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### Unit Spheres in a Pile

Posted: Mon May 15, 2017 2:57 pm UTC
There are an unknown number of unit spheres in a pile shaped like a right circular cone. All of the dimensions of the cone are known and the angle of the cone's base is equal to the sphere's angle of repose. How many spheres are in the pile?

### Re: Unit Spheres in a Pile

Posted: Mon May 15, 2017 3:05 pm UTC
The angle of repose is a property of the material, it's not enough to just say that they are spheres.

The packing density is variable. See
https://en.wikipedia.org/wiki/Sphere_packing

### Re: Unit Spheres in a Pile

Posted: Mon May 15, 2017 3:37 pm UTC
I did not know the term 'sphere packing'. When I googled it, I found this Wolfram page. It gives the densities for several different arrangements. Thank you for the help.

### Re: Unit Spheres in a Pile

Posted: Mon May 15, 2017 3:39 pm UTC
Worth noting that proving the "obvious" densest packing was actually so is a Gauss achievement. He is the best.

### Re: Unit Spheres in a Pile

Posted: Thu May 18, 2017 8:48 am UTC
Gauss only proved that for lattices. The full theorem was only resolved in 1998.

### Re: Unit Spheres in a Pile

Posted: Thu May 18, 2017 12:27 pm UTC
Oh right, I should have qualified that better.