Talk:Compound of four octahedra with rotational freedom

Coplanar planes angle
I added the angle at which coplanar faces arise, θ ≈ 44.47751(21859299)°. I realize I'm making an unsourced claim – I made the calculation myself on GeoGebra, so that probably doesn't meet Wikipedia's verifiability standards (although, you could verify my claim by building the model yourselves, I guess). On the other hand, most mathematical claims on the polyhedron pages (specially coordinates) on Wikipedia are equally unsourced, so I'm probably not committing that big of a sin. If someone finds a good source that arrives at my (or a similar) figure, or if they decide that the claim doesn't belong on the article, they can feel free to edit this in or out. – (talk) 22:37, 25 March 2020 (UTC)

Actually, it's possible to prove that $$\theta=2\tan^{-1}(\sqrt{15}-2\sqrt{3})$$. Taking the coordinates of any of the two faces that become coplanar from here (more unsourced claims!), we can calculate their centroids. Two of their coordinates will be identical, the third will be equal precisely when $$2\sqrt{3}\sin(\theta)=1+2\cos(\theta).$$ WolframAlpha is then able to kindly solve this for us. – (talk) 23:05, 25 March 2020 (UTC)