The purpose of this article is to show how fascinating geometrical ideas can be by introducing the reader to some polyhedral puzzles. Our intent is to present the material in much the order in which we ourselves discovered it. We would like you to experience some of the joy of discovery we have had, which means, of course, that you risk experiencing some frustration along the way before you finally achieve success in assembling your puzzles. However, don't despair since we will give you many hints along the way and, eventually, more complete instructions for the details involved in assembling the puzzles.
In Section 2 we will describe how to fold the tape required to make your puzzles. In Section 3 we will explain how to make the puzzle pieces for 9 models and challenge you to construct some of them without any further information. We also include in Section 3 one intriguing example of how the braided models may be used to visualise the answer to a combinatorial question in geometry. In Section 4 we give either more hints or complete instructions on how to actually assemble the remaining models. In Section 5 we suggest some variations on certain models you will already have built that, for one reason or another, seem to lack the symmetry you would expect them to have. In this last section we will challenge you to build the more perfect tetrahedron, octahedron and icosahedron without any further information other than the description, two illustrations and a picture.
Although this may be viewed as purely recreational mathematics, knowledge of the symmetry group for each model may be helpful in solving the puzzle. We ourselves are very much in favor of exploiting the mathematics connected with these fascinating models 1 and we are delighted that the editor of this journal has suggested that our article concerning some of the mathematics connected with these models should appear as a companion piece to the present article, in this same issue. That related article, entitled Symmetry in Theory - Mathematics and Aesthetics, abbreviated in this article to [Math], contains a fairly comprehensive list of references which you may consult if you wish to build other models. Of particular relevance is [HP5] of [Math]. We also include at the end of our companion article a brief history of how some of the folding procedures evolved, and how our friendship with the great mathematician and teacher, George Pólya, and with each other, resulted in mathematical collaborations concerning these models (and other topics).
We will concentrate on the directions for constructing polyhedral puzzles
(including the Platonic solids) which have regular triangles, squares, or
pentagons for faces. In order to be able to carry out the instructions and
build these puzzles you will need
1 On one occasion we were horrified to hear a mathematics teacher answer the question, "What did you do with the models you had the students build?" with "Oh, we hung them up!"