A Study of Dimensions

by Bill Price


Introduction

I am neither a physicist nor a mathematician. I am just an ordinary guy who happens to enjoy thinking about hyperspace and geometries of multiple dimensions, having independently "discovered" and explored the subject at the age of 16. Therefore, if you are a physicist or mathematician, please bear in mind that the things written in this essay are the musings of a sixteen-year-old Iowa farm boy who at the time had never heard of Georg Riemann, Charles Hinton, or Hermann Minkowski, and who started thinking about multiple-dimensional geometry entirely on his own.

Everything in the following essay is original; it is not a regurgitation of someone else's work. Even today I am not very well-read in the subject, and in 1966 I had not read anything. The only exception is the example cited from Gamow's book One, Two, Three,... Infinity, a book which I bought at a high school book sale in 1966, and the book which sparked my imagination on the subject of higher dimensions.


Original introduction, written in 1981, after having read Edwin Abbot’s "Flatland"

In June, 1966, at the age of sixteen, I did a lot of thinking about dimensions. I got so excited about the subject that I felt compelled to write down my "discoveries" for posterity. However, my own laziness kept me from finishing the project.

Recent interest in the subject has led me to review what I had learned before and to continue where I had left off.

When I was sixteen, I felt that I had come across something tremendously unique and important. Today I realize that it is nothing new. Others have written on precisely the same subject, and in a much more eloquent and readable manner. Nevertheless, I still feel the need to record these ideas in the form in which I first "discovered" them.

Hence, the following paper.

Anaheim, California

May, 1981


1. Understanding other dimensions

This paper is an investigation of dimensions.

Our 3-dimensional world is something that we take very much for granted. We think it nothing special that objects have the three dimensions of height, width, and depth. In fact, it is so natural for us to think in terms of three dimensions that it is very unnatural and illogical for us to think in any other terms. When in geometry we hear of planes, lines, and points, shapes that have only two, one, or zero dimensions, our 3-dimensional minds immediately protest. Absurd! How can anything be so thin as to have no thickness whatsoever? And yet that is what a plane, a figure of two dimensions, is supposed to be! And a line? Something that has no sort of thickness at all - something like an infinitely thin thread with only the dimension of length? Ridiculous! No matter how hard we try to imagine it, a line or a plane will always have some sort of thickness. And a point - infinitely small, with no size or measurable thickness at all in any direction? Obviously these are all theoretical fantasies created by mathematicians who need them in order to talk about geometry, a science which serves our 3-dimensional world of reality!

All of us instinctively think this way. It is a very mind-boggling thing to try to imagine an object with less than three dimensions! This is because we are 3-dimensional creatures living in a 3-dimensional world. We instinctively project a "3-dimensionality" into any object we conceive. All objects must have at least height, width, and depth. And what about other aspects? When we hear talk of "the fourth dimension" our minds really rebel! 2-dimensional things are at least halfway conceivable and are discussed in geometry books, but who could come up with the notion of a fourth dimension? Where would you place it? You can construct the axes of height, width, and depth, all at right angles with each other and all radiating out into a unique perpendicular direction, but where would you put a fourth axis? No matter how you twist or turn it, it's going to be pointing in some direction which is a combination of the others. If you look at the corner of a box, you see the edges running along in three directions, representing the three dimensions. What other physical dimension can there be in our world of reality? We come to the subjective conclusion: none whatsoever! There can only be three dimensions; no more and no less!

Thus we go along happily in our 3-dimensional world, thinking in 3-dimensional terms, and completely undisturbed by any absurd talk of more or less than three dimensions. And why shouldn't we be content? After all, everything in our 3-dimensional world functions perfectly, just as it should!

But let's stop and think for a moment. Why only three dimensions? What's so special about three? If I might persuade the reader to divorce himself momentarily from his 3-dimensional way of thinking, I would ask him: why only three? Why not four, five, or six? Or maybe only two? What I mean is, what is so logical or natural about three and only three dimensions? Try to look at it objectively, completely apart from the fact that we live in a 3-dimensional world - and I think you might be able to see what I mean.

What I mean is this: the "reality" of "only three dimensions, no more and no less" is only an illusion, the product of our biased minds. Yes, 3-dimensional objects exist, but so can 2-dimensional objects, 4-dimensional objects, or even 10-dimensional objects! What prevents them from existing? Of course, I have never seen such bizarre things myself, and neither has anyone else, but that in no way negates the possibility, and in fact, the very real possibility that such objects can exist.

What really got me to thinking about this was a very simple analogy I came across in a paperback book entitled "One, Two, Three, … Infinity" by George Gamow. Let’s imagine a world of only two dimensions, something like a huge thin piece of paper (actually, of course, with no thickness at all) inhabited by 2-dimensional men. Someone from our 3-dimensional world with the aid of a light bulb projects the image of a cube into the 2-dimensional world, much to the surprise of our 2-dimensional friends. What they see is not a cube as we know it, but two squares, one within the other, with lines connecting the corners of one to the other:

2-dimensional man looking at 2-D projection of 3-D cube

We of course fully understand what the object is supposed to be because we project depth into the 2-dimensional replica of the cube. We could, for instance, picture it as a small room with the small square in the middle representing the far wall, the sides being the right and left walls, the top part the ceiling and the bottom the floor. But what are 2-dimensional "men" able to make of it? All they understand is two dimensions, no more and no less. They think and function in a world of plane geometry. They certainly can’t project into the image the idea of depth as we know it. All they see is one large 2-dimensional square enclosing a smaller square with the corners connected. And if they were confronted with a figure like this:

a cube

they would merely see two squares intermeshed with the corners connected, or perhaps two parallelograms with corners connected:

a cube - one pair of parallelograms a cube - another pair of parallelograms

all, of course, within the same plane.

But we see the figure quite differently. We understand that the figure represents a 3-D cube and that the third dimension of depth is implied.

But try to explain that to our poor confused 2-dimensional friend! The idea of "depth" is inconceivable to him. He only understands length and width, two dimensions. Trying to imagine a third dimension in his flat world is about as frustrating as trying to draw a third line at right angles to a "+" on a piece of paper. On a piece of paper we of course cannot draw a third perpendicular line. We know, however, that a third one exists straight up and down at right angles with the surface of the paper, something which the poor little 2-dimensional man would never be able to imagine!

With the help of an analogy, then, we may be able to catch a glimpse of what a fourth dimension is like. Just as the 2-dimensional man tried in vain to find a third line perpendicular to the cross in front of him - no matter which way he placed it, it would not work - we, too, try in vain to imagine a fourth unique line perpendicular to the lines described by the three edges of a cube. But the answer is a line entirely outside our 3-D world, just as the line extending upwards from the surface of the paper is entirely out of the paper. Looking down at the 2-dimensional man we realize that the solution which he vainly seeks is in reality very real and simple. And so it is with our search for a fourth dimension. It is certainly there, and the solution is simple enough - it’s just that we can’t see it.

Someone at this point will object to my speaking of four or more dimensions. "How can you speak of such a thing if you don’t know if it even exists at all? How can you prove the existence of more than three dimensions?" Granted, I cannot "prove the existence" of a fourth dimension - not any more than the 2-D man in our analogy would be able to prove the existence of a third dimension. I firmly believe that other physical dimensions of space exist, but I cannot prove it or produce concrete evidence of it. What I would like to convey to my reader is the fact that such things are not an impossibility or an absurdity, but on the contrary, quite possible and completely in accordance with logic and common sense.

The best vehicle for conveying the idea of more than three dimensions is to do what we have done above: to use the analogy of a world minus (or plus) one dimension. By minus one dimension I mean that the analogy can be extended down to the level of 1 vs. 2 dimensions. Let us suppose, for instance, our 2-D man is trying to imagine what the third dimension is like. Since trying to conceive it in his down 2-D environment proves to be quite frustrating, he looks at the analogy of a one-dimensional "man" confronted with the second dimension. The one-dimensional man, he reasons, knows of only one dimension. His whole universe is merely a line. It is quite impossible for the one-dimensional creature to understand the existence of a line perpendicular to his universe, but the 2-dimensional man can see its reality quite clearly. The second line is very real indeed and its existence is quite a simple matter. "This," he concludes, "must be what the third dimension is like compared to our 2-D world. It sort of sticks out perpendicular to the universe in a very logical and simple direction, but completely inconceivable to us because it lies entirely outside of our world." And we, as 3-dimensional beings, who are very much familiar with "the third dimension", see that our 2-D friend’s conclusion is perfectly correct.

The analogy of "minus one dimension", then, provides us with a marvelous tool for visualizing things that are impossible for us to visualize. A line of the fourth dimension intersects our universe at a single point, extending infinitely in two directions in opposite "sides" of our 3-D world, just as a line perpendicularly intersects a plane at one point, or a line intersects another line perpendicularly at one point.

The use of these analogies brings up another interesting point. Earlier we mentioned the fact that a 2-dimensional object is impossible for us to imagine, since we invariably project a certain amount of thickness into it. But by analogy we can demonstrate to ourselves that we are continually confronted with "pure" planes. They are constantly before our very noses!

What do you suppose our 2-D friend sees when he looks at a square? Does he see a square as we see one?

a square

Remember, we are "outside" of the 2-D world and can look down upon a square or any 2-D figure from the outside. We simultaneously see every point of the square: its four corners, its sides, and every point it contains. Every point of the square is distinct for us. But what about the 2-D man? He, too, sees the square and understands its shape and construction, but with one very significant difference. He looks at it edgewise. He is inside of the same plane as the square. What he sees, then, is not actually a square at all! What he sees is a line segment - with "depth"! He is fully aware of this "depth" and understands it without any problems, since it is a part of his daily experience. But he cannot actually see this depth, this second dimension, because it is in the same direction as his eyesight. Each successive point of the depth he sees is covered by the point in front of it! Therefore, as he views the plane universe around him, he sees, not a plane as we understand and view it, but a line, a huge circle, each point of which represents an eternal succession of points radiating away from the direction of his eyesight. His eyes perceive a square not as we do:

a square

but as a line segment:

a line segment

(or

a line segment showing four points

depending on the angle from which he views it). It is his mind which adds the dimension of depth.

And what about ourselves? Do we as 3-dimensional beings fully understand the nature of 3-D space? When we look at any everyday object, do we see its three dimensions as they really are? The above analogy makes us aware of a simple but surprising fact. We view all 3-D objects in our universe "edgewise", just as the 2-D man looks at a square edgewise! One of the dimensions is continually pointing in the direction of our own eyesight. The result is that when we look at a cube or a sphere we do not actually see a cube or a sphere at all! What we see is a circle or a square, or any other purely 2-dimensional shape. Our brain, interpreting two slightly different images transmitted by our two eyes, tells us that the objects have depth. No matter where we look in our 3-D world, we see nothing but two dimensions. When we look at a box, we do not see a cube but a square (or a hexagon). When we look at an orange, we do not see a sphere but a circle. So, in this sense, 2-dimensional objects really are within our mental grasp!

Footnote written in 1996:

Although one would think that this is a very simple observation to make, one will always meet people who will argue with it. I remember one specific occasion of explaining this to a couple of rather intelligent colleagues, and was very surprised at their resistance to such a concept. They insisted that we observe the world around us in three dimensions, and that we can indeed see all three dimensions at once. They were right, of course, but only in the sense that the objects that we see have three dimensions, and that we can observe the effects of three dimensions, and can understand perfectly well how a system of three dimensions works. But they were wrong in believing that they can actually see three dimensions. The simple point that they were missing is that, although what we see are objects of three dimensions, the images themselves that are transmitted from our eyes to our brains, are purely of two dimensions, and only two dimensions. We can never actually physically see all three dimensions at once, and that is simply because we are inside the 3-D space observing everything "edgewise". We can be standing on a long straight flat street which stretches itself far out into the distance, and we think that we can actually see the depth. We know that the depth is there, and it is indeed there; however, what we are actually seeing, that is, the actual image that our retinas are transmitting, is a two-dimensional one, not a three-dimensional one. We do not have three-dimensional retinas which send a matrix of three dimensions to our brains! The image that our lenses cast upon our retinas is not three-dimensional! We always only see purely flat two-dimensional images! But alas, no amount of explaining on my part would convince my friends, however, who insisted that all three dimensions can be seen at once.

As an illustration to prove his point, one of my friends pointed out that upon observing a cube from an angle, we simultaneously see the three dimensions of length, width, and depth, as the edges of the cube radiate out from the corners. To him, that was ample proof of the apparent fact that three dimensions can be observed simultaneously. He did not realize, though, that this very illustration does not prove his assertion, but on the contrary, disproves it, and in fact disproves it in a very elegant manner.

The key to understanding why his illustration disproves his own assertion lies in the important fact that one needs to look at a cube from an angle in order to see clearly the three edges radiating out from any given corner. Since it is impossible to see any one given edge of a cube when the length of the edge is pointing directly at the observer, it is necessary to tilt the object at an angle to bring this "hidden" edge into view. What he ends up doing with this "hidden" edge, then, is "bringing it into view" by making it become a part of the two-dimensional image matrix defined by the plane which is perpendicular to an imaginary line extending straight out from the observer’s eye. After doing this, the "hidden" edge is no longer swallowed up by the third dimension which he can never see, and becomes a part of the two dimensions which he can always see.

We can belabor this even further by challenging our skeptical friends with yet another important observation. We all know, of course, that the three angles that meet at the corner of any cube are all 90-degree angles. Why is it, then, that we can never actually see three distinct 90-degree angles all at once? When we look at the corner of a cube "from an angle", what we actually see, that is, the image that is projected upon our retinas, is one of three angles, all of which measure more than 90 degrees! In fact, when we add them all together, we get 360 degrees, and not 270 degrees as should be expected. This is simply because the "tilting" of the object, which is necessary in order to observe all three angles at once, put the three mutually perpendicular axes of the cube into the two-dimensional plane of our vision, and as a result the angles are distorted. Of course, the angles are not actually distorted; they only appear to be. And that is the whole point of the matter. Every three-dimensional object that we see, whether we realize it or not, appears distorted, since we are looking at it "edgewise"! We can never physically see three dimensions simultaneously as they actually exist in reality.

End of footnote.

Also, unless an object is perfectly transparent, we can never see every point of it simultaneously, since the other side is always pointing away from us. This might seem to be an absurdly simple observation to make, but when we consider the analogy of the 2-D man looking at a square we realize something quite startling! The 2-D man sees the 2-D square as a 1-D line segment, but we can see a square as it really is, since we do not look at it edgewise. We can look down upon the square outside of the plane in which it is situated and simultaneously see every point of the square, its "front edge" as well as its "back edge" (from the vantage point of the 2-D man). Similarly, then, it should be possible for a four-dimensional "man" located outside of our 3-D space to view a box in such a way as to see all six sides at the same time! Not only would all six faces of the box be equally easy to observe, but the inside of the box would also be exposed to his 4-D eyesight. When we see a 3-D object, we see it "edgewise", but the 4-D man sees it as it really is. He sees all points simultaneously, because he is observing it from a remarkably new vantage point: "outside" of space. Likewise, rooms in a house would be as easy to survey as squares on a chessboard, regardless whether the doors are locked and the curtains drawn!


Dimensions, ch.2 | Dimensions, ch.3 | Dimensions, ch.4 | Dimensions, ch.5 | Hypercube formula | Decimals of Primes | Visual Definition of Primes | Calculator Trick | Decimal Equivalents of Fractions | Mental Calendar Trick | Kaleidodigit | Cobol Sudoku Solver | Math | Home Page


Copyright 1998 by Bill Price

graphic.gif (2326 bytes)