[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.]

[Illustration:  PLATE XII.  IMAGINARY COMPOSITION.  THE BALCONY]

[Illustration:  Figure 9.]

Suppose, for example, that we attempt to deal decoratively which such simple figures as the three lowest Platonic solids—­the tetrahedron, the hexahedron, and the octahedron. [Figure 12.] Their projection on a plane yields a rhythmical division of space, because of their inherent symmetry.  These projections would correspond to the network of lines seen in looking through a glass paperweight of the given shape, the lines being formed by the joining of the several faces.  Figure 13 represents ornamental bands developed in this manner.  The dodecahedron and icosahedron, having more faces, yield more intricate patterns, and there is no limit to the variety of interesting designs obtainable by these direct and simple means.

[Illustration:  Figure 10.]

If the author has been successful thus far in his exposition, it should be sufficiently plain that from the inexhaustible well of mathematics fresh beauty may be drawn.  But what of its significance?  Ornament must mean something; it must have some relation to the dominant ideation of the day; it must express the psychological mood.

What is the psychological mood?  Ours is an age of transition; we live in a changing world.  On the one hand we witness the breaking up of many an old thought crystal, on the other we feel the pressure of those forces which shall create the new.  What is nature’s first visible creative act?  The formation of a geometrical crystal.  The artist should take this hint, and organize geometry into a new ornamental mode; by so doing he will prove himself to be in relation to the anima mundi.  It is only by the establishment of such a relation that new beauty comes to birth in the world.