Density of the optimal sphere packing
When Kepler tried to answer the technical question of how to pack cannonballs in a ship he thought that the traditional way to pack oranges in a grocers stall was optimal.
That was known as Kepler Conjecture. It was very difficult to prove. At the end of the 20th century, Thomas Hales found a prove with the help of computers.
What is not so difficult to calculate is the density of the optimal packing of spheres. In fact, it will be very easy if we use some basic properties of the Rhombic Dodecahedron.
To calculate this density of packing we usually consider a 'unit cell', a polyhedron that tessellate space. Typically we use prims, in particular, cubes.
Inside this unit cell there will be some spheres or sphere parts. We must calculate the ratio between the volume of the part occupied by spheres and the volume of our unit cell.
Sometimes it is not easy to visualize which sphere parts are inside the unit cell.
With our knowledge about the Rhombic Dodecahedron this task could not be much easier.
Firstly, we can consider a Rhombic Dodecahedron as our unit cell. And inside this well known polyhedron there is only one ball.
Secondly, it is easy to calculate the volume of the unit cell and the volume of the sphere.
To show this situation I played with a semi transparent polyhedron and I put one red ball inside:
To calculate the volume of our Rhombic Dodecahedron we only needd to know that its volume is double the volume of the inscribed cube.
We can suppose that the inscribed cube has edge length equal 2. Then:
The volume of our unit cell is double the volume of the cube. We call this volume D.
Very easily we have found the volume of the unit cell.
Now, the volume of the sphere. Playing with a model we can realize that this inscribed sphere touches the edges of the inscribed cube, it is the midsphere of the cube.
We can calculate the radius of this sphere:
We call S the volume of the sphere :
Then, the density of the optimal sphere packing is:
Very easy calculation.
Johannes Kepler, 'The Six Cornered Snowflake: a New Year's gif' - Paul Dry Books, Philadelphia, Pennsylvania, 2010. English translation of Kepler's book 'De Nive Sexangula'. With notes by Owen Gingerich and Guillermo Bleichmar and illustrations by the spanish mathematician Capi Corrales Rodrigáņez.
D'Arcy Thompson, 'On Growth And Form' - Cambridge University Press, 1942.
Hugo Steinhaus, 'Mathematical Snapshots' - Oxford University Press - Third Edition.
Magnus Wenninger, 'Polyhedron Models', Cambridge University Press.
Peter R. Cromwell, 'Polyhedra', Cambridge University Press, 1999.
H.Martin Cundy and A.P. Rollet, 'Mathematical Models', Oxford University Press, Second Edition, 1961.
W.W. Rouse Ball and H.S.M. Coxeter, 'Matematical Recreations & Essays', The MacMillan Company, 1947.