The image of the 16-cell rotating about two orthogonal planes is inaccurate. The octahedral hull of the projection of the object into three-dimensional space should deform such that two of the vertices come sqrt(2)/2 times as far apart as in a regular octahedron when the two interior vertices reach their furthest apart points. I have an I.Q. of 129 and have personally visualized the polychoron rotating in four-dimensional space and being projected into three-dimensional space, and the last time I attempted to explain this mistake, my comment was ignored and deleted. I would imagine that changing the actual image would be difficult, but I would like my verbal description of the mistake to stay here instead. —Preceding unsigned comment added by 188.8.131.52 (talk) 01:13, 23 December 2010 (UTC)
I'd suggest you express your concern on the talk page of the user who created it, User_talk:JasonHise. It looks like he's not very active lately, but a better chance for a response there if he's busy since he'll get a notice when he logs in. OR better, I see his user pages links to his website with an email address . Tom Ruen (talk) 02:28, 23 December 2010 (UTC)
The o.p. describes the edge-first projection, which is shown in a chart lower on the page. "The two interior vertices reach their furthest apart points" when they coincide with "exterior" vertices. It's not surprising if the animation happens not to reach that particular configuration; its projection-vector traces a one-dimensional subset of S3, and if I were making such an animation I'd avoid the most symmetric forms (an arbitrary aesthetic preference). — I don't think I've ever stated my IQ in public, but it's higher than 129. :P —Tamfang (talk) 07:22, 23 December 2010 (UTC)
Tom, you're a card. Whatever that means. —Tamfang (talk) 01:15, 24 December 2010 (UTC)
Looking again, it appears to me that the animation does include the edge-first projection, but at a viewing-angle such that it's hard to judge the ratio in question. —Tamfang (talk) 18:01, 24 December 2010 (UTC)
Looks fine to me, and when tested (decades ago) my IQ was a lot higher than 129. It's easiest to see the rotations if you first wrap your mind around the similar tesseract, which is rotating in a similar way (an isoclinic double rotation) with a similar projection. Each looks like it's spinning on an axis in 3D. The two planes of rotation are the plane orthogonal to this axis (the xy-plane, say) and the zw plane where z is along the axis it's spinning on and w is the fourth dimension, projected with a perspective projection orthogonal to z.
There's no right way to do it. Think off all the ways to project a 3D cube into 2D (parallel; perspective; face-on; edge-facing; looking isometrically at a corner). It's at least twice as complex projecting into 3D from 4D. IMHO these animations do an excellent job of it, and rightly are among the most popular and best regarded mathematical images on WP.--JohnBlackburnewordsdeeds 01:45, 24 December 2010 (UTC)