It may be claimed that these two examples of a relation between ornament and mathematics are accidental and therefore prove nothing, but they at least furnish a clue which the artist would be foolish not to follow up.  Let him attack his problem this time directly, and see if number may not be made to yield the thing he seeks:  namely, space-rhythms which are beautiful and new.

We know that there is a beauty inherent in order, that necessity of one sort or another is the parent of beauty.  Beauty in architecture is largely the result of structural necessity; beauty in ornament may spring from a necessity which is numerical.  It is clear that the arrangement of numbers in a magic square is necessitous—­they must be placed in a certain way in order that the summation of every column shall be the same.  The problem then becomes to make that necessity reveal itself to the eye.  Now most magic squares contain a magic path, discovered by following the numbers from cell to cell in their natural order.  Because this is a necessitous line it should not surprise us that it is frequently beautiful as well.

[Illustration:  Figure 3.]

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.