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.