A fact from Prince Rupert's cube appeared on Wikipedia's Main Page in the Did you know column on 10 September 2012 (check views). The text of the entry was as follows:
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I can already think of a cute DYK hook: "Did you know that a cube can fit through itself?"
"Pieter Nieuwland found the largest possible cube that can pass through a hole in a unit cube" Also say who and when this fact was proved; seems important Ovinus (talk) 20:12, 30 June 2022 (UTC)[reply]
Finally found some time to work on this a little today. I found and added the original publication of Nieuwland's result but it's in Dutch so it's hard for me to tell exactly what is claimed. I did not find any source that mentioned Nieuwland at all but gave credit for the proof to anyone else; they all at least strongly suggest that Nieuwland knew his solution was optimal. —David Eppstein (talk) 06:11, 5 July 2022 (UTC)[reply]
"Many other polyhedra have been shown to have the Rupert property" Say "including all five Platonic solids" or something, just so that the reader knows it's not about some boring or obscure polyhedra. Ovinus (talk) 20:12, 30 June 2022 (UTC)[reply]
Presumably you mean convex polyhedra in the lead? Or is that equivalent to the general case by taking convex hulls? That doesn't seem right to me, but I'm afraid I can't visualize it. Ovinus (talk) 17:07, 5 July 2022 (UTC)[reply]
I meant convex. I remembered to say convex once in that sentence, but forgot the second time. I have no idea what should be true in the non-convex case (assuming we could even agree on one of the many definitions for non-convex polyhedra) but taking convex hulls doesn't look likely to work. —David Eppstein (talk) 18:25, 5 July 2022 (UTC)[reply]
See above. Presumably by Nieuwland, some time before he died, but I haven't found a clear explicit statement of that. I have no idea whether, if Nieuwland indeed had a proof, it would meet modern standards of rigor. —David Eppstein (talk) 06:11, 5 July 2022 (UTC)[reply]
"Cubes and all rectangular solids have Rupert passages in every direction that is not parallel to any of their faces" But in which direction is the "Rupert ratio" maximized? Ovinus (talk) 20:12, 30 June 2022 (UTC)[reply]
"optimal rectangle must always pass through the center of the cube" Pass through, sure... but centered on? Being explicit here would help. Ovinus (talk) 20:12, 30 June 2022 (UTC)[reply]
"or must be formed by an isosceles right triangle on one corner of the cube and by the two opposite points" I don't really get what this means. Formed? As in the triangle and the rectangle lie in the same plane and share vertices? Can't visualize this Ovinus (talk) 20:12, 30 June 2022 (UTC)[reply]
Rewritten. It doesn't mention the isosceles triangle any more, but it was intended to mean the 45-45-90 isosceles triangle that you get from cutting off a corner of a cube, at some distance from the corner. The base of this triangle is one edge of the rectangle, and its reflection across the center of the cube is the other edge. —David Eppstein (talk) 11:34, 5 July 2022 (UTC)[reply]
"If the aspect ratio is not constrained, the rectangle with the largest area..." What is the aspect ratio of this optimal rectangle? Ovinus (talk) 20:12, 30 June 2022 (UTC)[reply]
A fun and accessible article. For some reason the article got tens of thousands of views on June12; do you know why? Ovinus (talk) 20:12, 30 June 2022 (UTC)[reply]
Thanks for starting this review! I'm about to go on some international travel so I may be a bit less responsive than usual over the next week or two. No idea re the page views. As for DYK, it's already been listed, so a second nomination would be disallowed. —David Eppstein (talk) 20:26, 30 June 2022 (UTC)[reply]
I don't know. The Huber et al paper is not very explicit; their construction appears to converge to 1 but they don't prove that it's optimal. —David Eppstein (talk) 18:31, 5 July 2022 (UTC)[reply]
One last thing, regarding cite [23]: is it okay to cite ArXiV preprints? I'd suggest adding an inline statement like "In a preprint, Smith & Doe (2020) harvtxt error: no target: CITEREFSmithDoe2020 (help) found ... ", unless it's going to be published soon. But I'm not sure on the guidelines for this stuff. Ovinus (talk) 02:20, 6 July 2022 (UTC)[reply]
It's not a reliably published source, and (as its authors appear to be students) can't be squeezed in under the "established expert" clause of WP:SPS. I removed it. Pity, though, as I found the statistical evidence for some polyhedra not having the Rupert property to be interesting. —David Eppstein (talk) 04:53, 6 July 2022 (UTC)[reply]
Sad indeed... wondering whether it's appropriate to occasionally IAR and cite it (with inline attribution) for interesting but not highly questionable results. I think you accidentally removed some perfectly fine sources in that edit, the ones for the truncated tetrahedron. Once that's been addressed, I think I'll pass; am happy with the article. Ovinus (talk) 13:41, 6 July 2022 (UTC)[reply]
