# 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 I 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 explores the ways in which a particular trapezoid (or trapezium) can be used to tile the plane. The trapezoid in question is a ‘triamond’ – a shape composed of three equilateral triangles joined along their edges. It is also a ‘rep-tile,’ meaning that copies of it can be arranged to create a larger version of the same shape; this property is used to organize the composition. The overall trapezoid is divided into 4 colored trapezoids, each of which contains 4 smaller trapezoid-shaped regions filled with different tiling patterns. These are examples of the infinite number of patterns that can be generated using the basic trapezoid shape, chosen to show a range of tiling characteristics including: primitive-cell size, fault lines, vertex configurations, and symmetry types. Which ones have a primitive-cell size of 4? How many are fault-free? Hexagon-free? Are there tilings with only 1 vertex type? Any with 6? Which patterns have rotational symmetry? Mirror lines? No symmetry?

This work is based on a magic square of order 8, expressed in a visual format similar to a traditional quilt pattern. The magic square is an 8x8 array of numbers from 0 to 63, such that every row and column adds up to 252, the ‘magic constant.’ In addition, a path from 0-to-1, 1-to-2, …, 62-to-63 traces out a complete knight’s tour on the 8x8 grid. A knight’s-tour magic square of order 8 cannot be fully magic, so the diagonals do not sum to 252. The numbers in the square are represented in two ways – base 4 and base 2. Nested squares serve as the number places in the two base systems, and suggest the Log Cabin quilt structure. For each of the 64 squares, one half is shown in base 4 (red, yellow, green, and blue) and the other half in base 2 (black and white). The squares are oriented (by rotation) to create the appearance of concentric diamond-shaped rings, known as the ‘barn raising’ quilt pattern. A thin line traces out the knight’s tour, beginning with 0 in the upper-left corner.

This accordion-style book unfolds to reveal a decorative ‘screen,’ the front of which contains the 11 uniform tessellations of the plane, rendered with white lines on black. A uniform tessellation is made up of regular polygons, where all the vertex points have identical configurations. The 11 patterns are arranged in a grid format, with the 3 regular tessellations displayed in the central column (regular tessellations are uniform, but contain only one polygon type). When the centers of the polygons making up a particular tessellation are joined by a network of lines, a dual tessellation is created. For example, the dual of the square tessellation is another grid of squares -- it is self-dual. The triangular and hexagonal tessellations are mutually dual. The duals for the other 8 tessellations are neither regular nor uniform. The reverse side of the screen displays the duals of the 11 uniform tessellations. Each dual pattern is positioned directly behind its partner tessellation.