2. Multi-dimensional Geometric Units

Now that we have been exposed to a few concepts of dimensions and have been made aware of the possibility of other dimensions beyond our familiar three, let us go on to consider the nature of multi-dimensional geometric forms.

A point, a line segment, a square, a circle, a cube, a sphere - these all are concepts which do not give our imagination much trouble, because they have three dimensions or less. But what about 4-dimensional forms? What, for instance, would a 4-dimensional cube look like? How would it be constructed?

The answer to this question is also found by analogy.

1. First, let us start at the level of zero dimensions. With no dimensions we can only "construct" one "form", namely a point. A point has no length or depth, only location.


2. Adding one dimension to the concept of point, we can construct a line, which itself is merely an extension of a point. If we mentally picture a point moving infinitely in a certain direction, the trail created by this point produces a line, an infinite conglomeration of countless points, namely, all the points or locations which the "moving" point passes.

The only (finite) geometric shape possible in the 1-dimensional world is a line segment, an extension of a point to a certain length. If we look at a line segment as a "point having moved" we may identify:

  1. The original location of the point before extension
  2. The extension itself
  3. The final location of the point after extension

a line segment

Thus we find that a line segment, which I shall name the basic unit of one dimension, consists of:

  1. 2 points
  2. 1 line segment (which, of course, actually includes the two endpoints themselves)

3. Adding one more dimension we create a 2-D world. Here possibilities expand from points and lines to 2-D figures. If we take a line and extend it infinitely in a perpendicular direction, we create a plane. A plane, like a line compared to a point, is an infinite conglomeration of countless lines, something comparable to a piece of cloth being made up of many individual threads. In a plane we may construct not only points and line segments, but also line segments interconnected with each other in a 2-dimensional fashion. Our possibilities include squares, rectangles, parallelograms, pentagons, and an infinite variety of other shapes. Also, motion in two dimensions introduces the possibility of curves: circles, ellipses, or any sort of amoeba-like shapes. In this discussion, however, we would like to limit ourselves to what we have already named the basic unit. What is the basic unit for two dimensions? As a point extended a certain length created a line segment, the basic unit of one dimension, so a line segment extended the same length in a second dimension creates a square.

a square

We may look at a square as simply that - a square. But we may also look at it as a progression of extensions. We began with a single point, which extended itself into a line segment. The line segment then extended itself in a direction perpendicular to the original point extension, creating a square.

Looking at this "2-dimensionally extended point" then, we may identify the following components:

  1. 4 points (corners) - (2 from the original line segment before extension and 2 from the final line segment)
  2. 4 line segments (sides) - (the original line segment and the final one; plus the 2 line segments newly created by the extension of the two end points)
  3. 1 square (face)

4. Adding another dimension creates 3-dimensional space as we know it. Space is the infinite extension of a plane in a third dimension, something like the pages of an infinitely large book. Here 2-dimensional shapes can be put together 3-dimensionally, creating cubes, pyramids, diamonds, and so on. The concept of curves can be applied to 2-dimensional surfaces, producing cones, cylinders, spheres, or pear-shaped forms.

The basic unit of three dimensions would be, then, the third-dimensional extension of a square the same distance as its first two extensions, namely, a cube.

Examining a cube as a progression of extensions, we discover that it has:

  1. 8 points (corners) - (4 from the original square before extension and 4 from the final square)
  2. 12 line segments (edges) - (4 from the original square and 4 from the final one; plus 4 created by the extending of the four points of the square)
  3. 6 squares (faces) - (the original square and the final one; plus 4 square defined by the extension of the 4 line segments of the square)
  4. 1 cube

a cube

5. Moving on to four dimensions, common sense would tell us that this simple progression continues in an analogous manner. 4-D space, then, would be the infinite extension of 3-D space in a fourth dimension perpendicular to the three axes we have already defined. Imagining this is, of course, quite mind-boggling, and our only way out of this dilemma is to remember the frustrated 2-D man who is trying to visualize the extension of his own 2-D space. He sees that a plane is the sideways extension of a line and rightly concludes that 3-D space is a sideways extension of his own world in a direction unimaginable to him. From this analogy we conclude that 4-D space is a "sideways extension" of our own space in a direction inconceivable to our limited minds.

What, then, is a 4-D cube, the basic unit of four dimensions? Following the progression outlined in the previous paragraphs, the answer is quite simple. We start with a cube and extend it 4-dimensionally the same distance as before. The result is a 4-D "supercube", consisting of:

  1. 16 points (8 from the original cube and 8 from the final one)
  2. 32 line segments (12 from the original cube and 12 from the final one; plus 8 new ones created by the extension of the 8 points in the cube)
  3. 24 squares (6 from the original cube and 6 from the final one; plus 12 new ones created by the extension of the cubes’ 12 line segments)
  4. 8 cubes(!) (the original cube and the final one; plus 6 new ones created by the extension of the cubes’ 6 squares)

3-dimensionally, this could be depicted (on a 2-D sheet of paper!) by the following monstrosity:

a 4-D hypercube

At first glance this appears to be a meaningless jumble of lines. But with a little effort we can pick out four distinct pairs of cubes, some of which look strangely distorted:

a 4-D hypercube as a pair of cubesa 4-D hypercube as a pair of cubesa 4-D hypercube as a pair of cubesa 4-D hypercube as a pair of cubes

This agrees with our conclusion that a 4-D cube should contain eight 3-D cubes. Also, counting up the lines and corners we find that our predicted results match the figure. Now one can easily see that the figure is simply two identical cubes drawn in close proximity with all corresponding corners connected - something like a double exposure.

When we draw a cube, we do the same thing. First one square, then an identical square, and then connect the corners. The result, as above, may be viewed as pairs of squares:

a cube as a pair of squaresa cube as a pair of squaresa cube as a pair of squares

Now we of course know that they are indeed pairs of real squares, even though on the paper two of them are represented by pairs of parallelograms. A 2-D man observing our cube above would undoubtedly see them as parallelograms, but we know that the parallelogram shape is only due to distortion created by the projection of a 3-D object onto a 2-D surface. Likewise, the four pairs of cubes we see in the supercube are true cubes, the distortion being caused merely by the projection of a 4-D figure into 3-D space.

In the previous discussion we viewed a cube as an extension of a square. As a result of the extension we have two squares with everything between them, the "trail" described by the moving of the square. Part of this trail turns out to be two new pairs of squares, since a square contains four line segments, and an extended line segment creates a square. Perhaps this can be more easily understood with an illustration:

Square A:

 square A

when extended in a third dimension to square B

square A extended to square B

creates two new pairs of squares a, c, and b, d, by the extension of its own line segments 1, 2, 3, and 4:

three pairs of squares forming a cube

When looking at the finished product, we could just as easily say that the cube is not an extension of A to B, but an extension of b to d, or an extension of a to c. Each one of these extensions is at right angles with the other two, representing the three dimensions in which they are arranged. The parallel squares A and B are extended forward and backward, whereas the pair a and c are extended up and down, and the pair b and d left and right. A cube, then, is a harmonious interconnection of three pairs of parallel squares, each pair being diametrically opposed in one of three unique and mutually perpendicular dimensions.

Looking down one dimension we see that a square is the same thing on a 2-D scale: two pairs of line segments, parallel and diametrically opposed to each other in two perpendicular dimensions:

one pair of line segments and  a second pair of line segments

A 4-D supercube, therefore, is the same thing on the level of four dimensions. It is constructed of four pairs of "parallel" 3-D cubes (yes, one cube can be parallel to another in 4-D space, just as squares or lines can be parallel), each pair being diametrically opposed in one of four unique and mutually perpendicular dimensions! If we turn back to the diagram illustrating the 4-D supercube, we may see that this is what is represented.

Now consider the following arrangement of squares below:

an unraveled cube

If we were to cut the figure form the page, bending A, C, D, and E upwards so that their edges meet, and then bend F down parallel to B, we will have constructed a cube. C pairs with D, A with E, and F with B. Edge 2 coincides with edge 3, edge 5 with 6, 4 with 7, and so on.

Now let us consider the following arrangement of eight cubes:

an unraveled hypercube

Keeping the center cube F stationary, we should be able to bend cubes A, B, C, D, E, and G-H "upwards" into a fourth dimension, using the six faces of cube F as "hinges" upon which the other cubes are bent. Remember, just as squares A, C, D, E, and F immediately disappeared from the plane in which square B was located, swinging upward on "hinges" which were the 1-D sides of square B, likewise cubes A, B, C, D, E, G, and H immediately vanish from our 3-D space, swinging up into 4-D space on 2-D hinges, which are the six faces of the stationary cube F. Once the cubes are all at right angles 4-dimensionally with cube F, their faces will neatly mesh together much in the same manner as the edges of the squares came together while constructing the paper cube. At this point we merely "flip" cube H over so that it becomes parallel with cube F, enclosing the structure into a 4-D supercube! Which cubes are pairs, and thus parallel with each other? These are of course A and G, B and E, C and D, and F and H. A more challenging problem would be to determine which faces coincide with which faces from the adjoining cubes. After studying the problem a little we conclude that

a cube with the six faces identified

and so on.

At this point it would be interesting to make some observations on intersections and parallel forms. We know that two lines can be parallel or that two planes can be parallel. In the case of two lines, they are separated by a second dimension, and in the case of two planes, they are separated by a third dimension. 3-D space, by analogy, may also be parallel with another unique 3-D space in exactly the same manner, separated by a fourth dimension.

Two lines, apart from coinciding completely, may be parallel, skew, or intersecting. In a plane, two lines either are parallel or they intersect at some point. But in 3-D space two lines may also be skew, that is, non-parallel and non-intersecting.

Adding one dimension, we see that two planes in 3-D space are always either parallel or intersecting at some common line. But in 4-D space two planes have the extra option of being skew; that is, non-parallel and non-intersecting, just as two lines in space!

Also worth pointing out in this discussion is that two lines intersect at a common point, two planes intersect at a common line, and two 3-D spaces intersect at a common plane.

Now let’s have some fun with our 2-dimensional friends. They would like very much to see a 3-D cube face-to-face, but have no way of doing so unless we help them. The best we can do is to take a cube and to push it through their 2-D world for them to observe.

a 2-D man watching a 3-D cube passing through his 2-D world

What do they see? If we push it through, keeping it parallel with the plane of their 2-D environment, they first see a square which suddenly appears out of nowhere. As we continue to push it through, the square seems to remain motionless and unchanged. Then it disappears as suddenly and as mysteriously as it had appeared.

Even more interesting is to pass a sphere through their world for them to observe. What would they see this time? First a dot appears, which rapidly grows into a circle. The wider the circle becomes, the slower it grows. Having reached its maximum size, namely the diameter of the sphere itself, the circle slowly diminishes, shrinking more rapidly until it suddenly disappears altogether.

a sphere passing through a plane

By analogy we may conclude that a similar spectacle would unfold before our eyes should we desire to see a 4-D supercube or 4-D supersphere pass through our space. In the case of a 4-D supercube, we would first see a cube appear out of thin air, hover a bit, and then instantaneously disappear. It has just passed 4-dimensionally from one side of our 3-D world to the other, just as the cube passed through the plane. No motion was visible, because the only moving that took place was in a direction not found in our world of three dimensions.

A 4-D supersphere passing through our space would appear to us the same as a sphere passing through a plane as in the above example, the only difference being that one more dimension is added. First we would see a dot which immediately swells into a sphere, much like a balloon being inflated. The larger it grows, the slower it grows. After reaching its peak size (the diameter of the 4-D supersphere) the process reverses. The sphere shrinks, first slowly, but then with increasing rapidity, until it vanishes completely.

A 4-D sphere is a fascinating possibility to contemplate. It is, however, a very slippery concept to grasp and extremely difficult to visualize. When dealing with 4-D cubes we had edges and corners to help us nail down and mentally construct its form. But a 4-D supersphere, like a 3-D sphere or circle, has no corners, edges, or fixed points to help us. It is a continually curving surface. A circle is a line enclosed within itself, a sphere is a plane enclosed within itself, and a 4-D supersphere is therefore a unit of space enclosed within itself. All points within this curved 3-D space are equidistant from the center point of the supersphere, just as all points on a sphere are equidistant from its center and all pints on a circle are equidistant from its center. If we were able to warp our 2-D friend slightly and place him within the surface of a sphere, he would have nowhere to go but the surface of the sphere itself. No matter which direction he chooses to go, he ends up in the same spot. Being enclosed within the sphere’s surface, he is doomed to travel in endless circles, although he himself is not aware of any turning. Striking out in a direction which seems to him to be a straight line, he invariably comes back to his original starting place and is completely baffled as to how this could be.

Similarly, if one of us were "warped" 4-dimensionally and placed within the 3-D surface of a supersphere, he would find himself in exactly the same predicament as the 2-D man described above. Regardless of which direction he chooses to go, he finds himself mysteriously going in a circle. The space in which he is situated seems to be infinite in all directions, but is actually finite. In fact, we would even be able to measure its volume down to the last cubic inch. What to him seems to be an immeasurable expanse of space is in actuality a finite volume of space curved 4-dimensionally into itself, like the curved surface of a balloon.

One way to represent a 4-D supersphere is the description we have presented above. Another way is to picture it as an infinite progression of spheres of different sizes placed "on top of each other" 4-dimensionally. This can be illustrated by the following example. Suppose we take an orange and cut it in half. Then we take the two halves and cut them in half parallel to the first cutting. We then slice the fourths in to eighths and so on. If we were to cut the orange into an infinite number of parallel slices, each slice being so thin as to have no thickness whatsoever, we would have on a scale of "minus one dimension" the same picture was what we are trying to imagine in four dimensions. A chain of an infinite number of 3-D "slices" (spheres) strung together like beads on a string pointing in a fourth dimension would create for us a 4-D supersphere. In a way we already described this when we pictured the supersphere passing through our space. The sphere that we saw growing and then shrinking into nothing was actually an infinite sequence of sphere "slices", each of which became visible at a unique instant of time.

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Copyright 1998 by Bill Price

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