Margaret Kepner
I enjoy expressing mathematical concepts through attributes such as
color, geometric forms, and patterns. I have a background in
mathematics, which provides me with a never-ending supply of subject
matter, while my lifelong interest in art gives me a vocabulary and
references to use in my work. Recently, I have been experimenting with
geometric packing problems, particularly ones involving circles and
squares. Compound packings repeat the packing process at a smaller
scale and perhaps with a different shape. For example, a circle packed
with circles, which are in turn packed with squares, leads to a
compound packing with interesting patterns and relationships.
In this piece, 23 circles are optimally packed into a larger
circle. The circles have radii that are proportional. If the
circles are scaled so that the smallest has a radius of 1, the
next biggest one will have a radius of 2, etc., with the largest
one having a radius of 23. Even-numbered circles are black and odd
ones are white. Squares are packed into these 23 circles according
to size. For example, the 4-circle at the far right is packed with
4 white squares of the same size, while the 23-circle near the
center is optimally packed with 23 black squares. These two
packings are highly regular, while others are more chaotic. Small
red squares denote "sliders" that occur in packing a circle that
is a "rattler" in the larger circle packing.
This work is based on a packing of 23 unequal squares into a
larger square. This is one of the 12 Simple Perfect Squared
Squares (SPSS) of order 23. The squares' sides can be expressed as
integers with no duplicates. In this example, the square sizes
vary from 1 to 79, and the outer square is size 215. Odd-numbered
squares are black, evens are white. Based on their sizes, the
squares are ranked from 1 to 23, and packed with that number of
circles. Thus, the rank-22 square at the lower-left is packed with
22 uniform circles. One circle in this square is not held in
rigidly by its neighbors and can move slightly - it is known as a
“rattler.” The rattler circles in this piece are red. Squares with
maximum density packings have red backgrounds.