Tick Tock*

3D projection of early 6-simplex inflation

A 6-simplex can be conveniently projected onto 3D space so that the nodes occupy 7 of the 8 vertices of a cube. As initially shown in 2D along the axis through the omitted vertex, the 6-simplex appears as the planar view of a 5-simplex (fully connected hexagon) with the 7th vertex at the centre and its edges coincident with (half) edges of the 5-simplex [fig. 1]. Dragging on the graphic rotates the view in 3D.

By t+4, Tick Tock inflation of a 6-simplex produces a framework with 140 “boundary” tetrahedra connected in groups of 4 by 210 “joining units” each with 66 internal nodes—counting as “internal” 3 of the 5 nodes of 4 “corner” pentatopes which each share 1 edge with a boundary tetrahedron [fig. 5]. 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. 6].

The intervening figures are the 6-simplex seed's full Tick Tock expansion at t+1 [fig. 2], the site of a joining unit at t+2 where each node is an ancestor of 1 or 2 boundary units and the internal joining unit structure to come is only represented in their triangles [fig. 3], this step being examined in much greater detail on another page, and the emerging joining unit at t+3 [fig. 4].

To get some idea of the intermediate scale we also show a cluster of 6 joining units at t+3 surrounding the great grandchild of 1 of the 35 tetrahedra embedded in the original 6-simplex seed [fig. 7] and a cluster of 10 joining units surrounding the grandchildren of the 5 tetrahedra embedded in 1 of the 21 pentatopes produced at t+1 of the inflation of the 6-simplex seed, also at t+3 [fig. 8].

Viewed in their initial unrotated positions, the evolution of the lowest tetrahedron in the joining unit [figs. 3-5] is instructive. Tetrahedra persist across Tick Tock generations only by being replaced by their inverse each generation. Because the boundary tetrahedra both persist and are positioned to ensure that there are no other interconnections between joining units, it becomes practical to examine longer term 6-simplex inflation by following a single joining unit and its boundary tetrahedra stripped of those tetrahedra's links to other joining units, reducing the computing load by a factor of 200 and thus allowing an extra 5 ticks of not entirely predictable joining unit evolution to be determined with comparable computing resources.

Projection of a higher dimensional structure into 2D or even 3D space causes distortions like late afternoon shadows, only moreso. Of most relevance to Tick Tock, the projected 3D links should not be taken as implying anything about the lengths of graph edges which are by definition undefined. Even though our representations start out with all edge lengths notionally equal in coordinate space, that equality does not always survive 2 ticks, not even in the n+1D space used for calculations, let alone in 2/3D projections.

Of particular relevance here, tetrahedra frequently lose their 3rd dimension, turning into squares or other quadrilaterals with crossed diagonals, no matter how you rotate their 3D projections. Similarly, some thankfully small groups of nodes can all project onto the same 3D location, as has clearly happened disproportionately in 1 of the 6 junction units of the cluster shown in fig. 7 and even moreso to 2 of the 10 in fig. 8.