2018 Joint Mathematics Meetings

# Margaret Kepner

## Artists

## Statement

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 enjoy expressing mathematical concepts through attributes such as color, geometric forms, and patterns. Some topics I have explored include tessellations, combinatorics, groups, dissections, puzzles, and magic squares. I have found that the latter can have interesting properties beyond just being magic in the usual sense, including their capacity to “hold water” and to trace out “magic paths” that are also knight’s tours.

## Artworks

If a square column is erected on each cell of a magic square equal in height to the number in that cell, a topological surface is formed, and some relatively lower areas will "hold water.” This piece is based on two magic squares of order 8 that were the best submissions in a contest to determine squares holding the most water. Each one holds 797 units, but in slightly different ways. Numbers in the magic squares are expressed in base 8 via colors in a nested place system. The one’s place for each number is shown as a central form whose shape depends on its role in the “magic vessel" – a circle for a pond, a square for a barrier wall, etc. The two magic squares are spliced together using diagonal cuts to facilitate cell-by-cell comparisons.

A knight’s tour on an 8x8 grid is a path that visits every cell once and is made up entirely of knight’s moves. If a point on the tour is chosen as “1", continuing along the path assigning consecutive numbers produces an 8x8 array of numbers from 1 to 64. In certain cases, the array turns out to be “magic." Some closed knight’s tours can be numbered magically in several ways; a tour discovered by Jaenisch generates five different Magic Knight Tours. This piece shows these five MKTs expressed in a color-coded base 8 system. The numbering for each of the magic squares begins with the small black circle and ends with the larger one. The geometric path is shown in the remaining four squares, colored to correspond to the central magic square.