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I am studying the Kissing Number problem. We know the optimal kissing number for dimensions 1, 2, 3, 4, 8 and 24. For all of these, the optimal configurations stem from highly symmetrical lattices. The best known arrangements for dimensions 5, 6 and 7 also stem from lattices.

I am wondering, what are the best known non-lattice configurations for each dimension?

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    $\begingroup$ In dimension 3 at least it is possible to get the same center density and kissing number with a non-lattice construction. The idea is described in Conway & Sloane. You get the lattice by repeating a chosen 2D-layer with constant displacement from one layer to the next. But you get the same amount of kissing by using non-constant displacement (hence non-lattice). $\endgroup$ May 6, 2021 at 2:38
  • $\begingroup$ Thank you @JyrkiLahtonen! Do you know if the same can be done in higher dimensions? I suppose one could always start with a lattice and then shift some of the spheres by a small amount. Although not sure if there is any room to do that in the $E_8$ lattice? $\endgroup$ May 6, 2021 at 2:42
  • $\begingroup$ By the way, are you interested in configurations around a single sphere, or do you still want a pattern that can continue in the entire space? $\endgroup$ May 6, 2021 at 2:43
  • $\begingroup$ I just want a configuration around a single central sphere. $\endgroup$ May 6, 2021 at 2:44
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    $\begingroup$ My guess is that $E_8$ and Leech are so unique that there is no wiggle room, but I may be wrong about this. $\endgroup$ May 6, 2021 at 2:44

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