How can you use the triple tau as an mnemonic to calculate the properties of all Platonic solids?

Old English Royal Arch lectures assert that the intersections of the triple tau form a specific number of angles which can be arranged into five regular combinations 1. When reduced, the sum of these angles in right angles equals the proportions of the five Platonic bodies, which symbolically represent the four elements and the sphere of the universe 1.

The supplied sources record this symbolic connection, but they do not set out the explicit mathematical steps or mnemonic formula used to perform the calculation.