David Hall derived this amazing projection using the field derived from a 30-sided regular polygon. The actual van Oss projection is a 2D projection that Donald Coxeter highlights in his book, “Regular Polytopes”, and he gives the algebraic definition. David managed to encode that in the exact coordinates of the 30-gon field for vZome.
The story of how David managed that is quite remarkable. Hopefully I’ll be able to link to it once he has captured it in his own page.
The vZome file downloadable here is my own version derived from David’s, in order to highlight three special lines. If you open the downloaded file in vZome desktop, you can right-click on each of the red, yellow, and navy blue struts and select “build with this”. (Both navy blue struts are in the same orbit, so you only have to one of them.) Now you’ll see orbit dots (white) in the orbit triangle for these three orbits.
Next, if you right-click in the orbit triangle itself, you can select “configure…” and enable these three orbits as snap directions in the dialog that comes up.
Now you’re ready to turn on snap, make sure you have perspective off, and view the important projections: both of the van Oss projections, one with 4-fold symmetry, and two with simple mirror symmetries. The latter three lines of sight form a natural coordinate frame for this projection.
David and I hope to define an octahedral symmetry system that uses that frame. With that in place, we can determine how many distinct orbits are required for the projection.