# Margaret Kepner

I enjoy exploring the possibilities for conveying ideas in new ways, primarily visually. I have a background in mathematics, which provides me with a never-ending supply of subject matter. My lifelong interest in art gives me a vocabulary and references to utilize in my work. I particularly like to combine ideas from seemingly different areas. I coined the term “visysuals” to describe what I do, meaning the “visual expression of systems” through attributes such as color, geometric forms, and patterns. My creative process involves moving back and forth between a math concept that intrigues me, and the creation of visual images that interpret that concept in interesting ways.

This piece is based on a classic problem involving MacMahon’s colored cubes and utilizes the Tumbling Blocks quilt pattern as its underlying visual format. There are 30 unique cubes that can be formed with 6 colors, assigning one color per face with no repeats. MacMahon’s problem involves selecting a particular cube and then making a replica double-cube using 8 of the remaining 29 cubes. John Conway gave a solution to the problem for all cubes based on a particular matrix arrangement of the 30 cubes. Since the Tumbling Blocks format only presents 3 sides of each cube at a time, more than one view of the cubes is needed to describe the matrix. Each floating rhombus shows the matrix with all cubes oriented with one color-side facing ‘up.’

This work is a compound table of binary arithmetic operations (+, -, *, /, …) modulus 11. The numbers from 1 to 11 have been expressed as colors, and the results of the binary operations are shown in a tabular format consisting of an array of nested squares. The overall table is divided along the main diagonal; in the lower-left half, addition and subtraction results are shown in the center of each nested square, while in the upper-right half, these results are placed in the outermost square ring. Since 11 is a prime, many of the inverse operations have solutions. Where they do not, the ‘undefined’ result is shown as the color gray.

There are sixteen 2x2 matrices composed of the elements zero and one. These matrices are closed under matrix multiplication modulus 2. This piece is a visual representation of the multiplication table for these 16 matrices. Striped squares, arranged in a basketweave pattern, are used to represent the numbers, with the mostly-black squares corresponding to ‘ones’ and the mostly-white squares to ‘zeros.’ Six of the 16 matrices have inverses, and this subset forms a group. This order-6 group is non-abelian and is isomorphic to the dihedral group D3. The portion of the table representing the product of the D3 group falls into four sections at the center. The group elements are shown with a larger, 3-stripe pattern and a yellow background.