Tick Tock*

The Journey Begins

Over the early years of the new millennium, the idea that the universe we observe might at the most fundamental level be an evolving graph, i.e. a (vast) collection of nodes joined by a network of links, “edges” in language graph theory has borrowed by analogy from solid geometry, has started to gain traction. Of course, even with the support it is starting to get in some quarters, given the historical potency of the continuum hypothesis, this of course is an idea clearly in the category of “extraordinary claims requiring extraordinary evidence”, so it is unsurprising that a recent poster to NKS Forum simply could not accept it:

I found the section on networks and space strained and quite frankly I do not find the propositions believable.

While I have to concur that Wolfram's section on Evolution of Networks is not the strongest part of A New Kind of Science, that is a discussion for another page. What matters here is:

  1. that that weakness may be in a large part attributable to the difficulty we all seem to have in visualising simple graphs (of the graph theoretic kind), a difficulty that delayed me embarking on what has become Tick Tock as long as it did; and
  2. Wolfram was not the only person coming to that position in the same time frame, some from very different starting points.

Lee Smolin et al who are working on what they call “loop quantum gravity” and in some ways kicking against the string theory/m-theory tide, are motivated by the need to not just assume the existence of space-time-energy-matter as background, but to instead find a fundamental theory which some how explains their very existence. They claim evidence that there is in fact a network at the fundamental level, with nodes corresponding to a Planck scale quantum of space (volume) and edges corresponding to a related quantum of area (information), an idea which sits comfortably enough with what I understand of 3D solid geometry, save that it may not quite avoid implicitly assuming 3 dimensional space.

During the hiatis prior to the eventual publication of A New Kind of Science, I had got back into modeling higher dimensional space via close packing problems which have long been done to death by others but which were primarily intended to tackle what I felt was a gap in my ability to visualise the problem space. In the same period I was also gathering notes about “A New Kind of Gravity” and quickly convincing myself that the Hubble expansion must be being actively produced by deep space itself, because space itself, if not the matter and energy in it, is clearly not thinning out, nor was the implicit notion of the expansion somehow conserving the impetus from the Big Bang even vaguely credible, not even before it was observed that the bulk expansion may again be accelerating. And I had returned to that fundamental question of why there is something rather than nothing, my answer to which, beyond the almost mischievous “nothing is unstable”, was that simple graphs seemed like the best/only candidate for how things got started.[1]

But even with Wolfram and Smolin and me coming to evolving graphs from different directions, it was not until the discussion that ensued from Mark Suppes's 28 January 2004 post to NKS Forum on a “Mechanical Gravity Theory” whereby mass is but a measure of the consumption of space nodes by any massive object/particle, that I could no longer duck putting aside, at least for now, my until then open and active and space-time assuming cellular automata projects so I could do some real work on the evolution of simple graphs.

I wanted not just an evolving graph, but one which solved the perennial problem of time, of how to ensure through a purely local mechanism that from a global perspective all those local updates are well synchronised. And I had an idea! So on 24 February 2004 I wrote a first note to myself on the rapid (by human standards[2]) journey to what has become “Tick Tock”:

Given a sufficiently dense initial graph, first identyify all its irreducible polygons, e.g., if AB, BC, CD, DE, EA and AC are the only edges joining two of ABCDE, ABC and ACDE are irreducible polygons, but ABCDE is not.

For starters, it might be worth trying this with just triangles in lieu of general polygons and see what happens.

Each such polygon corresponds to a node in the next generation, with new edges joining it to each other such polygon with which it shares one edge (or more?).

Not only did the restriction to triangles promise and prove to be simpler to implement, but it made more and more sense in the light of as strong as possible an interpretation of “local”. It is important here to divest any question about what nodes and edges are “made of”. They need to be treated as being truly fundamental, so even the question of what they can “know” in order to determine the next generation needs to be understood in that context. For Tick Tock, edges only need to “know” which other nodes are linked directly to both their end nodes and nothing about anything further afield than one edge. The consequence that all three edges equally know of the existence of a triangle is not allowed to confuse the rule that each triangle produces exactly one node at the next tick.

My first naive cut of an algorithm to model Tick Tock only produced such crude statistics each tick as were readily available from the data stuctures used, in particular the number of nodes and edges in the graph, a frequency distribution of the number of edges connected at each node and, soon, a dump of the triangles which generated the last set of nodes for examination by hand and, later, by other algorithms. For input I started with just a number of nodes and a number of edges, the edges randomly selected from the full set of possible edges for the given number of nodes. For the densities I typically started with, e.g. 10 nodes and 20 edges, most often the graph would quickly disappear or stabilise into some simple structure. Just occasionally the algorithm would go AWOL so I had to quickly focus on some extra loop terminating conditions.

At that point I had no expectations for Tick Tock beyond helping me get started in visualising the possibilities of evolving graphs. I should have known better, but I did not anticipate that this simplest of possible mechanisms might actually produce some very elegant and satsifying behaviours.

I quickly observed the persistence of tetrahedra which my solid geometry familiarity with their being their own self inverse had led me to half expect. Tetrahedra with triangular “flags” attached to any number of edges also persist, and I soon discovered “linear exponential” growth between edge-joined tetrahedra—the first hint of Tick Tock “Inflation”.

Initially I was more concerned to comprehensively survey possible seeds in lieu of random generation, and so went on a tour of graph theory where many others have been before, before gradually coming to the realisation that beyond the detailed explanation of some emergent but still simple mechanisms, that what really mattered was the evolution of complete simplexes and, fortuitously, I was already generating them as my starting point before making random selections. The reason simplexes matter is that, the persistence of flagged tetrahedra apart, the only emergent mechanism which appears to create Tick Tock structure is the production “at” an edge contained in n+1 triangles of a brand new n-simplex, so n-simplexes and flagged tetrahedra are the only things in a Tick Tock graph after just one tick. But that does not mean outcomes become trivial. Even a seed pentatope, aka 4-simplex, aka 4D analogue of the tetrahedron, packs a surprise and beyond that things get very interesting indeed, hence this website.

[1] I guess it is an act of faith on my behalf to insist that things had to have a starting point somewhere, albeit however long was needed before our local Big Bang. That faith is based on a probably unfounded prejudice that an active world and a perpetual world appear to me to be mutually exclusive possibilities.

[2] One of the great lessons this perspective on our world provides is to see the timescale of humans as running extremely slowly relative to fundamental natural processes. The distinguishing characteristic of our world, from astronomical bodies to living organisms is that we persist against the odds for a time which is vast relative to the space we occupy.