Raul -
I opened this as a "parallel thread" to not interfere (mess) with
your original one ...
Using the 'brute force' approach you mentioned (took me a while, but
I'm now with you, after I looked up 'Under" (&.:) again)
Random Generator arguments:
[1] pair of points, two points: 2
[2] no of coordinates, dimension: 2, 3, 4, ...
[3] consider myriad point pairs, like: 1e6
[4] stay within unit square, cube, tesseract, etc: $ 0
So, moving up through dimensions, only that value [2nd parameter]
needs to change ...
* Unit Square (average distance)
(+/ % #) +/ &.: *: -/ ? 2 2 1e6 $ 0
0.521956
* Unit Cube
(+/ % #) +/ &.: *: -/ ? 2 3 1e6 $ 0
0.661962
* Unit Tesseract
(+/ % #) +/ &.: *: -/ ? 2 4 1e6 $ 0
0.777293
* Unit body (5th dimension) what's its name
(+/ % #) +/ &.: *: -/ ? 2 5 1e6 $ 0
0.878806
I'm definitely not an expert in probability, so I'm not sure whether
this makes any sense:
* Unit Line
(+/ % #) +/ &.: *: -/ ? 2 1 1e6 $ 0
0.333411
What I do know from experience is, that gut feeling and probability
don't go together (most of the time).
That said, I did some "plausibility checks" comparing avarage
distance with the unit "body" diagonal:
*Unit Line
0.33 % %: 1
0.33
* Unit Square
0.52 % %: 2
0.367696
* Unit Cube
0.66 % %: 3
0.381051
* Unit Tesseract
0.78 % %: 4
0.39
That looks somehow reasonable to me, means the equation probably
degrades gracefully in the case Dimension 1 (Line).
Furthermore, when going from Unit Square to Unit Cube, then from Unit
Cube to Unit Tesseract, I'd expect the average to increase according
to the increase of proportion of diagonals, i.e.
* Square -> Cube
0.52 * %: 3r2
0.636867
* Cube -> Tesseract
0.66 * %: 4r3
0.762102
which seems to approximately check with the distance average for
three, resp four dimensions.
Just a thought ...
-M
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