正四面体的稠密堆积

一类截角四面体的晶体组装和密度填充及方向熵力的作用

Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces

多面体和它们的排列方式从古希腊时代起就引起了人类的兴趣，并且是今天凝聚态物质的重要图案，适用于许多种类的液体和固体。然而，人们对多面体形状的构建块（如多面纳米颗粒和胶体）的结晶稳定相知之甚少。虽然已知硬颗粒仅由于熵而组织，并且在文献中报道了一些不寻常的相，但与多面体形状有关的熵力的作用还没有得到很好的理解。在这里，我们研究了一个家庭的截断四面体的热力学自组装，并报告了几个原子晶体的同构，包括金刚石，β-锡，高压锂，多面体形状从四面体到八面体变化。我们比较我们的研究结果与密度填充的截断四面体家庭通过数值压缩和报告一个新的空间填充多面体，这在以前的搜索中被忽视。有趣的是，自组装结构不同于密排填料。我们发现，自组装晶体结构可以理解为多面体的趋势，以最大限度地提高面对面的对齐，这可以概括为定向熵力。

一类稠密四面体填充的解析构造及对称性的作用

Analytical Constructions of a Family of Dense Tetrahedron Packings and the Role of Symmetry

正四面体（五种柏拉图立体之一）的密排包装的确定正引起人们的极大关注，包装记录被打破的速度之快和出现的迷人的包装结构证明了这一点。我们发现了密度为$ϕ= {12250/14319} = 0.855506...$的正四面体的密度已知填充.这些包装是特殊情况下的分析两个参数的家庭密集的周期性包装与四个基本细胞，我们已经在这里构建。从这个族，我们可以恢复一组最近的包装安排，由于Kallus { \it et al.} [arXiv：0910.5226]与密度$ϕ={100/117}=0.854700...$，这比我们的密堆积具有更高的对称性，我们还描述了一个程序，可以导致严格的上限的四面体堆积的最大密度，这可以帮助评估候选密堆积的堆积效率。

密堆积四面体的无序相、准晶相和晶相

Disordered, Quasicrystalline and Crystalline Phases of Densely Packed Tetrahedra

Ulam将所有坚硬的凸形都定义为比球形更致密的堆积，球形的最大堆积分数为π = π/ \sqrt 18 ~ 0.7405。对于许多形状，简单的晶格填充很容易超过这个填充分数。对于正四面体，这个猜想直到最近才被证明是正确的;通过几何构造获得了有序排列，其中，π = 0.7786，随后在数值上压缩到π = 0.7820。在这里，我们展示了四面体的包装比这好得多，而且是以一种完全出乎意料的方式。遵循概念上不同的方法，使用热力学计算机模拟，使系统自然地向高密度态发展，我们观察到，硬四面体的流体经历了一阶相变的十二方准晶，它可以被压缩到一个包装分数为0.8324。通过压缩准晶体的晶体近似物，我们得到的最高填充分数是ε = 0.8503。如果准晶的形成被抑制，系统将保持无序，堵塞，并压缩到ε = 0.7858。堵塞和结晶之前都是由熵驱动的过渡，从一个简单的流体的独立四面体到复杂的流体，其特征在于四面体排列在密集的局部图案，形成一个在过渡的结晶网络。我们报告的准晶代表了第一个例子的准晶形成的硬或非球形颗粒。我们的研究结果表明，粒子形状和熵可以产生高度复杂的，有序的结构。

二元球体堆积中的稠密四面体

On Dense Tetrahedra in Binary Sphere Packings

本文研究三维空间中不等球体堆积分解中四面体的密度，通过结合几何洞察与区间算术计算，改进了特定尺寸球体堆积密度的上界。

New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry

In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l^p_3$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. Thereby, we improve Zong's recent upper bound for the maximal density of translative packings of regular tetrahedra from $0.3840\ldots$ to $0.3745\ldots$, getting closer to the best known lower bound of $0.3673\ldots$ We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of densest packings of round spheres. The…

Upper bound of density for packing of congruent hyperballs in hyperbolic $3-$space

In \cite{Sz17-2} we proved that to each saturated congruent hyperball packing exists a decomposition of $3$-dimensional hyperbolic space $\mathbb{H}^3$ into truncated tetrahedra. Therefore, in order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyperball packings in truncated simplices. In this paper we prove, using the above results and the results of papers \cite{M94} and \cite{Sz14}, that the density upper bound of the saturated congruent hyperball (hypersphere) packings related to the corresponding truncated tetrahedron cells …

双曲空间中广义单纯密度函数的Horoball填充及其密度

Horoball packings and their densities by generalized simplicial density function in the hyperbolic space

本文的目的是确定双曲3-空间$\bar{\mathbf{H}}^3$中关于至少有一个对称面的完全渐近四面体的局部可解的horoball packing排列及其密度，其中horoball的理想中心产生完全渐近四面体的顶点.我们允许不同类型的horoball在不同的顶点。此外，我们推广了推广的双曲空间$\bar{\mathbf{H}}^n, ~(n \ge 2)$中的单纯密度函数的概念，并证明了在这个意义下，{ \it，即$\bar{\mathbf{H}}^3$的“全等球”填充的著名的Böröczky-Florian密度上界对完全渐近四面体不成立。}这个局部密排的密度是$\approx 0.874994$，可能令人惊讶地大于Böröczky-Florian密度上界$\approx 0.853276$，但我们的局部球排列似乎没有扩展到整个双曲空间。

Structure and Visualization of Optimal Horoball Packings in $3$-dimensional Hyperbolic Space

Four packings of hyperbolic 3-space are known to yield the optimal packing density of $0.85328\dots$. They are realized in the regular tetrahedral and cubic Coxeter honeycombs with Schläfli symbols $\{3,3,6 \}$ and $\{4,3,6\}$. These honeycombs are totally asymptotic, and the packings consist of horoballs (of different types) centered at the ideal vertices. We describe a method to visualize regular horoball packings of extended hyperbolic 3-space $\bar{\mathbb{H}}^3$ using the Beltrami-Klein model and the Coxeter group of the packing. We produce the first known images of these four optimal hor…

Close geodesics on regular tetrahedra in hyperbolic space

In this paper we present a necessary conditions, that simple close geodesics on regular tetrahedra in the 3-dimensional hyperbolic space must satisfy. Furthermore, we explicitly describe three classes of simple closed geodesics on regular tetrahedra in the hyperbolic 3-space. These are so-called 2-homogeneous, 3-homogeneous and (3,2)-homogeneous geodesics. Up to a rigid motion of a tetrahedron there exists a unique geodesic in each class.

球形填料II

Sphere packings II

早期的论文描述了一个程序来证明开普勒猜想的球包装。本文实现了该方案的第二步。球体填充导致$R^3$分解为多面体。多面体分为两类。第一类多面体称为准正四面体，其密度至多为正四面体的密度。其余类别中的多面体的密度最多为正八面体的密度（约0.7209）。

Congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoschemes in hyperbolic $3$-space

In \cite{Sz17-2} we considered hyperball packings in $3$-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing provides a decomposition of $\HYP$ into truncated tetrahedra. In order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyperball packings in truncated simplices. Therefore, in this paper we examine the doubly truncated Coxeter orthoscheme tilings and the corresponding congruent and non-congruent hyperball packings. We proved that related to the mentioned Coxeter tilin…

Simple closed geodesics on regular tetrahedra in spaces of constant curvature

In this survey results on the behavior of simple closed geodesics on regular tetrahedra in three-dimensional spaces of constant curvature are presented.

Simple closed geodesics on regular tetrahedra in Lobachevsky space

We obtained a complete classification of simple closed geodesics on regular tetrahedra in Lobachevsky space. Also, we evaluated the number of simple closed geodesics of length not greater than $L$ and found the asymptotic of this number as $L$ goes to infinity.

Densest plane group packings of regular polygons

Packings of regular convex polygons ($n$-gons) that are sufficiently dense have been studied extensively in the context of modeling physical and biological systems as well as discrete and computational geometry. Former results were mainly regarding densest lattice or double-lattice configurations. Here we consider all two-dimensional crystallographic symmetry groups (plane groups) by restricting the configuration space of the general packing problem of congruent copies of a compact subset of the two-dimensional Euclidean space to particular isomorphism classes of the discrete group of isometri…

Simple closed geodesics on regular tetrahedra in spherical space

On a regular tetrahedron in spherical space there exist the finite number of simple closed geodesics. For any pair of coprime integers $(p,q)$ it was found the numbers $α_1$ and $α_2$ depending on $p$, $q$ and satisfying the inequalities $π/3< α_1 < α_2 < 2π/3$ such that on a regular tetrahedron in spherical space with the faces angle $α\in \left( π/3, α_1 \right)$ there exists unique, up to the rigid motion of the tetrahedron, simple closed geodesic of type $(p,q)$, and on a regular tetrahedron with the faces angle $α\in \left( α_2, 2π/3 \right)$ there is no simple closed geodesic of type $(p…

Regular Totally Separable Sphere Packings

The topic of totally separable sphere packings is surveyed with a focus on regular constructions, uniform tilings, and contact number problems. An enumeration of all regular totally separable sphere packings in $\mathbb{R}^2$, $\mathbb{R}^3$, and $\mathbb{R}^4$ which are based on convex uniform tessellations, honeycombs, and tetracombs, respectively, is presented, as well as a construction of a family of regular totally separable sphere packings in $\mathbb{R}^d$ that is not based on a convex uniform $d$-honeycomb for $d \geq 3$.

Stability of the simplex bound for packings by equal spherical caps determined by simplicial regular polytopes

It is well known that the vertices of any Euclidean simplicial regular polytope determine an optimal packing of equal spherical balls. We prove a stability version of optimal order of this result.

Density upper bound for congruent and non-congruent hyperball packings generated by truncated regular simplex tilings

In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular tetrahedron tilings. These are derived from the Coxeter simplex tilings $\{p,3,3\}$ $(7\le p \in \mathbb{N})$ and $\{5,3,3,3,3\}$ in $3$ and $5$-dimensional hyperbolic space. We determine the densest hyperball packing arrangements related to the above tilings. We find packing densities using congruent hyperballs and determine the smallest density upper bound of non-congruent hyperball packings generated by the above tilings.

双曲$n$ -空间中全渐近正则单形的Horoball填充

Horoball packings to the totally asymptotic regular simplex in the hyperbolic $n$-space

在\cite{Sz11}中，我们推广了扩展双曲空间$\bar{\mathbf{H}}^n, ~(n \ge 2)$中的单密度函数的概念，其中我们允许{ \it不同类型的全等球}以全渐近四面体的各个顶点为中心。利用这一新的观点，本文研究了双曲型$n$ -空间$\bar{\mathbf{H}}^n$中关于全渐近正四面体的局部可解的方球填充排列及其密度，其中方球的理想中心产生全渐近正四面体的顶点.我们将证明，在这个意义上，{ \it众所周知的“全等球”填充$\bar{\mathbf{H}}^n$的Böröczky密度上界不保持有效的$n\ge4$，}但这些局部最优球安排没有扩展到整个$n$维双曲空间。此外，我们确定了上述局部最优horoball填充的密度的显式公式，允许horoball在不同的类型。

The Quadrahelix: A Nearly Perfect Loop of Tetrahedra

In 1958, S. Świerczkowski proved that there cannot be a closed loop of congruent interior-disjoint regular tetrahedra that meet face-to-face. Such closed loops do exist for the other four regular polyhedra. It has been conjectured that, for any positive ε, there is a tetrahedral loop such that its difference from a closed loop is less than ε. We prove this conjecture by presenting a very simple pattern that can generate loops of tetrahedra in a rhomboid shape having arbitrarily small gap. Moreover, computations provide explicit examples where the error is under $10^{-100}$ or $10^{-10^{6}}$. T…

Hyperball packings in hyperbolic $3$-space

In earlier works \cite{Sz06-1}, \cite{Sz06-2}, \cite{Sz13-3} and \cite{Sz13-4} we have investigated the densest packings and the least dense coverings by congruent hyperballs (hyperspheres) to the regular prism tilings in $n$-dimensional hyperbolic space $\HYN$ ($ 3 \le n \in \mathbb{N})$. In this paper we study a large class of hyperball (hypersphere) packings in $3$-dimensional hyperbolic space that can be derived from truncated simplex tilings (e.g. \cite{S14}, \cite{MPSz}). It is clear, that in order to get a density upper bound for the above hyperball packings, it is sufficient to determi…

Ball packings with high chromatic numbers from strongly regular graphs

Inspired by Bondarenko's counter-example to Borsuk's conjecture, we notice some strongly regular graphs that provide examples of ball packings whose chromatic numbers are significantly higher than the dimensions. In particular, from generalized quadrangles we obtain unit ball packings in dimension $q^3-q^2+q$ with chromatic number $q^3+1$, where $q$ is a prime power. This improves the previous lower bound for the chromatic number of ball packings.

Geodesic ball packings generated by regular prism tilings in $\mathbf{Nil}$ geometry

In this paper we study the regular prism tilings and construct ball packings by geodesic balls related to the above tilings in the projective model of $\mathbf{Nil}$ geometry. Packings are generated by action of the discrete prism groups $\mathbf{pq2_{1}}$. We prove that these groups are realized by prism tilings in $\mathbf{Nil}$ space if $(p,q)=(3,6), (4,4), (6,3)$ and determine packing density formulae for geodesic ball packings generated by the above prism groups. Moreover, studying these formulae we determine the conjectured maximal dense packing arrangements and their densities and visua…

正多胞体、球填充与阿波罗尼奥斯截面

Regular polytopes, sphere packings and Apollonian sections

本文研究了由任意维度正多胞体诱导的多胞体球填充的几何与算术性质，证明了所有整多胞体皆为晶体学的且共有11个晶体学正多胞体，引入阿波罗尼奥斯截面概念以确定相关截面填充，并计算了所有正多胞体的Möbius谱。

$\SLR$空间中无限正棱柱镶嵌的非周期测地球填充

Non-periodic geodesic ball packings to infinite regular prism tilings in $\SLR$ space

在\cite{Sz13-1}中，我们定义并描述了$\SLR$空间中的{ \it正则无限或有界} $p$-棱柱镶嵌.证明了对于整数参数$p,q;~3 \le p$，$ \frac{2p}{p-2} < q$，存在无穷多个正则无限$p$ -π面-面棱柱镶嵌$\cT^i_p(q)$和无穷多个正则有界$p$-π非面-面棱柱镶嵌$\cT_p(q)$.此外，在\cite{MSz14}和\cite{MSzV13}中，我们通过$\cT_p(q)$的指数为2的旋转子群确定了它的对称群，记为$\mathbf{pq2_1}$，并研究了相应的测地线和平移球填充.本文研究了正则无穷或有界$p$ -棱柱镶嵌的结构，证明了对于每个参数，其基图的侧曲线都是欧氏圆的弧。此外，我们还研究了由$\SLR$几何中的正无穷$p$ -正无穷面-面棱柱镶嵌$\cT^i_p(q)$导出的全等正非周期棱柱镶嵌的非周期测地球填充。我们发展了一个确定上述非周期最优测地球填充的密度的方法，并将此算法应用于它们。我们寻找上面的参数$p$和$q$，其中堆积密度尽可能大。现在，我们得到了$(p, q) = (29,3)$的最大密度$\approx 0.626606$，它大于相应的周期测地球填充在群$\mathbf{pq2_1}$下的最大密度.在我们的工作中，我们将使用由E. {Moln在\cite{M97}。

Packings of Regular Pentagons in the Plane

We show that every packing of congruent regular pentagons in the Euclidean plane has density at most $(5-\sqrt5)/3$, which is about 0.92. More specifically, this article proves the pentagonal ice-ray conjecture of Henley (1986), and Kuperberg and Kuperberg (1990), which asserts that an optimal packing of congruent regular pentagons in the plane is a double lattice, formed by aligned vertical columns of upward pointing pentagons alternating with aligned vertical columns of downward pointing pentagons. The strategy is based on estimates of the areas of Delaunay triangles. Our strategy reduces th…

n维交叉多面体的填充密度

The packing density of the $n$-dimensional cross-polytope

本文研究n维正交叉多面体的填充密度，针对n≥7的情况，采用改进的Blichfeldt方法，证明了其填充密度的上界。

A colouring problem for the dodecahedral graph

We consider vertex colourings of the dodecahedral graph with five colours, such that on each face the vertices are coloured with all the five colours. We show that the total number of these colourings is 240. All such colourings can be obtained from any given such colouring, by permuting the colours, and possibly applying central symmetry with respect to the centre of the regular dodecahedron. For any such colouring, the colour classes form the vertex sets of five regular tetrahedra. These tetrahedra together form one of the two compounds of five tetrahedra, inscribed in the regular dodecahedr…

Hyperball packings related to octahedron and cube tilings in hyperbolic space

In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular octahedron and cube tilings. These are derived from the Coxeter simplex tilings $\{p,3,4\}$ $(7\le p \in \mathbb{N})$ and $\{p,4,3\}$ $(5\le p \in \mathbb{N})$ in $3$-dimensional hyperbolic space $\mathbb{H}^3$. We determine the densest hyperball packing arrangement and its density with congruent and non-congruent hyperballs related to the above tilings in $\mathbb{H}^3$. We prove that the locally densest congruent or non-congruent hyperball configuration belongs to the regular truncate…

$\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$空间中正则棱柱镶嵌的球填充和测地球填充

Volumes and geodesic ball packings to the regular prism tilings in $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ space

在第二作者之前的工作\cite{Sz13-1}中研究了$\SLR$空间中的规则棱镜和棱镜镶嵌之后，我们考虑了与这些镶嵌及其对称群$\mathbf{pq2_1}$相关的测地球填充问题。$\SLR$是八个Thurston几何之一，可以从所有行列式为1的$2\times 2$实矩阵的三维李群导出。本文考虑$\SLR$（甚至$\mathbf{SL_{\mathrm{2}}R})$）中的测地球面和球，如果它们的半径为$ρ\in [0, \fracπ{2})$，并确定它们的体积。此外，我们考虑了上述空间的棱柱，计算了它们的体积，定义了测地球填充的概念及其密度。我们开发了一个程序，以确定密度的dennial测地线球填充的平铺，或在本文中更准确地说，为他们的生成群$\mathbf{pq2_1}$（整数旋转参数$p,q$ ; $3\le p, \frac{2p}{p-2} <q$）。我们寻找上面的参数$p$和$q$，其中堆积密度尽可能大。现在我们的记录是$0.567362$ for $(p, q) = (8, 10)$。这些计算似乎是重要的，因为我们不知道最佳球包装，即在双曲空间$\HYP$。我们只知道密度上界0.85326，通过$\HYP$的horoball填充到其理想的正则单纯形平铺来实现。令人惊讶的是，对于在\cite{MSzV13}中的相同基团$\mathbf{pq2_1}$下的所谓的平移球…

莫利四面体的两个逆猜想反例及一个新猜想

The Converse Problem for the Morley Tetrahedron: Counterexamples, Conjectures, and Partial Results

本文给出莫利四面体两个逆猜想的反例，提出新猜想并证明其在多种特殊情形下成立，但一般情形仍开放。

Maximally Dense Disc Packings on the Plane

Suppose one has a collection of disks of various sizes with disjoint interiors, a packing in the plane, and suppose the ratio of the smallest radius divided by the largest radius lies between $1$ and $q$. In his 1964 book Regular Figures (MR0165423), László Fejes Tóth found a series of packings that were his best guess for the maximum density for any $1 > q > 0.2$. Meanwhile Gerd Blind in (MR0275291, MR0377702) proved that for $1 \ge q > 0.72$, the most dense packing possible is $π/\sqrt{12}$, which is when all the disks are the same size. In Regular Figures, the upper bound of the ratio $q$ s…

New dense superball packings in three dimensions

In this paper we construct a new family of lattice packings for superballs in three dimensions (unit balls for the $l^p_3$ norm) with $p \in (1, 1.58]$. We conjecture that the family also exists for $p \in (1.58, \log_2 3 = 1.5849625\ldots]$. Like in the densest lattice packing of regular octahedra, each superball in our family of lattice packings has $14$ neighbors.

变长宽比矩形中全等圆的稠密填充

Dense Packings of Congruent Circles in Rectangles with a Variable Aspect Ratio

我们使用计算实验，以找到一个给定的数量n的非重叠全等圆可以包装成最小面积的矩形。没有对矩形的形状做出任何假设。发现的大多数填料具有通常的规则正方形或六边形图案。然而，对于测试范围内的1495个n值，n =< 5000，具体地，对于n = 49、61、79、97、107、. 4999，我们证明，最优不可能实现这样的定期安排。证据表明，当n趋于无穷大时，包含最佳六边形排列的圆的矩形的极限高宽比趋于2-sqrt（3），如果极限存在的话。

确定球面多边形排列的密度上限

Bounding the density of spherical polygon packings

通过在光滑流形上的优化确定正则球面多边形的可能最优排列，利用扩展的Lovász theta数和代数准则，结合辛群SO(3)上的调和分析，将问题转化为半正定规划可解的三角和平方问题。

New Lower Bound for the Optimal Ball Packing Density in Hyperbolic 4-space

In this paper we consider ball packings in $4$-dimensional hyperbolic space. We show that it is possible to exceed the conjectured $4$-dimensional realizable packing density upper bound due to L. Fejes Tóth (Regular Figures, 1964). We give seven examples of horoball packing configurations that yield higher densities of $0.71644896\dots$, where horoballs are centered at ideal vertices of certain Coxeter simplices, and are invariant under the actions of their respective Coxeter groups.

中心对称凸体的位似堆积

Homothetic packings of centrally symmetric convex bodies

本文研究中心对称凸体的位似堆积问题，证明随机半径位似堆积的接触图具有(2,2)-稀疏性，并证明存在余贫集使得任意(2,2)-稀疏平面图可实现为无应力位似堆积的接触图。

Five-neighbour packings of centrally symmetric convex discs

In an old paper of the author the thinnest five-neighbour packing of translates of a convex disc (different from a parallelogram) was determined. The minimal density was $3/7$, and was attained for a certain packing of triangles. In that paper it was announced that for centrally symmetric convex plates (different from a parallelogram) the analogous minimal density was $9/14$, and was attained for a certain packing of affine regular hexagons, and a very sketchy idea of the proof was given. In this paper we give details of this proof.

Horoball packings related to hyperbolic $24$ cell

In this paper we study the horoball packings related to the hyperbolic 24 cell in the extended hyperbolic space $\overline{\mathbf{H}}^4$ where we allow {\it horoballs in different types} centered at the various vertices of the 24 cell. We determine, introducing the notion of the generalized polyhedral density function, the locally densest horoball packing arrangement and its density with respect to the above regular tiling. The maximal density is $\approx 0.71645$ which is equal to the known greatest ball packing density in hyperbolic 4-space given in \cite{KSz14}.

Computing upper bounds for the packing density of congruent copies of a convex body

In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in $\mathbb{R}^n$; this theorem is a generalization of the linear programming bound for sphere packings. We illustrate its use by computing an upper bound for the maximum density of packings of regular pentagons in the plane. Our computational approach is numerical and uses a combination of semidefinite programming, sums of squares, and the harmonic analysis of the Euclidean motion group. We show how, with some extra work, the bounds so obtained can be made rigor…