# Dallas Clement

I have been interested in mathematics and art for a long time, but only recently have I tried combine these two disciplines. My current work is an investigation into the abstract paintings or pictures that result from the Cayley table

of a group. I decided to work with these mathematical objects after repeatedly failing to convey their innate beauty to my friends. The rigidity of the groups structure forces me to play with the colour combinations or the texture of the paint to bring out the natural beauty in these objects. Modern technology allows me to represent much larger groups than would be possible with oil paint. By creating the pictures using a computer program, I am able to try out many combinations of groups and colours. My paintings and pictures use the layout that Cayley used, featuring no row or column header.

This work was inspired by conversations with my friend. I would go on and on about how beautiful groups were,

and she would barely listen. I decided to convey the beauty of groups by creating a series of artworks that depicted the Cayley table of a group. A Cayley table depicts the result of the group operation between elements of the group. In this case, the group elements are symmetries of a regular polygon with 225 sides. The artwork was created using the gap and python programming languages. Every element in the group is represented in the picture by a square with a unique colour.

This was the first work where I used a group as inspiration. The inspiration for this group is the Dihedral group

with 6 elements, which is the smallest non-abelian group. A group is not abelian if there are group elements a,b such that ab does not equal ba. This can be seen in the painting by comparing the colour is the square (row = 2, column =3) and the colour in (row = 3, column = 2). The elements represent the symmetries of an equilateral triangle. The painting was completed using both paint brushes and painting knifes that give the painting texture and add depth to the colours.

This work was inspired by conversations with my friend. I would go on and on about how beautiful groups were,

and she would barely listen. I decided to convey the beauty of groups by creating a series of artworks that depicted the Cayley table of a group. A Cayley table depicts the result of the group operation between elements of the group. In this case, the group is a Dicyclic group with 900 elements. I find the myriad of shapes, curves and patterns that appear in this picture remarkable because the group can be defined very succinctly as Dic(225) = < a, x | a^450 = 1, x^2 = a^n, x^-1 a x = a^-1>. Every small box has a unique colour, which represents a given element of the group.