Robbins constant

In geometry, the Robbins' constant, named after David P. Robbins, is the average distance between two points selected at random within a unit cube. Robbins' constant can be expressed as ^[1]

${\displaystyle {\frac {4+17{\sqrt {2}}-6{\sqrt {3}}-7\pi }{105}}+{\frac {\ln(1+{\sqrt {2}})}{5}}+{\frac {2\ln(2+{\sqrt {3}})}{5}}.}$

Its numerical value is approximately ^[2]

${\displaystyle 0.66170718226717623515583.}$

References

1. Robbins, David P.; Bolis, Theodore S. (1978), "Average distance between two points in a box (solution to elementary problem E2629)", American Mathematical Monthly, 85 (4): 277–278, doi:10.2307/2321177.
2. Plouffe, Simon, "The Robbins constant", Miscellaneous Mathematical Constants