# Talk:Schönhardt polyhedron

This review is transcluded from Talk:Schönhardt polyhedron/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: Dedhert.Jr (talk · contribs) 02:37, 19 February 2024 (UTC)

I'll be reviewing this article, although I have edited it before, with the reason below. Following comments:
The lead summarizes the whole article, but one problem is the source of its historical discovery. In order to make the readers, especially if they particularly are in favor of researching even more about this polyhedral article, can we simply link it more specifically rather than just at the beginning of the journal? Also, is it possible now to add {{infobox polyhedron}}, as the article describes the number of vertices, edges, faces, and types of this polyhedron, together with the net from Branko's source? Dedhert.Jr (talk) 04:16, 19 February 2024 (UTC)
Construction: This section fully describes how the Schonhardt polyhedra are constructed, but it does not mention what is the angle of one of two equilateral triangles rotated. I do think we can use the first source in the lead to mention it, or was it fully restricted by WP:PRIMARY so we should find other sources that are cited? Dedhert.Jr (talk) 04:16, 19 February 2024 (UTC)
Properties: Each Schönhardt polyhedron has six vertices, twelve edges, and eight triangular faces: Why "each"? This section also mentions the convex hull, but I do not have a good visualization about this after I read this; is it possible to explain more specifically? Also, the source does not mention that both the Schonhardy polyhedron and its convex hull are combinatorially equivalent to the regular octahedron [1]. I wonder if is there some mensuration of this polyhedron, such as surface area or volume. Is there a symmetry group? I hope they can be included here. Dedhert.Jr (talk) 04:16, 19 February 2024 (UTC)
Properties/Impossibility about triangulation: It is impossible to partition the Schönhardt polyhedron into tetrahedra whose vertices are vertices of the polyhedron. Umm, "vertices of the polyhedron"? What? Does it refer to every polyhedron in general? I don't get it. Dedhert.Jr (talk) 04:16, 19 February 2024 (UTC)
Properties/Jumping polyhedron: The cited source in this section does explain the definition of "jumping polyhedron". However, it does not mention the Cauchy rigidity theorem. Did I miss something here? Dedhert.Jr (talk) 04:16, 19 February 2024 (UTC)
Related constructions: Rambau (2005) has a dead link, and I cannot check the verifiability. Also, the second paragraph mentions there is no internal diagonal of the resulting polyhedron by such construction. Frankly, why does Czasar tetrahedron suddenly appear here? Does it solely explain what are other examples of polyhedrons with diagonal entirely on the outside? Dedhert.Jr (talk) 04:16, 19 February 2024 (UTC)
Applications: The section explains its usage in the proof of NP-complete problem, but I wonder why there is an image of tensegrity here? If that's the case, can we add the applications in tensegrity as well? Note that I have edited the article before, trying to tidy up one reference in the caption image. One problem is the source is Lulu, a self-publisher; is it fine? Dedhert.Jr (talk) 04:16, 19 February 2024 (UTC)
Overall, the article has met some criteria according to WP:GACR, with some comments of suggestions to improve the article and meet the criteria even more. Will respond to them swiftly. Dedhert.Jr (talk) 04:16, 19 February 2024 (UTC)

Some partial responses:
Added material from Schönhardt (1928) to the "Jumping polyhedra" and "Related constructions" sections, including Schönhardt's observation that some forms of this polyhedron are shaky. The addition to "Related constructions" also expands on the history behind Schönhardt's discovery.
Added an infobox and a net. I drew the net differently from Grünbaum's, which I am not entirely sure of (it looks like it has a triangular-prism state with some of its diagonals non-folded). After initially making a similar mistake I checked with paper that the net I drew really does work.
Added a reference in "jumping polyhedra" to Cauchy's theorem; the reference explicitly compares rigid polyhedra, jumping polyhedra, and flexible polyhedra, although it does not specifically mention the Schönhardt polyhedron as jumping.
Re the tensegrity image: I moved this to the section on jumping polyhedra as it is closely related, after finding more related material. That source looked ok to me (I think really published by the "Natural Philosophy Alliance" and using Lulu only as a convenience) but I swapped it out for others that were more clear on the geometry and history of the tensegrity prism.
—David Eppstein (talk) 19:56, 19 February 2024 (UTC)
@David Eppstein Okay. Let me know if you have completed all of the suggestions above. Dedhert.Jr (talk) 00:49, 20 February 2024 (UTC)
Sure. I'm just replying to the ones I've already gotten to, when I get to them. —David Eppstein (talk) 01:46, 20 February 2024 (UTC)
Some more responses:

Construction: Clarified that the rotation angle is variable, sourced to Schönhardt.
Re properties: copyedited the first paragraph of this section, I hope to make it more clear. But mensuration or symmetries would require a specific choice of the shape, not possible here because there are free parameters in the choice of the shape. Even if we fix the special 30° rotation angle we would still have a choice in the distance between the planes of the two equilateral triangles. We could choose those to be a unit distance apart, or choose the convex prism edges to be of unit length, giving two specific choices for the shape, but those choices are arbitrary and unsourced. It is the combinatorial structure, not the precise measurement, that is important.
Re "vertices of the polyhedron": every polyhedron can be triangulated by adding more triangulation vertices that are not vertices of the polyhedron. In the case of the Schönhardt polyhedron this can be done by adding one more vertex at the centerpoint. It is only when restricting to the six given points that triangulation is impossible. Copyedited to clarify.
Fixed the Rambau deadlink. Re why Czasar polyhedron: because it satisfies the same property as the Schönhardt polyhedron of having no interior diagonals.
@Dedhert.Jr: I think that's at least something in response to each of your comments, but if I missed anything please let me know. —David Eppstein (talk) 07:25, 21 February 2024 (UTC)

@David Eppstein Okay. Thank you for addressing all of them. Just passing by and reviewing the article again. But I think there is something I want to add in the lead. It says that it is "the simplest polyhedron that cannot be triangulated into tetrahedra without adding new vertices". I wonder if we can put some explanation of the context of the phrase "simplest" here in both the lead and the property section? Also, maybe we can add another paragraph, so there are two paragraphs in the lead, in which to describe briefly the related construction in the article's body; it is not mentioned anyway. Dedhert.Jr (talk) 07:35, 21 February 2024 (UTC)
Ok, I'll work on the lead and ping you again. Real life has been keeping me busy lately so it might be a couple days before I have the time and energy for that. —David Eppstein (talk) 06:56, 22 February 2024 (UTC)
Nah, that's okay. Take your time to do your priority first, then you can continue this work later. I have some other busy things to do here. Dedhert.Jr (talk) 12:11, 22 February 2024 (UTC)
@Dedhert.Jr: Ok, lead expanded. —David Eppstein (talk) 08:36, 25 February 2024 (UTC)
Okay. I do not see anything missing here. So,  passing for GA. Felicitations. Dedhert.Jr (talk) 10:33, 25 February 2024 (UTC)