Maximum Volume of Unit Radius Sphere in Fractional Dimensions

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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


Historical Note

This result has been attributed to David Breyer Singmaster.


Sources