A remarkable 8 × 8 magic square, found by the designer on the internet ^* and appropriated for the thematic base of his program, has a multiplicity of magic properties: Not only does the sum of the numerals of each horizontal, vertical, and diagonal line of the square add up to 260, but each quadrant exhibits the same magic property of the numerals in each of their corresponding lines adding up to 130-albeit, the only quadrant with a consecutive set of numbers is the lower left quadrant whose numerals run from 25 through 40-in contrast to the upper right quadrant with its extreme sequences of numerals 1 through 8 and 57 through 64.

The designer subdivided each of the magic square's 64 numerated squares into 64 sub-squares and systematically located counters (a circle inscribed in the modular sub-square) on them-in numbers that correspond to the given square's numeral. (The reader is invited to reconstruct the numeration of the magic square from the image that the designer has derived from it.)

According to the program's rules for locating counters, each numeral of the magic square must be addressed consecutively. First, each new counter is placed in the same location of the numerated square's 8 × 8 grid as that numerated square is located in the full magic square's 8 × 8 grid. E.g., one counter, representing numeral 1, is placed in the top right corner of numerated grid of square 1, because this numerated square is situated in the top right corner of the magic square. Second, while, in any new square that is being addressed, a new counter is located in the same manner as just described, all previously located counters are duplicated (retaining their same relative positions on the grid) in the square being currently provided with its requisite number of counters.

All new counters are drawn in bold line, an emphasis that is eliminated as these counters are duplicated in all subsequent numerated squares. Because of the difference in tone between the new counters and the duplicated ones, the set of new counters stands out as a group; and it is seen that these counters by the nature of their programmed positions-define their own 7 × 7 grid-configuring 49 square bays.

There are other fascinating visual outcomes of this design. First, the squares that are nearly solidly filled in the top right quadrant successively break down from quadrant to quadrant into a nearly full-fledged textured mesh in the bottom left quadrant. On closer inspection, it is seen in this breakdown that there is a visual, reciprocal inversion: Relative to a focal point of reciprocal exchange at the center of the format, solid area by solid area dissolves correspondingly into textured area by textured area. Finally, while the larger, numerated squares pair into vertical rectangles that checker through the design, the small counters coincidingly pair vertically; and these couplets checker as well: two degrees of fractalization. In respect to the original numbers, another reciprocal inversion (one that does not have visual consequence in the design) extends throughout each quadrant; for the sum of the numerals in any two squares that reciprocate with one another through the center of a quadrant is always 65 (e.g., squares 1 and 64)-which is one half of the sum (130) of each quadrant's operative magic lines and one quarter of the sum (260) of the magic square's operative magic lines.

*) Credited to Yoshi Moriyama; see A. Hirayama and G. Abe, title translated from Japanese as Researches in Magic Squares, 1983.