Maximum Volume of Unit Radius Sphere in Fractional Dimensions

Theorem

The volume of a unit sphere in $x$-dimensional Euclidean space for real $x$ occurs when $x$ is given as:

$x = 5 \cdotp 25694 \, 64048 \, 60 \ldots$

This sequence is A074455 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The corresponding volume at that dimension is given by:

$V = 5 \cdotp 27776 \, 80211 \, 13400 \, 997 \ldots$

This sequence is A074454 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

This theorem requires a proof. In particular: Requires significant work to even define the concepts. 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.

Historical Note

This result has been attributed to David Breyer Singmaster.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5 \cdotp 256 \, 946 \, 404 \, 860 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5 \cdotp 25694 \, 64048 \, 60 \ldots$