Suppose you are asked to create a larger sphere using smaller spheres. You can create only something that can be just contained inside a large sphere. If the smaller spheres are closely packed and stacked as layers above and below a central layer, such that it has a plane of symmetry, what is the minimum number of smaller spheres required? Has 'pi' any role in it?
If the diameter of the central layer is 7 spheres (22/7 represents the whole number fraction for 'pi' correct to 2 digits), will that represent the above structure having minimum number of spheres?
The minimum number of spheres required to create a larger sphere with perfect symmetry is twelve. This is known as the "Close Packing of Spheres" problem, and it has been mathematically proven that a three-dimensional arrangement of equal spheres can only achieve perfect symmetry with 12 spheres arranged around a central sphere. (see https://www.wikiwand.com/en/Circle_packing)
This arrangement is known as a face-centered cubic (FCC) lattice, and it is the most efficient way to pack spheres in three dimensions, as it provides the highest packing density.
The number "pi" ($\pi$) does play a role in the calculation of the sphere's volume, as the formula for the volume of a sphere is $V = \frac{4}{3} \pi r^3$, where $r$ is the radius of the sphere. However, it does not play a direct role in the calculation of the number of spheres required for close packing, which is determined purely by geometrical considerations.