Any other Caltrops?
There is a caltrop on 76 points.
Points 1, 13, 25, 29, 41, and 53 are as follows:
{0.0833`, 0.0833`, 0.4930122817942774`}
{0.32530527130128584`, -0.20709494964790603`, 0.32530527130128584`}
{0.28875291001058745`, 0.28875291001058745`, 0.28875291001058745`}
{-0.2142`, 0.40369721678726284`, -0.2142`}
{-0.07272969962634213`, 0.35355339059327373`, -0.35355339059327373`}
{0.07587339432355446`, 0.44185`, -0.23402687345687453`}
This caltrop generates a solid of constant width. The polyhedron has 150 unit diagonals, is self-dual, and has tetrahedral symmetry.