Tick Tock*

3D projection of early 5-simplex inflation

A 5-simplex can be conveniently projected onto 3D space so that the nodes occupy the 6 vertices of an octahedron [fig. 1]. Initially shown in 2D along a third diagonal edge perpendicular to the 2 visible diagonal edges linking 4 of the 6 nodes, the other 2 nodes being at the ends of the perpendicular edge. There is no implied node where those three edges pass at the centre of the 3D projection. Dragging on the graphic rotates the view in 3D.

By t+3 [fig. 4], Tick Tock inflation of a 5-simplex produces a framework with 30 “hub” tetrahedra connected by 90 “spokes” each spoke doubling its hub to hub length each subsequent tick. The graph produced by joining 1 node from each joining unit directly to the four other nodes representing its boundary units is also shown [fig. 5] alongside the cluster of spokes surrounding one hub tetrahedron [fig. 6].

The octahedral projection from 6D to 3D superimposes many nodes on other nodes, simplifying the structure somewhat without losing the main message. There really are 30 hubs in the data, but 24 project in pairs onto the outer visible 12 and the remaining 6 all project onto the visible central hub.

Just as the original 5 embedded tetrahedra persisted through the explosion of the pentatope and the 30 early hub tetrahedra here anchor the emerging structure, so in the more complex inflation of the 6-simplex, 140 boundary tetrahedra are connected in groups of 4 by 210 joining units, the evolution of which under the Tick Tock rule is much more complex that the simple doubling of spoke length here.