Reye Configuration
A Mathober 25 challenge! Day 20 was Reye Configuration, which has two versions, one with points at infinity, and one for finite 3D space. First I built just a hexahedron, then realized that would never naturally have this structure. The amazing thing is that there is this point (L here) coplanar in every way you want.
From MathWorld: the configuration consists of 16 lines and the same 12 points such that four lines pass through every point and three points lie on every line. https://mathworld.wolfram.com/ReyeConfiguration.html