Tick Tock*

Evolution from a pentatope seed

A pentatope can be conveniently projected onto 3D space so that the nodes occupy 5 of the 6 vertices of an octahedron [fig. 1]. As initially shown in 2D along the axis through the omitted vertex, the pentatope appears as the planar view of a tetrahedron (fully connected square) with the 5th vertex at the centre and its 4 edges coincident with (half) diagonal edges of the flattened tetrahedron [fig. 1]. Dragging on the graphic rotates the view in 3D.

By t+3, Tick Tock evolution of the pentatope has effectively blown in apart into 5 “fully flagged” tetrahedra—graphs which regenerate themselves each tick [fig. 4]. The octahedral projection from the original 5D coordinate space used to model the pentatope to 3D causes 1 pair of nodes to be projected onto the same 3D coordinates at t+1, 5 pairs to do likewise at t+2 and still 1 at t+3 and thereafter.

The actual appearance of five fleeing fully flagged tetrahedra [figs. 4 & 5] is very much a product of the use of 5D coordinate space to model the evolving graphs and the projection method back into 3D space. If we revert to all edges being equal and build one fully flagged tetrahedron by hand we get something that might be a little easier to make sense of [fig. 6].

The persistence of variously flagged tetrahedra was the first unexpected discovery made with Tick Tock because, while the potential self inverting persistence of naked tetrahedra was in more obvious, less so was the fact that applying the Tick Tock rule to a third triangle attached to an edge of a tetrahedron means that edge has 3 triangles and thus produces another triangle on the successor tetrahedron.

Tick Tock evolution from a pentatope seed is driven by 2 of 3 local mechanisms, the 3rd of which drives the beginnings of “inflation” and all 3 of which combine to produce even greater complexity.