Michael Macrone
Articles & Essays


The Fourth Dimension

A Dream About Time and Space

By Michael Macrone

On Mathematics

MATHEMATICS IS THE study of a language that comprises symbols, expressions, and axioms—the letters, phrases, and syntactic suppositions by which it renders ideas readable. Mathematics is a philosophy, a system with rules of deduction and induction, logic and intuition. Mathematics provides order to imagination and structure to reason.

It would seem as though we had before us, as a reward for all our toils, a country still undiscovered, the horizons of which no one has yet seen, a beyond to every country and every refuge of the ideal that man has ever known, a world so overflowing with beauty, strangeness, doubt, terror, and divinity, that both our curiosity and our lust of possession are frantic with eagerness. —Nietzsche, Ecce Homo

The world is larger than any single vision of it, larger than the union of all our personal worlds. The world is not ours to make but to encounter. We perceive the world from either within or without the “normal mode of vision.” There are many ways of escaping the confined perspectives of daily existence; mathematics is one.



Mathematical dimension is the number of perpendicular directions in a space. A point has dimension zero because it has no direction. A line is one-dimensional; it runs in one direction, backwards or forwards. A plane is two dimensional; every point may be uniquely described as the endpoint of a sum of multiples of two fixed perpendicular vectors, or directed line segments. All points in the plane may thus be associated with number-pairs—this is the significance of the Cartesian and polar coordinate systems. The world of mundane physical experience is three-dimensional, recognizing directions front and back, left and right, up and down.


Dimension is by definition a limitation; all numbered dimensions exist within an infinite dimension, a dimension where the possibility of motion is completely unlimited. Living in a finite dimension means being restricted from moving into the other dimensions. The higher the dimension, the greater the freedom of motion.


As the mathematician approaches the limits already achieved by study, the colder and thinner becomes the air and the fewer the contacts with the affairs of every day. The Promethean fire of pure mathematics is perhaps the greatest of all in man’s catalogue of gifts; but it is not most itself, but least so, when, immersed in the manifoldness of phenomenal life, it is made to serve purely utilitarian ends. —Claude Bragdon, Four-Dimensional Vistas (1916)

Extrapolation is a tool we use to learn about the world beyond our own. It is a rule of induction which assumes that the transcendental spaces (hyperspaces) operate under rules analogous to those in known spaces. The assumption of such order in hyperspace is ultimately unfounded; no one has actually visited higher dimensions. But extrapolation is an indispensible technique for the mathematician, who after all suspends notions of “applicability” when theorizing. It is of interest only after the modeling process whether the model is adequate in application. Mathematic models are deliberate constructions, imaginative concoctions; they do not mirror a reality but rather create one.

The ability to visualize, to comprehend intuitively, the “wholly other” world of higher space is given in each century only to a few chosen seers. For the rest of us, we must approach hyperspace indirectly, by way of analogy. —Martin Gardner, “The Church of the Fourth Dimension” (1969)

The Fourth Dimension

The expression, the fourth dimension, offers a shock to the mind accustomed to practical handling of matter, because all our experiences of measurement are ultimately founded upon matter possessing but three dimensions, so that we have great difficulty in accepting the reality of a direction not contained in our space or our matter but definitely at right angles to every line that can be drawn within the matter and space which contain all our ordinary experiences. —Bragdon, A Primer of Higher Space (1913)

The fourth dimension contains one direction perpendicular to every direction in three-space. It is a direction that, without superseding normal experience, we cannot imagine. Using extrapolation, we may put the unknown in terms of the known.

Suppose by analogy that we lived in a two-dimensional space, as shapes in the plane, drawings and figures, infinitely thin. We would have no notion of motion up or down; we would perceive motion only left to right, or front to back, or in combination of these. If some three-dimensional alien descended upon our world, we would only see a cross-sectional image, or “slice” of the being, that is, its intersection with our plane world, its extension into the directions we know.


Suppose then that this being were in the shape of a sphere; it would be very much like the “spheres” in the plane, that is, circles: loci (collections) of points equidistant from a fixed point. The plane slices, or manifestations, of a sphere are concentric circles; as the sphere passes through our planar world, we flat creatures would see a waxing then waning progression, growth then shrinkage—from a point to a great circle and then back to a point.

The dimension of time is our only access to this transcendental entity—we are allowed to see only a succession of pieces of it, and never the thing all at once—never the thing fixed in time.

And so it would be for us if a sphere in four-dimensional space were to pass through the three-space we inhabit. We would see only a succession of “slices” (concentric spheres), a sequential pattern of growth and regression.


A Dream About Time

IT IS INSTRUCTIVE TO THINK of the role of time in the mundane world in terms of spatial manifestation—change. The mathematics of dimensionality suggest that change might be a sequence of successive cross-sections of a discrete, transcendental phenomenon, which, if we could expand our limited perception, would appear in toto, timeless and whole. Could we then not better understand that phenomenon?

Time present and time past
Are both perhaps contained in time future,
And time future contained in time past.
If all time is eternally present
All time is unredeemable.
What might have been is an abstraction
Remaining a perpetual possibility
Only in a world of speculation.
What might have been and what has been
Point to one end, which is always present. —T. S. Eliot, “Burnt Norton” (1936)

And Space

Beings living in a plane can only see the boundaries of other two-shapes and not penetrate within them. To see the boundary of a shape, the 2-D individual must “walk around” it. To see inside, he must be let in through a rip, a door, in the shape.


We in the third dimension, however, may perceive at once the whole of the shape in the plane and its contents.

So too would a 4-D individual see our houses and their furnishings at once, our bodies and hearts simultaneously; from the fourth dimension, our space is entirely penetrable, our most hidden secret laid bare: nothing is concealed.

Two Left-handed Mittens

Suppose you were a planar being in a predicament: winter is setting in and all you have for protection is pair of left-handed mittens. Of course, there’s nothing you can do, short of turning one of the mittens inside-out, to reverse its orientation; no matter how you turn it in the plane, it remains left-handed.


Luckily, a benevolent three-entity lends a helping hand: he picks one of your mismatched mittens out of the plane, into the third dimension, turns it around, and gives you back a mirror-image right-handed mate to your other mitten. You could Interpret what you see of the matter only as heavenly intervention: a useless mitten disappears and seconds later, a needed one appears in its place.

The scenario would be similar in higher space: a left-handed three-mitten cannot be reoriented into a right-handed mitten. However, if a benevolent four-entity took pity, he could pull it out of our space, into the fourth dimension, reverse it, and give us back a right-handed mitten. What seems a miracle is explained by mathematics.


The image in a mirror is an unattainable, symmetric reorientation of our space: our world lifted from the third dimension and inverted. To get through the looking glass, Alice went through the fourth dimension.

Thomas Banchoff

A man who devoted his life to it could perhaps succeed in picturing to himself a fourth dimension. —Henri Poincaré, Science and Hypothesis (1905)

TWELVE YEARS AGO, Professor Thomas Banchoff, currently chairman of the mathematics department at Brown, developed a way to show us through the mirror. Ever since his childhood encounter, in a Captain Marvel comic book, with the fourth dimension, a place of dreamscapes and exotic power, the topic has fascinated Banchoff; it has become his avocation.

The fourth dimension occupied Banchoff’s reflections throughout his school days. At the University of Notre Dame, he wrote a term paper on the fourth dimension and the Holy Trinity. His teacher couldn’t understand it. Before that, at high school in hometown Trenton, Banchoff pursued his interests by reading what books he could find and by writing on topics such as the mystical implications of encounter with the fourth dimension, and the fourth dimensional nature of heaven. He was searching for models to explain the supra-normal and the paranormal, models to enlighten spiritual philosophy.

Photo of Thomas Banchoff by Rob Seidenberg

Tom Banchoff is a geometer; that is, he specializes in the properties and measures of configurations in space: sets of points, loci, shapes, and solids. In his courses, therefore, he often employs visual images to illustrate and enlighten theory. He says: “A geometric model gives us something to look at, a way to see into an algebraic problem.” The right picture will do more: “A certain view of the model may provide crucial insight; it may suggest new relationships, new proofs, and new methods of proof.”

Mathematics is founded on the vital interplay of algebraic intuition (by which we organize material logically) and geometric intuition (by which we organize material visually). Banchoff does his best work by playing algebraic or combinational theory off of geometric models.

“I’ve always been concerned with different ways of modeling things.” Models are at once a means of rendering the abstract visible and of breaking open and reorganizing our way of interpreting phenomena. They illuminate that which lies beyond “normal” experience, clarifying the obscure and picturing the unseen.

Banchoff, in his fascination with a world one step beyond our own, needed to develop adequate models, find the right pictures, to get a footing in that world. To show what the fourth dimension looks like, he needed to do better than static drawings of its shadows.


A life drawing is really the drawing of shadows. Whenever we depict a solid object on a flat page, we are presenting a shadow, or projection, of it.


By extrapolation 4-D solids cast 3-D shadows on our space, of which we may construct models, or which we may draw.



In 1969, Banchoff learned of the thesis project of Charles Strauss, a Brown Ph.D. candidate in computer science, who had modeled highly complex piping configurations for industrial design. His work required models of three-dimensional geometrical configurations changing in time and therefore depending on four variables; the computer program Strauss produced provided a way of visualizing projections from higher space. He had a powerful technique and was looking for new problems—Banchoff had the problems and needed a new technique.

The two took the hardware at hand, an 1130 computer with a storage tube, and concocted the requisite algorithms to yield a working image generator. Later, a Vector General video scope hooked into a META-4 computer, jacked up with a parallel processor and SIMAL-E matrix multiplier (constructed at Brown by Hal Webber), allowed more sophisticated animation, by which Banchoff and Strauss could show moving pictures of the 3-D projections, the shadows, of 4-D objects.

The Vector General has recently been replaced by a RAMTEK video screen, which operates very much like a television tube. It produces images less quickly then the Vector General, but allows for a finer picture and for solid colors with more subtle gradation.

The most effective picture of the fourth dimension emerges from multiple perspectives. Using animation, Banchoff has produced films displaying a sequence of shadow-pictures; these displays give the viewer a continual procession of perspectives of a given model and enable him to “see” what the fourth dimension “looks like.”

Banchoff features films of computer generated images at his frequent lectures on hyperspace and computer animation; using moving models, he has allowed his audiences entry into the fourth dimension.

The Hypercube

“The Hypercube” film has become standard at Banchoff’s presentations. A hypercube, or tesseract, is the 4-D analogue of a 3-D cube. In the film are depicted animated representations of a rotating tesseract and pictures of its “space slices.” The images are pictures of shadows—2-D depictions of the projections of a hypercube onto three-space.


As a square might be seen as the evolution of a line segment through two-space, the cube as the evolution of a square into three-space, so the hypercube might be seen as the evolution of a cube through four-space. We construct squares from four (two times two) congruent line segments, cubes from six (two times three) congruent squares, and thus hypercubes from eight (two times four) congruent cubes.


The Fourth Dimension and Art

EARLY IN 1975, THE Washington Post published a feature on Banchoff on the front page of its “Style” section including a stylish photo of him standing before Salvador Dali’s The Crucifixion (Corpus Hypercubus), holding a model of the “unfolded hypercube.” The Dali, which Banchoff calls his “favorite Surrealist painting,” depicts Christ nailed with cubes to just such a decomposed four-cube. Dali was alerted to the article, and soon after phoned Banchoff; ever since, the two have met almost every year to chat and trade notes.

Theories of the fourth dimension were naturally attractive to the Surrealists, but their work largely postdates the moment of impact of hyperspace with the art world. The fourth dimension captivated French artists and thinkers at the beginning of the twentieth century, spurred by the volatility of mathematical and physical revisions. Inspired by the current writings of Guillaume Apollinaire and the mathematician Poincaré, early Cubists such as Gleizes and Metzinger began applying the “new geometry” to their theories of art. Surrealist author Alfred Jarry, Pablo Picasso, and Marcel Duchamp (among others) began practically applying notions of hyperspace to their work. Duchamp, in his notes for The Bride Stripped Bare By Her Bachelors, Even (Large Glass), refers frequently to the fourth dimension of space; in fact, the “bride” is a four-dimensional creature, and the glass “a mirror of the fourth dimension.” His Nude Descending a Staircase depicts a series of solid cross-sections of the nude, like superimposed time-lapse photos, forming a “4-dim’l continuum.” The Analytical Cubist works of Picasso and Georges Braque break down three-space and compose a series of multi-angular images, essentially creating a four-dimensional vision of the subject. In four-space, all angles of a three-solid are at once perceptible.

Russian Futurists and Suprematists, such as Peter Ouspensky and Kazimir Malevich respectively, independently developed their approach to the fourth dimension prior to 1920. Russian artists and thinkers were more ignorant of geometric models than were the French; they were captivated more by the possibilities of unfettered vision promised by an intuition of four-space. In the philosophical work Tertium Organum, Ouspensky presents the fourth dimension as “a key to the enigmas of the world”; with an understanding of hyperspace, the artist is able to tap into the “noumenal world,” the spiritual manifold of which our "phenomenal world” is but a shadow. James Billington, in The Icon and the Age: An Interpretative History of Russian Culture (1968), identifies in the writings of Ouspensky and Malevich “the belief that man—when fully aware of his true powers—is capable of totally transforming the world in which he lives.”

Ouspensky corresponded with an American draftsman/architect named Claude Bragdon. Bragdon, like the Russian (and W.B. Yeats, Piet Mondrian, and Wassily Kandinsky) had been interested in the cultish mysticism of Theosophy, by which the natural world is interpreted as essentially spiritual. Also like Ouspensky, he carried his spiritual fixations into his mathematical speculations. Bragdon published a series of books which, aside from their pseudo-mathematical speculations on mystical phenomena, promoted a widespread conception of the fourth dimension. Extrapolations upon the fourth dimension had appeared somewhat earlier in popular literature, beginning with the publication of the groundbreaking Flatland.

Writings on the Fourth Dimension

FLATLAND BY “A SQUARE,” né Edwin Abbott Abbott, is the largest selling paperback in the Dover Press catalogue. It sells more copies in Providence than anywhere else, a fact largely the result of the missionary work of Professor Banchoff. He invariably wields a copy of the book at his lectures, shaking it at the audience in exhortation.

Flatland, a fantasy about extrapolation and a vision of the fourth dimension, was first published in 1884. There had been such speculative works before, and there have been many since; Abbott’s book is special because it took the speculations and made them literature. A fantasy about a flat world inhabited by regular polygons is Abbott’s springboard to a Swiftian vivisection of the prejudices of his age; extrapolation, exploration beyond the self, is Abbott’s theme.

Abbott was a jack-of-all-trades and a schoolmaster, a noted scholar of Shakespeare and Bacon, a mathematician and a philosopher. You might today find Flatland in the science fiction section of the bookstore, but it is fantasy that barely masks philosophy.

The philosophy of extradimensionality has a history which stretches back to ancient Greece. Plato’s Republic, book VII, begins with the famous allegory of prisoners, chained to the wall of a cave, confusing the passing shadows for the bodies that cast them. Socrates postulates that mundane objects are actually the shadows of transcendental essences, which mathematics can locate in the fourth dimension. Aristotle as well toyed with the idea of hyperspace but rejected it as impossible. References to the fourth dimension also appear in the mathematical works of Euclid.

No serious speculation was accorded the fourth dimension until the nineteenth century, although there had been passing references, for example, in the writings of Descartes, Henry More, and Kant. In 1827, the geometer Möbius theorized on rotation in the fourth dimension; later, tracts were forthcoming from a scatter of mathematicians, physicists, philosophers, and mystics: Johann Zollner, Herman Shubert, Alexander Dumas, S. Newcomb, and others.

The two greatest modelers of the fourth dimension after Abbott were C.H. Hinton and Henry Parker Manning: the former taught at Princeton and the latter at Brown. Hinton began publishing in Scientific American and Harpers at the turn of the century and wrote his most enduring work, The Fourth Dimension, in 1904. Hinton’s A New Era of Thought was a model for Peter Ouspensky’s Tertium Organum, both exploring implications of Platonic and Kantian notions of essence. H.P. Manning’s Geometry of the Fourth Dimension (1914) and The Fourth Dimension Simply Explained (1909), which he edited, were the definitive works on the topic in their time.

Since the First World War, speculations on the fourth dimension have cropped up in science fiction, occult writings, the works of Borges and Castaneda, comic books, and television programs, as well as in mathematics and physics texts. Clearly, the idea of hyperspace appeals to a broad range of fancy, engaging the imagination of almost anyone who wonders about the limits of time and space.

Where It Leads

The fact that the number of persons who are experiencing abnormal states of consciousness, many of which are easily accounted for by the assumption of the world of four dimensions, is increasing year by year, would indicate that the “four-dimensional consciousness” is just beginning to appear in a more general way in human evolution. —C.L.B. Schuddemagen, The Forum vol. L (1913)

MATHEMATICS SEEKS metaphor. It is a code, a system of models, endowed with modes of postulation. The mathematician seeks expression through this language, expression of the nature of space and the nature of time. When the world is brought into alignment with mathematical models, a door of perception has been opened: we may extrapolate a broader view of our situation, a view entirely less confused or obscured.

endcap-345x333 The question of the fourth dimension is a question of our most fundamental perceptual dispositions. The fourth dimension is the resting place of that which we by nature desire: vision without barriers, perception without dilution. Mathematical imaginings are human imaginings, bestowed thus with the power of providing a picture of that which lies beyond us.

Michael Macrone is concentrating in semiotics and mathematics and occasionally kicks booty.

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First published in Issues (February, 1982)

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