Tasha went to Abby's lab by public transit and gave Abby the $10 she had successfully bummed off a bum whose bum was filled with a substance she could not, as an professed asexual, positively identify within a 95% degree of certainty, a statistical threshold that while arbitrary was yet suggestive of scientific authority and officiousness. Abby accepted the $10. As to whether or not Abby said 'thank you' or Tasha replied 'you're welcome' is anyone's guess, and in any case not relevant to the overarching narrative of time traveling singers, axolotls, Garnabby, legal procedural side plots, and homosexual encounters.
As a researcher in a well-funded institute of theoretical physics whose principal investigators all had massive multi-million dollar grants, Abby decided to put the $10 toward the purchase of a box or two of frozen calzones, her secret guilty pleasure. Most of the pleasure was in the aftermath: hot Italian food farts, moist and sulfurous to a redolent degree.
Of all the equations scribbled across five blackboards, one in particular had Abby in a state of consternation. Firstly, Abby was aware of the elementary fact that given a 1-dimensional interval of length 1, the average distance between two points chosen at random was 1/3. Secondly, she was aware of the fact that given a 2-dimensional square of side length 1, the average distance between two points chosen at random was (5*Ln(Sqrt(2) + 1) + Sqrt(2) + 2)/15, an irrational number of flatulent complexity. WolframAlpha said it was equal to 0.521405433164720678330982356607243974914031567779 00834179621051875050789330481583186792813292526145 24679141951460963171979156799..., which nearly caused Abby to orgasm.
And what's more, Abby was also aware that given a 3-dimensional cube of edge length 1, the average distance between two points chosen at random was (4 + 17*Sqrt(2) - 6*Sqrt(3) - 7pi + 21*Ln(4*Sqrt(6) + 4*Sqrt(3) + 7*Sqrt(2) + 7))/105. WolframAlpha faithfully rendered this as 0.661707182267176235155831133248413581746400135790 95360480894422947958464613859763130665248076810712 01517097753107594109724786805..., at which Abby moaned in pleasure. Tasha, watching all this from behind a curtain, was quite jealous.
But what of the average distance between two points chosen at random from inside a 4-dimensional hypercube of edge length 1??? The octuple integral she had set up to represent this value was taking up much space on the blackboard and wasn't going to solve itself, but Abby maxed out at quintuple integrals. She had, after all, only gone to community college for half a semester. Abby did not want to run a billion Monte Carlo simulations to approximate its value, she needed the exact answer, replete with its odious and foul logarithms, square roots, pi, and other such putridly flatulent irrational constants.
Oh well. Nothing to do about it but heat up a calzone, she mused.