We use this logic to form an icosahedron and subdivide each of its faces into n^{2} equilateral triangles for some n≥1 about 4.
This yields a simplex with 20n^{2} faces and 10n^{2}+2 vertices.
We move the vertices radially so that they all have distance 1 from the origin.

We follow this plan for recording the data of the configuration.

This is the code.