Definition:Robbins Constant

Definition

The Robbins constant $R$ is defined as the mean distance $D$ between $2$ points chosen at random from the interior of a unit cube:

\(\ds R\)

\(=\)

\(\ds \frac {4 + 17 \sqrt 2 - 6 \sqrt3 - 7 \pi} {105} + \frac {\map \ln {1 + \sqrt 2 } } 5 + \frac {2 \, \map \ln {2 + \sqrt 3} } 5\)

\(\ds \)

\(\approx\)

\(\ds 0 \cdotp 66170 \, 71822 \, 67176 \, 23515 \, 582 \ldots\)

This sequence is A073012 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Also see

• Mean Distance between Two Random Points in Unit Cube

Source of Name

This entry was named for David Peter Robbins.

Historical Note

The Robbins constant arises from a question posed by David P. Robbins in $1977$ in the American Mathematical Monthly in the context of a general cuboid of arbitrary dimensions.

The published solution, submitted by Theodore S. Bolis applied the result to a general unit cube and presented the result in a format similar to that given here.

Sources

• 1978: David P. Robbins and Theodore S. Bolis: E2629: Average distance between two points in a box (Amer. Math. Monthly Vol. 85: pp. 277 – 278)  www.jstor.org/stable/2321177