Tick Tock*

Coordinates reveal boundary hub/joining unit framework

Representing the vertices of an (n-1)-simplex by unit vectors on each of n mutually orthogonal dimensions makes it a lot easier to visualise the initial evolution of the framework of boundary hubs and joining units/spokes which provide the large scale symmetric structure of the inflating graph generated by the Tick Tock rule from a simplex seed.

First, recognise that the n dimensions are equivalent, so whenever we can identify a structure in the first k dimensions, where k≤n, that structure is repeated each of the nCk times for the nCk ways we can choose k from n. For all n>3 a tetrahedron is formed by the first 4 nodes, with equivalent tetrahedra for each other combination of 4 seed nodes, so we initially only need to consider 1 tetrahedron which, while staying the same coordinate size across each tick, accelerates away from the origin exponentially:[1]

   t         t+1        t+2        t+3          t+4            t+5
1 0 0 0    1 1 1 0    3 2 2 2    7 7 7 6    21 20 20 20    61 61 61 60
0 1 0 0 -\ 1 1 0 1 -\ 2 3 2 2 -\ 7 7 6 7 -\ 20 21 20 20 -\ 61 61 60 61 -\
0 0 1 0 -/ 1 0 1 1 -/ 2 2 3 2 -/ 7 6 7 7 -/ 20 20 21 20 -/ 61 60 61 61 -/
0 0 0 1    0 1 1 1    2 2 2 3    6 7 7 7    20 20 20 21    60 61 61 61

This representation also makes obvious the inversion of the tetrahedron each tick.

Thus the nC4 t=0 tetrahedra contained within an (n-1)-simplex seed become the first set of boundary hubs in the inflating simplex. At the bottom of the scale, a single tetrahedron is thus effectively its own singular “bounding hub” and the 5 tetrahderal facets in a pentatope survive, one each in its 5 exploded parts. But it is the 5-simplex and beyond where boundary hubs and joining units/spokes start to provide an overall symmetrical framework for Tick Tock inflation.

The other ends of each joining unit/hub are provided by the direct descendants of the n-2C4 t+1 tetrahedra within the nC2 simplexes generated by the nC2 edges of the original seed simplex, by virtue of each such edge forming a triangle with each other node of the seed simplex. Again we can safely concern ourself with the edge joining the first 2 nodes and the tetrahedron formed by joining that edge to the next 4 nodes, positioning their tetrahedral nodes from t+1 at:

    t+1            t+2            t+3             t+4                t+5
1 1 1 0 0 0    3 3 1 1 1 0    9 9 3 2 2 2    27 27 7 7 7 6    81 81 21 20 20 20
1 1 0 1 0 0 -\ 3 3 1 1 0 1 -\ 9 9 2 3 2 2 -\ 27 27 7 7 6 7 -\ 81 81 20 21 20 20 -\
1 1 0 0 1 0 -/ 3 3 1 0 1 1 -/ 9 9 2 2 3 2 -/ 27 27 7 6 7 7 -/ 81 81 20 20 21 20 -/
1 1 0 0 0 1    3 3 0 1 1 1    9 9 2 2 2 3    27 27 6 7 7 7    81 81 20 20 20 21

For the 5-simplex (n=6) 6C2 = 6C4, so the number of t=0 and t+1 hubs is equal, with 15 of each and their “joining units” are symmetrical spokes. In general, the number of t+1 hubs is n-4C2 times the number of t=0 hubs, so for n>6 the joining units are asymmetrical.

It is important to notice here that at t=1 (1 1 1 0 0 0)-(1 1 0 1 0 0) is an edge of both a t=0 hub tetrahedron and a t=1 hub-containing simplex. Again for n=6 it is simple, that hub-containing simplex is another tetrahedron, but for n>6 that edge belongs to n-4C2 of n-2C4 tetrahedra within the simplex, 3 of 5 for n=7, the 6-simplex. That t+1 edge is thus effectively the site from which a spoke/joining unit is generated over the following 2 ticks, nodes that belong to the spoke/joining unit first appearing at t+3, prior to which each node belongs to one or more boundary hubs.

The expected number of boundary hub tetrahedra and of joining units/spokes can thus be calculated for each simplex seed:

dimsseedt hubst+1 hubsjoining unitshubs per joining unitt+3 nodes per joining unit
4tetrahedron1
5pentatope5The 30 “flags” are failed t+1 hubs/joining units
65-simplex151590 “spokes”24
76-simplex35105210410
87-simplex70420420720
98-simplex12612607561135
109-simplex210315012601656

At t+3, the boundary hub tetrahedra are completely separated and the first n-2C3 internal nodes have been generated inside joining units. The difference in joining unit topology between those t+3 nodes linked to the t=0 hub and those linked to the n-4C2 t+1 hubs, for the 6-simplex and higher, results in joining unit nodes with different topologies, a difference which grows impressively in subsequent generations, but which the coordinate system discussed here does not do such a good job of representing as their persistent connections to exponentially separating hub units tend to stretch joining units into coordinate spaces which make the shape of the actual network harder to recognise.


[1] When using the procedure described elsewhere of locating a new node at the sum of the coordinates of the vertices of the triangle at t which produced the node at t+1. A tetrahedron centred on the origin does not move, though it still inverts each tick:

 1  1  1       1  1 -1
 1 -1 -1  =>   1 -1  1
-1  1 -1  <=  -1  1  1
-1 -1  1      -1 -1 -1