# Maximum Volume of Unit Radius Sphere in Fractional Dimensions

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## Contents

## 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

- 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): $5 \cdotp 25694 \, 64048 \, 60 \ldots$