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- W2999644219 abstract "We define two notions of generation between the various optimal packings ${cal Q}_m^K$ of $m$ congruent disks in a subset $K$ of ${Bbb R}^2$. The first one that we call weak generation consists in getting ${cal Q}_n^K$ by removing $m-n$ disks from ${cal Q}_m^K$ and by displacing the $n $ remaining congruent disks which grow continuously and do not overlap. During a weak generation of ${cal Q}_n^K$ from ${cal Q}_m^K$, we consider the contact graphs ${cal G}(t)$ of the intermediate packings, they represent the contacts disk-disk and disk-boundary. If for each $t$, the contact graph ${cal G}(t)$ is isomorphic to the largest common subgraph of the two contact graphs of ${cal Q}_n^K$ and ${cal Q}_m^K$, we say that the generation is strong. We call strong generator in $K$, an optimal packing ${cal Q}_m^K$ which generates strongly all the optimal ${cal Q}_k^K$ with $k < m$. We conjecture that if $K$ is compact and convex, there exists an infinite sequence of strong generators in $K$. When $K$ is an equilateral triangle, this conjecture seems to be verified by the sequence of hexagonal packings ${cal Q}_{Delta (k)}^K$ of $Delta (k)=k(k+1)/2$ disks. In this domain, we also report that up to $n=34$, the Danzer graph of ${cal Q}_n^K$ is embedded in the Danzer graph of ${cal Q}_{Delta (k)}^K$ with $Delta (k-1)leq n < Delta (k)$. When $K$ is a circle, the first five strong generators appears to be the hexagonal packings defined by Graham and Lubachevsky. When $K$ is a square, we think that our conjecture is verified by a series of packings proposed by Nurmela and al. In the same domain, we give an alternative conjecture by considering another packing pattern." @default.
- W2999644219 created "2020-01-23" @default.
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- W2999644219 date "2008-02-20" @default.
- W2999644219 modified "2023-09-25" @default.
- W2999644219 title "Generation of Optimal Packings from Optimal Packings" @default.
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- W2999644219 doi "https://doi.org/10.37236/113" @default.
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