Inside an icosahedron

Geometry Level pending

Points A and B are chosen uniformly at random inside two regular icosahedra of edge lengths 2 and 20 respectively, both centered at the origin. (i.e. Point A is in the smaller one and B in the larger one.)

What is the probability that the distance between A and B is less than 5?

Give your answer to 3 decimal places.

Useful formula : V i c o s a h e d r o n = 5 ( 3 + 5 ) 12 a 3 V_{icosahedron} = \frac{5(3+\sqrt5)}{12} a^3 where a = a = Edge length


The answer is 0.03.

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Jan 11, 2017

Consider any sphere of radius 5 centered at point A.

Since, no matter what point you choose inside the smaller icosahedron, this entire sphere will be contained in the larger icosahedron, the probability will be given by the ratio of the volume of this sphere to the volume of the icosahedron. ( Note : The fact that the smaller one is shaped like an icosahedra is immaterial as long as the sphere above is guaranteed to be inside the larger icosahedron)

  • V s p h e r e = 4 3 π 5 3 V_{sphere} = \frac{4}{3}\pi 5^3
  • V i c o s a h e d r o n = 5 ( 3 + 5 ) 12 2 0 3 V_{icosahedron} = \frac{5(3+\sqrt5)}{12} 20^3

So, the probability the distance between the two points is less than 5 is given by:

P = V s p h e r e V i c o s a h e d r o n = 0.03 P = \frac{V_{sphere}}{V_{icosahedron}} = \boxed{0.03}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Tricky, tricky, tricky! The problem initially looks a lot worse than it really is.

Michael Mendrin - 4 years, 4 months ago

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Hahaha... Yup! :0)

Geoff Pilling - 4 years, 4 months ago

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