# 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.

A geometric dissection is a subdivision of a shape into pieces that can be reassembled to create a different shape. This design is based on dissections of nine squares into regular polygons. For each square, the sub-pieces are colored and the corresponding rearranged shape is shown using the outlines of the pieces. For example, the leftmost square is cut into five colored sub-parts; these pieces can be used to form an octagon, as the figure directly to the left illustrates. Visually, the fractured squares resemble the random patchwork patterns that compose traditional Crazy Quilts. In this case, however, the pieces are not so “crazy” – they have a clear purpose. The center square shows a three-way dissection: square, hexagon, and octagon.

The Catalan numbers are a sequence of positive integers that provide answers to certain combinatorial questions. For example, in how many ways can a polygon with n+2 sides be cut into triangles? A pentagon (setting n=3) can be triangulated in five different ways, so the 3rd Catalan number is five. Other types of problems also lead to the Catalan numbers: counting binary trees, balancing parentheses, finding paths through a grid, and shaking hands in a circle. This piece is composed of diagrams representing five different problems, for which the solutions yield the 3rd Catalan number (five). The solution sets for each problem are displayed in diagonal bands. The columns indicate correspondences between elements in different solution sets.

A Primitive Pythagorean Triple (PPT) is a set of 3 integers, with no common divisor, that corresponds to the sides of a right triangle. All PPTs can be generated through matrix transformations using just three 3x3 matrices. Applying these matrices to the “seed” triangle (3:4:5) yields 3 new PPTs, the 1st generation in a ternary tree structure. The 2nd generation consists of 9 PPTs, and the 3rd has 27. This work explores three steps of the generating process. Triangles are scaled to have a unit hypotenuse. Each row in the 3x3 grid of black triangles corresponds to a generation, while columns relate to branches of the underlying tree structure. The color of each triangle is defined by the sequence of matrix transformations determining it.