Tick Tock*

“Tick Tock” is a simple program[1] acting on the nodes and edges of a graph to produce a new graph each “tick”, the evolving series of graphs over a number of ticks behaving as a complex system, at least for modestly complex starting graphs. Tick Tock was discovered in 2004 by Tony Smith.

The local Tick Tock mechanism can be defined completely in two English sentences:

Or, in a simple enough graph transformation diagram:[2]

triangle => node, shared edge => new edge

Or, in four formal lines:[3]

Being as simple as possible was one key criteria in selecting this mechanism. An even more important criterion was that no single element at one tick can simply persist at the next tick. Given that overriding criterion, every other viable rule for an evolving series of graphs is necessarily more complex than Tick Tock,[4] although that should not be taken to imply any expectation that a mechanism quite as simple as Tick Tock actually powers our world.

While previously described simple programs, such as cellular automata and Turing machines, may make understated assumptions about the universe they operate in, often presupposing spatial dimensions; graphs and networks have been identified as inherently simpler. It is vital to resist even asking the question as to what the nodes and edges might be “made of”—the essential concept being that the nodes and edges are truly elementary—even while, to make them accessible to our world, we most often model them in a digital computer using patterns of bits.

It is also well known that spatial metrics emerge naturally from certain kinds of graphs. This applies to many graphs produced by Tick Tock even though it was not a criterion for selecting the rule. Instead the rule was chosen because it was seen to potentially solve some of the problematics of time, most particularly how to coordinate global updating while only acting locally.

Tick Tock met that criterion and delivered some other suggestive results which could not have been anticipated before doing the experiments:

None of this should be taken to suggest that Tick Tock is expected to ever produce Class 4 or even spontaneous Class 3 behaviour, although it is anticipated that it might sometimes appear to amplify introduced randomness.

Much of the analysis of Tick Tock results draws on natural geometric representations of a graph’s topology. ABC and ABD as defined above form triangles. The simplest persistent graph and thus stable pattern is a tetrahedron, reflecting the fact that a tetrahedron is its own inverse—the 4 faces at one tick becoming the 4 nodes at the next, with the 6 edges rotated orthogonally in this one special case. But a tetrahedron is also generated whenever an edge is shared by exactly 4 triangles, and there it starts to get interesting.

[1] A “simple program” in the sense defined in A New Kind of Science [Wolfram, 2002].

[2] If your broswer does not display this Scalable Vector Graphics (SVG) diagram, you need at least an SVG viewer.

[3] If your browser has trouble rendering symbols here, particular the special characters used in formulae you may need to upgrade to a browser which is more compliant with newer web standards. This site has been checked with current versions of Mozilla and Safari. Even then, the unicode character ∣ which doubles for “such that” is not a named HTML entity and so had to be numerically coded.

[4] As defined, edge PQt+1 could be thought of as an “orthogonal” successor to edge ABt, but, as we will see, PQ is most often one edge in an n-simplex, with the 1-simplex/simple edge/n=1 case being of limited significance.