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### Unit Spheres in a Pile

Posted: **Mon May 15, 2017 2:57 pm UTC**

by **jewish_scientist**

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**

by **doogly**

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**

by **jewish_scientist**

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**

by **doogly**

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**

by **LucasBrown**

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**

by **doogly**

Oh right, I should have qualified that better.