Table of examples of pointwise extremal configurations on three-dimensional sphere discussion in the arXiv preprint "Extremal spherical polytopes and Borsuk's conjecture "

In our simulation, we apply gradient flow to the diameter functional in order to find local minima. The results of this simulation reveal that these local minima are anti-self-polar (ASP), meaning that their convex hulls have the property of being ASP. This finding sheds light on the relationship between the diameter functional and ASP configurations.

Simulation procedure: The objective of the simulation is to obtain configurations derived from k-stacked tetrahedra configurations \(T_k\), which is a set consisting of the north pole and \(k\) tetrahedra on the 3-sphere \(\mathbb{S}^3\). Four calibrated tetrahedra \((T_1, T_2,T_3, T_4)\) are chosen, and up to 4 points are deleted from each of them to form the initial sets. The gradient flow is then applied to this initial sets.

The table presents 65 configurations from the diameter gradient flow, sorted based on the number of points, the number of edges in its convex hull, maximum number of vertices in a facet, number of facets with maximum vertices, number of triangles in diameter graph,and number of complete 4-node subgraphs in the diameter graph . These configurations are distinguished by the isomorphism type of their diameter graphs. These configurations are novel in that they are not simply pyramid extensions of configurations found on the 2-sphere. To see configurations on the 2-sphere with at most 10 vertices, follow this link. Some additional configurations on \(\mathbb{S}^3\) derived from the configurations in the table below can be found by following this link