Geomagic square

A geometric magic square, often abbreviated to 'geomagic square', is a generalization of magic squares invented by Lee Sallows in 2001. A traditional magic square is a square array of numbers (almost always positive integers) whose sum taken in any row, any column, or in either diagonal is the same target number. A geomagic square, on the other hand, is a square array of geometrical shapes in which those appearing in each row, column, or diagonal can be fitted together to create an identical shape called the target shape. As with numerical types, it is required that the entries in a geomagic square be distinct. Similarly, the eight trivial variants of any square resulting from its rotation and/or reflection, are all counted as the same square. By the dimension of a geomagic square is meant the dimension of the pieces it uses. Hitherto interest has focused mainly on 2D squares using planar pieces, but pieces of any dimension are permitted.

Figure 1 above shows a 3 × 3 geomagic square. The 3 pieces occupying each row, column and diagonal pave a rectangular target, as seen at left and right, and above and below. Here the 9 pieces are all decominoes, but pieces of any shape may appear, and it is not a requirement that they be of same size. In Figure 2, for instance, the pieces are polyominoes of consecutive sizes from 1 up to 9 units. The target is a 4 by 4 square with an inner square hole.

Surprisingly, computer investigations show that Figure 2 is just one among 4,370 distinct 3 × 3 geomagic squares using pieces with these same sizes and same target. Conversely, Figure 1 is one of only two solutions using similar-sized pieces and identical target. In general, repeated piece sizes imply fewer solutions. However, at present there exists no theoretical underpinning to explain these empirical findings.

The pieces in a geomagic square may also be disjoint, or composed of separated islands, as seen in Figure 3. Since they can be placed so as to mutually overlap, disjoint pieces are often able to tile areas that connected pieces cannot. The rewards of this extra pliancy are often to be seen in geomagics that possess symmetries denied to numerical specimens.

Besides squares using planar shapes, there exist 3D specimens, the cells of which contain solid pieces that will combine to form the same constant solid target. Figure 5 shows an example in which the target is a cube.

A well-known formula due to the mathematician Édouard Lucas characterizes the structure of every 3 × 3 magic square of numbers. Sallows, already the author of original work in this area, had long speculated that the Lucas formula might contain hidden potential. This surmise was confirmed in 1997 when he published a short paper that examined squares using complex numbers, a ploy leading to a new theorem that correlated every 3 × 3 magic square with a unique parallelogram on the complex plane. Continuing in the same vein, a decisive next step was to interpret the variables in the Lucas formula as standing for geometrical forms, an outlandish idea that led directly to the concept of a geomagic square. It turned out to be an unexpected consequence of this find that traditional magic squares now became revealed as one-dimensional geomagic squares.

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