So what other spheres or near spheres can be made? A good answer from those with a little geometry are the regular polyhedra. These are the 3d shapes where every face, vertex and edge looks exactly the same as every other.These shapes provide the climax of Euclid’s Elements, and the proof that they are the only ones is up there with Hamlet, the ceiling of the Sistine Chapel or the Taj Mahal as one of the greatest human achievements, but you can make this one yourself from paper:There is a reasonable argument that each of these must be close to a ball due to the high symmetry. This is a little stretched for the tetrahedron, but we will give it a pass. Can we do anything else with these pieces? To make a sphere we want to have the same amount of curvature everywhere. This does not require that every shape created have the same number of sides and the same angles at the corners, but that seems to be a useful place to start. Then at least every face has the same total curvature. In addition as all the angles on an individual curvahedra piece are the same, the angles around a corner should be the same. These conditions lead us to the Catalan solids, or Archimedean duals, the shapes where the model looks exactly the same from every face. Thus every face in a Catalan solid is the same. Here is one made with Curvahedra:
but that is not a sphere, what went wrong?
Looking at an individual pieces we see that the problem is, the curvature is not spread equally over the shape, one end is more bent.
The same polyhedron with flat sides shows where this might come from. The edges of the shape are not all the same length (but all curvahedra edges are).That does leave two Catalan polyhedra though, the rhombic dodecahedron and rhombic triacontahedron (with 12 and 30 sides). For these every edge is the same length.
They use 3 and 4 connectors (dodecahedron) and 3 and 5 connectors (triacontahedron) arranged to make rhombs
This gives us two new balls!
The triacontahedron is particularly pleasing, being a large ball,, as you can see when it is compared to the classic ball:
So by reasoning we were able to discover something new, but have we found everything? Maybe other close to perfect spheres are possible, what can you find?