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  "body": "\n## Definition\n\nLet $n \\ge 1$.\nA *kissing configuration* in the [[Euclidean space|0XPR1E85W0]] $\\R^n$ is a finite set $X \\subset \\R^n$ such that:\n\n1. $\\abs{x} = 2$ for every $x \\in X$;\n2. $\\abs{x - y} \\ge 2$ for all distinct $x, y \\in X$, where $\\abs{\\cdot}$ is the [[Euclidean distance|0NPPGHQNXQ]].\n\nThe *kissing number* $\\tau(n)$ is the largest size of a kissing configuration in $\\R^n$.\n\n## Remarks\n\nThe conditions say that the [[closed balls|0Q0H13G8ZB]] of radius $1$ centred at the points of $X$ touch the unit ball at $0$ and have pairwise disjoint interiors.\nExample: in $\\R^2$, the six points $2(\\cos(k\\pi/3), \\sin(k\\pi/3))$ for $k = 0, \\dots, 5$ have norm $2$, and neighbours are at distance exactly $2$, so they form a kissing configuration; hence $\\tau(2) \\ge 6$.\nNon-example: seven points of norm $2$ in $\\R^2$ cannot form one.\nTwo of them make an angle less than $60^\\circ$ at $0$, and are then at distance less than $2$.\nEquivalently, $\\tau(n)$ is the largest number of points on the unit sphere $S^{n-1}$ with pairwise angles at least $60^\\circ$.\nThe *lattice kissing number* is the largest size of such a configuration when $X$ is the set of shortest nonzero vectors of a lattice, scaled to length $2$.\n",
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