Tick Tock*

Evolution from simple structures

Tetrahedra are more than just the basic stable patterns under Tick Tock^[1], they also effectively repel each other, so driving “inflation”.

On this page we illustrate three local mechanisms which are prevalent in the inflation of the higher simplexes:

1. When 2 tetrahedra share a face, they separate completely in 3 ticks. The starting graph here is a pentatope with one edge omitted, a complete pentatope producing 5 rather than 2 flagged tetrahedra in 3 ticks.
2. When 2 tetrahedra share an edge, a new tetrahedron replaces the shared edge and in turn shares a pair of opposite edges with each of the 2 successors to the original 2 tetrehedra. Those new joining edges do likewise next tick.
3. When a triangle has a tetrahedon on each edge, it is replaced by a temporary “flag junction” node at t+1 which is common to new flags connected to each of the separating tetrahedra, the junction node but not the flags disappearing at t+2.

The first and third of these can be seen as different views of basically the same mechanism which drives the explosion of the pentatope and is commonly seen in the inflation of the 6-simplex and higher where it effectively reduces longer term entanglement.

Tick Tock is not always easy to project meaningfully into coordinate space, not even for the simpler graphs shown here. In particular, it may take time to recognise that a simple square, rectangle or trapezium with crossed diagonals is the topological equivalent of a tetrahedron.

In the graphs shown here, most tetrahedra appear initially in crosed diagonal view, so you need to rotate them by dragging the figure to get more recognisable tetrahedra. In some cases on the middle row, tetrhedral coordinates generated by Tick Tock fall in a plane, so rotation doesn't help and you need to use your imagination.

In the particular case of growth of a “spoke” in 5-simplex inflation, which the middle row relates to, the third dimensions of the end “hubs” are orthogonal so this projection problem is easier to avoid.

[ Table of contents! ] [ n-simplex overview! ]

^[1] As well as a plain tetrahedron persisting over Tick Tock evolution, so also does any tetrahedron adorned with a triangular “flag” on each of from 1 to all 6 of its edges.