Number of Multidimensional Spheres that can touch One Other Sphere

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Theorem

Let $S_n$ be a spheres of $n$ dimensions with a given radius $r$.

Let $\map T n$ denote the number of instances of $S_n$ that can touch one other such instance of $S_n$.


The sequence of $\map T n$ begins as follows:

$n$ $\map T n$
$0$ $0$
$1$ $2$
$2$ $6$
$3$ $12$
$4$ $24$
$5$ $40$
$6$ $72$
$7$ $126$
$8$ $240$
$9$ $272$

This sequence is A001116 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof



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