On page 475 of NKS Wolfram asserts: “nothing fundamental is lost by requiring that all the nodes in a network have exactly the same total number of connections to other nodes.” Clearly, the essential simplicity and power of Tick Tock would be lost if we were to use Wolfram’s mapping from simple graphs to trivalent graphs,^{[1]} so we don’t!

Simple graphs are by definition simple. The combinatoric possibilities of even simple graphs can become quite a distraction and have been well studied, although they are a topic I should eventually return to.

The Tick Tock mechanism generates a new simple graph each tick from an old simple graph through the simplest possible productive rule which does not directly retain nodes nor edges of the old graph.

Early experiments with Tick Tock on a range of random and small seed graphs soon demonstrated that its potential to generate new structure, and thus to inflate, is dependent on the existence of edges which are common to 4 or more triangles. Wolfram’s mapping to trivalent graphs would obscure this essential result of Tick Tock.

For each particular number of nodes in a simple graph, there is a unique maximally connected graph known, unsurprisingly, as a simplex. Each simplex is known by a number which is 1 less than the number of nodes, thus a 5-simplex contains 6 maximally connected nodes. Those less than 5 have common names:^{[2]}

n | name | nodes | edges | triangular faces | tetrahedral facets |
---|---|---|---|---|---|

0 | node | 1 | |||

1 | edge | 2 | 1 | ||

2 | triangle | 3 | 3 | 1 | |

3 | tetrahedron | 4 | 6 | 4 | 1 |

4 | pentatope | 5 | 10 | 10 | 5 |

5 | 5-simplex | 6 | 15 | 20 | 15 |

6 | 6-simplex | 7 | 21 | 35 | 35 |

7 | 7-simplex | 8 | 28 | 56 | 70 |

The order (n) of a simplex is in fact the number of cartesian dimensions needed to accommodate a “solid” with equivalent topology. However, for the purposes of visualisation, it is generally easier to map the nodes of a simplex into n+1 orthogonal dimensions:^{[3]}

1 0 0 0 ⋯ ⋯ 0 0 1 0 0 ⋯ ⋯ 0 0 0 1 0 ⋯ ⋯ 0 ⋮ ⋮ 0 ⋯ 0 1 0 ⋯ 0 ⋮ ⋮ 0 0 ⋯ ⋯ 0 1 0 0 0 ⋯ ⋯ 0 0 1

Such a mapping guarantees all edges of the simplex are of equal length (√2).

A mapping of Tick Tock into cartesian space can maintain integral coordinates by 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. Such a procedure also produces equal edge length at t=1 from a simplex seed, but not always for t>1. (This coordinate representation assists appreciation of the overall structure produced from higher simplex seeds.)

From an edge connecting n+1 triangles, it is easy to see that Tick Tock produces an n-simplex. It has thus been reasonable to focus study of Tick Tock on the evolution of n-simplexes for n>2.

One early interesting discovery is illustrated by 2 tetrahedra sharing a common edge. As well as the 2 tetrahedra persisting, the edge also generates a new tetrahedron which is joined to each of the other 2 through an opposite pair of its edges.

This process repeats with respect to those 2 new joining edges at the next tick, and similarly at subsequent ticks, producing an exponentially expanding line of edge connected tetrahedra.

On other pages we explore what a few ticks do to simplex seeds. The density plot at right is particularly relevant to the discussion above of the representation of Tick Tock inflation of, in this case, a 6-simplex seed for 6 ticks.

The plot is a simple projection of the 7D representation space used for the 6-simplex onto a 2D plane. Because of the symmetry in the way the dimensions are used, it doesn't actually matter which 2 of the 7 dimensions we project the data onto.

The axes span from 0 to 3^{t-1}, in the t+6 case shown 0-243. (0, 0) is at top left. The number of nodes projecting onto one node of the 2D plane in this case is one of 18 values between 10 and 720, those discrete values represented by 18 equally spaced grey levels.

2D projection GIFs for t+7 provide even more detail than the t+6 GIF here, through the basic pattern clearly persists.

^{[1]} This reflects a wider concern I have with the weight that Wolfram sometimes gives his “Principle of Computational Equivalence” which I see, irrespective of its theoretical truth, as sometimes being counterproductive to other goals of A New Kind of Science. With respect to graphs, Wolfram continues in his discussion of Space as a Network:

With two connections, only very trivial networks can ever be made. But if one uses three connections, a vast range of networks immediately become possible. One might think that one could get a fundamentally larger range if one allowed, say, four or five connections rather than just three. But in fact one cannot, since any node with more than three connections can in effect always be broken into a collection of nodes with exactly three connections (…) what this means is that it is in a sense always sufficient to consider networks with exactly three connections at each node.

My concern is that the over use of such notional equivalences, as I claim is the case here, can serve to hide potentially useful results.

^{[2]} There are no prizes for recognising this table as forming a part of Pascal’s Triangle.

^{[3]} The main exception is for graphs which contain no higher simplex than a tetrahedron because there is more than one simple way of representing the vertices of a regular tetrahedron with integer coordinates in 3D space, starting with each set of 4 alternate vertices of the unit cube:

1 0 0 0 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 1 1 0 0 0