Talk:Polyhedron

message: I have a web page at wordpress.com (URL:http://polyhedron100.wordpress.com) with a variety of Nolidean Polyhedra including some nice Crown Polyhedra in wood. I am hoping I might have a link to them here at the polyhedron page as there is no mention of these types of polyhedra elsewhere at wikipedia. Please advise if this would be acceptable? Thanks and Take Care User:Bertimusminimus 10:15, 29 November 2014  — Preceding unsigned comment added by Bertusminimus (talk • contribs)  
:Hi again. These are really beautiful, they certainly deserve a wider audience. I will post a link on my facebook page for sure. But they are not normally regarded as polyhedra, because the surfaces are bounded. If they are regarded as toroidal nolids, even then they are not usually understood as polyhedra: for example two (coincident) edges may share the same two vertices, which is not allowed in conventional polyhedron theory, but only if one chooses to extend the theory specially. As a footnote, a "crown polyhedron", sometimes called a stephanoid, is a particular kind of axially-symmetric (pyramid/prism symmetries) polyhedron. In the end, your creations are at heart beautiful symmetrical mathematical sculptures, and best appreciated as such. Not perhaps what you wanted to hear, but I hope it helps clarify things for you. — Cheers, Steelpillow (Talk) 16:47, 29 November 2014 (UTC)
thread_title: Link to Web Page

message: The text of this article allows us to regard a polyhedron as a 2-dimensional CW-complex as well as a 3-dimensional one. However the definition of the Euler-characteristic implicitly assumes that it is taken as a 2-dimensional CW-complex (i.e. the definition of the Euler characteristic given here is true for polyhedral surfaces, but it isn't true, if the interior volume is considered also to be part of the polyhedron. Perhaps it would be useful to tell this in the article (see Euler_characteristic#Topological_definition).
89.135.8.194 (talk) 22:40, 25 August 2015 (UTC)
replies: text: What the text allows and what it should say are very different things. There is no need to discuss the niceties of CW-complexes in an article which does not mention them. In particular, the Euler characteristic as defined for a polyhedron references only vertices, edges and faces: contrary to what you say, whether or not there is an interior has no relevance. A figure decomposed into say tetrahedral cells is no longer just a polyhedron but a more general topological object - and of no relevance to an article on polyhedra. — Cheers, Steelpillow (Talk) 10:05, 26 August 2015 (UTC)

text: And that shows clearly what I mean. The conventional definition of the Euler characteristic for a polyhedron is given in the article. It does NOT invoke the body and therefore does NOT address the polyhedron as a ball. You are using a different definition applicable in different and more advanced circumstances. — Cheers, Steelpillow (Talk) 11:14, 26 August 2015 (UTC)

text: It is there to be read. The section refers to "any simply connected polyhedron (i.e. a topological sphere)". — Cheers, Steelpillow (Talk) 11:33, 26 August 2015 (UTC)

text: I think the confusion was caused more by the introduction of a ball in this context. Although more complete it brings complexities seldom addressed at this level and best left out. Accordingly, I have edited the general remarks about topological characteristics to confine the discussion to the surface. Any better? — Cheers, Steelpillow (Talk) 11:42, 26 August 2015 (UTC)

text: I don't think that would be accurate. One can perfectly well be considering the surface of a solid polyhedron. The phrase "polyhedral surface" deliberately covers both possibilities. — Cheers, Steelpillow (Talk) 14:10, 26 August 2015 (UTC)

text: You mean, as in "The topological class of a polyhedron is defined by its Euler characteristic and orientability"? The problem we face is that most if not all mainstream sources associate the Euler characteristic in this way, whether or not they note the stricture about its surface. What I have tried to do with my recent edits is to lead the reader from this widely-stated but not wholly rigorous picture to something closer to the truth. But on Wikipedia one cannot say "reliable sources are not rigorous" unless there is a reliable source telling us this explicitly. We are stuck with some measure of woolliness and I am not sure how the current text can be improved on. — Cheers, Steelpillow (Talk) 15:28, 26 August 2015 (UTC)
thread_title:  Euler characteristic 

message: There has been a bit of editing back-and-forth about duality.  I'm inclined to prefer David Eppstein's version (though actually I might even prefer something like this: 
"For every convex polyhedron there exists a dual polyhedron, having ....  (Abstract polyhedra also have abstract polyhedral duals, with the same properties.  However, some definitions of non-convex geometric polyhedra may not have duals meeting the same definition.)").  There is one minor thing that concerns me about it: at that stage of the article, the class of convex polyhedra has not been introduced yet.  (Possibly this is an indication that convexity should be mentioned earlier.)  --JBL (talk) 18:29, 11 February 2017 (UTC)
replies: text: I will not enter further discussion with an editor who accuses me of lying and shouts at me in his edit comments. By rights I should be taking this straight to WP:ANI. I will say here that I have cited the case for nonconvex polyhedra in both articles - Wenninger's is a popular and well-regarded text from a reputable academic publisher. — Cheers, Steelpillow (Talk) 18:46, 11 February 2017 (UTC)
replies: text:  If you want to take David Eppstein to ANI, go for it, but it has nothing to do with choosing among a few options for how to word a sentence. --JBL (talk) 18:58, 11 February 2017 (UTC)
replies: text: Yes, that's exactly the problem. — Cheers, Steelpillow (Talk) 19:01, 11 February 2017 (UTC)
replies: text:  I cannot decipher this comment. --JBL (talk) 19:12, 11 February 2017 (UTC)

text: [edit conflict] Since this response was not particularly informative, let me describe what I think Steelpillow's position is (he can obviously correct me if he would like to actually participate in the discussion rather than blustering). It is that the theory of abstract polyhedra provides a valid form of duality for almost all instances of what people call polyhedra (true) and therefore that all uses of the word "polyhedra" in our articles (unless otherwise qualified) should be assumed to mean abstract polyhedra. Under this interpretation, the sentence "all polyhedra have duals" is true, because what it really means is "all abtract polyhedra have duals". My own position, on the other hand, is that most readers are likely to come to the article with a naive conception of what it means to be a polyhedron (involving something embedded into Euclidean space with flat sides), and that the sentence "all polyhedra have duals" is likely to seriously mislead these readers into thinking that all non-convex Euclidean things with flat sides have dual Euclidean things with flat sides, something that generally isn't true. If we say things that we can reasonably expect to lead to false beliefs in our readers, we are lying to them. (This, by the way, is what Steelpillow thinks of as incivility: pointing out situations where what we write may cause readers to have false beliefs.) To avoid lying to the readers, I would prefer to qualify the statement by saying which kinds of polyhedra have duals: convex polyhedra have convex duals, and abstract polyhedra have abstract duals, but other kinds of polyhedra may not have duals within those other classes of polyhedra. —David Eppstein (talk) 19:02, 11 February 2017 (UTC)
replies: text:  This summary accords with my sense of things.  May I ask specifically about your feelings about the alternate wording I proposed, and the question of whether the location of the discussion of convexity in the article is problematic for it?  --JBL (talk) 19:12, 11 February 2017 (UTC)
replies: text: It's a bit more cumbersome than my attempt but I have no objections to it. You're absolutely right about convexity not having been introduced yet; I think it would make sense to move the convexity section above "Characteristics", since it provides readers with an important class of examples to use in understanding the more technical parts of the Characteristics section. The same problem also happens earlier in Characteristics, in the "Topological characteristics" subsection, which mentions convex polyhedra without their having been introduced yet. So this ordering issue needs to be addressed regardless of what we decide about duality. —David Eppstein (talk) 19:18, 11 February 2017 (UTC)
replies: text:  Well, I made a second attempt but apparently now Steelpillow feels that the source Steelpillow added to support "all polyhedra" in fact supports "uniform polyhedra" but *not* "convex polyhedra."  Sigh.  --JBL (talk) 16:24, 12 February 2017 (UTC)

text: So tell us what your position is, as I invited you to do. Or, maybe more usefully, explain why you think readers will not be misled by your preferred sentence stating that all polyhedra have duals, without qualification. Not why the sentence is true (we all know there is an interpretation for which it is true), but why it is non-misleading. —David Eppstein (talk) 20:06, 11 February 2017 (UTC)

text: As we already discussed on Talk:Dual polyhedron, see Figure 1 (right) of my paper  Steinitz theorems for simple orthogonal polyhedra, a cube with a smaller cubical divot taken out of one edge. It has simple-polygon faces, each edge joining two faces, and no geometric dual, because there are two faces that share two edges and it is not possible to have two different geometric line segments between the corresponding two dual vertices. Like most of the other examples we're talking about, you can throw away the geometry to get an abstract polyhedron, which has a dual, but the dual is not a geometric polyhedron. —David Eppstein (talk) 01:17, 12 February 2017 (UTC)
replies: text: Oh, sorry, I had misinterpreted that at a cubic hole in a face instead of in an edge, and hadn't seen the paper/figure itself. Maybe if also adding the constraint that two faces may only share one edge. Don't know, or maybe there are counterexamples to everything I can come up with… Either way, I think having an abstract dual which doesn't make sense geometrically is something that probably ought to be mentioned in some form. Κσυπ Cyp   10:48, 12 February 2017 (UTC)
replies: text: Some people do indeed add a constraint like that, even for abstract polyhedra. See e.g. Whiteley, who discusses the stronger rule that, when faces share more than one vertex, there can be only two and they must share an edge that's also on both faces. One drawback to such rules is that unless you're careful you end up with only the same topological structures as convex polyhedra, just with different geometries (so as abstract polyhedra there would be no point to calling them non-convex). Anyway, I agree that we should mention the existence of non-convex geometric polyhedra with no non-convex geometric dual. That has been my point all along, and I don't see why it has become so controversial. —David Eppstein (talk) 18:35, 12 February 2017 (UTC)
replies: text:  I rather liked this version -- in particular, I feel like it separates the claim that the most common classes of polyhedra do have duals from the more subtle issues.  I would set about finding supporting sources if you two agreed with using it as a base.  It still needs convexity to be mentioned earlier, of course.  I can't decide if just cutting and pasting the convexity section before this one is the right thing; maybe make convexity an (early) subsection of the big Characteristics section? --JBL (talk) 14:22, 13 February 2017 (UTC)

text:  , you've been active on this article and talk page as well; would you care to weigh in? --JBL (talk) 14:42, 13 February 2017 (UTC)

text:  Here is a nice survey of Grunbaum and Shepard from 1969; on page 260 we find the statement that all [convex] polytopes have [convex] duals, as well as the statement that polar reciprocation provides the dual.  (Of course Grunbaum's text also would serve as a source.) --JBL (talk) 14:39, 13 February 2017 (UTC)

text: May I suggest that the "what is a polyhedron?" definitional issue should be addressed here, but that the subsequent duality issue is best addressed where it began, at Talk:Dual polyhedron? It is not helpful to have parallel discussions on the same issue. Do folks have a problem with that or is there a better way to structure the content discussion, say keeping both issues on the same page? Now that the ANI issue is basically settled, I will return to these discussions once I know where they are taking place. — Cheers, Steelpillow (Talk) 16:51, 12 February 2017 (UTC)
thread_title:  Duality 

message: Comments moved to Talk:Polyhedron#Duality and citation below, to give better visibility and coherence. — Cheers, Steelpillow (Talk) 12:04, 18 February 2017 (UTC)
thread_title: Proper citations

message: A relevant citation of a reliable source cannot be summarily removed based on editorial opinion. It needs to be discussed and consensus established first. If that is to be done then let us revert to an earlier version of this section and stop making unwarranted edits. — Cheers, Steelpillow (Talk) 15:14, 12 February 2017 (UTC)
replies: text: Demanding that other people discuss while refusing to engage substantively is not constructive.  David Eppstein has repeatedly asked you some very simple substantive questions; can you answer them?  --JBL (talk) 16:17, 12 February 2017 (UTC)
replies: text: A content dispute is one thing, but bad behaviour by editors is quite another. I am at present happy to engage on the second - on whether or not a relevant citation of a reliable source can be summarily dismissed, and if not then whether the content it supports may be allowed to contradict it. Are you willing to accede that all editors here should abide by WP:POLICY and edit per reliable sources and not per their personal opinions? Are you willing to acknowledge that a pertinent passage, supported by a reliable citation, should not be summarily deleted or perverted just because an editor disagrees with it but can themself produce no better citation? — Cheers, Steelpillow (Talk) 16:51, 12 February 2017 (UTC)

text: I will answer David once I know that he can assume good faith and keep a civil tongue in his head, there is no mileage in a slanging match. I will say to you that I think the "what is a polyhedron?" definitional issue should be addressed here, but that the duality issue is best addressed at Talk:Dual polyhedron. It is not helpful to have parallel discussions on the same issue. Do you have a problem with that or do you think there is a better way to structure the content discussion, say keeping both issues on the same page? — Cheers, Steelpillow (Talk) 16:51, 12 February 2017 (UTC)
replies: text: This attempt at organizing the discussion, while laudable, still seems to miss the point: we should not insist on having only a single definition for a polyhedron and standardize on that one definition throughout our articles on the subject. To do so would violate WP:NPOV. Rather, to the extent that multiple competing definitions have been covered in reliable sources, we should describe them all here, and describe their dualities at the other article. —David Eppstein (talk) 17:29, 13 February 2017 (UTC)
replies: text: Yes, that would need to be a part of the discussion - which definitions do we present and how? I offer my own view on that in a new discussion below. — Cheers, Steelpillow (Talk) 18:21, 13 February 2017 (UTC)
thread_title:  reliable source 

message: Does anyone else think this subsection is a bit odd?  The first half is an (uncited) discussion of Platonic solids, giving an unusual way to write their volume.  The second is also somewhat odd -- yes, the divergence theorem could be used to compute volumes, but it's not the first thing I would think to tell someone about volumes of polyhedra. --JBL (talk) 14:49, 13 February 2017 (UTC)
replies: text: Yes, the second section is certainly not trivial, something most readers would ignore. Tom Ruen (talk) 15:16, 13 February 2017 (UTC)

text: I see Steelpillow restored it on 2009/Dec/9 , and original anonymous editor added it on 2009/Nov/19  with talk discussion here Talk:Polyhedron/archive2#section_added. Tom Ruen (talk) 15:23, 13 February 2017 (UTC)
replies: text: I restored a wholesale delete by an IP editor in part because there was an ongoing discussion about it and deleting material wholesale while it is under discussion is seldom helpful. I have no opinion beyond my comments made back then. — Cheers, Steelpillow (Talk) 15:43, 13 February 2017 (UTC)

text:  I was the one who removed it back then, moving a copy to the talk page. I'm not a deletionist at all and don't delete material wholesale when it is worthy of discussion. Tom Ruen (talk) 16:47, 13 February 2017 (UTC)
replies: text:  No worries, I was just going by my edit diff that you posted. — Cheers, Steelpillow (Talk) 16:59, 13 February 2017 (UTC)
thread_title:  [[Polyhedron#Volume|Characteristics#Volume]] 

message: BTW, since some readers here seem to believe that Cromwell's Polyhedra is the bible of polyhedra, and should be used in preference to all other sources, it may be worth repeating what I found on Peter McMullen's MathSciNet review, :
Which is not to say that it's a bad source by any means, just that (like any other source) its author is human, and we should not be uncritical in our reading of it. —David Eppstein (talk) 01:18, 21 February 2017 (UTC)
replies: text: Thank you. If I meet one of those Cromwell evangelists I will let them know. The key thing about criticism is that it needs to be backed up by a reliable source of that criticism, as you have done here. However, I am unclear as to the relevance of the above criticisms to the present article? WP:PSTS obliges us to take notice of such a widely-read book and the suggestion that we look to Cromwell as a starting point seems a reasonable one. — Cheers, Steelpillow (Talk) 11:01, 21 February 2017 (UTC)
thread_title:  Reliability of Cromwell 

message: Under "Generalisations of polyhedra" we have a section Polyhedron#Hollow-faced or skeletal polyhedra, while under "Alternative usages" we have a section Polyhedron#Polyhedra as graphs.  Both refer to the idea of disregarding to some degree the faces of polyhedra.  Is there a reason not to merge these two sections?  If not, are there strong feelings about which of the two locations is better? --JBL (talk) 20:31, 4 March 2017 (UTC)
replies: text: I'd support graphs as a subsection to the first. Tom Ruen (talk) 14:23, 5 March 2017 (UTC)
replies: text:  Ok, I have merged the two.  --JBL (talk) 21:17, 7 March 2017 (UTC)
thread_title:  Skeletal polyhedra versus polyhedra as graphs 

message: I found this odd-ball article Polynomial-time algorithm for approximating the volume of convex bodies that doesn't seem like it deserves it's own page. Maybe it ought to be merged here? Jason Quinn (talk) 11:03, 1 August 2017 (UTC)
replies: text: It's a significant topic in geometric algorithms, worthy of its own article. Also, these algorithms work even for smooth convex bodies, so polyhedron is a bad merge target. —David Eppstein (talk) 15:59, 1 August 2017 (UTC)
thread_title:  Merge Polynomial-time algorithm for approximating the volume of convex bodies? 

message: Considing that historically the Ethruscans are thought to be the Villanovians that upgraded their civilization after contacts with the Greeks (see books by italian scholar Mario Torelli), and given the impressive indifference of Italic people, the Romans included, for subjects related to mathematics, it's conceivable that the Ethruscan dodecahedron wasn't an original result.  — Preceding unsigned comment added by Ygmarchi (talk • contribs) 14:51, 24 August 2017 (UTC)
thread_title:  History 

message: Why doesn't the illustration at the top of the page have the best-known solid ?5.34.81.117 (talk) 01:41, 26 September 2017 (UTC)
replies: text: I would think that having one convex regular polyhedron should already be representative of that limited class. I am also not sure if the cube is really significantly more well-known than the tetrahedron, which is after all easily described in layman's terms as a triangular pyramid. Double sharp (talk) 10:47, 26 September 2017 (UTC)
thread_title:  cube 

message: The following paper in Synthese may be of interest to editors of this page:  . --JBL (talk) 18:20, 12 July 2021 (UTC)
replies: text: Paywalled. Try this link — Cheers, Steelpillow (Talk) 18:49, 12 July 2021 (UTC)
replies: text:  Wikipedia editors often can get access to paywalled sources via the Wikipedia Library.  In this particular case, any editor of sufficient standing (account at least 6 months old, 500+ edits including at least 10 in the last month, not currently blocked) has access to the "library bundle".  The bundle includes ProQuest, and this article can be accessed there. --JBL (talk) 19:07, 12 July 2021 (UTC)
thread_title:  On varying definitions 

message: The lede first explains what a polyhedron is, before attempting in the next paragraph to explain what a convex polyhedron is. But that attempt is hopeless. Instead of explaining the property of convexity, it instead references the "convex hull", with no attempt to explain what convexity actually is. Furthermore, it demands a "finite" point set. Sure that is a common definition in convexity theory (with plenty of RS in that context), but infinite polyhedra can be constructed in several ways, some of which may lead to convex constructions. (Consider for example constructing four Archimedean spirals from the corners of a square and in towards the centre. Mark the points of a projective measure along each spiral, such that the measure converges on the centre after infinitely many iterations. Now connect the points of adjacent measures to form a triangulating zigzag between each pair of measures. Straighten up all edges, push the middle of the square down to form a shallow bowl, and glue six together to make a convex polyhedron with infinitely many vertices. In short, project one of Dan Erdely's spidron-ised polyhedra onto a sphere.) So I'd suggest going back to an earlier and more general definition of convexity, and say instead that "A convex polyhedron is one in which a line may be drawn, connecting any two interior points, without intersecting the boundary." Thus, we actually explain the convexity property to the reader. Note that the "interior" exclusively is mentioned, along with "can be constructed", because a degenerate two-dimensional construction has no such points and therefore no such line can be drawn. Any objections/improvements? — Cheers, Steelpillow (Talk) 08:51, 5 August 2023 (UTC)
replies: text: The lead is supposed to be vague and intuitive, not to provide a precise definition, but it is also supposed to be not misleading. You can make things that have piecewise linear boundaries and infinitely many points, but it is dubious whether they can be called polyhedra. Your example is non-polyhedral at the limiting center point of the spiral, for instance. Your example meets none of the definitions of polyhedra in the definitions section. —David Eppstein (talk) 14:00, 5 August 2023 (UTC)
replies: text: There is nothing intuitive about "explaining" convexity by buzzing readers with hulls while omitting to actually explain convexity. Far more intuitive to offer the line-between-two-points description. I take your point about finiteness; Coxeter and Cromwell both give such definitions, though in other contexts they contradict themselves and it is moot whether they regard the infinite variety as a sub-class or an extension, so yes, keep that out of the lede. — Cheers, Steelpillow (Talk) 15:24, 5 August 2023 (UTC)
replies: text: I added "a convex polyhedron is a polyhedron that bounds a convex set", as an intuitive definition. It is deliberately vague rather than being more specific about connecting interior points by line segments, as you suggested, because it is very difficult to state that more specific formulation without pre-judging the definitional issue of whether a polyhedron is just the boundary or whether it includes the interior as well. You may well talk about "buzzing readers with hulls" but that is a very important aspect of that topic, important enough I think to mention in the lead here. We may not expect all readers to understand it but if we wrote our article at a level already understood by all readers we would have very incomplete coverage of its topic. —David Eppstein (talk) 21:10, 5 August 2023 (UTC)
replies: text: I don't see that a convex "set" is any more intuitive than a convex "hull", if one singularly fails to explain what "convex" means in the first place. We can at least expect novices to know what points and lines are. The line between interior points does not prejudge whether the interior is part of the polyhedron or not. How about writing it as, "A convex polyhedron is one in which a line may be drawn, connecting any two interior points, while remaining wholly within the interior." The remark about convex hulls will then be more intelligible to said novices. — Cheers, Steelpillow (Talk) 07:28, 6 August 2023 (UTC)

text: This seems like confusing pedantry. I would say something along the lines of "Any two points inside a convex polyhedron can be connected by a line segment contained wholly in the interior." Edit: Erm, not sure why I started writing before reading  Steelpillow's almost identical comment. Whoops. –jacobolus (t) 07:34, 6 August 2023 (UTC)
thread_title:  Convex polyhedron in the lede