The Platonic solids!
Tetrahedron, cube, octahedron, icosahedron, dodecahedron. Each face is identical and each edge of each face is also identical, being either a equilateral triangle, square or regular pentagon. This special set of objects has fascinated people for many centuries and found its way into many different systems of thought in both the arts and sciences.
Here I show the construction of the five Platonic solids from cereal box cardboard. The tools you need are at least two cereal boxes or similar thin cardboard box, scissors, a pen or marker, cutouts of a triangle, square and pentagon as shown in my other Instructables, some tape, and something sharp to score cardboard with.
Step 1: Trace Out the Nets
"net" is cardboard cutout geometry jargon for the pattern of the shape you're folding up from the cardboard. It is much easier to show than to explain here, the images show how the pentagon cutout, square cutout and triangle cutout are used to generate the five platonic solids. Nets are extremely useful types of information--they're a compact tool to make three dimensional things from two dimensional things! If you want to know how to make some 3d thing from 2d, and it's part of the geometry jargon lexicon, just google the thing you want and the word "net", e.g. "icosahedron net". Also not that nets are not unique. There are many ways to trace out shapes in 2d to get the same 3d result. I chose the nets shown here for various reasons, thought and part of what I'm trying to communicate in this document is a favored set of nets that I think are conducive to building useful things.
Step 2: Trace Nets and Cut Out
Cereal boxes don't come apart as easily as you might think, and they do rip pretty easily. So care must be taken to cut out flat panels as large as possible with scissors as you initially harvest those boxes, rather than just ripping them apart like an angry kitty. Trace all the nets, cut out. And yes you could have skipped the previous step and just traced out on cardboard, but paper is nicer to draw on, and this way you can laminate the paper thing and keep it for making more in the future, and decorate it and label it with information on how to make more.
Step 3: Score and Fold
This takes a little bit of practice to get right. Use a sharp thing like the point of a fork or a ballpoint pen or skewer and a ruler to score all the fold lines of all the objetcts
Step 4: Assemble, Tape, Photograph
Now fold them up as shown, tape very gently along the various joints. Arrange them nicely, photograph and share. The thing you just did is construct the five Platonic Solids, so that's what you tell the world.
Step 5: Understand and Visualize Relations
Why do we actually want to make these things in the year 2016 when technology has moved so much since Plato? What is the value here? One reason I'm really glad I rediscovered the Platonic solids later in life is that I find understanding how they relate is a powerful tool for building practical things with minimal needed documentation. Two things I want to point out here are how the objects are the duals of each other and how the tetrahedron and octahedron stack together. For more information on the "dual" relationship, see the wikipedia page here: https://en.wikipedia.org/wiki/Platonic_solid, this is also a good jumping point for a million other things you can do with these shapes as your basis. Also, painting these is a good idea.
Step 6: Applications
Here I show a gallery of applications I've found for these solids. Again I like doing technology building with these because of the simplicity of specification. For instance if I say "3 inch cube" or "3 inch tetrahedron" that carries a ton of useful structural information in a very compact way and also leaves the door open for many possible implementations, different scales and materials. The scalability is critical! The fact that the tetrahedron/octahedron lattice can be built at any scale opens the possibility of the construction of self-fabricating fractal structures, useful for assembly of all things at all scales.