PLATONIC SOLIDS

THERE ARE MANY different polyhedra in geometry, but for the purposes of this book, I have only included images of the Platonic Solids.

The Platonic Solids are the five basic regular convex polyhedra comprising a single type of regular polygon faces.

Tetrahedron—six edges, four faces, four vertices.

Hexahedron/cube—twelve edges, six faces, eight vertices.

Octahedron—twelve edges, eight faces, six vertices.

Dodecahedron—thirty edges, twelve faces, twenty vertices.

Icosahedron—thirty edges, twenty faces, twelve vertices.

Applying polyhedra in the assembly process

Each origami unit will represent an edge, face, or vertex of a polyhedron; usually an edge. For instance, in Figure 1, above, each unit represents one edge of an icosahedron, as marked by the red lines. Five units will meet at each vertex, three at each face. Identifying what part of a polyhedron the unit represents, as well as what polyhedron is being represented, will make the assembly simpler. While the illustration above is from a Kusudama, this same process applies to Wire Frame modulars, although the interwoven units makes the pattern less obvious.

The connection between polyhedra

The Kusudamas are not just a warm-up for the folding process; they are to familiarize you with the basic polyhedra behind the more complicated Wire Frames. Once you understand the dodecahedron on the left in Figure 2 (below), the model on the right will be easier to understand. In the first few Wire Frame models in this book, potential underlying polyhedra will be given, but later in the book I will leave it to you to interpret the models’ symmetry, and will focus more on axial weaving and overarching assembly techniques. Note also that some models have a more direct resemblance to an Archimedean Solid. Archimedean Solids have all the same conditions as Platonic Solids, except that they allow for the polyhedron to have more than one kind of regular convex polygon. (Because of their symmetry, however, they exclude models with dihedral group symmetry, such as prisms.)

All the models in this book can still be linked back to a Platonic Solid, so I won’t include images of the Archimedean Solids group. (If you are interested, these can be found in a variety of mathematical textbooks and online sources.) There are other sets of polyhedra as well, each bound by various parameters, such as the Catalan, Kepler-Poinsot, and Johnson Solids. For this volume, however, a basic understanding of the Platonic Solids and how they can be applied in assembling polyhedral modular origami will be sufficient.