# Ratios of Sizes of Mutually Inscribed Multidimensional Cubes and Spheres

Jump to navigation
Jump to search

## 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

This theorem requires a proof.In particular: Formulae for the volumes of $n$ dimensional squares and circles need to be established first.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1964: David Singmaster:
*On Round Pegs in Square Holes and Square Pegs in Round Holes*(*Math. Mag.***Vol. 37**: pp. 335 – 337) www.jstor.org/stable/2689251

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $9$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $9$