# Piotr Pawlikowski

mathematics teacher

Adam Mickiewicz Secondary School, Kluczbork

Kluczbork, Poland

For many years I have been making polyhedron models. I implement different techniques, but I get the best results using card stock paper and glue. For a few years I have also been using computer software (mainly Great Stella) for creating nets. Some time ago, the manager of our local museum encouraged me to organize an exhibition of my models there. My exhibition attracted many visitors and it was the moment when I started to think about my activity as a kind of mathematical art. In polyhedra especially in their compounds) simple shapes such as triangles, squares, pentagons etc. form highly complex and tricky structures. The beauty of mathematics can be seen in my models. Looking at them you can also feel some tension between simplicity and complexity, both of which turn out to be the two sides of the same coin. Although building card stock paper polyhedron models remains the main area of my interest, I also make models using other techniques - origami and kirigami.

Squares and Octagons

13.5 x 13.5 x 13.5 inches

glued card stock (160g/m2)

2012

The sculpture is built from 60 squares and 30 octagons. From the mathematical point of view this is a uniform compound of 5 small rhombihexahedra (the constituent has a Wenninger number 86). Polygons intersect each other in such an incredible way that the whole structure consists of 2160 facelets and make it one of the most complex uniform compounds of unifrom polyhedra. Octagons are difficult to be noticed at first sight. Apart from squares, you can see square and triangular openings as well and only careful study of the whole model allows detecting the second type of polygons. Nets for the model were derived from the Stella 4D software.

60 A

6 x 6 x 6 inches

160 g/m2 card stock

2013

The sculpture consists of 60 identical parts which are in the shape of a letter A. They intersect themselves (each letter cuts seven others) in a tricky and intriguing way, but all the parts remain flat. The model is assembled without glue. The base for this construction was one of the stellations of a strombic hexecontahedron (a dual to Archimedean rhombicosidodecahedron). The model is also interesting from the combinatorial point of view. The pieces are in 5 colours. There are 10 different triplets of 5 elements and each triplet can be circularly arranged in two different ways. It makes a total of 20 possible arrangements. In 60 A you can see each of them (and triplets composed from the same colours are on the opposite sides of the model). The data of the base polyhedron were obtained using Stella 4D.