The Geometry of the Imagination

For you creative types or for those of you who find your imaginations limited, here's a formal exercise* to help you stretch your brain.

Imagine a point. In geometry, a point is merely a location. It has no size or dimension. Let's call it P.

Simple enough. Now, imagine a point moving from one location to another. Two points describe a line that extends through both of them infinitely in two directions; it has length. Call it Line AB. Length is the first dimension.

Add a third point that is not in line with AB, call it C. Or, imagine Line AB moving in a perpendicular direction through Point C. That describes a plane which, like the line extends infinitely in two dimensions: length and width. Call it Plane ABC:

So far so good. Now, imagine a fourth point which does not lie on Plane ABC. Visualize Plane ABC moving as a whole through this point. This fourth point adds the element of depth, describing the third dimension of space.

So far, our little formal exercise has been fairly intuitive. Every school kid understands how this works because, for one thing, our bodies describe a space—as does our expanding universe. Now, however, things become less obvious.

Imagine another point, a fifth point through which this three-dimensional space itself (which, I must remind you, like its subsidiary dimensions is infinite in every direction) moves as a whole. This is difficult to do because we have no way of formally representing this motion.

Let's simplify. The most efficient description, or model, of a finite space (or a solid) is a tetrahedron, a three-sided pyramid. Four points.

Imagine holding a solid tetrahedron in your hand and tossing it in the air and catching it. See it tumbling through space, up and back down. Now, visualize the entire tumbling arc, from toss to catch. This describes an event in time, the fourth dimension.

Of course, we must expand our model as we have each of our other dimensions and recognize that time is infinite and unbounded. And, as with our other dimension defining points, lines, and planes, we recognize (a) that time exists both within and throughout space and (b) that infinite three-dimensional space also moves through eternally extended time—much like our odd-shaped triangular ball above.

Have we stretched our imaginations? Yes. Are our brain muscles feeling sore like our arm muscles after lifting weights or our leg muscles after running? Sure. Dare we dare to imagine more? Let's.

We've imagined a model of infinite space moving through eternally extended time. The first thing we intuit about this space-time model, though, is that time seems to move in only one direction. Forward. Even sitting still, perfectly motionless, we move forward through time (and, technically, space! See my serial post entitled Being v. Becoming under Pages in the right-hand column). We time-travel (the relevant post from my series) at a relative rate towards what we call the future, never backwards toward the past. Why is this? It is for the same reason a point moving from A to B defines only a line, or a moving line describes only a plane, or a moving plane delimits space. As spatial beings, our experience of time is dimensionally limited. We exist in space-time and our perceptions are limited by its geometry.

But (and the "but" is always the engine of the imagination, isn't it) can we imagine what might exist outside of time? Can we conceive of a fifth dimension? A further point that we cannot intuit, one beyond space, time, and space-time, but through which infinite space and eternally extended time might move as whole?

Let's go back to our model: the tossed and tumbling tetrahedron. We've watched as it grew from a mere point through the dimensions to a line to a plane to a space. Now we've seen its arc from beginning to end describe an event in time. We've noticed that this event only happens in a forward sequence from toss to catch. (And we recall that it is at once happening in space-time and, for our purposes, is a finite model of infinite space moving through infinite time.) If we press a bit and let our imaginations run, we can theorize what a dimension beyond this event might have to be like. This would be a dimension where the toss has not yet happened and yet it has already happened and, what's more, that it is always already happening—forward and backward.

Just as a point dissolves into a line, and a line into a plane, etc., time dissolves into an eternal now (to borrow a phrase [but not a concept] from 20th Century theologian Paul Tillich). As we've noted, there is an infinite length of time: as space expands infinitely, time extends eternally. But this is different from an eternal now in which this infinite extension has already happened.

In this fifth dimension, space does not exist, yet it is infinite and everywhere. Time has not yet begun, yet it extends eternally, and is always already happening—forward and backward.

Now, to try to keep our feet on the ground, we must ask whether such a fifth dimension exists, or is it only something we've imagined? Well, if it's something we can only imagine, then our exercise has been successful! We've expanded our minds, and opened new dimensions of thought.

The question whether there is some sort of fifth dimension beyond time and space and space-time is probably not one we can answer. We must leave it to our imaginations. Though it is fair to say that such a dimension is a possible inference from the geometric argument of dimensional expansion. (As are further dimensions in which eternities themselves dissolve.)

It could be, however, that we're asking the wrong question. Existence might be something that can happen only within the limits of the space-time dimensions. It is not a relevant category of the eternal now. It obtains only where there are limited durations of concrete spaces. It pertains only to realms where what we know as dimensions happen. Existence itself only exists where there is, in fact, geometry.

Imagine that!

----------
*  FN. I was fortunate in college to attend a master class in creative writing conducted by Allen Ginsberg and William S. Burroughs at my university (UNC-Chapel Hill). Mr. Ginsberg led off the session with a meditation encouraging us to breathe deeply and while doing so ply our imaginations in a similar manner as described above.