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Topic: Tetrahedron Formation.
Replies: 1   Last Post: Dec 21, 2012 4:25 PM

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Avni Pllana

Posts: 546
Registered: 12/6/04
Re: Tetrahedron Formation.
Posted: Dec 21, 2012 4:25 PM
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> If I break a stick into 6 pieces, what is the
> probability that they
> can form a tetrahedron?
> In general, if I break a stick into (n*n+n)/2 pieces,
> what is the
> probability that they can form an n-simplex?
> I know that the longest piece must be no more than
> 0.5 of the lengh of
> the original unbroken stick.
> Thanks in advance.

Hi Patrick,

let the stick be of unit length. We generate five random numbers uniformly distributed between 0 and 1, and after we sort them in ascending order we have some r1, r2, r3, r4, r5. Let a=r1, b=r2-r1, c=r3-r2, d=r4-r3, e=r5-r4, f=1-r5 be the edges of a possible tetrahedron. The edges should satisfy the triangle inequalities for the four faces of tetrahedron. A program based on this reasoning for 1000000 trials yields an approximate probability p = 0.0185 .

Best regards

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