Ratios of Sizes of Mutually Inscribed Multidimensional Cubes and Spheres

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Theorem

Consider:

a cube $C_n$ of $n$ dimensions inscribed within a sphere $S_n$ of $n$ dimensions
a sphere $S'_n$ of $n$ dimensions inscribed within a cube $C'_n$ of $n$ dimensions.


Let:

$A_{cn}$ be the $n$ dimensional volume of $C_n$
$A_{sn}$ be the $n$ dimensional volume of $S_n$
$A'_{cn}$ be the $n$ dimensional volume of $C'_n$
$A'_{sn}$ be the $n$ dimensional volume of $S'_n$.


For $n < 9$:

$\dfrac {S_n} {C_n} > \dfrac {C'_n} {S'_n}$

but for $n \ge 9$:

$\dfrac {S_n} {C_n} < \dfrac {C'_n} {S'_n}$


That is, for dimension $n$ less than $9$, the $n$ dimensional round peg fits better into an $n$ dimensional square hole than an $n$ dimensional square peg fits into an $n$ dimensional round hole, but for $9$ and higher dimensions, the situation is reversed.


Proof


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Sources

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