Robbins constant

The Robbins constant , named after David P. Robbins , is a geometric constant that specifies the expected Euclidean distance between two points of the three-dimensional unit cube , which are drawn randomly , independently and uniformly . The constant has the following value: ${\displaystyle [0,1]^{3}\subset \mathbb {R} ^{3}}$

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

Your decimal expansion begins with (sequence A073012 in OEIS ):

${\displaystyle 0{,}66170718226717623515583\ldots }$

Explanation

The gray area consists of the points with , whose area is. ${\displaystyle (x_{1},x_{2})}$ ${\displaystyle (x_{1}-x_{2})^{2}\leq t}$ ${\displaystyle F(t)}$

It should be briefly indicated why such a complicated expression is used here. Ultimately, it's about the integral

${\displaystyle \int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}}}\mathrm {d} x_{1}\mathrm {d} x_{2}\mathrm {d} x_{3}\mathrm {d} y_{1}\mathrm {d} y_{2}\mathrm {d} y_{3}}$,

the calculation of which can be carried out using probabilistic approaches as follows. If and are the randomly drawn points, the probability distribution of must be determined to determine the expected distance sought . The probability distribution has the form , as can be seen from the adjacent sketch. The associated probability density is obtained by deriving: . The probability density of the sum is then the convolution , which results in complicated integrals. The associated probability distribution describes the sum of the squares of the coordinate differences, but we need the square root of this sum. The probability distribution sought is therefore with the associated density . After all, the integral is the expected value we are looking for. The complex calculations are carried out in the work given below with Maple support, whereby the even more complicated case of a cuboid is treated instead of the unit cube. ${\displaystyle (X_{1},Y_{1},Z_{1})}$ ${\displaystyle (X_{2},Y_{2},Z_{2})}$ ${\displaystyle {\sqrt {(X_{1}-X_{2})^{2}+(Y_{1}-Y_{2})^{2}+(Z_{1}-Z_{2})^{2}}}}$ ${\displaystyle F(t):=P(\{(X_{1}-X_{2})^{2}\leq t\})}$ ${\displaystyle F(t)=1-(1-{\sqrt {t}})^{2},\quad t\in [0,1]}$ ${\displaystyle f(t)={\tfrac {\mathrm {d} F(t)}{\mathrm {d} t}}={\tfrac {1}{\sqrt {t}}}-1}$ ${\displaystyle (X_{1}-X_{2})^{2}+(Y_{1}-Y_{2})^{2}+(Z_{1}-Z_{2})^{2}}$ ${\displaystyle f*f*f}$ ${\displaystyle G(t)=\int _{0}^{t}(f*f*f)(s)\mathrm {d} s}$ ${\displaystyle H(t)=G(t^{2})}$ ${\displaystyle h(t):={\tfrac {\mathrm {d} G(t^{2})}{\mathrm {d} t}}=2t\cdot (f*f*f)(t^{2})}$ ${\displaystyle \int th(t)\mathrm {d} t}$

Individual evidence

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
2. ^ Simon Plouffe : The Robbins Constant. Miscellaneous Mathematical Constants.
3. ^ Johan Philip: The probability distribution of the distance between two random points in a box