Number of Multidimensional Spheres that can touch One Other Sphere

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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Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $12$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12$