The new solids described in this paper are the product of a hybrid between two families of solids. One is ancient – the family of the platonic solids that consists of the five well-known regular polyhedra. The other is fairly new, named only recently – the developable roller family. A developable roller is a convex solid whose surface consists of a single continuous developable face. These are two very different families in many ways. Being polyhedrons, the platonic solids have flat faces that provide them with a stable base. The shape of the faces is simple and repetitive, and their arrangement in space has a high degree of symmetry that makes the solids look the same when viewed from different points of view. All of these give the platonic solids qualities of simplicity, stability, order and permanence. Developable rollers are different. Their single, twisted faces make them baseless, thus unstable. Any slight deviation from a delicate equilibrium causes them to start rolling. Their single faced, curved and unbounded surfaces create a sense of entities that have no beginning or end. Their special surface shape also determines the way they move, and shapes their characteristic, and surprising, meandering rolling motion. All of these make the developable rollers dynamic, unstable, and hard to predict solids.
In 2017, the question arose in my mind, whether it was possible to create bodies that combine the qualities of these two different families? The simple and uniform structure of the Platonic bodies has given me the key to resolving this question, mainly the fact that at each of their vertices, the same number of equal length, equally spaced edges meets. This allowed me to fit to the vertices parts of identical conical surfaces that I later joined together to create one surface, which eventually became a “tight garment” that wrapped the entire polyhedron. The “sewing” process was interesting and challenging in itself because there are many possible ways of joining together the surface parts and most do not give the desired result (a continuous individual face). Even after solving the problem geometrically, it was still difficult for me to visualize what the end result would look like. Only when the 3D printed models were in my hands, could I begin to visually grasp the unique and complex structure of each one of them. In my view, my original question was answered positively. The new solids do maintain a balance between the qualities of the two original families. They fully preserve the vertex system of the Platonic solids thus when you focus on the vertices only, you can easily identify in each body the Platonic solid it circumscribes. On the other hand, their surfaces and their complex edge patterns have the dynamic and mysterious qualities attributed to developable rollers.
The links below show animations and videos of some platonicons: