# 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.

Some years ago 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. I intend to continue to explore the expression of my ideas in a range of media including prints, books, and textiles.

This piece is based on six different tilings of the Z-pentomino and is influenced by traditional Japanese patterns. Each of the 12 pentominoes will tile the plane -- all but one of them in infinitely many ways. If, however, reflections are not allowed and only directly congruent tilings are considered, the Z-pentomino tiles in six, and only six, distinct ways. The resulting tilings are reminiscent of Japanese sashiko pieces, which typically feature white stitching in geometric patterns on indigo-colored cloth. This piece presents the six tiling patterns in “sashiko style” blocks, using color differences to emphasize the various roles the Z-pentomino plays. For example, examination of the darker ”stand-up” Zs and their immediate neighbors reveals intrinsic differences among the six patterns. In addition, at the center of each block a minimal group of Zs is highlighted that will tile by translation alone.

The traditional quilt pattern “Nine Patch” is based on a 3x3 grid of nine squares, usually colored in a checkerboard fashion. This piece uses the pattern as a point of departure and includes other references to “nine-ness.” The basic 9-patch pattern is generalized to produce additional “odd” patch formats including one-patch, 25-patch, 49-patch, and 81-patch squares. These in turn are displayed in an overall 9x9 grid. The small outer squares provide the key for determining the coloring of each patch square. For example, the central 81-patch square is in the yellow row and the purple column, resulting in a pattern of small yellow squares against a purple background. The 9-patch structure can be found at several different scales in this piece. It is the basis for 9 symbols that are used to represent the elements in the order-9 group tables for C9 and C3 x C3. These tables are displayed via small monochrome squares that float in front of the overall grid of patch squares.

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. Circles are used for zeros and squares represent ones; initially the shapes are black on white. A “super” zero resulting from reduction modulus 2 (1+1 = 2 → 0) is shown larger than a simple zero, and has a dot at its center. Six of the 16 matrices have inverses, and this subset forms a group. The portion of the table representing the product of these six matrices is shown with black and white inverted, and falls into four sections at the center. This order-6 subgroup is non-abelian, and therefore must be isomorphic to the dihedral group D3. The yellow shading of the shapes in the subgroup table highlights the identity elements. The intensity of the shading relates to cycles and reveals the mapping of the six matrices to the elements of D3.