Einstein's Cave

     My first experience of computer-generated four-dimensional geometry was
 through Thomas Banchoff, then chair of the mathematics department at Brown
 University. I visited Banchoff and his collaborator/programmer Charles
 Strauss in the summer of l979, and was permitted time on the special,
 parallel processor computer they had built. It was one of the most profoundly
 inspiring experiences of my life.

    After years of hoping to visualize the four-dimensional cube, there it
 was. By moving a joystick, I could turn the hypercube in my hand. The
 computer recomputed the position of the object thirty times a second so that
 the joystick was exquisitely sensitive to the touch, and the illusion of
 actually handling this beloved and mysterious object was very strong indeed.
 It was as though I had plunged my hands through the computer screen to touch
 it.  There was no question of this figure's being a complicated
 three-dimensional object; it behaved in ways I had never seen, flipping
 inside out as it turned around, dissolving and reconstituting before my eyes.
 Like quicksilver in the palm of my hand, its movements were endlessly
 fascinating.

    In a trance, I drove home the five hours from Providence to upstate New
 York where I was spending the summer.  For three nights, I woke up frequently
 from dreams of the images that I had seen on Banchoff's computer: the green
 screen, the quivering geometric figure. It seemed as if these images were
 imprinted on my mind. I had seen the fourth dimension directly. Here at last,
 was the secret I needed to fulfill my ambitions. (fig 1)


    Time is not the fourth dimension.  The fourth dimension is a spatial
 dimension represented by a line  perpendicular to each of three other
 perpendicular lines. This fourth perpendicular line leads out of the space
 defined by the other three and never intersects them. The space-time of
 Relativity physics (where time is the fourth dimension, or more
 precisely, one of the four dimensions) is an application of
 four-dimensional geometry. We can understand what this distinction between
 geometry and applicationmeans by considering a map of the United States.  The
 East/West axis runs horizontally across the page, along what by convention is
 the X axis, and the North/South axis runs vertically along what is generally
 called the Y axis. When giving map coordinates, we usually mention the x axis
 first and the y axis second. But we do not say that North is the second
 dimension. We know that it is the second dimension mentioned when looking at
 these kinds of maps, but there are many situations when the second dimension
 is attributed to something other than North, that the second dimension
 existed before maps of this kind were invented, and that there would be a
 second geometric dimension if maps like this had never been invented. In
 other words, we do not confuse the application with the underlying geometry.

    Even though space-time physics may be the most common application of
 four-dimensional geometry to date, the fourth dimension exists as a geometric
 entity represented by four mutually perpendicular lines independent of that
 application. In fact, the fourth dimension of space was known for generations
 before the invention of space-time physics by Einstein in l906.

    August Mobius seems to have been the first mathematician to speculate
 about a fourth spatial dimension. In 1827, noticing that a silhouette of a
 right hand can be turned into a silhouette of a left hand by passing the
 figure through a third dimension (one higher than the figure itself), Mobius
 states that a space of four dimensions would be needed to turn right-handed
 three-dimensional crystals into left-handed crystals.  In the l850s, two
 papers by the Swiss mathematician Ludwig Schlafli determined all the regular
 four-dimensional figures, including the star-type figures, computed their
 numerical and metrical properties, obtained both the four-dimensional pattern
 of hypercubes and the fifth dimensional cube, and invented an elegant means
 of describing these figures. However, his work remained little known, and
 credit for many on these discoveries was erroneously given to the American
 mathematician William Stringham. Stringham's l880 paper in American
 Journal of Mathematics was less complete than Schlafli's work, but it
 did include, and for the first time, illustrations of many four-dimensional
 figures. (fig 2) These visualizations propelled the study of
 higher-dimensional figures as nothing had done before, captivating the
 imagination of mathematicians and non-mathematicians alike. Also about this
 time, the mathematician and philosopher Charles Hinton became convinced that
 people could be trained to see the fourth dimension of space and that an
 intellectual and spiritual liberation would follow such experience.  Hinton's
 technique was to study the changing appearance of color-coded hypercubes -the
 four-dimensional analog of the cube - as they passed through (were sliced by)
 the three-dimensional space of our normal experience.

    In l884, the English clergyman and educator Edwin Abbott published
 Flatland: A Romance of Many Dimensions. Based on a simplified
 version of Hinton's technique, two-dimensional creatures are visited by a
 sphere from the land of three dimensions. Flatlanders attempt to comprehend
 the third dimension from the succession of two-dimensional slices of the
 sphere that appear in their Flat Land. Abbott's parable is of such charm and
 coherence that even today it initiates people into higher-dimensional
 speculations.

    At the beginning of the twentieth century, Henri Poincare, even then
 France's most esteemed mathematician and thinker, repeatedly discussed the
 possibility of perceiving the fourth dimension, giving credibility to the
 earlier speculations. In 1902, in Science and Hypothesis, he
 speculated that geometry does not exist a priori in the mind,
 nor is it derived from the experience of the senses. "None of our sensations,
 if isolated, could have brought us to the concept of space; we are brought to
 it solely by studying the laws by which those sensations succeed  one
 another.... Well, in the same way that we can draw the perspective of a
 three-dimensional figure on a plane, so we can draw that of a
 four-dimensional figure on a canvass of three (or two) dimensions...[And by
 studying the "group" of these perspectives] we may say that we can represent
 to ourselves the fourth dimension." (page 58 -70 of the Dover edition).

    At the same time Poincare was writing, the period when modern art was
 being invented, this idea of a fourth dimension of space was widely known by
 artists. Research by art historian Linda Henderson proves conclusively, with
 page after page of documentary evidence, that the desire to create
 four-dimensional art was the motivation and philosophical support for those
 daring leaps into abstract art and modernism in general.

     Henderson's work shows that the fourth dimension was all the rage in
 Paris by the second decade of the twentieth century.  She has discovered an
 issue of the literary and art magazine Comoedia from  l912, the
 first page of which is divided evenly between a review of "le Salon des
 Independants" and Gaston de Pawlowski's Voyage au pays de la quatrieme
 dimension. Moreover, in 1909, when Scientific American
 magazine sponsored an essay contest for "the best popular explanation of the
 Fourth Dimension," they received 245 entries from all over the world.

    The artists who invented modern art did not depend on watered-down
 popularizations of four-dimensional geometry, however. Henderson tell us
 about Maurice Princet, a mathematician working as an actuary for an insurance
 company, who taught the new four-dimensional geometry to his artist friends
 -Metzinger, Gleizes, Appollinaire, Delaunay, Duchamps, Gris, Villon, and
 others including, probably, Picasso. Further, in 1903 E. Jouffret published
 in France an elementary text of four-dimensional  geometry that included
 illustrations. These "perspective cavaliere" illustrations look much like
 early Cubist works such as  Marcel Duchamp's Nude Descending a
 Staircase(fig 3 & 4). In fact, Duchamp quotes Jouffret in his notes.

    After investigating the development of every modern art movement in
 Europe, Russia, and America, Henderson concludes:

During the first three decades of the twentieth century, the fourth dimension
 was a concern common to artists in nearly every major modern movement:
 Analytical and synthetic cubists (as well as Duchamp, Picabia, and Kupka),
 Italian Futurists, Russian Futurists, Suprematists, and Constructivists,
 American modernists in the Stieglitz and Arensberg circles, Dadaists, and
 members of De Stijl.  While the rise of Fauvism and German Expressionism
 preceded the first artistic application of higher dimensions by the Cubists,
 Matisse himself later demonstrated a passing interest in the subject.  And,
 even though the German Bauhaus was not an active center of interest in the
 fourth dimension, it, too, was touched by the idea through the propagandizing
 of Van Doesburg, Kandinsky's own awareness of the idea, and the growing
 interest in Germany in the space-time world of Einstein.

                            Henderson.  page 339

      Poincare, as we have seen, thought it was theoretically possible to see
 the fourth dimension but only by somebody "who will devote his life to it."
 (page 51) This pessimistic appraisal was reinforced in l947, by the great
 geometer H. S. M. Coxeter who wrote: "Only one or two people have ever
 attained the ability to visualize hyper-objects as simply and naturally as we
 ordinary mortals visualize solids." (Regular Polytopes p. 119) But only a
 quarter of a century later the psychologist of perception Heinz Von Foerster,
 in reporting on perception experiments done at the University of Illinois at
 Champaign/Urbana in 1971 and 1972, states unequivocally that any subject can
 learn to see four-dimensionally to the extent that they "show no surprise
 about the results of legal four-dimensional maneuvers, can spot
 inconsistencies in figures,...can perform tasks in four-space, can anticipate
 the results of rotations in four-space...can manipulate one object behind
 another or into another in four-space, as in a 'hyper-Soma cube', etc." (lab
 reports from the Biological Computer Laboratory, University of Illinois,
 Urbana. 71.2 and 72.2)

    The twenty-five-year span between Coxeter's pessimistic statement and Von
 Foerster's bounding optimistic statement about the human capability of
 visualizing four dimensions was the period during which computers were
 invented; Von Foerster's subjects were training on a real-time interactive
 computer, and carrying out precisely the study program Poincare imagined.
 Unfortunately, because of a lack of funding, Von Foerster could not gather a
 large enough sample to publish final results, but informally he has stated
 that over and over again his subjects could satisfy his elaborate protocol,
 and could see the fourth dimension according to his strict definitions. On
 the disk that comes with this book is my program Hyper, which
 re-creates Von Foerster's training system.

    It is a human capability to be as comfortable seeing four
 spatial dimensions as seeing three. For 100 years, sensible people have tried
 to learn to do this. Just in the last few years have computers been able to
 aid in this effort. And computers, by presenting us for the first time with
 the actual moving images projected in our three dimensional space, are
 miraculously effective learning tools. Only after you see four-dimensional
 space, can you hope to make four-dimensional art. Such four-dimensional works
 of art, if taken to heart, can change the way we see.


    A two-dimensional object can rotate about a point; a three-dimensional
 object can rotate about a line; a four-dimensional object can rotate about a
 plane.  It is this extra dimension of freedom of movement that distinguishes
 a four-dimensional object from a merely intricate three-dimensional object
 and reveals its true properties. Consequently, it is planar rotation that
 should be the focus of our attention when considering four-dimensional
 figures. To be sure, from the point of view of the person accustom only to
 observation in three dimensions, the properties of planar rotation are
 mysterious, even paradoxical (shapes appear and disappear, turn inside out,
 flex and reverse); but these paradoxes become the very means by which we see
 the fourth dimension.

    Since planar rotation is the single most important property of
 four-dimensional figures, and is, in fact, the only way we can distinguish
 four-dimensional figures from merely intricate three-dimensional figures,
 works of art that propose to be four-dimensional must embody planar rotation.
  Anything else is like trying to live in an architect's rendering: titles
 notwithstanding, the work will not provide the opportunities advertised; you
 can not sleep in the drawing marked bedroom or eat out of that little box
 labeled refrigerator.

    I have discovered that by using a combination of two-dimensional and
 three-dimensional objects taken together as a single entity, the visual
 information of planar rotation can be presented, and the fourth dimension can
 be brought out of the computer. The strategy is to depict hypercubes,
 individual cells of which are made with painted lines on canvas and others of
 which are sculpted in welded steel rod of the same thickness and color as the
 painted lines. The three-dimensional cells parallax (have a changing aspect
 as you walk around them).  But the two-dimensional cells (which are perceived
 as three-dimensional elements) do not parallax; they seem to remain fixed and
 only move relative to the sculpted cells.  The viewer sees a figure that
 partially rotates and partially does not, a property of planar rotation
 familiar to us from the computer-generated rotations.

         Rather than spend a year or two making smaller works, I wanted to
 celebrate my discovery with a gigantic work. After three months of drawing I
 exploded with an 8 1/2 ft by 27 ft work that I called
 Fourfield. (figs  5,6,7,8).

    All the paradoxes of planar rotation are vividly present in
 Fourfield.  Cells are hidden behind planes, and a slight
 movement on the part of the viewer spins a space out of a place where there
 was only flatness.  Parts of geometric figures taken to be rigid structures
 move relative to each other; they flex even though the art work has no moving
 parts. Sometimes the painted cell is more spatial than the sculpted cell; it
 often  appears to be the moving one, providing that special relative motion
 of planar rotation where some cells of a hypercube move clockwise and others
 counter-clockwise.  Finally, these seemingly rigid geometric objects pass
 through each other with the paradoxical ease of shadows of a skeletal cube
 swimming through each other.



    To find a car in the parking lot only a space of two dimensions is needed,
 and to conceptualize the parking lot as a space of three dimensions is a
 needless complication. However to pick apples from a tree with a two
 dimensional map is also hopeless: the instruction to pick the apple on the
 outside on the left means nothing unless the height of the branch to be
 picked is specified. And rather than a messy pile of two dimensional maps,
 one for each height of branch, it is more direct and simple to have a three
 dimensional model of the tree, even if that model is molded out of air by the
 person instructing the picker. People seem to need to deal in  "reality"
 rather than "models." The parking lot is "really" two dimensional, though of
 course nothing physical can have only two dimensions. Likewise no one leans a
 ladder against an apple tree without knowing that it is "really" three
 dimensional.  Now Physics teaches that many objects in the universe are best
 described by a geometric model that is fourdimensional. We can consider that
 model real, and being human it must become real for us if we are to work with
 it efficiently.

    Consider the shadows on the wall of Plato's cave. These shadows resulted
 from dancers behind the viewer's backs, dancing in front of a fire. the
 spectators are forbidden to turn and face the dancers. But with a
 sophisticated mathematical system that could keep track of the changing size
 of the fire and its distance from the wall, the viewer's changing distance
 from the wall, the curvature of the wall itself, the changing size of the
 dancers and their changing proximity to the wall and to the fire--if we could
 keep track of all this, then the phantasmagorical images on the wall could be
 seen to result from the integral beings of dancers.  They have no trouble
 accepting the dancers as real rather than as mere bookkeeping devices.  Why?
 They see only their shadows and must learn to see their full
 three-dimensional forms from their moving shadows alone.  Yet they accept the
 dancers as real nevertheless because the dancers are the integral (the
 symmetrical) origin of the otherwise disorganized shadows, and because these
 is nothing sacrosanct about a geometry of just two dimensions.

    We can speculate with confidence about the experience of the dwellers in
 Plato's cave because, outside the cave, we make a similar leap of faith. The
 light from objects that reached our eyes comes in a rapid succession of two
 dimensional sheets, but we "see" the three dimensional objects not their
 changing two dimensional colored shapes.  It is true for us that out visual
 experiences are backed up by tactile ones, and we can later feel the solidity
 of what we see. But we must recognize that the human body has no mechanism
 for directly seeing the "reality" of the third dimension of objects.

    Now, in Einstein's cave, the shadows we observe are three-dimensional;
 they have mass, exist in time, and obey physical laws.  But these shadows,
 too, are phantasmagorical.  Their shapes, masses and internal clocks are
 different for just about every viewer, owing to relativistic effects.  For
 some viewers, the physical attributes of the shadows are vastly
 different, and sprouts grow into huge oak trees that wither from old age all
 in the time it takes to drink a cup of coffee.  To be faithful to our
 philosophical traditions, we should say that the four-dimensional, integral,
 symmetrical origins of these multiple and contradictory shadows are real and
 that this reality is present to us via its projections.  We must break the
 habit of considering the four-dimensional entities to be merely artificial
 constructs used in physics.

    In the caves, we are forbidden by authority to turn and face the dancers
 directly, but in fact authority has no real power over us in this matter. We
 have the ability to see the dancers in their full dimensionality -to accept
 the cultivated experience of seeing the fourth dimension as being "out
 there," and it is our choice to do so. Failing to make this choice handicaps
 our ultimate understanding of reality. Our ability to apply four-dimensional
 geometry as a useful template for experience connects us to the multiplicity
 of spaces and points of view that implode upon us every day. If culture can
 teach us to see the third dimension as real, then just a little more culture
 can teach us to see the fourth dimension as also real.



e the third 