When I was about 12, my dad and I built four stellated polyhedra (star-shaped, many-sided platonic solids). Dad, a mathematician and math teacher, used the book shown below as a resource and then worked out his own dimensions and angles (you can see his notations right in the book). We hung the three largest polyhedra from the kitchen ceiling and they remained up there all the years I was growing up. I’m a bit heartbroken that I can’t find a photograph of them installed there. We simply hung them from cheap white cup hooks, using invisible dental floss which we knotted and glued inside the final glue joint of the model. This is what passes as decor in a math household, I guess. They were beautiful, colourful and modern, and kids and adults loved them. My friend Doug, who owns way more of these platonic solids than I do, recently said that he feels good just looking at them, and I know what he means.
The book Dad used, still one of the best resources for building geometric models, is Mathematical Models, 2nd Edn, by H. M. Cundy and A.P. Rollett, Oxford University Press, 1961. I am really surprised that you can’t find plans for these online, so that you don’t have to be a mathematician to make them. Has anyone ever found plans? Or perhaps kits? Dad and I made the models with only a geometry set, his calculations, mayfair cover stock, an X-acto knife, white glue, hand-mixed tempera poster paint (as they used to call it) and dental floss. I highly recommended this as a father-daughter project, not to mention as decor. A note on the colours: the faces aren’t painted randomly. The colours help you to see what’s going on in each model. In the model above, for example, each colour indicates a separate triangular pyramid – for a total of five intersecting pyramids. In the polyhedron in the far right of the top photo, each colour indicates either a star shape or a large triangle. The actual choice of colours, though, was subjective. Not surprisingly the polyhedron at top right is Dad’s colour scheme, while the girly one above was mine.