Inscribed sphere - Kepler Conjecture

[Berea81]

Newbie to the forum here. Hoping y'all can help with something that's been bugging me for a while now.

I would like to know the relationship between two characteristic radii in a close packing of equal spheres. The first radius of interest is that of the equal sphere's themselves (r1). The second radius (r2) is that of the largest inscribed sphere which would fit inside the void space created between the equal spheres of radius r1. Or as a friend put it what's the biggest (spherical) grape you could fit inside a pyramid of oranges without disturbing the pyramid.

I'm also interested in the smallest 'grape' (r3) that would fit within the close packing but be in contact with three different 'oranges'.

Ideas?

 

[tiny-tim]

Welcome to PF!

Hi Berea81! Welcome to PF!*

(try using the X₂ button just above the Reply box )

… what's the biggest (spherical) grape you could fit inside a pyramid of oranges without disturbing the pyramid.

I'm also interested in the smallest 'grape' (r3) that would fit within the close packing but be in contact with three different 'oranges'.

I think the best approach would be to draw the lines joining the centre of each small sphere to the centre of each large sphere that it touches.

So each line would have length r₁ + r₂, and if you know the layout of the large spheres, it should be easy to find that length.