It was during my master degree. One day, my friend Isabelle told me about polyhedron origami: You start with a certain amount of square paper pieces, all folded in the same way. This gives you modules that you assemble into the required solid. To see how, you can refer to this video. Although it is in french, it is enough visual to see how it works.
The solid you obtain are not convex polyhedron. They are sometimes referred to as "stellated polyhedron", but this is not exactly a stellation. In fact, the polyhedron is augmented with regular pyramids on all its sides. What makes them fascinating to me is how they embody the duality of the platonic solids...
The solid you obtain are not convex polyhedron. They are sometimes referred to as "stellated polyhedron", but this is not exactly a stellation. In fact, the polyhedron is augmented with regular pyramids on all its sides. What makes them fascinating to me is how they embody the duality of the platonic solids...
Origami
Those 5 constructions are examples of modular origami. This branch of origami uses several pieces of paper to construct the final model. The solidity of the resulting polyhedron is quite impressive compare to the fragility of the original paper. It makes me think of crystal structures: matter becomes stronger when well organized.
The modules used here are not to be mixed up with the most common Sonobé modules. They are quite alike (maybe a variation?), but faster and easier to fold. Also I was not able to find any other models using all the same unit modules to build the five solids in a similar form (i.e. for example all convex or all augmented the same way).
The modules used here are not to be mixed up with the most common Sonobé modules. They are quite alike (maybe a variation?), but faster and easier to fold. Also I was not able to find any other models using all the same unit modules to build the five solids in a similar form (i.e. for example all convex or all augmented the same way).
The five platonic solids
It can be proven that there exist exactly five convex regular polyhedron: the platonic solids. They are named after Plato as they are prominent in his philosophy. However, it's Theatetus, a contemporary of Plato, who was the first to construct all of them. He may even been at the origin of the proof showing that there are only five regular polyhedron.
One interesting fact is that if you exchange vertices and faces in a platonic solid, you obtain an other platonic solid: its dual.
One interesting fact is that if you exchange vertices and faces in a platonic solid, you obtain an other platonic solid: its dual.
Duality
What I especially like with those origami construction is how it show the duality of the polyhedrons. First, by looking at the table, you may notice that dual solids require the same amount of modules: 6 for the tetrahedron, 12 for both the cube and octahedron and 30 for both the dodecahedron and icosahedron. This makes the dual origami perfectly balanced. I once used this to make a nice platonic mobile where two dual origami where suspended on the same branch, the tetrahedron hanging for himself at the bottom of the structure.
If you compare the picture of the origami and the corresponding platonic solid, you may notice that the origami is not the augmented of the homonym polyhedron, but of its dual. So where is the solid? It is actually the convex envelop of the origami, that is the smallest convex shape containing it. More simply, the apices of the pyramids on the origami correspond to the vertices of the homonym solid.
Look at the images of the platonic solids and their duals. Now imagine pyramids whose bases are the faces of the inner blue solid and apices are the vertex of the outer solid in front of it. This is your origami polyhedron making the link between the dual polyhedrons!
If you compare the picture of the origami and the corresponding platonic solid, you may notice that the origami is not the augmented of the homonym polyhedron, but of its dual. So where is the solid? It is actually the convex envelop of the origami, that is the smallest convex shape containing it. More simply, the apices of the pyramids on the origami correspond to the vertices of the homonym solid.
Look at the images of the platonic solids and their duals. Now imagine pyramids whose bases are the faces of the inner blue solid and apices are the vertex of the outer solid in front of it. This is your origami polyhedron making the link between the dual polyhedrons!