problem book of Croft, Falconer and Guy in 1994.
Supergroups of some hyperbolic space groups are classified as a continuation of our former works. Fundamental domains will be integer parts of truncated tetrahedra belonging to families F1 - F4, for a while, by the notation of E. Molnár et al. in $2006$. As an application, optimal congruent hyperball packings and coverings to the truncation base planes with their very good densities are computed. This covering density is better than the conjecture of L.~Fejes~Tóth for balls and horoballs in $1964$.
In this paper, we describe and visualize the densest ball and horoball packing configurations belonging to the simply truncated $3$-dimensional hyperbolic Coxeter orthoschemes with parallel faces. These beautiful packing arrangements describe and show the very interesting structure of the mentioned orthoschemes and the corresponding Coxeter groups. We use for the visualization the Beltrami-Cayley-Klein ball model of $3$-dimensional hyperbolic space $\boldsymbol{H}^3$ and the pictures were made by the Python software.
Density upper bound for congruent and non-congruent hyperball packings generated by truncated regular simplex tilings
本文研究了截断正四面体镶嵌的全等和非全等超球（超球）填充。这些是从$3$和$5$维双曲空间中的Coxeter单纯形镶嵌$\{p,3,3\}$ $(7\le p \in \mathbb{N})$和$\{5,3,3,3,3\}$导出的。我们确定了与上述镶嵌相关的密度超球填充安排。我们利用全等超球求出了堆积密度，并确定了由上述拼接生成的非全等超球堆积的最小密度上界。
Planar pseudo-triangulations, spherical pseudo-tilings and hyperbolic virtual polytopes
我们希望提请注意两种理论之间有趣和有希望的相互作用。一方面，\textbf{pseudo-triangulations}理论对于隐式地解决卡彭特规则问题是有用的，并且后来被证明为图嵌入提供了一个很好的工具。另一方面，它是双曲虚多面体理论，它产生于一个古老的关于凸体的唯一性猜想（A. D. Alexandrov的问题：假设一个常数$C$在任何地方（非严格地）分离光滑三维凸体$K$的主曲率半径。那么$K$必然是一个半径为$C$的球。两个关键思想是：通过从平面伪三角形到球面伪平铺，我们避免了非点顶点。相反，我们使用伪二进制。给出了球面嵌入Laman加1图的一个定理。构造双曲多面体的困难问题可以归结为求球面嵌入图的问题。
In this paper we consider the ball and horoball packings belonging to $3$-dimensional Coxeter tilings that are derived by simply truncated orthoschemes with parallel faces. The goal of this paper to determine the optimal ball and horoball packing arrangements and their densities for all above Coxeter tilings in hyperbolic 3-space $\mathbb{H}^3$. The centers of horoballs are required to lie at ideal vertices of the polyhedral cells constituting the tiling, and we allow horoballs of different types at the various vertices. We prove that the densest packing of the above cases is realized by horob…
Optimal Ball and Horoball Packings Generated by Simply Truncated Coxeter Orthoschemes with Parallel Faces in Hyperbolic $n$-space for $4 \leq n \leq 6$
