ɸ among the integers

The integers form an ordered series, with properties which may strike one as trivial. For instance, any given element [n] of the series occurs once and only once, and in a specifiable location, i.e., the nth place. Thus, e.g., 5 occurs only in the 5th place and nowhere else in the series of integers. These characteristics practically constitute a definition of the series of integers. Listed off in a series, each of the integers has two immediate neighbors, N+1 and N-1. But what if we imagine each integer N as having precisely N “neighbors”?

To determine this, we’ll sketch the first few stages of a process by which the integers are imagined as “generated” out of each other. This thought-experiment is based on John Conway’s in On Numbers and Games and on Donald Knuth’s exposition of Conway’s notions, in which numbers are imagined as being born discretely, one at a time. Conway’s constructions take us very quickly to the infinitieth step and well beyond. We, however, will stay safely within the finite. We are after a different phenomenon.

The integers will generate each other by a specific mechanism. Each element [N] of the series is entitled to have precisely N “links” to other elements of the series. We will see that this produces an interesting pattern.

In the graphics below, the colors here are not indicative of any absolute differences. The element [N] whose "turn" it is will be shown in gold. Each new element will in turn link to other elements, generating new elements as necessary until it has the number of connections specified for it by the rule that it have precisely N such connections.

When an element first is generated, it will be shown linked in red to the element that has generated it. Pre-existing links between an element [N] whose turn it is and other elements -- that is links which come into being before a given turn -- will be shown in green. (This means that green links will always be to elements less than N.) New links to pre-existing elements will be shown in blue. New links to new elements, i.e., "generating" links, will be in red, as mentioned. Blue and red links will therefore be to elements greater than N. (Any links to an element whose turn is not current will be in black.)

We begin with element [1].

This first element [1] needs to be connected to precisely one other element. We will establish these connections in order; so the first element [1] now “generates” a second element [2]. This generation (shown in red) also suffices as the “link” between [1] and [2].



This second element [2], in turn, needs to be connected to two other elements of the series (two, precisely because it is the second element). One connection already exists (shown in green) – its connection with [1]. It therefore generates a further element of the series, element [3], and is thus linked to both [1] and [3].



“The Tao produced One; one produced Two; Two produced Three. And three produced the Ten Thousand Things,” says the Tao Te Ching (ch. 42). And if we read “the ten thousand things” as “more than one,” this is what happens.

[3], as the third element of the series, requires connections to three other elements. It is already linked to [2] (shown in green), but it needs two more links. It cannot link to [1] (because [1] already has the single connection which exhausts its quota). So [3] generates, and is thereby linked with, [4] and [5].



[4] in turn requires four connections. One exists already, to [3]. It cannot link to [1], nor can it link to [2], because [2] already has two connections, but it can link to [5], which is does forthwith. (In the graphic, a new link to a pre-existing element is shown in blue.) That makes two links. To get it to its requisite four, [4] then generates [6] and [7].



By now you are probably starting to see how things work. We cannot skip on to [6] or [7] yet – it is not their turn. Before we get to them, we must see to [5].

[5] already has two (green) links – to [3] and to [4]. It needs three more. [6] and [7] are both available, so [5] links (in blue this time, because these elements are already there) to [6] and to [7], and then, to finish off its fifth link to which it is entitled, it generates (in red) a new element, [8].






Now we may proceed with [6]. [6] has pre-existing links to [4] and [5]. It forges two new links, to [7] and to [8] respectively, and then generates elements [9] and [10].






[7] has links to [4], [5], and [6]. It makes links to [8], [9] and [10] and then generates a new link to [11].






[8] has pre-existing links to [5], [6], and [7]. It establishes links to [9], [10], and [11] and then makes new links to [12] and to [13].



You have almost certainly glimpsed something in the graphics by now. Whenever an element generates a new element (the ones in red here) (as opposed to linking to an element that has already been generated), it generates either one or two of them. In our graphics, these are always put to the right of the “parent” element whose turn it is. These pile up, until the turn moves back down to the bottom of the next stack, and the piling-up starts over. This process always leaves a top element in any given pile, and those numbers probably look familiar. [1], [2], [3], [5], [8], [13]. . . that’s right, you in front waving your hand! It is indeed the Fibonacci series. Very good.

The Fibonacci series, as you know, generates each next term by summing the two previous terms. So (3+2)=5, (5+3)=8, (8+5)=13, and so on. The ratio between any two adjacent terms converges, as the series goes on, upon the golden ratio, that splendid number also known by the letter phi, ɸ, the ratio defined by (a+b)/a = a/b, whose decimal expression is 1.618033988. . ., and which is also expressible as (1+√5)/2, the value of the lovely continued fraction 1 + (1/(1+(1/(1+(1/1+…)))), and any number of other nifty tricks.

I have in fact traced our linked-integer series a few steps further, just for fun, but as you can imagine, the graph gets pretty tangled and hard to follow after this stage. However, the Fibonacci pattern remains for as far as I followed, and I can’t see why it wouldn’t. (The number of members comprising each stack is of course a Fibonacci number; the next stack after the one I have depicted will be eight numbers tall and be topped with [21].) I presume this pattern is a trivial implication of the mathematics involved, and that this way of deriving the series is well-known in relevant circles. I don’t move or read in those circles; all I can report is that in those popular-mathematics accounts with which I am familiar, this way of deriving the golden section is not described. (I'd love to hear of any references.) I find it of interest because it shows how the series emerges from the overlay of two ways of construing the integers – as cardinals (one, two, three…) and as ordinals (first, second, third…). It is, moreover, interesting to see the Fibonacci numbers emerge from a set of well- (and minimally-) defined relations (a.k.a. “links”, above).

As mentioned, new elements are generated either singly or in pairs. Fans of recreational mathematics may find it interesting to trace the pattern that emerges in the way these variations continue to play out.