Margaret Kepner

This piece is derived from a family of binary operation tables expressed in a visual format. The term “ELOP” comes from “elementary operations” and “mod 4” means modulus 4 arithmetic is used. The symbols representing the underlying numbers are triangle slices; for example, the number “3 mod 4” would be shown as a 3/4 slice of a triangle. Six operation tables are shown in a compound table of nested squares, with inverse operations paired together and both possible operand orders expressed. The original design consists of solid shapes, and the overall effect is an array of jagged pinwheels. In this version, however, only the outlines of the shapes are shown. The square web of white lines at the top represents the compound table with addition/subtraction in the inner squares, multiplication/division in the middle square rings, and exponentiation/roots in the outer square rings. In the bottom square, with a web of black lines, the nesting order of the operations has been reversed.

This work is a visual presentation of the 5 non-isomorphic groups of order 8: C8, C2 x C4, C2 x C2 x C2, D4, and Q8. It employs a visual vocabulary derived from a traditional quilt pattern, Drunkard’s Path. Each of the small shapes used in the design is a quarter circle in a square, scaled so that its area equals the square’s residual area. Combining 4 orientations and 2 colorings yields 8 basic shapes. These are used to form the quilt pattern shown in the 4 large squares at the top, bottom, and sides. The same basic shapes are combined to generate the group tables appearing in the remaining 5 large squares. The 3 Abelian groups appear along the anti-diagonal, with the cyclic group, C8, in the center. The same basic shape represents the identity elements in all tables, and color accents are used to highlight their positions. Except for C8, the group elements in each table have been arranged so that a normal subgroup of order 4 appears in the upper left quadrant of the table.

This structure of this design is a based on traditional quilt patterns, (Flying Geese, Wild Goose Chase), while its content relates to the Fundamental Theorem of Arithmetic. The integers from 1 to 256 are the “geese,” and the prime decomposition of each integer is shown using colored triangles. There are 8 columns of numbers, starting with a black triangle representing 1 at the upper left. Solid triangles are used for primes, and each prime is assigned a unique color: 2 = red, 3 = gold, 5 = yellow-green, …, 19 = magenta. As larger primes are needed, more colors are created by adding white to the basic 8 hues. Composite numbers are represented by subdivided triangles. Since 6 = 2 x 3, it is half red and half gold. Powers of primes are shown using horizontal shades of the base color. Background colors and lines provide additional information. This design is a visual table, allowing number patterns and properties to be studied -- for example, the distribution of the prime numbers.