From: verdy_p (email@example.com)
Date: Sun Jan 10 2010 - 15:11:37 CST
There's an additional condition: the vertical height of the center of gravity, MUST be equal independantly of the
face on which the dice finishes its course. Otherwise the dice will not jump down and up equally, and the faces that
is farther from the center of gravity will have lower chance to be selected as the base laying on the table (because
there is higher chance that the dice continues to roll.
The problem is how the cinetic energy (when the center of gravity is lower) can be converted into potential energy
(when the center of gravity is higher).
There's another condition and it is related to the distibution of the next selected faces when the dice continues
rolling from one face to the next ones: the cinetic energy is oriented by the moment vector of rotation and by the
horizontal speed vector: there must be equal distribution in all directions from one face to the others. This
implies the required symetry of polyedrons (even if you can make it more balanced by using circular faces.
But you're right, we are not required to cut the circular faces on a perfect sphere: it could be an ellipsoid
(ovoidal), but then the faces will not be perfectly circular (except those centered on the line of foci). Once again
the minimum cinetic energy to convert to potential energy to go from one face to any other one, will not be balanced
: it will be smaller for going from a face which is not on the focus line, to the border of a face nearer the focus
line of the ellipsoid.
The example of the 7-face dice with two opposed pentagonal face does nto use in fact 5 square faces, the dice is
clearly not cut within a vircle but within an ellipsoid (the line of foci is the axe that goes throught the centers
of the two pentagones). But, it's easy to test that the two pentagonal faces have higher chances of being selected
when the dice stops rolling, even of the distances from each face to the center of gravity are equal.
> Message du 10/01/10 21:43
> De : "Marcin 'Qrczak' Kowalczyk"
> A : "verdy_p"
> Copie à : "William J Poser" , firstname.lastname@example.org
> Objet : Re: Seven-sided die (was Re: This just in)
> 2010/1/10 verdy_p :
> > Hmmm.. due to the spacial distribution of weight, the center of gravity will be WITHIN the half-sphere,
> Not necessarily if the pyramid is long enough, and/or the half sphere
> is actually less than half. Then resting on the top of the half sphere
> would be an unstable equilibrium.
> Instead of half sphere we can also have 7 faces with wider angles,
> such that the dice laying on one of them has the center of gravity
> above a point lying outside the face. Then the only stable resting
> positions are on the remaining 7 faces.
> Marcin Kowalczyk
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