By Larry Hecht
I can conceive in the mind of six objects, whose relationship to one another I wish to investigate. Their character as real objects does not interest me, but only that quality which makes them distinct, thinkable. They are, thus, objects in thought. I will label them with the number designations 1 to 6, though I might equally denote them by letters, or any other symbols which allowed me to keep them distinct in my mind. I am interested in discovering the number of different ways these six distinct objects can be formed into pairs. Their representation by numbers, allows a convenient means of investigating this. I first list all the pairs of 1 with the other 5, then all the pairs of 2, and so forth. The result is summarized in the table:
12
13 23
14 24 34
15 25 35 45
16 26 36 46 56
== == == == ==
5 4 3 2 1
Counting the number of pairs in each column and summing them, produces 5 + 4 + 3 + 2 + 1 = 15 pairs.
In another form of representation, I can imagine the six objects as points on a circle, and portray their pairing as the straight lines connecting any two. Drawing them produces a hexagon, and all the straight lines that may be drawn between its points. Counting all the connecting lines, we find 15, the same as the number of pairs above! The mind rejoices in the discovery of the equivalence of the two representations.
Closer examination of the second form of representation, now reveals also a difference with the first. In the first, nothing distinguished one pair from the next, except the symbols used to designate them. In the second, we discover three distinct species of relationships among pairs, each characterized by a different length of connecting line. We have (i) the six lines forming the sides of the hexagon; (ii) the six somewhat longer lines connecting every other vertex (i.e., 13, 24, etc.); (iii) the three longest lines connecting diametrically opposite vertices (14, 25, 36).
Where, before, the mind celebrated the sameness, it now rejoices at the difference of the two forms of representation, and is impelled to look for its cause. We hypothesize that the difference must reside in a property of the spatial mode of representation. We may reflect that, from the manifold ways we might have chosen to arrange our six points in space, we chose to place them on the circumference of a circle, equally spaced. An arbitrary arrangement of six points in a plane would have produced another, less-ordered relationship among the pairs. Another arangement, a spiral perhaps, would have produced a richer ordering.
Thus, from the positing of relationship among things in the mind, we moved to two modes of representation of that relationship, then to their sameness and difference, then to the causes of that difference. Having hypothesized that the latter is the result of the spatial form of representation, we are next led to explore the variety of such representations.
Of the great variety of possibilities, we choose now to rise above the plane, in order to examine the relationship among six points in three-dimensional space, the familiar backdrop for our visual imagination. Just as the circle aided us in ordering the points in the plane, here its counterpart, the sphere, comes to our aid. Six points, spaced evenly around the surface of a sphere, form the vertices of the Platonic solid known as the octahedron. We can picture two of its six points at the north and south poles of a globe, and four more forming a square inscribed in the circle of the equator. Connecting each point to its nearest neighbor, we find the 12 lines which form the 8 equilateral triangles, which are the octahedron’s faces. But we have not yet connected the six points in all the ways which space allows. Each point can yet be connected to its opposite, forming 3 more lines, which are diameters of the circumscribing sphere. Behold, again, the 15 paired relationships of six objects, now clothed in a new ordering, this time of two species!
We may now compare the three modes of representation our mind has invented to investigate these pairings:
1) By number, which produced the series 1 + 2 + 3 + 4 + 5 = 15.
2) In planar space, using the circle, which produced the three species of lines connecting the points of the hexagon.
3) In space, using the sphere, which produced the two species of lines connecting the vertices of the octahedron.
In turn, each of these modes of representation suggests new investigations. For example, with respect to the first (i.e., number), we may inquire into the pairwise combinations of other numbers of things, from which we soon discover that, in general, for “n” things, the number of pairs that can be formed is equal to n(n-1)/2, and we may next inquire, what is the expression for combinations three-wise, four-wise, or n-wise?
With respect to the second (the distribution of points on a circle and their combinations), we discover that there exist species beyond the regular polygons, which are known as the star (or Poinsot) polygons. These cannot be generated out of any arbitrary number of points, but only when the number of points, and the order in which we take them, are relatively prime to each other (that is, have no common divisor). The first of the star, or Poinsot, polygons, appears when we take five points on a circle, and connect every second one until the figure closes (that is, 1 to 3, 3 to 5, 5 to 2, 2 to 4, and 4 to 1). The result is the star pentagon, or pentagram, which is conveniently described as 5/2. We can then discover the 7/2 and 7/3, the 8/3, the 9/2 and 9/4, and so forth.
With respect to our third mode of representation of the pairwise combination of things (the distribution of points on a sphere), a new ordering principle arises: that a perfectly even distribution is only possible in the cases of 4, 6, 8, 12, and 20 points. When we investigate these, we find species of pairwise combinations called edges, diagonals, diameters, and some others, the greatest variet of species occurring in the 20-point figure.
Now, let us reflect on the higher ordering principle: All of the representations we have given, even the spatial, are creations of mind, products of the arithmetic or visual imagination. Yet, so real do these creations of the mind seem to us, we may be tempted to marvel at them as if they had some existence outside of the mind. (“But Platonic solids are {real}. I can build them!” you say. Perhaps you never have. Anyone who has tried, soon discovers a, sometimes gooey, massiness where massless points are supposed to be, a very finite thickness to the infinitely thin lines of the edges, and a, sometimes wrinkly, bulk to the massless surfaces. Even three-dimensional space, the forgiving medium of all our constructions, which seems so certain, so real, is only the ingenious work of the mind, the visual imagination. All are products of the mind.)
But when, in nature, the mind discovers forms just like these we have just created (thought), put there not by us, but by something like to us in mind, yet much vaster, then may we truly marvel, and reflect: What makes nature makes us. What we make in mind, think, is then nature — and may be so in a higher form than what we perceive outside us. (The proof of this truth, well-known to readers of this publication, need not be repeated here.) So in the ordering, number, space, and mind, the mind stands at both ends of the series, as both creator of its own images, and perceiver of others; the one is called imagination, the other, reality. Yet they are both real, as we just showed, and even both imagined, in so far as the perceived external is {known} only through the images of mind.
With such considerations, true science begins.