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