Polyhedra in Poland

By Pawlikowski, Piotr | Mathematics Teaching, September 2007 | Go to article overview

Polyhedra in Poland


Pawlikowski, Piotr, Mathematics Teaching


Piotr Pawlikowski gives a personal account of his relationship with polyhedra.

Polish students learn about Platonic solids in the final year of secondary grammar school, but in such a way that few perceive the beauty of this branch of mathematics. An average student associates polyhedra with calculating lengths and angles in a cube, cuboid or pyramid. But for me 3-D geometry is the branch of mathematics that shows the beauty of tangible physical forms - as well as the beauty of mathematical reasoning.

My personal adventure with polyhedra began in 1996 during an annual conference of SNM (Polish Mathematics Teachers' Association). Until then, despite being a mathematics graduate and a professionally active teacher, my knowledge about polyhedra did not go beyond the above-mentioned stereotypes. During the conference, I met Jan Baranowski and saw his collection of polyhedra and 3-D puzzles. For the first time, I saw physical models of Archimedean solids and I learned about the existence of the Kepler-Poinsot solids. When I got home, I tried to make some simple models of polyhedra. The first attempts were rather ineffective, but in the course of time I became so skilled that building models turned into something more than just a hobby. The book which allowed me to take my first steps was Mathematical Models (Cundy and Rollett, 1981). In a short time, I had built all the polyhedra presented in that book. I also developed my interest by joining an SNM working group 'Warsztat Otwarty' (Open Workshop), where I met a lot of great people who inducted me into the land of polyhedra! My small collection of polyhedra gradually grew into Archimedean solids, Kepler-Poinsot solids, stellations of an icosahedron, compounds of Platonic polyhedra and many others.

I also discovered for myself other articles and books, among them Polyhedron Models (Wenninger, 1971). At the end of 1999, I decided to add something extraordinary to my collection. I started to build a model of polyhedron number 117 in Wenninger's book. Unfortunately, it soon appeared that the diagrams presented there weren't precise enough, and it was impossible to build a good model. I wrote about it in an email to the author. Not only did Father Magnus kindly answer my letter, but he also sent me as a gift some works which included the precise instructions to allow me to make two of the most complicated models described in his book (W117 and W118). After about five months, I could boast about having made models of two the most complex uniform polyhedra. At that time, I had not yet come across Polyhedra (Cromwell, 1999), in which the author warned the reader that models of polyhedra were addictive: making a few models of a particular family causes an irresistible desire to make the rest! This is no problem if it happens to be the Platonic polyhedra or the Archimedean ones, but in my case it meant the uniform polyhedra. This family of 75 polyhedra (not counting the endless series of prisms and pyramids) includes the Platonic, Archimedean and Kepler-Poinsot solids, and also S3 other nonconvex polyhedra. Most of the latter have a very complex structure. At that time, I made about 45 models from this group. During the course of another two years, I built the rest of the models. As far as I know, the collection of mine is one of the three complete uniform polyhedra sets existing in the world. (One of the other two is in the Science Museum in London.)

At the beginning of 2002, in a local museum in my hometown, Kluczbork, I presented an exhibition of my collection. During three weeks, over 3300 people visited the museum to see the exhibition. In autumn of the same year, a similar number of people (but over the course of only three days) saw my models at the Lower Silesian Science Festival in Wroclaw. Along with my models of uniform polyhedra, there were also polyhedra models made by my students. They prepared a complete set of 92 regular-faced convex polyhedra (called Johnson's solids). …

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