A Study in Dimensions

3. Computing Components of Basic Units

Now that we have toyed a little with the idea of four-dimensional figures, let us go back a bit and pick up on a train of thought we began to develop earlier. In the discussion of basic units of various dimensions we began to tabulate some data on "component parts" or basic units. A line segment, for instance, has two points and one line segment. A cube has 8 points, 12 line segments, 6 squares, and one cube. Arranging this data in the form of a table we arrive at the following:

		Number of dimensions in component part
		0	1	2	3	4
Number of dimensions in basic unit	0	1
	1	2	1
	2	4	4	1
	3	8	12	6	1
	4	16	32	24	8	1

The data for the first four levels - that is, 0 through 3 - we were able to fill in from our own familiar realm of experience. We know what a square looks like, or what a cube looks like, and we are able to count the number of edges or corners without any trouble. The data for a 4-D basic unit we were able to deduce logically from the progression already begun. We may further conclude that the data for basic units of more than four dimensions continues to follow this pattern.

For instance, let us consider a 5-D basic unit. Here we are confronted with a figure constructed within five dimensions, consisting of five pairs of 4-D supercubes diametrically opposing each other in five unique and mutually perpendicular dimensions. It can also be thought of as an extension of a 4-D supercube into a fifth dimension, creating four other pairs of 4-D supercubes as a result of its "trailing". Without going into the details of these trailings, which the reader may work out for himself on the basis of previous discussion, we find that the 5-D supercube contains:

32 points (corners)
80 line segments (edges)
80 squares (faces)
40 cubes
10 4-D supercubes
1 5-D supercube

Filling this information into our table we obtain the following:

	0	1	2	3	4	5
0	1
1	2	1
2	4	4	1
3	8	12	6	1
4	16	32	24	8	1
5	32	80	80	40	10	1

Upon examining the data in the table we make the following observations:

Numbers in the 0 column are merely powers of 2; namely, 2 to the power of the basic unit dimension level. This is of course because each time a new dimension is added to a basic unit, the number of endpoints is doubled (the basic unit before extension plus after extension).
The number 1 extends downward to the right, representing, of course, the obvious fact that a square contains only one square, a cube contains only one cube, and so on.
All other numbers in the table are obtained by doubling the number immediately above and adding the number diagonally above to the left. For example, the 24 (representing the 24 squares in a 4-D supercube) can be found simply by doubling 6 and adding 12. The 40 in the table is arrived at by doubling the 8 and adding the 24. We might be a little hesitant in concluding that this is actually taking place, but if we investigate the reasons, it becomes readily apparent that this procedure is indeed correct.

In saying that the 6 is doubled, we mean simply that the number of squares in a 4-D cube is at least double that found in a 3-D cube, since a 4-D cube is an extension of a cube, and therefore the number of faces is of course doubled.

6 faces

a hypercube showing doubling of faces from 3-D cube to 3-D cube

6 faces

By adding 12 we mean that twelve additional squares are created by the extensions of the twelve line segments found in one 3-D cube.

from each edge

a hypercube showing new faces as trailings of cube edges

to corresponding edge - 12 new squares

This simple mathematical progression provides us with an easy way of filling in the rest of the table for as many dimensions as we desire. Without drawing any models or twisting our brains we may fill in the data for a supercube of 6 dimensions by calculating from the data of the previous line:

	0	1	2	3	4	5	6
5	32	80	80	40	10	1
6	64	192	240	160	60	12	1

(2⁶ = 64; 80 x 2 + 32 = 192; 80 x 2 + 80 = 240; 40 x 2 + 80 = 160; 10 x 2 + 40 = 60; 1 x 2 + 10 = 12)

In this manner we may extend the table to as many dimensions as we choose:

	0	1	2	3	4	5	6	7	8	9	10
0	1
1	2	1
2	4	4	1
3	8	12	6	1
4	16	32	24	8	1
5	32	80	80	40	10	1
6	64	192	240	160	60	12	1
7	128	448	672	560	280	84	14	1
8	256	1024	1792	1792	1120	448	112	16	1
9	512	2304	4608	5376	4032	2016	672	144	18	1
10	1024	5120	11520	15360	13340	8064	3360	960	180	20	1

Now let's examine these numbers for a moment. We already know how they are obtained, but can we make further discoveries as to the nature of this mathematical progression? One thing we might notice, for instance, is that in row 2 we have two 4's side by side; three rows down and one column to the right are two 80's side by side; and three rows down and one column to the right appear two 1792's side by side. Is there any special reason for this? Another question we might ask has to do with the size of the numbers as they progress towards the right. In row 6, for instance, the numbers peak in column 2 (240) and then decrease in size. In row 9 they peak in column 3 and then decline. What is the nature of this rising and falling? Can it be determined in any way other than calculating from the numbers provided in the previous row?

If we look at the ratios of one number to another as they progress across a row we find something rather interesting. In row 6, for example, the ratio of 192 to 64 is 6 to 2; the ratio of 240 to 192 is 5 to 4; the ratio of 160 to 240 is 4 to 6; and so on according to the following pattern:

192 / 64 = 6 / 2
240 / 192 = 5 / 4
160 / 240 = 4 / 6
60 / 160 = 3 / 8
12 / 60 = 2 / 10
1 / 12 = 1 / 12

Now let us examine any other row at random - let's say row 4:

32 / 16 = 4 / 2
24 / 32 = 3 / 4
8 / 24 = 2 / 6
1 / 8 = 1 / 8

And now row 5:

80 / 32 = 5 / 2
80 / 80 = 4 / 4
40 / 80 = 3 / 6
10 / 40 = 2 / 8
1 / 10 = 1 / 10

It is now apparent that the ratios follow a distinct pattern, namely

x / 2, (x-1) / 4, (x-2) / 6, ... 1 / 2x

where x represents the number of dimensions found in the basic unit in question. Checking the other rows in the table, the reader will find this to be true in every case. This now answers the questions we have raised above. 4 and 4 follow in succession in row 2 since the ratio is x/2 where x=2. 80 follows 80 in row 5 because the ratio reaches the point where numerator and denominator are identical: 4/4; ((x-1)/4 where x is 5). In row 8 two 1792’s appear together since the same phenomenon occurs - numerator and denominator are equal, 6/6 ((8-2) / 6). The increasing and the decreasing of the numbers as they progress to the right is now distinctly seen in the predictable ratio pattern which occurs. The numbers rise where the ratio has a larger numerator, and they decrease in size when the ratio’s denominator exceeds the numerator. A peaking takes place at a point where the numerator and denominator "pass each other" on their separate ascending and descending paths.

Now we have found a neat way of filling in data for one row of the table without having the previous rows filled in beforehand. Simply insert 2^xin column zero, and then multiply that number and each resulting number by the ratio

x/2, (x-1)/4, (x-2)/6, … 1/2x

until the number 1 is obtained and therewith the end of the row.

For example, without having row 11 filled in for us, we may determine that row 12 follows the sequence

4096, 24576, 67584, 112640, 126720, 101376, 59136, 25344, 7920, 1760, 264, 24, 1

Based on the knowledge of these progressive ratios, we may go yet one step further and put together a formula which enables us to fill in the value for any box in the table at random without having to fill in the values for any other boxes at all. This formula is:

	2^(x-y)x!
n =	_________
	y!(x - y)!

where n represents the value we are seeking, x the number of dimensions in the basic unit, and y the number of dimensions in the particular component part we have chosen. What this boils down to is the fact that in a basic unit of x dimensions, there are n number of y-dimensional component parts, according to the formula stated above.

Let us test it out. Suppose we have no table before us and we would like to know how many 2-D faces there are in a 6-dimensional supercube. Plugging in the values, we have:

n = 2^(6-2) 6! / (6-2)! 2! = (2⁴) ^.1^. 2^.3^.4^.5 ^.6 / (1^.2 ^.3 ^.4) ^.1^.2 =

(16)^.5 ^.6 / 2 = 240

Checking back on the table we find that a 6-D supercube has indeed 240 faces.

Referring back to the table it may be seen that our basic units very rapidly become quite complex structures as soon as we add a few dimensions. A cube, having only three dimensions, may quite simply be represented 2-dimensionally on a sheet of paper by a mere twelve lines:

a cube

and a 4-D supercube by 32 lines:

a 4-D hypercube

However, a 10-dimensional supercube has 5120 edges and would require a mesh of 5120 carefully interconnected lines to be represented 2-dimensionally! From this it is clear that such a representation is entirely impractical and would appear only as a black splotch due to the high density of drawn lines. Remember, such representations are only aids to help us understand the structure of super-dimensional figures which we cannot visualize, and do not in any way pretend to be an accurate representation of the figures as they really are. An analogy "minus two dimensions" would be the 1-dimensional representation of a cube for 1-D beings living in a line universe. Reducing a 3-D cube to 2 dimensions, we obtain the image

a cube

which then may be projected down to a 1-D "screen" in the following manner:

the image of a 3-D cube projected onto a 1-D line segment

The result is a line segment divided by eight endpoints, each representing, of course, the eight corners of a cube. A 1-D man would view this model as a rather odd-looking telescoped line segment, which, when carefully studied, would represent to him the various components of a 3-D cube. We know, however, that a cube is not a telescoped line segment, but something quite different. In the same way, our pitifully weak representation of 4-D supercubes and of cubes of higher dimensions, which appear to be multiple exposures of various cubes and squares, tend to give us a rather jumbled picture of what they really are.