Consideration of a Fourth Dimension

30 03 2010

Sometimes the most mind-busting line of thought yields the most obvious answer.  It is healthy for us to question our assumptions, and we might enjoy a rather fantasy-driven mental experiment in speculative thought, but sometimes, much to our chagrin, we must consider the likelihood that whatever conclusion we come to will be no different than the one described by a simpleton, only in a far more complicated fashion.  This, I think, is the stumbling block of modern String Theory.  People stretched and wrestled with their minds to arrive at an astounding mathematical explanation of the universe and everything, assuming the universe had a couple dozen dimensions not previously known.  The objective conclusion would have been that String Theory is therefore likely incorrect, but one hates to go to all of that trouble to prove that the formulas on the document for the theory are worth less than the coffee stain caused by the cup that was accidentally placed atop it.  In the end, the spectator claps his hands and leaves the theater feeling entertained.  The performer continues to act his part, hours and days after the curtain falls, in some vain hope that he can convince himself that the whole thing was real.

But it is entertaining to play the game, and so we shall.

The intersection of two line segments is a point.  Each line is a one-dimensional object, and the point is a zero-dimensional object.  The essence is that the intersection resulted in an object with one less dimension than the objects that intersected.

The intersection of two plane segments is a line.  The plane segments are two-dimensional, and the line is one dimensional.  Therefore, the intersection has one less dimension than the objects that intersected.

What, then, would we expect from two intersecting three-dimensional objects, like cubes?

Oh, well, the intersection of cubes forms another cube, if they are arranged just right.  At the very least, they form another three-dimensional object of some sort.  Therefore we conclude that the intersection of two three-dimensional objects is another three-dimensional object.  This breaks with the previous pattern.  If the analogy had held true, then we would have expected a plane segment to be formed by the intersection of two three-dimensional objects.

This might seem strange, at first, until we consider this analogy in reverse.

In this example, the intersection of two plane segments (rectangles) results in another two-dimensional object, just as the intersection of cubes was a cube.  Likewise, we can repeat this scenario with the one-dimensional line segments.

Two line segments, when intersecting end on end, form another one-dimensional line segment between them.  So, in these three examples, the intersection had the same number of dimensions as the objects that intersected.

What is the difference?  The difference is in the dimension of the space around the objects. Two dimensional objects intersect two-dimensionally in a two-dimensional environment.  If that environment has one more dimension than the objects involved, then they intersect with one less dimension than the objects.  Two plane segments yield another plane segment if their universe is two-dimensional, but if the space around them is three-dimensional, then they intersect to form a line, most often.  Two line segments form a line segment between them if their universe is also one-dimensional.  If their universe is at least two-dimensional, then they usually intersect to form a zero-dimensional object.  Add a dimension to the space and you lose a dimension from the intersection.  What we conclude, then, is that we can imagine three-dimensional volumes intersecting to form other three-dimensional volumes only.  This would suggest three-dimensional space.  If space were four-dimensional, then two solid volumes would intersect most often to form a plane.  We cannot even conceive of it.

Space is always the first thing that we take for granted.  We see the objects and miss the space.  We measure the space with objects.  We assume, naturally, that the space around us is three-dimensional because the objects within it are three-dimensional.  We measure an object’s height, depth and width, and we perceive three dimensions.  We do the same for empty space, but we measure that space with a three-dimensional object.  We imagine that empty volume as an invisible cube, but in so doing, we equate filled space with the attributes of empty space.  As in the examples above, a plane can conceivably exist within a three-dimensional volume or a two-dimensional area.

Could empty space be four-dimensional?  In attempting to discover this, we might be tempted to arrange the two blocks side-by-side.

By pressing their flat surfaces together, we have formed a two-dimensional intersection.  Have we thus demonstrated a four-dimensional space?  Well…no.

As it turns out, we can easily form a one-dimensional intersection between two-dimensional objects in two-dimensional space, so long as they both have a flat side and we press them together.  A similar situation exists for the line segment.

If the two segments are pressed end-to-end, they can form a zero-dimensional intersection in one-dimensional space.  Therefore, abutment is the exception that proves the rule.  Simply putting two cubes together side-to-side does not demonstrate four-dimensional space.  It’s a lot like two bubbles stuck together.  Between them is formed a flat circle.  This is a two-dimensional product of three-dimensional intersection, in a way, but it is only abutment, and it does not demonstrate four-dimensional space.

In reality, there is no practical example of 3-D intersection.  Matter is 3-D, but it cannot really intersect.  It occupies space, and no two objects can occupy the same space at the same time.  Their molecules can move about and intermingle, but they cannot really intersect.  Otherwise, we would have nuclear fusion.  Hence, we might be tempted to wonder if our model for the intersection of 3-D objects is limited only by our imagination.

The stinging rebuke that reality sends us in reply is that if empty space really contained a fourth dimension, then all we would have to do is rotate our 3-D object to see the glaring absence of the unused dimension.  This is an inescapable fact.  String Theorists, desperately needing far more than four dimensions, use the argument of scale.  For example, a string is functionally a one-dimensional object, even though it does have all three dimensions.  Paper is functionally a two-dimensional object, even though it, too, has all three dimensions.  Some would have us believe that our whole universe is functionally a 3-D object though it has, say, twenty-four dimensions, give or take a few.  They would have us believe that we are all flat as a pancake in more ways than we might suspect.  But, while this may be a convenient view, it is not a very objective one.  Disproving it would be easy if the claim were made only of occupied space, which we call objects.  But, they make the claim of the empty space also, so that we do not have room to rotate our object and observe it edge-on.  This is not unlike a magician strategically arranging his audience to one side of him so that they cannot see beyond the smoke and mirrors.

Yet, there still remains the fact that we cannot even imagine what 3-D intersection in 4-D space would look like.  We can imagine 3-D intersection in 3-D space, even though it does not really exist in nature, but we cannot fathom the other.  The implication is obvious.

One last thing worth considering is the effect of compression.  Push the ends of a functionally one-dimensional string together and it releases the pressure by increasing along an extra dimension ( it bends).  The same could be said of a functionally two-dimensional piece of paper.  So what happens when you compress a 3-D object from all sides?  It might compress like a gas, or stay rigid, but either way, the matter doesn’t lap over into an extra dimension, substantially.  It is not just functionally three-dimensional.  The Theorists would say that there is no such room for that, anyway.  We cannot very well compress empty space in this way, so we cannot cause this effect upon the whole universe.  With devices such as the hadron collider, people still haven’t managed to invoke this effect.

Three dimensions extend unimaginably far into the depths of space.  Some would have us believe that there are others, which hardly extend beyond the width of an atom.  The length of the available space may be too small for a rotation, that we might see it, and it might be too small to bend matter into it.  However, it cannot be completely filled by matter, or it would be, for all practical purposes nonexistent, having no freedom of movement.  Therefore, at the very least, we should be entitled to insist on some kind of potential for overlap.  That being the case, we should expect to see two tennis balls effortlessly pushed into each other, overlapping on one of the extra dimensions which some people are so fond of.

In the meantime, I choose to invoke Ockham’s razor.

The ability to harness a fourth dimension would yield some pretty incredible power, not the least of which is teleportation, walking through walls and invisibility.  If it could be found, then I would rather a corrupt human race did not find it.  Even so, I suggest that these are attributes of spirits, not physical forms.  That being the case, I do not expect to find more than the standard three dimensions in this world.  This is the most uninteresting conclusion, but it is possibly the most rational.




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