# 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 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.

A magic square of order 8 is an 8x8 array of numbers from 1 to 64, where the rows, columns, and diagonals all sum to the same constant. If a square column is erected on each cell equal in height to the number in that cell, a topological surface is obtained, and some relatively lower areas will "hold water.” This particular square holds the maximum amount of water possible for an 8x8 magic square (797 units). Numbers in the magic square are expressed in base 8 via colors, with the one’s place for each number shown as a central form whose shape depends on its role in the “magic vessel.” For example, the shape for the cell in row 3, column 3 is a square (for barrier wall), while to left a circle indicates that cell is a pond containing water.

A knight’s tour on an 8x8 grid, such as a chessboard, 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 the start and labeled “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 “semi-magic” with the rows and columns adding up to the same constant. Some closed geometric knight’s tours can be numbered magically in several ways; a tour discovered by Jaenisch generates five different Magic Knight Tours. This piece shows one of these MKTs expressed in a color-coded base 8 system. The numbering for the magic square begins with the small black circle and ends with the larger one.