Your physics teachers have been lying to you.
All throughout the day we are engaged in motion through space. We move forward. We move backward. We do the lambada.
But, to understand the fundamental structure of motion, we must also consider motion with regard to time. This dimension -- motion with regard to time -- is the fourth dimension. A quick example, if I may.
A man may reside on 10th street, on 6th avenue, on the third floor of the I_B Hotel.
Now, if one was to meet that man at 1:00 A.M at his room, their fixed position would be: (10 Street, 6th Avenue, 3rd Floor).
Also important is the time that person was at the hotel, which would be the forth piece of information, which would be 1:00 A.M.
See The fourth dimension (no, really) for an unabridged version.
Imagine a point - a singularity. That's the 0th dimension. Only one point exists. Everything within this universe exists at exactly the same place.

Now run a coordinate axis through the point. Suddenly you've got a line. Everything within this coordinate system has a single coordinate - we'll call it the x coordinate, and we'll call the axis the x axis. That's the 1st dimension. You probably know it as a number line from algebra.

Now run a new axis perpendicular to the x axis. We'll call it the y axis. Now everything addressable has two coordinates, an x and a y coordinate. This is 2-space, a plane, the 2nd dimension. We can put points anywhere on this infinite plane.

Now we're going to run an axis perendicular to the x and y axes. We'll call this new axis z. We now have 3-space, the 3rd dimension. We can address points in space - we can describe a cube or a sphere.

Now we will induce a mindfuck. Run a fourth axis perpendicular to your x, y, and z axes. We'll call this the w axis. Impossible, you say? Can't find a directon that will place this new line perpendicular to all the others? That's because you probably cannot conceive of this direction. You can't move in it, you can't see it, and you probably can't think of it in anything but an abstract way because you have no experience with it. Depending on which physicists you listen to, this dimension is either too damn small (see string theory), only gravity propagates in its direction (will try to find reference), or it doesn't physically exist. This is the fourth dimension. It may or may not have physical relevance, but mathematically it's completely valid, and even necessary for modelling certain things in 3-space such as temparature in a solid (anything that's a function of 3 variables can describe a four-space curve).

You can continue adding as many new dimensions as you please, without compromising validity. AFAIK the greatest number of physical dimensions that have been seriously speculated is seven (again, they are too damn small for you to interact with), but the magic of math lets you deal with n-space. Mathematically, adding more dimenstions is sufficiently trivial that textbooks may introduce extra dimensions without fanfare.

See also: Flatland, Flatland: A Romance of Many Dimensions, vector calculus

As we are beings of the third dimension we can both see and understand the 0th dimension, 1st dimension, and 2nd dimension. We can understand the 3rd dimension, in which we live but cannot see it, as we can the other lower dimensions. And as we are relative to the third dimension we cannot, in theory, conceive of a dimension beyond that which is three-dimensional.

It has been speculated that this 4th dimension is time in that all things in our world are either restricted by time or functions of it. However, in order to both see and understand the 3rd dimension we would have to no longer be restricted by time (for it is time that restricts us from being at both one point in space and another simultaneously).

Without the restriction of time we could literally be in all places at once, and that would not be time but more the absence of time, or eternity. Eternity is something we cannot comprehend as humans because we operate inside of time, just like the creatures in flatland could not comprehend height and depth, so we cannot comprehend eternity, the fourth dimension.

A group of tadpoles swim around in their very shallow puddle of a home. Their world is two dimensional. They can move widthwise, they can move lengthwise. They have no concept of depth, or height, but they are quite aware that it exists. They can see the sky above them, they can feel the warmth of the sun in the water. However, they cannot full understand the third dimension, the dimension of depth.

We humans are accustomed to the three dimensional world. We understand the depths of water, the altitudes of air, even the deepness of space. But what if we, like the tadpoles, are missing something that we cannot comprehend?

It is frequently believed that the fourth dimension is Time. Some would believe that time is simple to comprehend, that we can measure it, count it, and witness it. However, we are really only feeling the effects of time. Unlike the other three dimensions, we cannot simply point at part of time. We cannot traverse time, as we can length, width, and depth. Time is always in the present, we cannot move forward through time, we cannot move backwards through time.

We already have a sort of influence of time. Time is not absolute. Gravity has an effect on time, making it slower the closer you are to the surface of a large mass. The difference is very minute, but if you were to travel away from the Earth at near lightspeeds for some time, and then return, you will find that you have aged normally, but time on Earth has passed many years. Therefore, time is not just a static constant, but is dependant on where you are, and the speed you are moving.

It isn't possible to imagine a fourth dimension physically. Perhaps it is possible that we may one day be able to travel through time. Then, the 4th dimension may be open for us to explore. Or it could be quite possible that there is no other dimension, and that what we see is all there is. Or, after traversing time, there may be some higher plane, another level of existence, a fifth dimension.

Works cited and consulted:
Stephen Hawking, A Brief History of Time
Stephen Hawking, The Cambridge Lectures
Ken Grimes and Alison Boyle, "The Universe Takes Shape" Astronomy Magazine

One fine day while attending college, I was trying to exercise my mind for calculus by visualizing 3-dimensional objects (we were working on multivariable equations). I concentrated on giving the objects mass, texture, and color, and after a few minutes grew quite comfortable doing this.

I began to wonder if I could perhaps visualize a four-dimensional shape...something I'd been trying to do for years. This train of thought was prompted by a mental exercise in which I attempted to unfold a cube into panels, then fold it back up again.

As I mentally considered the unfolded cube, I remembered a graphic from a book I own called The Fourth Dimesion, by Rudy Rucker; the graphic shows a three-dimensional unfolded cube. Holding this figure in my mind as clearly as I could, I attempted to "fold it up" into a hypercube.

I let the figure do what it wanted to in my mind's eye--shift and bend and change its orientation several times. I could practically feel the gears grinding in my brain, but for some reason I felt fairly confident that something would come of this visualization exercise.

For one brief, shining second, what I consider to be a glimpse into the fourth dimension became available to me.

I was basing my visualization on the three dimensional analogy: if you look at a cube from the vantage point of the third dimension from one side, you see a two-dimensional square. Therefore, I figured that if you looked at one "side" of a hypercube from the vantage point of four-dimensional space, you might see a three-dimensional cube--but you'd see every surface of this cube simultaneously. (Kinky!)

My little splinter of enlightened vision showed me a glowing figure, one with astonishing depth yet still somewhat out-of-focus. This doesn't sound all that spectacular, but the actual experience was one of the most bizarre things I've ever felt. The hypercube faded in my mind's eye after a moment.

I have read that if one is trying to visualize in four dimensions, a sphere -- actually, a hypersphere, is easier than a cube to get a decent mental picture of. In my exercise, I tried to consider what a hypersphere might look like. In my calculus book, there's a chapter on three-dimensional surfaces being used to represent four-dimensional shapes.

The equation for a function of three variables giving a set of spherical level surfaces is given by: x2 + y2 + z2 = r2. Thinking along these lines, I tried to imagine the shape that these level surfaces might represent. The intuitive idea that came out of this thought process was a set of concentric solid spheres, varying in radius from infinitely small (a point) to the radius specified in the equation.

I was picturing a series of solid metallic spheres, all superimposed upon one another, yet occupying the same region of space without getting in one another's way. I then tried to "view" the situation from a fourth-dimension perspective: that is, have the spheres fully superimposed, yet still be able to see each sphere from all sides at once even as it was nested inside another sphere.

I can now do this visualization with some ease, though I don't know if it makes any real sense mathematically.

I remember reading once an anecdote concerning a jewel thief with the ability to travel in the fourth spatial dimension. By doing this, he was able to reach up and around the window of the diamond store and steal his loot. He did this by utilizing another spatial direction, a direction that is not left, right, up, down or a combination of any of these, but something entirely different that allowed him to reach into an area of space where the window simply did not exist.

My four-dimensional sphere visualization used the idea that I was looking at the object from a point where ALL the three-dimensional level surfaces were visible--I was not limited by the three-dimensional wall of every part of the object, because I was existing in a space where I had access to the "other" direction.

As interesting as this was, the most amazing realization I made while in the midst of my pondering was that I could understand why time and a fourth spatial dimension could be considered analogous entities. I could imagine that each concentric 3D sphere either existed at the point of a different four-dimensional spatial variable, or that each sphere merely existed for a different time interval.

A phrase coined by Guillaume Apollinaire to describe the particular style of the cubists.

At the start of the 20th century there were two popular interpretations of the fourth dimension. One was that it is time. This is fairly easy to imagine, and remains a common meaning for the term today. The second is that the fourth dimension is another spatial dimension. This is much harder, if not impossible, to visualize. Nevertheless, several illustrators attempted to. One of the most popular illustrations of this was the four-dimensional hypercube. Just as we can form a cube by folding a cross-shaped piece of paper consisting of six squares, so it was believed one could form a four-dimensional "hypercube" by folding a similar arrangement of seven cubes. (see Crucifixion, Dali, 1954) A mathematician, E. Jouffret, attempted to depict four-dimensional objects by drawing their projections on a two-dimensional plane.

These representations were fairly well established when Picasso and Braque invented cubism. While it would be incorrect to claim that they formed the basis of cubism, it would be equally incorrect to say they had no influence on it. In the years before he started cubism, Picasso met regularly with a group of friends who called themselves la bande à Picasso. Guillaume Apollinaire, a member of la bande describes the influence of the fourth dimension on cubism in his book Les Peintres Cubistes.
"Until now, the three dimensions of Euclid's geometry were sufficient to the restiveness felt by great artists yearning for the infinite... The new painters do not propose, any more than did their predecessors, to be geometers. But it may be said that geometry is to the plastic arts what grammar is to the art of the writer. Today, scholars no longer limit themselves to the three dimensions of Euclid. The painters have been led quite naturally, one my say by intuition, to preoccupy themselves with the new possibilities of spatial measurement which, in the language of modern studios, are designated by the term fourth dimension."1
In this statement, he recognizes the graphical representation of the fourth dimension in cubist art, and also the mental influence. The fourth dimension is representative of the infinite possibilities that the cubists sought. Apollinaire reaffirms this in La Peinture nouvelle: "The art of the new painters takes the infinite universe as its ideal, and it is to the fourth dimension alone that we owe this new norm of the perfect..."2 Another member, Maurice Princet had extensively studied Poincaré's writings and is widely recognized as having exposed the cubists to his work. Poincaré's book l'Science et l'Hyposthese, written in 1902, popularized four-dimensional geometry and is often linked to the cubists work through Princet. First hand accounts tell of Princet discussing problems of perspective and simultaneously representing objects from multiple viewpoints. This would be a consequence of the fourth dimension being a spatial one which acts as an "astral plane", from which an object of the usual three dimensions can be viewed from all sides simultaneously. (Just as in our three dimensional world we can see the entirety of a two dimensional object at once).

We can see this effect quite clearly in Picasso's paintings. In Les Demoiselles d'Avignon, 1907, the crouching woman's body is seen from behind while her head is seen from the front. Similarly, the two central standing figures are shown in a frontal view, but their noses are painted in profile. The painting also shows influences of Jouffret's projections of a four-dimensional ikosatetrahedroid on a plane. The rightmost woman's upper body fits into a diamond grid that is extremely similar to Jouffret's projections of 1903. (This is more clearly seen in Standing Nude with Joined Hands (Study of Proportions), 1907.) The faceting which has become synonymous with Picasso's name is also very similar to Jouffret's drawings, in which he superimposes his projections on top of each other in an attempt to display multiple sides of a polyhedron simultaneously. Though this may have been more a similarity in appearance than a direct depiction of the fourth dimension, it was certainly a rejection of three-dimensional perspective.

Later cubists, particularly Metzinger and Gleize show an even greater influence from Princet's lectures. Unlike Jouffret, who conceded to project his four-dimensional figures into two dimensions, Metzinger believed that the mind was capable of perceiving all four at once. In a style similar to Picasso's he shows figures from various perspectives in the same painting, though often with less fragmentation than Picasso used. In Le Gouter (1911), he varies perspectives as the viewer looks from one side of the woman's face to the other. The left side of her face is seen in a frontal view, her nose in a three-quarters view, and her right eye in profile. Of even greater interest is the bowl from which she eats. The left side of it is seen from the side, while the right side is seen from above. This is almost exactly the same problem posed by Princet in 1910:

"You represent by means of a trapezoid a table, just as you see it, distorted by perspective, but what would happen if you decided to express the table as a type? You would have to straighten it up onto the picture plane, and from the trapezoid return to a true rectangle. If that table is covered with objects equally distorted by perspective, the same straightening up process would have to take place with each of them. Thus the oval of a glass would become a perfect circle."3
Princet, in turn, took this from Poincaré, who writes, "We can even take of the same four-dimensional figure several perspectives from several different points of view. We can easily represent to ourselves these perspectives, since they are only three dimensions. Imagine that the various perspectives... succeed one another..."4 In 1880, a mathematician, W.I. Stringham, attempted to illustrate such perspectives. In 1910, two painters also made use of Stringham's work. The vase in Gleizes's Woman with the Phlox and the fruit in Le Fauconnier's Abundance bear a striking resemblance to Stringham's figures.

The futurists also used the term fourth dimension, but not in the same way the cubists did. While the cubist fourth dimension was spatial, the futurists' was temporal. Boccioni describes this in his Plastic Dynamism (1913).
"...Instead of the old-fashioned concept of sharp differentiation of bodies, instead of the modern concept of the Impressionists with their subdivision, their repetition, their rough indications of images, we would substitute a concept of dynamic continuity as unique form. And it is not by accident that I say form and not line, since dynamic form is a species of fourth dimension in painting and sculpture, which cannot exist perfectly without the complete affirmation of the three dimensions that determine volume: height, width, depth."5
This "dynamic form" he writes about is easily seen in his work. In The City Rises, 1910, the elongated brushstrokes create a blurred motion that is similar to the kind created when a photograph is taken of a moving object. His sculpture Unique Forms of Continuity in Space, 1913 perfectly captures the essence of motion, as the figure seems to liquefy and move forward through itself. Later futurist works, most notably Balla's "dynamisms" show a scene over the course of a period of time with much less distortion, and clearly demonstrate the idea of the fourth dimension being time. The futurist method was so effective that even Picasso reconsidered the possibilities of the fourth dimension. Around the same time as Balla's paintings, it was reported by Kahnweiler that Picasso "considered setting his pictures in motion using a clockwork mechanism or producing a series of works which could be shown in rapid succession."6

Marcel Duchamp also used the Fourth Dimension (and other topics from Poincaré's books). In 1911 he began meeting regularly with Princet, who, as mentioned above was a leading proponent of art of the "new geometries". A good example of the fourth dimension in his work is The Bride Stripped Bare by Her Bachelors, Even, 1915-1923 (often called Large Glass) In this work, Duchamp's goal was to depict the bride as four-dimensional, and the bachelors in three dimensions. The shapes which make up the bachelors' machine are textbook examples of geometric solids seen in one-point perspective. The bride, however, is composed of parabolic and hyperbolic forms which Duchamp considered idealized and typical of a four dimensional object's projection in three dimensions. Duchamp arrived at the fairly logical conclusion that because a three-dimensional object has a two-dimensional shadow, a four-dimensional object must have a three-dimensional shadow. This is the same reasoning that Jouffret used in his projections, and these were certainly Duchamp's inspiration. Although it is not readily apparent in Large Glass, Duchamp did extensive research into four-dimensional perspective. He likens the three-dimensional projection to "the method by which architects depict the plan of each story of a house"7 and continues to discuss how the four-dimensional object is constructed: "A 4-dim'l figure is perceived (?) through an ∞ of 3-dim'l sides which are the sections of this 4-dim'l figure by the infinite number of spaces (3-dim'l) which envelope this figure."8 Though these methods cannot be practically applied, they show the devotion which Duchamp gave to this idea. Some scholars have even gone as far as to claim that Duchamp's famous "ready-mades" have their roots in Poincaré's work. In an essay on mathematical thought, Poincaré describes how the unconscious mind cannot supply "ready-mades", but that it does constantly sift through ideas which the conscious mind can then select from. A related suggestion is that the photos he took of these ready-made objects from various perspectives were the result of his fascination with projecting higher-dimensional objects into lower-dimensional spaces.

The spread of the fourth dimension continued, and the depiction of it became more abstract as art did. The de stijl artists in Holland interpreted the fourth dimension as negative space. Van Doesburg used shades of gray to represent negative space, and the primary colors as positive space. This interpretation differs from earlier ones because it does not try to represent the fourth dimension a as a physical reality, but conceptually. Mondrian's appreciation for mathematics led him to his unique style of representing the fourth dimension. He believed that his use of colored planes "by both their dimensions (line) and values (color), can express space without the use of visual perspective."9 By eliminating perspective while maintaining the appearance of a three dimensional space, Mondrian has indirectly represented the fourth dimension. In a sense, color was Mondrian's fourth dimension. Van Doesburg continued using the fourth dimension after Mondrian abandoned it. He did this in a sort of natural continuation of Mondrian's work, by combining colored planes in three dimensional compositions. In Color Construction in the Fourth Dimension of Space-Time, 1924 he draws colored planes in perspective to create the four dimensional view which Mondrian denounced in 1918. Around this time, Van Doesburg also began experimenting with the fourth dimension as it was interpreted in Einstein's relativity theory, which was confirmed in 1919 and quickly gained popularity. Van Doesburg also applied the fourth dimension to architecture. In a plan for a house which he drew in 1923, he combined his three-dimensional colored planes with the idea of a hypercube. In this way he combined Mondrian's abstract notion of the fourth-dimension with the original, concrete notion of it. Van Doesburg explains:
"The new architecture is anticubic, in other words, its different spaces are not contained within a close cube. On the contrary, the different cells of space (balcony volumes, etc., included) develop excentrically, from the center to the periphery of the cube, so that the dimensions of height, width, depth, and time receive a new plastic expression. Thus, the modern house will give the impression of floating, suspended in air, in opposition to the natural force of gravity... The new architecture takes account not only of space but also of time as an architectural value. The unity of space and time gives architectural vision a more complete aspect."10
By eliminating gravity, Van Doesburg eliminated the idea of an absolute coordinate system in architecture. No longer was one direction defined as "down" and opposed by "up", nor did the words "left" or "right" have meaning. All directions were equal, and only their relative orientation to each other mattered. The complex shape of his buildings also required motion in time to view. Van Doesburg was also the only major artist of the time to embrace Einstein's relativity theory. While it may seem natural the popularization of relativity theory would lead to a popularization of the fourth-dimension in art, the actual result was just the opposite.

Why? Well it's hard to say. The driving goal of using the fourth dimension had always been to make an art form that was somehow more ideal, more perfected than previous works. With the realization of the fourth dimension, this idea had in some ways been confirmed, but the thrill of pursuing it was lost. Another thing that should be mentioned is the frequent interpretation of relativity as the basis for cubism. Picasso denied any scientific roots to cubism but his recollections often contradict history as well as each other. Einstein, however, flat out states, "This new artistic ‘language' (of cubism) has nothing in common with the Theory of Relativity."11 He seems to reject any connection between science and art, stating, "In science, the principle of order which creates units is achieved through logical connection while, in art, the principle of order is anchored in the unconscious."12

Works Cited:
1 Apollinaire, Les Peintres Cubistes, 1912, p. 15. Cited in Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art, 1983, p. 75
2 Apollinaire, La Peinture Nouvelle. Cited in Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art, 1983, p. 75.
3 Delaunay, 1957, p. 146. Cited in Miller, Einstein Picasso, 2001, p. 114.
4 Poincaré, La Science et l'Hypothese, 1902, p. 89
5 Boccioni, "Plastic Dynamism", 1913. in, Futurist Manifestos, ed. Apollonio, trans. Brain, Flint, Higgitt, Tisdall, p. 93
6 Wolter-Abele, "How Science and technology changed art", History Today vol.46 no.11 , November 1996, p. 64
7 Duchamp, A l'infinitif, 1966. Cited in Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art,1983, p. 139.
8 Ibid.
9 Mondrian, "The New Plastic Painting", 1917. In The New Art – The New Life: The Collected Writings of Piet Mondrian, 1986, ed., trans. Holtzman, James, p.38
10 Van Doesburg, "L'Evolution de l'architecture moderne". Cited in Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art,1983, p. 325
11 Einstein, letter to Paul Laporte, in Laporte, "Cubism and Relativity with a Letter of Albert Einstein", from Leonardo, vol.21 no. 3, 1988, p. 313.
12 Ibid.

node your homework...err, my homework
There is a difference between the fourth spatial dimension and the temporal dimension which some number as the fourth.

Someone who can perceive and manipulate the temporal dimension can prove it to you by disappearing and reappearing in a different place - in reality he simply goes forward in time and space at an increased rate so that to you it appears he has disappeared and then reappeared.

Someone who can perceive and manipulate the fourth spacial dimension can prove it to you by performing open heart surgery without opening anything. (Assuming, of course, he's a doctor -- otherwise you might be happier if he stayed away from your heart...) In the fourth dimension, someone can see and manipulate all three spacial dimensions that we are aware of. Take this Flatland analogy: a triangle trapped in a square can see no way out, but a sphere can simply grab the triangle and place him outside.

For good measure, you might want to find someone who can prove he truly occupies and perceives two dimensions by having him jump on a Mobiüs strip and returning with his heart in the wrong side of his chest. That would be cool. (Alternately a four-d person could do this with a Klein bottle).

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