[Illustration:  Figure 15.  DIRECT VIEW AXES SHOWN BY HEAVY LINES TILTED VIEW APEXES SHOWN BY CIRCLES THE 16-HEDROID IN PLANE PROJECTION]

For if the author has been successful in his exposition up to this point, it should be sufficiently plain that the geometry of four-dimensions is capable of yielding fresh and interesting ornamental motifs.  In carrying his demonstration farther, and in multiplying illustrations, he would only be going over ground already covered in his book Projective Ornament and in his second Scammon lecture.

Of course this elaborate mechanism for producing quite obvious and even ordinary decorative motifs may appear to some readers like Goldberg’s nightmare mechanics, wherein the most absurd and intricate devices are made to accomplish the most simple ends.  The author is undisturbed by such criticisms.  If the designs dealt with in this chapter are “obvious and even ordinary” they are so for the reason that they were chosen less with an eye to their interest and beauty than as lending themselves to development and demonstration by an orderly process which should not put too great a tax upon the patience and intelligence of the reader.  Four-dimensional geometry yields numberless other patterns whose beauty and interest could not possibly be impeached—­patterns beyond the compass of the cleverest designer unacquainted with projective geometry.

[Illustration:  Figure 16.]

The great need of the ornamentalist is this or some other solid foundation.  Lacking it, he has been forced to build either on the shifting sands of his own fancy, or on the wrecks and sediment of the past.  Geometry provides this sure foundation.  We may have to work hard and dig deep, but the results will be worth the effort, for only on such a foundation can arise a temple which is beautiful and strong.

In confirmation of his general contention that the basis of all effective decoration is geometry and number, the author, in closing, desires to direct the reader’s attention to Figure 17 a slightly modified rendering of the famous zodiacal ceiling of the Temple of Denderah, in Egypt.  A sun and its corona have been substituted for the zodiacal signs and symbols which fill the centre of the original, for except to an Egyptologist these are meaningless.  In all essentials the drawing faithfully follows the original—­was traced, indeed, from a measured drawing.

[Illustration:  Figure 17.  CEILING DECORATION FROM THE TEMPLE OF DENDERAH]

Here is one of the most magnificent decorative schemes in the whole world, arranged with a feeling for balance and rhythm exceeding the power of the modern artist, and executed with a mastery beyond the compass of a modern craftsman.  The fact that first forces itself upon the beholder is that the thing is so obviously mathematical in its rhythms, that to reduce it to terms of geometry and number is a matter of small difficulty.  Compare the frozen music of these rhymed and linked figures with the herded, confused, and cluttered compositions of even our best decorative artists, and argument becomes unnecessary—­the fact stands forth that we have lost something precious and vital out of art of which the ancients possessed the secret.