symbol, it is the very thing itself. Whence this weltering tide of
life arose, and whither it flows, we know not; but that it is governed
by mathematical law all of our knowledge in every field confirms. Were
it not so, knowledge itself would be impossible. It is because man is
a counting animal that he is master over all the beasts of the earth.
Number is the tune to which all things move, and as it were make
music; it is in the pulses of the blood no less than in the starred
curtain of the sky. It is a necessary concomitant alike of the sharp
bargain, the chemical experiment, and the fine frenzy of the poet.
Music is number made audible; architecture is number made visible;
nature geometrizes not alone in her crystals, but in her most
intricate arabesques.
If number be indeed the universal solvent of all forms, sounds,
motions, may we not make of it the basis of a new æsthetic--a loom on
which to weave patterns the like of which the world has never seen? To
attempt such a thing--to base art on mathematics--argues (some one
is sure to say) an entire misconception of the nature and function of
art. "Art is a fountain of spontaneous emotion"--what, therefore,
can it have in common with the proverbially driest, least spontaneous
preoccupation of the human mind? But the above definition concludes
with the assertion that this emotion reaches the soul "through various
channels." The transit can be effected only through some sensuous
element, some language (in the largest sense), and into this the
element of number and form must inevitably enter--mathematics is
"there" and cannot be thought or argued away.
[Illustration: PLATE XI. IMAGINARY COMPOSITION: THE PORTAL]
But to make mathematics, and not the emotion which it expresses, the
important thing, is not this to fall into the time-worn heresy of
art for art's sake, that is, art for form's sake--art for the sake of
mathematics? To this objection there is an answer, and as this answer
contains the crux of the whole matter, embraces the proposition by
which this thesis must stand or fall, it must be full and clear.
What is it, in the last analysis, that all art which is not
purely personal and episodical strives to express? Is it not the
_world-order_?--the very thing that religion, philosophy, science,
strive according to their different natures and methods to express?
The perception of the world-order by the artist arouses an emotion to
which he can give vent only in terms of number; but number is itself
the most abstract expression of the world order. The form and content
of art are therefore not different, but the same. A deep sense of this
probably inspired Pater's famous saying that all art aspires toward
the condition of music; for music, from its very nature, is the
world-order uttered in terms of number, in a sense and to a degree not
attained by any other art.
This is not mere verbal juggling. We have suffered so long from an
art-phase which exalts the personal, as opposed to the cosmic, that
we have lost sight of the fact that the great arts of antiquity,
preceding the Renaissance, insisted on the cosmic, or impersonal
aspect, and on this alone, just as does Oriental art, even today.
The secret essence, the archetypal idea of the subject is the
preoccupation of the Oriental artist, as it was of the Egyptian,
and of the Greek. We of the West today seek as eagerly to fix the
accidental and ephemeral aspect--the shadow of a particular cloud upon
a particular landscape; the smile on the face of a specific person, in
a recognizable room, at a particular moment of time. Of symbolic art,
of universal emotion expressing itself in terms which are universal,
we have very little to show.
The reason for this is first, our love for, and understanding of,
the concrete and personal: it is the _world-aspect_ and not the
_world-order_ which interests us; and second, the inadequacies of
current forms of art expression to render our sense of the eternal
secret heart of things as it presents itself to our young eyes.
Confronted with this difficulty, we have shirked it, and our ambition
has shrunk to the portrayal of those aspects which shuffle our poverty
out of sight. It is not a poverty of technique--we are dexterous
enough; nor is it a poverty of invention--we are clever enough; it is
the poverty of the spiritual bankrupt trying to divert attention by a
prodigal display of the smallest of small change.
Reference is made here only to the arts of space; the arts of
time--music, poetry, and the (written) drama--employing vehicles more
flexible, have been more fortunate, though they too suffer in some
degree from worshipping, instead of the god of order, the god of
chance.
The corrective of this is a return to first principles: principles so
fundamental that they suffer no change, however new and various their
illustrations. These principles are embodied in number, and one might
almost say nowhere else in such perfection. Mathematics is not the
dry and deadly thing that our teaching of it and the uses we put it
to have made it seem. Mathematics is the handwriting on the human
consciousness of the very Spirit of Life itself. Others before
Pythagoras discovered this, and it is the discovery which awaits us
too.
To indicate the way in which mathematics might be made to yield the
elements of a new æsthetic is beyond the province of this essay, being
beyond the compass of its author, but he makes bold to take a single
phase: ornament, and to deal with it from this point of view.
The ornament now in common use has been gathered from the dust-bin
of the ages. What ornamental _motif_ of any universality, worth, or
importance is less than a hundred years old? We continue to use the
honeysuckle, the acanthus, the fret, the egg and dart, not because
they are appropriate to any use we put them to, but because they are
beautiful _per se_. Why are they beautiful? It is not because they
are highly conventionalized representations of natural forms which
are themselves beautiful, but because they express cosmic truths. The
honeysuckle and the acanthus leaf, for example, express the idea
of successive impulses, mounting, attaining a maximum, and
descending--expanding from some focus of force in the manner universal
throughout nature. Science recognizes in the spiral an archetypal
form, whether found in a whirlpool or in a nebula. A fret is a series
of highly conventionalized spirals: translate it from angular to
curved and we have the wave-band; isolate it and we have the volute.
Egg and dart are phallic emblems, female and male; or, if you prefer,
as ellipse and straight line, they are symbols of finite existence
contrasted with infinity. [Figure 1.]
[Illustration: Figure 1.]
Suppose that we determine to divest ourselves of these and other
precious inheritances, not because they have lost their beauty and
meaning, but rather on account of their manifold associations with a
past which the war makes suddenly more remote than slow centuries have
done; suppose that we determine to supplant these symbols with others
no less charged with beauty and meaning, but more directly drawn from
the inexhaustible well of mathematical truth--how shall we set to
work?
We need not _set_ to work, because we have done that already, we are
always doing it, unknowingly, and without knowing the reason why. All
ornamentalists are subjective mathematicians--an amazing statement,
perhaps, but one susceptible of confirmation in countless amusing
ways, of which two will be shown.
[Illustration: Figure 2.]
Consider first your calendar--your calendar whose commonplace face,
having yielded you information as to pay day, due day, and holiday,
you obliterate at the end of each month without a qualm, oblivious to
the fact that were your interests less sordid and personal it would
speak to you of that order which pervades the universe; would make you
realize something of the music of the spheres. For on that familiar
checkerboard of the days are numerical arrangements which are
mysterious, "magical"; each separate number is as a spider at the
center of an amazing mathematical web. That is to say, every number
is discovered to be half of the sum of the pairs of numbers which
surround it, vertically, horizontally, and diagonally: all of the
pairs add to the same sum, and the central number divides this sum by
two. A graphic indication of this fact on the calendar face by means
of a system of intersecting lines yields that form of classic grille
dear to the heart of every tyro draughtsman. [Figure 2.] Here is
an evident relation between mathematical fact and ornamental mode,
whether the result of accident, or by reason of some subconscious
connection between the creative and the reasoning part of the mind.
To show, by means of an example other than this acrostic of the days,
how the pattern-making instinct follows unconsciously in the groove
traced out for it by mathematics, the attention of the reader is
directed to the design of the old Colonial bed-spread shown in Figure
3. Adjacent to this, in the upper right hand corner, is a magic
square of four. That is, all of the columns of figures of which it is
composed: vertical, horizontal and diagonal add to the same sum: 34.
An analysis of this square reveals the fact that it is made up of
the figures of two different orders of counting: the ordinary order,
beginning at the left hand upper corner and reading across and down in
the usual way, and the reverse-ordinary, beginning at the lower right
hand corner and reading across and up. The figures in the four central
cells and in the four outside corner cells are discovered to belong
in the first category, and the remaining figures in the second. Now
if the ordinary order cells be represented by white, and the reverse
ordinary by black, just such a pattern has been created as forms the
decorative motif of the quilt.
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.
[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.
[Illustration: Figure 11.]
Ornament in its primitive manifestations is geometrical rather than
naturalistic. This is in a manner strange, that the abstract and
metaphysical thing should precede the concrete and sensuous. It would
be natural to suppose that man would first imitate the things which
surround him, but the most cursory acquaintance with primitive art
shows that he is much more apt to crudely geometrize. Now it is
not necessary to assume that we are to revert to the conditions of
savagery in order to believe that in this matter of a sound æsthetic
we must begin where art has always begun--with number and geometry.
Nevertheless there is a subtly ironic view which one is justified in
holding in regard to quite obvious aspects of American life, in the
light of which that life appears to have rather more in common with
savagery than with culture.
[Illustration: Figure 12.]
[Illustration: Figure 13.]
The submersion of scholarship by athletics in our colleges is a case
in point, the contest of muscles exciting much more interest and
enthusiasm than any contest of wits. We persist in the savage habit of
devouring the corpses of slain animals long after the necessity for it
is past, and some even murder innocent wild creatures, giving to their
ferocity the name of sport. Our women bedeck themselves with furs and
feathers, the fruit of mercenary and systematic slaughter; we perform
orgiastic dances to the music of horns and drums and cymbals--in
short, we have the savage psychology without its vital religious
instinct and its sure decorative sense for color and form.
But this is of course true only of the surface and sunlit shadows of
the great democratic tide. Its depths conceal every kind of subtlety
and sophistication, high endeavour, and a response to beauty and
wisdom of a sort far removed from the amoeba stage of development
above sketched. Of this latter stage the simple figures of Euclidian
plane and solid geometry--figures which any child can understand--are
the appropriate symbols, but for that other more developed state of
consciousness--less apparent but more important--these will not do.
Something more sophisticated and recondite must be sought for if we
are to have an ornamental mode capable of expressing not only the
simplicity but the complexity of present-day psychology. This need not
be sought for outside the field of geometry, but within it, and by
an extension of the methods already described. There is an altogether
modern development of the science of mathematics: the geometry of
four dimensions. This represents the emancipation of the mind from
the tyranny of mere appearances; the turning of consciousness in a
new direction. It has therefore a high symbolical significance as
typifying that movement away from materialism which is so marked a
phenomenon of the times.
Of course to those whose notion of the fourth dimension is akin to
that of a friend of the author who described it as "a wagon-load
of bung-holes," the idea of getting from it any practical advantage
cannot seem anything but absurd. There is something about this form
of words "the fourth dimension" which seems to produce a sort of
mental-phobia in certain minds, rendering them incapable of perception
or reason. Such people, because they cannot stick their cane into it
contend that the fourth dimension has no mathematical or philosophical
validity. As ignorance on this subject is very general, the following
essay will be devoted to a consideration of the fourth dimension and
its relation to a new ornamental mode.
[Illustration]
II
THE FOURTH DIMENSION
The subject of the fourth dimension is not an easy one to understand.
Fortunately the artist in design does not need to penetrate far into
these fascinating halls of thought in order to reap the advantage
which he seeks. Nevertheless an intention of mind upon this
"fairy-tale of mathematics" cannot fail to enlarge his intellectual
and spiritual horizons, and develop his imagination--that finest
instrument in all his chest of tools.
By way of introduction to the subject Prof. James Byrnie Shaw, in an
article in the _Scientific Monthly_, has this to say:
Up to the period of the Reformation algebraic equations of
more than the third degree were frowned upon as having no
real meaning, since there is no fourth power or dimension.
But about one hundred years ago this chimera became an actual
existence, and today it is furnishing a new world to physics,
in which mechanics may become geometry, time be co-ordinated
with space, and every geometric theorem in the world is a
physical theorem in the experimental world in study in the
laboratory. Startling indeed it is to the scientist to be told
that an artificial dream-world of the mathematician is
more real than that he sees with his galvanometers,
ultra-microscopes, and spectroscopes. It matters little that
he replies, "Your four-dimensional world is only an analytic
explanation of my phenomena," for the fact remains a fact,
that in the mathematician's four-dimensional space there is
a space not derived in any sense of the term as a residue of
experience, however powerful a distillation of sensations or
perceptions be resorted to, for it is not contained at all in
the fluid that experience furnishes. It is a product of the
creative power of the mathematical mind, and its objects are
real in exactly the same way that the cube, the square, the
circle, the sphere or the straight line. We are enabled to see
with the penetrating vision of the mathematical insight that
no less real and no more real are these fantastic forms of the
world of relativity than those supposed to be uncreatable or
indestructible in the play of the forces of nature.
These "fantastic forms" alone need concern the artist. If by some
potent magic he can precipitate them into the world of sensuous images
so that they make music to the eye, he need not even enter into the
question of their reality, but in order to achieve this transmutation
he should know something, at least, of the strange laws of their
being, should lend ear to a fairy-tale in which each theorem is a
paradox, and each paradox a mathematical fact.
He must conceive of a space of four mutually independent directions; a
space, that is, having a direction at right angles to every direction
that we know. We cannot point to this, we cannot picture it, but we
can reason about it with a precision that is all but absolute. In such
a space it would of course be possible to establish four axial lines,
all intersecting at a point, and all mutually at right angles with one
another. Every hyper-solid of four-dimensional space has these four
axes.
The regular hyper-solids (analogous to the Platonic solids of
three-dimensional space) are the "fantastic forms" which will prove
useful to the artist. He should learn to lure them forth along them
axis lines. That is, let him build up his figures, space by space,
developing them from lower spaces to higher. But since he cannot enter
the fourth dimension, and build them there, nor even the third--if he
confines himself to a sheet of paper--he must seek out some form of
_representation_ of the higher in the lower. This is a process with
which he is already acquainted, for he employs it every time he makes
a perspective drawing, which is the representation of a solid on
a plane. All that is required is an extension of the method: a
hyper-solid can be represented in a figure of three dimensions, and
this in turn can be projected on a plane. The achieved result will
constitute a perspective of a perspective--the representation of a
representation.
This may sound obscure to the uninitiated, and it is true that the
plane projection of some of the regular hyper-solids are staggeringly
intricate affairs, but the author is so sure that this matter lies so
well within the compass of the average non-mathematical mind that he
is willing to put his confidence to a practical test.