hyper-cube on the paper of this page, that is, on a space of two
dimensions. Let us start as far back as we can: with a point.
This point, a, [Figure 14] is conceived to move in a direction w,
developing the line a b. This line next moves in a direction at right
angles to w, namely, x, a distance equal to its length, forming
the square a b c d. Now for the square to develop into a cube by a
movement into the third dimension it would have to move in a direction
at right angles to both w and x, that is, out of the plane of the
paper--away from it altogether, either up or down. This is not
possible, of course, but the third direction can be _represented_ on
the plane of the paper.
[Illustration: Figure 14. TWO PROJECTIONS OF THE HYPERCUBE OR
TESSERACT, AND THEIR TRANSLATION INTO ORNAMENT.]
Let us represent it as diagonally downward toward the right, namely,
y. In the y direction, then, and at a distance equal to the length
of one of the sides of the square, another square is drawn, a'b'c'd',
representing the original square at the end of its movement into the
third dimension; and because in that movement the bounding points of
the square have traced out lines (edges), it is necessary to connect
the corresponding corners of the two squares by means of lines. This
completes the figure and achieves the representation of a cube on a
plane by a perfectly simple and familiar process. Its six faces
are easily identified by the eye, though only two of them appear as
squares owing to the exigencies of representation.
Now for a leap into the abyss, which won't be so terrifying, since
it involves no change of method. The cube must move into the fourth
dimension, developing there a hyper-cube. This is impossible, for
the reason the cube would have to move out of our space
altogether--three-dimensional space will not contain a hyper-cube. But
neither is the cube itself contained within the plane of the paper;
it is only there _represented_. The y direction had to be imagined and
then arbitrarily established; we can arbitrarily establish the fourth
direction in the same way. As this is at right angles to y, its
indication may be diagonally downward and to the left--the direction
z. As y is known to be at right angles both to w and to x, z is at
right angles to all three, and we have thus established the four
mutually perpendicular axes necessary to complete the figure.
The cube must now move in the z direction (the fourth dimension)
a distance equal to the length of one of its sides. Just as we did
previously in the case of the square, we draw the cube in its new
position (ABB'D'C'C) and also as before we connect each apex of the
first cube with the corresponding apex of the other, because each of
these points generates a line (an edge), each line a plane, and
each plane a solid. This is the tesseract or hyper-cube in plane
projection. It has the 16 points, 32 lines, and 8 cubes known to
compose the figure. These cubes occur in pairs, and may be readily
identified.[1]
The tesseract as portrayed in A, Figure 14, is shown according to the
conventions of oblique, or two-point perspective; it can equally be
represented in a manner correspondent to parallel perspective. The
parallel perspective of a cube appears as a square inside another
square, with lines connecting the four vertices of the one with those
of the other. The third dimension (the one beyond the plane of the
paper) is here conceived of as being not beyond the boundaries of the
first square, but _within_ them. We may with equal propriety conceive
of the fourth dimension as a "beyond which is within." In that case
we would have a rendering of the tesseract as shown in B, Figure 14:
a cube within a cube, the space between the two being occupied by six
truncated pyramids, each representing a cube. The large outside cube
represents the original generating cube at the beginning of its motion
into the fourth dimension, and the small inside cube represents it at
the end of that motion.
[Illustration: PLATE XIII. IMAGINARY COMPOSITION: THE AUDIENCE
CHAMBER]
These two projections of the tesseract upon plane space are not the
only ones possible, but they are typical. Some idea of the variety of
aspects may be gained by imagining how a nest of inter-related cubes
(made of wire, so as to interpenetrate), combined into a single
symmetrical figure of three-dimensional space, would appear
from several different directions. Each view would yield new
space-subdivisions, and all would be rhythmical--susceptible,
therefore, of translation into ornament. C and D represent such
translations of A and B.
In order to fix these unfamiliar ideas more firmly in the reader's
mind, let him submit himself to one more exercise of the creative
imagination, and construct, by a slightly different method, a
representation of a hexadecahedroid, or 16-hedroid, on a plane. This
regular solid of four-dimensional space consists of sixteen cells,
each a regular tetrahedron, thirty-two triangular faces, twenty-four
edges and eight vertices. It is the correlative of the octahedron of
three-dimensional space.
First it is necessary to establish our four axes, all mutually
at right angles. If we draw three lines intersecting at a point,
subtending angles of 60 degrees each, it is not difficult to
conceive of these lines as being at right angles with one another
in three-dimensional space. The fourth axis we will assume to pass
vertically through the point of intersection of the three lines,
so that we see it only in cross-section, that is, as a point. It is
important to remember that all of the angles made by the four axes
are right angles--a thing possible only in a space of four dimensions.
Because the 16-hedroid is a symmetrical hyper-solid all of its
eight apexes will be equidistant from the centre of a containing
hyper-sphere, whose "surface" these will intersect at symmetrically
disposed points. These apexes are established in our representation by
describing a circle--the plane projection of the hyper-sphere--about
the central point of intersection of the axes. (Figure 15, left.)
Where each of these intersects the circle an apex of the 16-hedroid
will be established. From each apex it is now necessary to draw
straight lines to every other, each line representing one edge of the
sixteen tetrahedral cells. But because the two ends of the fourth axis
are directly opposite one another, and opposite the point of sight,
all of these lines fail to appear in the left hand diagram. It
therefore becomes necessary to _tilt_ the figure slightly, bringing
into view the fourth axis, much foreshortened, and with it, all of the
lines which make up the figure. The result is that projection of the
16-hedroid shown at the right of Figure 15.[2] Here is no fortuitous
arrangement of lines and areas, but the "shadow" cast by an
archetypal, figure of higher space upon the plane of our materiality.
It is a wonder, a mystery, staggering to the imagination,
contradictory to experience, but as well entitled to a place at the
high court of reason as are any of the more familiar figures with
which geometry deals. Translated into ornament it produces such an
all-over pattern as is shown in Figure 16 and the design which adorns
the curtains at right and left of pl. XIII. There are also other
interesting projections of the 16-hedroid which need not be gone into
here.
[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.
It is for the restoration of these ancient verities and the discovery
of new spatial rhythms--made possible by the advance of mathematical
science--that the author pleads. Artists, architects, designers,
instead of chewing the cud of current fashion, come into these
pastures new!
[Illustration]
[Footnote 1: The eight cubes in A, Figure 14, are as follows:
abb'd'c'c; ABB'D'C'C; abdDCA; a'b'd'D'C'A'; abb'B'A'A; cdd'D'C'C;
bb'd'D'DB; aa'c'C'CA.]
[Footnote 2: The sixteen cells of the hexadehahedroid are as follows:
ABCD: A'B'C'D': AB'C'D': A'BCD: AB'CD: A'BC'D: ABC'D: A'B'CD': ABCD':
A'B'C'D: ABC'D': A'B'CD: A'BC'D: AB'CD': A'BCD': AB'C'D.]
HARNESSING THE RAINBOW
Reference was made in an antecedent essay to an art of light--of
mobile color--an abstract language of thought and emotion which should
speak to consciousness through the eye, as music speaks through the
ear. This is an art unborn, though quickening in the womb of the
future. The things that reflect light have been organized æsthetically
into the arts of architecture, painting, and sculpture, but light
itself has never been thus organized.
And yet the scientific development and control of light has reached a
stage which makes this new art possible. It awaits only the advent of
the creative artist. The manipulation of light is now in the hands
of the illuminating engineers and its exploitation (in other than
necessary ways) in the hands of the advertisers.
Some results of their collaboration are seen in the sky signs of upper
Broadway, in New York, and of the lake front, in Chicago. A carnival
of contending vulgarities, showing no artistry other than the most
puerile, these displays nevertheless yield an effect of amazing
beauty. This is on account of an occult property inherent in the
nature of light--_it cannot be vulgarized_. If the manipulation of
light were delivered into the hands of the artist, and dedicated
to noble ends, it is impossible to overestimate the augmentation of
beauty that would ensue.
For light is a far more potent medium than sound. The sphere of sound
is the earth-sphere; the little limits of our atmosphere mark the
uttermost boundaries to which sound, even the most strident can
possibly prevail. But the medium of light is the ether, which links
us with the most distant stars. May not this serve as a symbol of the
potency of light to usher the human spirit into realms of being at the
doors of which music itself shall beat in vain? Or if we compare the
universe accessible to sight with that accessible to sound--the
plight of the blind in contrast to that of the deaf--there is the same
discrepancy; the field of the eye is immensely richer, more various
and more interesting than that of the ear.
The difficulty appears to consist in the inferior impressionability
of the eye to its particular order of beauty. To the average man
color--as color--has nothing significant to say: to him grass is
green, snow is white, the sky blue; and to have his attention drawn to
the fact that sometimes grass is yellow, snow blue, and the sky green,
is disconcerting rather than illuminating. It is only when his retina
is assaulted by some splendid sunset or sky-encircling rainbow that
he is able to disassociate the idea of color from that of form and
substance. Even the artist is at a disadvantage in this respect, when
compared with the musician. Nothing in color knowledge and analysis
analogous to the established laws of musical harmony is part of the
equipment of the average artist; he plays, as it were, by ear. The
scientist, on the other hand, though he may know the spectrum from
end to end, and its innumerable modifications, values this "rainbow
promise of the Lord" not for its own beautiful sake but as a means
to other ends than those of beauty. But just as the art of music
has developed the ear into a fine and sensitive instrument of
appreciation, so an analogous art of light would educate the eye to
nuances of color to which it is now blind.
[Illustration: PLATE XIV. SONG AND LIGHT: AN APPROACH TOWARD "COLOR
MUSIC"]
It is interesting to speculate as to the particular form in which this
new art will manifest itself. The question is perhaps already answered
in the "color organ," the earliest of which was Bambridge Bishop's,
exhibited at the old Barnum's Museum--before the days of electric
light--and the latest A.W. Rimington's. Both of these instruments were
built upon a supposed correspondence between a given scale of colors,
and the musical chromatic scale; they were played from a musical score
upon an organ keyboard. This is sufficiently easy and sufficiently
obvious, and has been done, with varying success in one way or
another, time and again, but its very ease and obviousness should give
us pause.
It may well be questioned whether any arbitrary and literal
translation, even though practicable, of a highly complex, intensely
mobile art, unfolding in time, as does music, into a correspondent
light and color expression, is the best approach to a new art of
mobile color. There is a deep and abiding conviction, justified by the
history of æsthetics, that each art-form must progress from its
own beginnings and unfold in its own unique and characteristic way.
Correspondences between the arts--such a correspondence, for
example, as inspired the famous saying that architecture is frozen
music--reveal themselves usually only after the sister arts have
attained an independent maturity. They owe their origin to that
underlying unity upon which our various modes of sensuous perception
act as a refracting medium, and must therefore be taken for granted.
Each art, like each individual, is unique and singular; in this
singularity dwells its most thrilling appeal. We are likely to miss
light's crowning glory, and the rainbow's most moving message to the
soul if we preoccupy ourselves too exclusively with the identities
existing between music and color; it is rather their points of
difference which should first be dwelt upon.
Let us accordingly consider the characteristic differences between
the two sense-categories to which sound and light--music and
color--respectively belong. This resolves itself into a comparison
between time and space. The characteristic thing about time is
succession--hence the very idea of music, which is in time, involves
perpetual change. The characteristic of space, on the other hand, is
simultaneousness--in space alone perpetual immobility would reign.
That is why architecture, which is pre-eminently the art of space, is
of all the arts the most static. Light and color are essentially
of space, and therefore an art of mobile colour should never lack a
certain serenity and repose. A "tune" played on a color organ is only
distressing. If there is a workable correspondence between the musical
art and an art of mobile color, it will be found in the domain of
harmony which involves the idea of simultaneity, rather than in
melody, which is pure succession. This fundamental difference between
time and space cannot be over-emphasized. A musical note prolonged,
becomes at last scarcely tolerable; while a beautiful color, like the
blue of the sky, we can enjoy all day and every day. The changing hues
of a sunset, are _andante_ if referred to a musical standard, but to
the eye they are _allegretto_--we would have them pass less swiftly
than they do. The winking, chasing, changing lights of illuminated
sky-signs are only annoying, and for the same reason. The eye longs
for repose in some serene radiance or stately sequence, while the ear
delights in contrast and continual change. It may be that as the eye
becomes more educated it will demand more movement and complexity, but
a certain stillness and serenity are of the very nature of light,
as movement and passion are of the very nature of sound. Music is a
seeking--"love in search of a word"; light is a finding--a "divine
covenant."
With attention still focussed on the differences rather than the
similarities between the musical art and a new art of mobile color,
we come next to the consideration of the matter of form. Now form
is essentially of space: we speak about the "form" of a musical
composition, but it is in a more or less figurative and metaphysical
sense, not as a thing concrete and palpable, like the forms of space.
It would be foolish to forego the advantage of linking up form with
colour, as there is opportunity to do. Here is another golden ball to
juggle with, one which no art purely in time affords. Of course it is
known that musical sounds weave invisible patterns in the air, and to
render these patterns perceptible to the eye may be one of the more
remote and recondite achievements of our uncreated art. Meantime,
though we have the whole treasury of natural forms to draw from, of
these we can only properly employ such as are _abstract_. The reason
for this is clear to any one who conceives of an art of mobile color,
not as a moving picture show--a thing of quick-passing concrete
images, to shock, to startle, or to charm--but as a rich and various
language in which light, proverbially the symbol of the spirit, is
made to speak, through the senses, some healing message to the soul.
For such a consummation, "devoutly to be wished," natural forms--forms
abounding in every kind of association with that world of materiality
from which we would escape--are out of place; recourse must be had
rather to abstract forms, that is, geometrical figures. And because
the more remote these are from the things of sense, from knowledge and
experience, the projected figures of four-dimensional geometry would
lend themselves to these uses with an especial grace. Color without
form is as a soul without a body; yet the body of light must be
without any taint of materiality. Four-dimensional forms are as
immaterial as anything that could be imagined and they could be made
to serve the useful purpose of separating colors one from another,
as lead lines do in old cathedral windows, than which nothing more
beautiful has ever been devised.
Coming now to the consideration, not of differences, but similarities,
it is clear that a correspondence can be established between the
colors of the spectrum and the notes of a musical scale. That is,
the spectrum, considered as the analogue of a musical octave can
be subdivided into twelve colors which may be representative of
the musical chromatic scale of twelve semi-tones: the very word,
_chromatic_, being suggestive of such a correspondence between sound
and light. The red end of the spectrum would naturally relate to the
low notes of the musical scale, and the violet end to the high, by
reason of the relative rapidity of vibration in each case; for the
octave of a musical note sets the air vibrating twice as rapidly as
does the note itself, and roughly speaking, the same is true of the
end colors of the spectrum with relation to the ether.
But assuming that a color scale can be established which would yield
a color correlative to any musical note or chord, there still remains
the matter of _values_ to be dealt with. In the musical scale there is
a practical equality of values: one note is as potent as another. In
a color scale, on the other hand, each note (taken at its greatest
intensity) has a positive value of its own, and they are all
different. These values have no musical correlatives, they belong to
color _per se_. Every colorist knows that the whole secret of beauty
and brilliance dwells in a proper understanding and adjustment of
values, and music is powerless to help him here. Let us therefore
defer the discussion of this musical parallel, which is full of
pitfalls, until we have made some examination into such simple
emotional reactions as color can be discovered to yield. The musical
art began from the emotional response to certain simple tones and
combinations, and the delight of the ear in their repetition and
variation.
On account of our undeveloped sensitivity, the emotional reactions
to color are found to be largely personal and whimsical: one person
"loves" pink, another purple, or green. Color therapeutics is too
new a thing to be relied upon for data, for even though colors
are susceptible of classification as sedative, recuperative and
stimulating, no two classifications arrived at independently would be
likely to correspond. Most people appear to prefer bright, pure
colors when presented to them in small areas, red and blue being
the favourites. Certain data have been accumulated regarding the
physiological effect and psychological value of different colors, but
this order of research is in its infancy, and we shall have recourse,
therefore, to theory, in the absence of any safer guide.