# Piotr Pawlikowski

mathematics teacher

Adam Mickiewicz Secondary School, Kluczbork

Kluczbork, Poland

For many years I have been doing polyhedron models. I use different technics, but the best results I get in cardboard and glue. For a few years I have been using a computer software (mainly Great Stella) for creating nets. At first, I did not recognize my hobby as an art. I was just adding a new models to my collection. Once a manager of our local museum encouraged me to organize an exihibition of my models in Kluczbork's museum. It attracted many visitors and it was a moment when I started to think about my activity as some kind of mathematical art. In polyhedra (especially in their compounds) simple shapes – triangles, squares, pentagons etc. form highly complex and tricky structures. In my models one can see the harmony and the beauty of mathematics. Looking at them one can also feel some tension between simplicity and complexity. Both turn out to be the two sides of the same coin.

Triangles and Squares

14 x 14 x 14 inches

glued cardboard (160g/m2)

2003

80 triangles and 60 squares form this highly complex and beautiful structure. From the mathematical point of view this is a uniform compound of 20 tetrahemixahedra (THH). Its constituent is interesting because it is the simplest non-covex uniform polyhedron and the only uniform polyhedron with an odd number of faces (7 – 4 triangles and 3 squares). Compound of 20 THH is intriguing because it is the only uniform compound of uniform polyhedra which cannot be obtained by adding symmetry to a group in which the basic polyhedron is uniform. The faces of each THH are so intricately facetted that the whole model consists of 1620 facelets. Nets for the model were derived from the Great Stella software.

Squares and Triangles I

13.5 x 13.5 x 13.5 inches

glued cardboard (160g/m2)

2004

This highly symmetrical structure is built from 90 squares and 40 triangles. This is a uniform compound of 5 small rhombicuboctahedra. Its constituent is one of the Archimedean solids. It is a remarkable polyhedron because there exists a „twin brother” of it which is not uniform. The compound is interesting too. Most of the Archimedean solids do not form any uniform compounds. This one is the most complex uniform compound of not only Archimedean solids, but also of all convex uniform polyhedra with non-prismatic symmetry (the whole model has 1080 external parts). Nets for the model were derived from the Great Stella software. A complex shape created simply with squares and triangles.

Squares and Triangles II

22 x 22 x 22 inches

glued cardboard (160g/m2)

2007

90 squares and 40 triangles arranged in the other way than in „Squares and Triangles I” form this wonderful shape. These polygons cut one another in so incredible way that the whole structure consists of 3540 facelets. But it is not only complex, but it is fully symmetrical too. Mathematically this is a uniform compound of 5 great rhombicuboctahedra (M. Wenninger number 85) and this is the most complex of all uniform compounds of uniform polyhedra. This is a true example of extreme connections between simplicity and complexity. Nets for the model were derived from the Great Stella software.