Tick Tock*

Deterministic breaking of symmetry

Symmetry increases in cellular automata

It is a given of cellular automata in particular that a symmetric rule applied to symmetric starting conditions cannot generate anything asymmetric. Rather, as illustrated at right, symmetric configurations sometimes arise from asymmetric or less symmetric precursors, and thereafter the new symmetry persists for as long as it remains isolated.

Yet the application of the seemingly perfectly symmetric Tick Tock rule to a clearly fully symmetric 6-simplex seed for just 3 ticks produces 210 symmetrically arranged “joining units” each with only partial internal symmetry.

Before we look at how we might understand this apparent symmetry breaking, it may help to describe in some detail how Tick Tock starts to inflate from a 6-simplex seed, as much to account for the symmetry growth from 7 identical nodes to 210 joining units connected via 140 “boundary hubs” as for the symmetry loss within the joining units.

The ancestory of the 210 joining units can be identified within the 6-simplex seed by arbitrarily choosing 2 of its 7 nodes to be “shared edge” nodes, 2 more to be “tetrahedral” nodes and the remaining 3 to by “pentatope” nodes. There are 7C2 = 21 ways to choose the 2 shared edge nodes from the original 7 and 5C2 = 10 to then choose the 2 tetrahedral nodes from the remaining 5. While 21 × 10 = 210, as required, there are only 21 real edges at t=0 and no discernible loss of symmetry at t+1, but in view of the inherent dynamic of an edge shared between a pentatope and a tetrahedron to generate asymmetry, as discussed in considerable detail below, might it be that the best way to understand the 21 edges of the original 6-simplex seed is as superpositions of the 210 edges the original 21 generate at t+1?

Each of the 21 edges in the seed is clearly common to 5 triangles, the third vertices of which are each of the other 5 nodes in the seed, and thus each edge will generate a new pentatope at t+1, so we get 21 new pentatopes at t+1, while the 7C4 = 35 tetrahedral facets of the 6-simplex seed spawn their own direct descendants.

The initially surprising thing here, and a large part of the story of Tick Tock is that both these mechanisms — edge ⇒ pentatope and tetrahedron ⇒ tetrahedron — apply to the same set of edges. It may help here to see that it is edge ⇒ simplex[1] which actually generates the new edges and that tetrahedron ⇒ tetrahedron just reflects certain arrangements within those new edges where they connect to form tetrahedra and triangles which were not contained within a single edge spawned simplex but rather reflected the prior topology of the spawning edges. It is an emergent property of Tick Tock, illustrated in some detail on another page, that tetrahedra persist while triangles not contained within tetrahedra break up, leaving a persistent trail of triangular “flags” attached to the edges of the descendants of any tetrahedra that shared an edge with the original triangle.

Early study of Tick Tock focused on simple measures of the number of nodes and edges generated each tick, then breaking those numbers down by simple measures of local topology surrounding those nodes and links. Yes, it is now possible to provide formulae which will predict these raw numbers in each column for any larger t, and yes the raw numbers do keep roughly quadrupling each tick, but, no, the numbers in the following table do not tell us all there is to know about the structure generated, although they do contain a few strong clues.

edges per nodenodestickedgestriangles per edge
Total nodes by number of edges and edges by number of triangles at each tick of Tick Tock inflation from a 6-simplex seed.

In the particular case of the 6-simplex seed, each of the original 7C2 edges which generate 7C2 pentatopes at t+1 can be seen as serving simultaneously as the shared edge between 5C2 tetrahedron-pentatope pairs, the successor structures of all of which persist in accordance with these emergent local mechanisms. It is also important to be aware in the discussion that follows that each edge of a pentatope is common to 3 of its 5 tetrahedral facets.

At t=1 we have 35 nodes, 1 for each of the 7C3 = 35 triangles in the 6-simplex seed, and 210 edges, 5C2 = 10 in each of the pentatopes generated by each of the 7C2 = 21 6-simplex edges. Each of the 35 nodes terminates 12 edges, up from the 6 each node terminated in the original seed. There are as yet no signs of assymetry at the local topology level which determines what is generated at the next tick. Each of the 35 t=1 nodes also belongs to 30 of the 350 triangles, up from 15, 16 of the 140 tetrahedra, down from 20, and 3 of the 21 pentatopes, down from 15. The fact that 30 does not divide 350 exactly, nor 16 divide 140 hints at some loss of internal evenness, but nothing which is yet visible to measures of local technology.

Each of the 210 t+1 edges shares the same local topology, which they must have in order to serve as the grandparents of the 210 identical joining units which emerge at t=3, so just how does the first hint of local assymetry arise between t+1 and t+2?

Each t+1 edge belongs to 1 pentatope and 4 tetrahedra, 3 of which are facets of the pentatope. So each edge uniquely joins each pentatope to 1 of the 35 direct descendents of the 35 tetrahedral facets of the 6-simplex seed, a singular position equivalent to the virtual positions which we have already discussed the original 21 edges as each holding 10 times over. It turns out to be such pentatope-tetrahedron joining edges which drive the inflation from the 6-simplex seed, each producing a new pentatope at the next tick, a pentatope which shares an edge with 4 tetrahedra, for it is in the way a pentatope shares 4 of its 10 edges with tetrahedra that the asymmetry of joining units begins to appear.

If we apply Tick Tock to a constructed seed graph consisting of just the pentatope and a tetrahedron sharing an edge which is the t+1 site of each joining unit, a 6-simplex joining unit does emerge, but cluttered by 2 separated fully flagged tetrahedra, corresponding to the 2 of 5 tetrahdral facets within the t=1 pentatope which did not share the shared edge, plus 5 flags on each of the 3 pentatope tetrahedra which did. The clutter can easily be recognised as products of the explosion of a simple pentatope seed. To construct the simplest seed which generates just one joining unit and its four bounding hub tetrahedra, unencumbered by such clutter, we need the t+2 configuration pentatope joined to 4 tetrahedra.

In isolation at the local level, a new pentatope generated from an edge joining an old pentatope to an old tetrahedron will always have 1 edge common with the descendent of that old tetrahedron and 3 equiavlent edges common with the 3 tetrahedra descended from the 3 of the 5 tetrahedral facets of the old pentatope which shared the original shared edge. The descendant of the old tetrahedron shares 1 edge and thus 2 nodes of the new pentatope, so the descendents of the tetrahedra from the old pentatope must shared 3 edges and the 3 remaining nodes, which form a triangle in the new pentatope. So each such new pentatope becomes asymmetrically divided between 1 shared edge, a triangle of 3 shared edges and the 6 remaining unshared edges. Such an arrangement is of course not totally asymmetric as can most easily be visualised in 3 dimensions by locating the odd edge on a perpendicular through the centre of the triangle formed by the other three nodes, the tetrahedron attached to the odd edge being best considered as off in another dimension, but shown in fig. 1 as flattened into 2 dimensions. Dragging on the graphic rotates the view in 3D.

The first sign of this asymmetry appears at t+2 because each of the 210 pentatopes generated at t+2 share that singular edge with the direct descendents of the 35 tetrahedral facets in the seed 6-simplex and share the “opposite” triangle of edges with the immediate descendants of 3 of 5 tetrahedral facets of a t+1 pentatope. Yet at t+2 none of the 350 nodes belongs to any of the joining units, despite how distracting it might appear that 350 nodes should somehow equal 210 joining units plus 140 boundary hubs.[2] The reality is that all of those 350 nodes belong to tetrahedra which are direct ancestors of boundary hub tetrahedra, in the case of the 3/5ths which are triangle nodes each node belonging to 2 such tetrahedra, nodes whose double use at t+2 accounts for the extra 210 needed to form the 140 fully separated boundary hub tetrahedra from t+3.

But still at t+2 there are no nodes which can be labelled joining unit, or rather none other than the 4 near edges of the 4 boundary hubs which connect to each joining units. At t+3 and beyond, each boundary hub connects 6 joining units, 1 at each edge, so each boundary hub node effectively belongs to 3 joining units. For those who need such detail, this means each of the 140 t=0 tetrahedron descendant nodes at t+2 serves as a boundary node to 3 joining unit sites while the other 210 each similarly serve 6, thus providing the 3 × 140 + 6 × 210 = 1680 (= 3 × 560, t>2) end nodes to the required 4 × 210 boundary hub links. The 3 + 1 boundary tetrahedra and the pentatope formed purely from their edges at t+2 that can be seen in fig. 2 delimit the site of a joining unit. Fig. 2 is topologically equivalent to fig. 1 but rather than being located in 3D coordinate space by hand, the nodes are located using Tick Tock evolution from the simplex unit vector coordinate system which provides the basis of other presentations on this website of 3D projections of results from a 6-simplex seed.

At each such joining unit site the next tick generates 4 new pentatopes, 1 at each tetrahedron-pentatope shared edge. The 10 triangles in the t+2 pentatope at the joining unit site generate 10 nodes which are finally separate from the boundary hub nodes. Those 10 nodes exhibit 3 distinct local topologies, only 1 of the 10 having 12 edges as boundary hub nodes have and the other 9 having 8 edges, but being separable into a group of 3 and another of 6 only by the minimally differing numbers of 12-connected and 8-connected nodes they are linked to. However the number of distinct local topology categories grows quite rapidly for t>3, at least within joining units. At the same time all difference disappears between the local topology of the 35 original hub tetrahedra and the 105 whose tetrahedral ancestory can only be traced to t+1.

The inflation atom of a pentatope with 4 bounding tetrahedra can be clearly seen to generate 4 new pentatopes, to expose 5 new tetrahedra corresponding to the tetrahedral facets of the old pentatope, and 6 embedded triangles each bounded by 3 tetrahedra, the 10 nodes of the t+1 pentatope each being a node of 2 of the 5 tetrahedra, 9 of those 10 nodes being in 2 embedded triangles, the 10th being in 3 of the 4 new pentatopes, each of the 9 being in 1. If we take the liberty of considering nodes at one tick to have successors in the next tick where they belong to a child tetrahedron in the same relative position to some persistent stucture joined to an edge of that tetrahedron, then each t+2 pentatope can be seen as getting 3 nodes from the t+1 pentatope plus 2 from a bounding tetrahedron.

Beyond t+2, each boundary hub edge generates a new pentatope each tick so that its successor edge continues to join a boundary hub tetrahedron and a joining unit pentatope. Meanwhile, deeper inside each joining unit fractally nested asymmetry becomes more and more complex, with the number of distinct classes of node topolgy growing rapidly, but that is a topic for another page.

[1] A pentatope is a 4-simplex as a tetrahedron is a 3-simplex. Mention of a “pentatope” here is specific to its importance in understanding Tick Tock evolution from a 6-simplex seed, while “simplex” is prefered when more general statements can be made.

[2] This just emphasises what should be an obvious fact that it is not the nodes which matter to evolution under Tick Tock, individual nodes having no true descendants. It is the pattern of their connections that matters.