# 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
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.

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
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.