Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.13978

An Improved Lower Bound for Packing 203 Equal Spheres in a Unit Cube

Peng Lu
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper reports an improved numerical lower bound for packing 203 equal spheres in a unit cube. Let r denote the maximum common radius of equal non-overlapping spheres that can be packed in a unit cube. For N = 203, the configuration currently listed in Packomania has a feasible radius of approximately r = 0.088145402159. Starting from this public configuration, we perform further numerical optimization and obtain a new feasible configuration with r > 0.088145634693. The final configuration is checked by an independent verification program using 100-digit arithmetic. All 20503 sphere-pair constrains and all 1218 sphere-wall constraints are satisfied. The coordinates of the new configuration and the related programs are provided as supplementary material.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Peng Lu. 2026-09-22. An Improved Lower Bound for Packing 203 Equal Spheres in a Unit Cube. https://arxiv.org/abs/2610.13978

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Many Antipodal Pairs Force Many Neighboring Pairs

Let $X=\{x_1,\dots,x_n\}\subset \mathbb{R}^2$ be a finite set of points of diameter at most $1$. It is natural to expect that if many pairs $(x_i,x_j)$ lie at distance close to $1$ from each other, then some clustering phenomenon must occur, implying that a significant number of these pairs are also very close to each other. %For $0<\varepsilon<1$, we call a pair $(x_i,x_j)$ $\varepsilon$-antipodal if $\|x_i-x_j\|\ge 1-\varepsilon$ and $\varepsilon$-neighboring if $\|x_i-x_j\|\le \varepsilon$. We prove that there exists a universal constant $c>0$ such that for all $0<\varepsilon<1$, whenever $n$ is large enough, we have: \[ \big|\{(i,j):\|x_i-x_j\|\le \varepsilon\}\big| \geq c\cdot \varepsilon^{1/2}\cdot \big|\{(i,j):\|x_i-x_j\|\geq 1-\varepsilon\}\big|. \] This confirms a recent conjecture of Steinerberger, who asked whether the $\varepsilon^{1/2}$ ratio is the best possible. We also study a two-parameter version of Steinerberger's question by considering the number of pairs at distance at most $\varepsilon_1$ and at distance at least $1-\varepsilon_2$. We show that in this case the optimal ratio is $\varepsilon_1^2\cdot\varepsilon_2^{-3/2}$ when $\varepsilon_1\leq\varepsilon_2$, and $\varepsilon_1^{1/2}$ otherwise. The proof proceeds by introducing an auxiliary graph associated with the set $X$ and reducing the problem to bounding the largest eigenvalue of its adjacency matrix. Our main result is the outcome of human--AI interactions using ChatGPT 5.4.

math.MG↗
arXiv2026

Monge's transport problem in synthetic Lorentzian spacetimes

We prove the existence of solutions to the Lorentzian Monge problem on synthetic Lorentzian spacetimes that satisfy the forward timelike measure contraction property $TMCP^+(K,N)$. The cost for this problem is the time separation function $\ell(x,y)$ (the Lorentz distance), which represents the maximum amount a particle can age when traveling from $x$ to $y$. The solution is based on the needle decomposition technique, and amounts to a reduction of the full problem to its one-dimensional counterparts. As an application of needle decomposition, we introduce the blended $TMCP$ condition, and obtain a quantitative refinement of the synthetic Hawking singularity theorem. We additionally prove a timelike nonbranching reversibility theorem for $TCD_p(K,N)$ spacetimes, showing that forward $p$-essentially nonbranching spacetimes are backward $p$-essentially nonbranching.

math.MG↗
arXiv2026

Metric foundations of geometry

A metric space is called all-set-homogeneous if every isometry between two of its subsets extends to an isometry of the whole space. We classify all-set-homogeneous length spaces: besides the classical examples, they include the universal metric trees of finite valence.

math.MG↗