Circle Packing Surfaces – 2D Calculations
In geometry, circle packing is an arrangement of non-overlapping circles within a containing space and would apply to any balls sitting on a flat surface such as tanks, pools and reservoirs. The associated packing density is the proportion of the surface covered by the circles.
In two-dimensional space, Joseph Louis Lagrange proved in 1773 that the highest-density lattice arrangement of circles is the hexagonal packing arrangement in which the centres of the circles are arranged in a hexagonal lattice (staggered rows like a honeycomb), and each circle is surrounded by 6 other circles. The density of this arrangement is
Therefore, by using this density percentage of 90.7%, it is relatively easy to divide the area of a circle into the area of a surface you want to cover. Then, apply the packing density to get a very accurate calculation of how many balls you will need. Note this does not allow for the part balls around the edges or disruption of the hexagonal packing but larger areas contribute to far less than a 1% error.
Sphere Packing Volumes – 3D Calculations
Sphere packing is an arrangement of non-overlapping spheres within a containing space. In three-dimensional space, there are three packing types for identical spheres: cubic lattice, face-centered cubic lattice, and hexagonal lattice. Carl Friedrich Gauss proved in 1831 that Hexagonal packing is the densest possible among all possible lattice packings using the following formula:
Whilst the best possible packing density is about 74%, the theoretical worst is about 60% before you stop filling the space or start ignoring gravity, and randomly-poured sphere packing efficiency is around 64% which is traditionally used for playpen balls.
As per the 2D Circle Packing calculation above, it is relatively easy to divide the volume of a sphere into the volume you want to fill multiplied by the efficiency of 64% and get a very accurate idea of how many balls you will need.
ROSPA recommend that ball pools should have a maximum depth of 450mm in a toddler area and 600mm in a junior area to minimise the danger of accidents from concealment. Moreover, ball pools should not be entered directly from a slide. Also, balls should be a minimum diameter of 70mm to prevent choking and ball pool surfaces should have continuous level bases.