The left hand drawing in Figure 4 represents the smallest aggregation of numbers that is capable of magic square arrangement. Each vertical, horizontal, and corner diagonal column adds up to 15, and the sum of any two opposite numbers is 10, which is twice the center number. The magic path is the endless line developed by following, free hand, the numbers in their natural order, from 1 to 9 and back to 1 again. The drawing at the right of Figure 4 is this same line translated into ornament by making an interlace of it, and filling in the larger interstices with simple floral forms. This has been executed in white plaster and made to perform the function of a ventilating grille.
Now the number of magic squares is practically limitless, and while all of them do not yield magic lines of the beauty of this one, some contain even richer decorative possibilities. But there are also other ways of deriving ornament from magic squares, already hinted at in the discussion of the Colonial quilt.
[Illustration: Figure 4.]
[Illustration: Figure 5.]
Magic squares of an even number of cells are found sometimes to consist of numbers arranged not only in combinations of the ordinary and the reverse ordinary orders of counting, but involving two others as well: the reverse of the ordinary (beginning at the upper right hand, across, and down) and the reversed inverse, (beginning at the lower left hand, across, and up). If, in such a magic square, a simple graphic symbol be substituted for the numbers belonging to each order, pattern spontaneously springs to life. Figures 5 and 6 exemplify the method, and Figures 7 and 8 the translation of some of these squares into richer patterns by elaborating the symbols while respecting their arrangement. By only a slight stretch of the imagination the beautiful pierced stone screen from Ravenna shown in Figure 9 might be conceived of as having been developed according to this method, although of course it was not so in fact. Some of the arrangements shown in Figure 6 are closely paralleled in the acoustic figures made by means of musical tones with sand, on a sheet of metal or glass.
[Illustration: Figure 6.]
[Illustration: Figure 7.]
The celebrated Franklin square of 16 cells can be made to yield a beautiful pattern by designating some of the lines which give the summation of 2056 by different symbols, as shown in Figure 10. A free translation of this design into pattern brickwork is indicated in Figure 11.
If these processes seem unduly involved and elaborate for the achievement of a simple result—like burning the house down in order to get roast pig—there are other more simple ways of deriving ornament from mathematics, for the truths of number find direct and perfect expression in the figures of geometry. The squaring of a number—the raising of it to its second power—finds graphic expression in the plane figure of the square; and the cubing of a number—the raising of it to its third power—in the solid figure of the cube. Now squares and cubes have been recognized from time immemorial as useful ornamental motifs. Other elementary geometrical figures, making concrete to the eye the truths of abstract number, may be dealt with by the designer in such a manner as to produce ornament the most varied and profuse. Moorish ceilings, Gothic window tracery, Grolier bindings, all indicate the richness of the field.
[Illustration: Figure 8.]