Talk:Dual polyhedron

Untitled
The statement "If a polyhedron has an element passing through the center of the sphere, it will have an infinite dual." makes no sense, since it there is no definition (or even explanation) what "an infinite dual" means. No definition of "polyhedron" admits vertices or edges or faces "at infinity.

Generalize to dual polytopes?
Many of the concepts described in this article also apply to n-dimensional polytopes. Would it make sense to extend this article so that it applies to both? There are quite a good number of articles on 4-dimensional polytopes (see polychora and uniform polychora), which would make good use of a more general description of duals.&mdash;Tetracube 22:14, 17 February 2006 (UTC)

Good to add something! Choices? Tom Ruen 23:10, 17 February 2006 (UTC)
 * 1) Rename to Dual polytope and expand with divided sections by dimension? (starting with regular polygons as self-duals)
 * 2) Create Dual polychoron article and move/expand the 4D/honeycomb content there?
 * 3) Keep Dual polyhedron, add Dual polychoron, and add Dual polytope with dimensional article references?


 * I prefer to put them together, since otherwise there will be a lot of needless repetition. So option 1 sounds good to me.&mdash;Tetracube 01:54, 18 February 2006 (UTC)

Sounds good! I wonder who else is watching?

Incidentally, there's LOTS of links to this article. I've found it useful elsewhere to link expanded by article/headers. In this case like dual polyhedron and dual tiling and dual polychora and dual honeycomb for dimensional subsections. That works well as long as headers are not changed! Tom Ruen 04:08, 18 February 2006 (UTC)


 * As these articles pad out, I think this begins to make for rather large pages. For example, consider listing all the interesting dual pairs in every dimensionality all on a single page! .


 * I'd like to keep this page fairly simple, and add a Dual polytope page for the general theory, e.g. to do justice to the duality of abstract polytopes. Dual polychora? well, I'd keep it as part of Dual polytopes until it all gets too big, then float it off on its own. -- Steelpillow 21:26, 4 June 2007 (UTC)

Consensus to rename to Dual polytope?
YES
 * 1) Tom Ruen 02:43, 18 February 2006 (UTC)
 * 2) Dshin 19:39, 23 March 2006 (UTC)
 * 3) SLWoolf (talk) 04:50, 28 October 2008 (UTC)
 * 4) Tamfang (talk) 01:10, 29 October 2008 (UTC)
 * 5) Tetracube (talk) 04:57, 29 October 2008 (UTC) &mdash; we can use dual polyhedra to explain to concept first, and then lead on to polytopes. This should keep the article accessible to the general reader.

NO
 * 1) Steelpillow 21:26, 4 June 2007 (UTC)

Please, no
There is much in this article at an elementary level, such as the important Dorman Luke construction. To bury this in a discussion of polytopes generally is IMHO unhelpful to the majority of students who wish only to find out about dual polyhedra. If you guys move it, I will have to pull the elementary stuff back here - so by all means move or copy across the relevant stuff, but please save me the trouble of recreating this article over. -- Cheers, Steelpillow (Talk) 20:30, 29 October 2008 (UTC)


 * What about putting the polytopes stuff (including any polytope-specific generalizations) at the end of the article, after the elementary stuff? The concept of dual polytopes is a generalization of dual polyhedra, after all. We can add a section, maybe entitled "Generalization to higher dimensions", and put the polytope-related material under it.&mdash;Tetracube (talk) 20:38, 29 October 2008 (UTC)


 * The difficulty here is that the theory of topological and abstract dualities is primarily worked out for arbitrary dimensionality, i.e. for polytopes generally. Here, it would be better to treat polyhedra as a follow-on topic. There is a Wikipedia policy which allows some repetition of material, where it enables each article to stand on its own. In the present case, the treatment of the common material would tend to diverge in the two articles - the one becoming a little simpler, and the other more advanced. -- Cheers, Steelpillow (Talk) 20:56, 29 October 2008 (UTC)


 * In passing, I'm amused at the term polytope-specific, since polytope is the general case and polyhedron is specific. —Tamfang (talk) 07:24, 30 October 2008 (UTC)

Generalization of the Dorman Luke construction to n dimensions
The Dorman Luke construction of the dual essentially makes use of the fact that the facets of the dual polytope are precisely the dual of the vertex figures suitably enlarged. For polyhedra, it does not really matter whether it's the vertex figure or its dual, at least not for regular/uniform polyhedra, since regular polygons are self-dual. In higher dimensions, however, this quickly becomes obvious (e.g., the vertex figure of a 24-cell is a cube, yet its dual has octahedral cells; similarly, the vertex figure of the 600-cell is an icosahedron, but its dual has dodecahedral cells). Even in 3D, however, we do see a subtle hint that it's not simply the vertex figures, but the dual of the vertex figures, that form the dual polyhedron: take the vertex figures of the cube, for example. They are triangles, but oriented in the dual position to the orientation of the faces of the octahedron. Taking their duals gives us the faces of the octahedron in the correct orientation.

By this, it should be obvious that the Dorman Luke construction of the polyhedral dual is easily generalized to higher dimensions: given an n-polytope, finding its dual amounts to finding the duals of its vertex figures. Since vertex figures are (n-1)-dimensional, we simply recursively apply this process until we reach the trivial case (polygons).

I'd add this to the article, except that I'm not sure if this violates original research or not. :-)&mdash;Tetracube (talk) 20:36, 6 August 2008 (UTC)


 * None of this is OR. However, one needs to be very careful in generalising - Luke's construction was originally described only for highly symmetrical polyhedra. Wenninger generalised it to some extent, but it does not apply to polyhedra in general. It is in fact a special case of the more general reciprocity (also called polarity or, incorrectly in this instance, projective duality) of any polyhedron about any sphere. Yes the Dorman Luke construction can be generalised to higher dimensions, but at this level of difficulty it is more productive to generalise polar reciprocation. FYI, see my essay on Vertex Figures. -- Cheers, Steelpillow (Talk) 21:21, 7 August 2008 (UTC)

Transverse Homonym
Has anyone noticed that all references, citations, and Wiki links to my transverse homonym are missing from these articles? Even his/her/their biography(ies) is (are) missing from WikipediA. Laburke (talk) 19:48, 1 November 2011 (UTC)
 * I guess that's because he was (IIRC) an otherwise undistinguished school teacher, whose construction was only ever published by other authors. &mdash; Cheers, Steelpillow (Talk) 20:54, 1 November 2011 (UTC)
 * Too bad, but people do seem to go on about his "generalization". Was he so undistinguished that they don't want to honor him by raising it to a "conjecture"? Perhaps his own schools were too undistinguished for that ;) As for school teachers in general, Eriugena was a distinguished philosopher yet he was killed by the quills of his students. Thanks for answering so quickly and BTW, I "see" polyhedra as central point(s) joining apices. It's a chemist sort of thing. Laburke (talk) 03:33, 2 November 2011 (UTC)
 * The generalisation of his construction seems to have been first described by Wenninger. Both Luke's original and Wenninger's generalisation are simplifications of construction methods in projective geometry that had been known for many years - specifically the polarisation of a polygon in a conic section (here a circle). The appealing feature of these simplifications is that they are suited to the classroom, not that they have any theoretical novelty. BTW, Socrates was a distinguished philosopher but killed himself because he was condemned to do so by his fellow citizens. &mdash; Cheers, Steelpillow (Talk) 22:42, 2 November 2011 (UTC)

self and not
A sentence in Square pyramid prompts me to wonder: What's the simplest convex polyhedron which has V=F but is not self-dual? —Tamfang (talk) 05:33, 21 May 2010 (UTC)


 * Gyrobifastigium looks like a good candidate. Tom Ruen (talk) 21:25, 1 November 2011 (UTC)


 * First off, I realised that the solutions must come in pairs - dualising any solution obtains a second solution (because V and F dualise to each other). I then asked around. Someone pointed out that in the table of heptahedra, the 4th from the left in row 5 and the 2nd from the left in the last row are dual solutions. I checked the rest, then the simpler hexahedra, etc, and his solution appears to be correct. &mdash; Cheers, Steelpillow (Talk) 22:30, 2 November 2011 (UTC)

Duality
I was quite surprised about the sentence "regular polygons are geometrically self-dual". The dual of a cube is a octahedron and I wouldn't consider these to be congruent figures - which is required in the definition of self-duality. Can anyone explain how congruence of polyhedra is defined in this case, so that it makes sense at all? — Preceding unsigned comment added by 128.176.180.52 (talk) 11:15, 12 February 2013 (UTC)


 * Regular polygons are self-dual, most regular polyhedra are not (only the regular tetrahedron is). Hope this helps. &mdash; Cheers, Steelpillow (Talk) 20:49, 12 February 2013 (UTC)

Quick question: I am trying to understand the sentence "Starting with any given polyhedron, the dual of its dual is the original polyhedron." I thought that to construct the dual of a polyhedron, all you do is place a (dual) vertex at the centroid of each face of the original polyhedron, and then connect the (dual) vertices that correspond to adjacent (original) faces. Well, if our original polyhedron is concave, then the dual of its dual would not be the original polyhedron. (For example, say we start with a concave hexagonal prism with end faces shaped like an hourglass -- (0,0), (6,0), (4,4), (6,8), (0,8), (2,4), with z-values at 0 and 4. The dual polyhedron would be a convex hexagonal bipyramid with equatorial vertices at (3,0), (5,2), (5,6), (3,4), (1,6), (1,2) at a z-value of 2, and end vertices at (3,4) at z-values of 0 and 4.  Finally, the dual-dual would be a convex hexagonal prism (I could find the vertices, but not right now), and thus not similar to the original polygon.)  My question is this:  Am I understanding the dual incorrectly or are there unstated conditions on the original statement? I would appreciate some clarification here. (FYI, I am a high school math teacher, so I understand most of this, but I am not a specialist.) 138.199.105.15 (talk) 21:16, 15 September 2023 (UTC) B.E.


 * See previous sentence: "Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra". The dual of the dual preserves the abstract structure of the original – its system of vertices, edges, faces, and which pairs of these things touch each other – but (when the duality is performed in a setting where geometric duality does not work) loses its geometric structure.
 * The centroid method you describe is very special, basically only works in cases where there is a lot of symmetry, and even in those cases shrinks the result (so that when you take the dual of the dual you get the same shape but never the same size). The centroid method does not work more generally for convex polyhedra, even though they always have a geometric dual. Instead, in that case the "polar reciprocation" method described in our article is more general, and does preserve geometry: the polar of the polar is the original geometric polyhedron. —David Eppstein (talk) 21:49, 15 September 2023 (UTC)
 * I would add, there is a difference of meaning between "the dual" and "a dual". Depending on the level of abstraction (among other things), there may be one unique dual (as in the abstract formalism) or there may be many geometric "realizations" of that abstraction. It is all too common to take some standardised construction of a dual and refer to it as "the" dual, giving the false impression that the geometrical result is unique. At high school level, the best general approach is probably polar reciprocity via the Dorman Luke construction, as introduced by Cundy & Rollett's Mathematical Models and treated more fully in Wenninger's Dual Models. The abstract structural formalism is often associated with topology or "rubber-sheet geometry", and may be sidled up to via the duality of the regular figures and their Schlaefli symbols, where {a b} is dual to {b a}. &mdash; Cheers, Steelpillow (Talk) 07:50, 16 September 2023 (UTC)


 * Thank you so much to both of you! I hope I am understanding things correctly.  To summarize, the dual of a polyhedron is not necessarily unique (depending on how abstractly similar two shapes have to be for us to call them "the same").  Also, the "centroid" method only really works if there is a lot of symmetry, and even then the dual of the dual is not equal to the original shape, nor even necessarily similar in the high school geometry sense, but should be topologically equivalent in that it will have the same number of vertices, edges, and faces, and they will be connected in the same way.  A better construction would be the "polar reciprocation" method, especially the Dorman Luke construction, which gives results that better preserve the geometry.  I hope I interpreted your comments correctly, please let me know if I am misunderstanding anything.  Thanks again!  138.199.105.15 (talk) 20:56, 18 September 2023 (UTC) B.E.
 * The only clarification I would make is that it is more accurate to say "depending on how geometrically alike two shapes have to be for us to call them 'the same'". &mdash; Cheers, Steelpillow (Talk) 18:48, 19 September 2023 (UTC)

r1 · r2 = r02
Under Polar reciprocation it says:
 * If r0 is the radius of the sphere, and r1 and r2 respectively the distances from its centre to the pole and its polar, then:
 * r12 · r22 = r02.


 * Shouldn't the formula read:
 * r1 · r2 = r02 ,
 * in accordance with Inversive geometry?

--Episcophagus (talk) 17:25, 2 December 2014 (UTC)


 * Yes indeed. Now fixed. &mdash; Cheers, Steelpillow (Talk) 17:46, 2 December 2014 (UTC)

Assessment comment
Substituted at 19:56, 1 May 2016 (UTC)

Definition
This has been removed from the lead as "outright false": In geometry, every polyhedron is associated with a second dual polyhedron, where the vertices of one correspond to the faces of the other. This is the fundamental definition of dual polyhedra and can be found in most any book on the subject. There is an equivalent definition of dual graphs, which is not being questioned. So I am not quite sure why it is being questioned here. What exactly is "false" about it? For example the theory of duality covers non-convex polyhedra as well - the Kepler-poinsot stars being a case in point - so one needs to make that clear. As ever in polyhedron theory, results derived for convex forms sometimes copy across to the non-convex and sometimes they don't.&mdash; Cheers, Steelpillow (Talk) 21:29, 8 February 2017 (UTC)
 * It is just not true that, for every reasonable definition of a geometric polyhedron, there exists a dual geometric polyhedron. For instance, take a big cube and stick a small cube into the middle of one of its faces, causing that face to become an annulus. What is the dual? It would have to be two octahedra joined at a single vertex. But one may reasonably define a polyhedron in a way that requires it to have a manifold boundary; the two cubes stuck together do satisfy this requirement but the two octahedra don't. The Kepler–Poinsot stars are a special case and obey a different and very weird definition of polyhedron where the faces are allowed to cross and not enclose a volume. It would be a mistake to assume that they provide the only way for a polyhedron to be non-convex. —David Eppstein (talk) 21:30, 8 February 2017 (UTC)
 * If the face is an annulus then the figure is no longer a polyhedron in elementary theory, for example it breaks Euler's formula. That was determined in the nineteenth century and affirmed by abstract theory in the 21st. See also the article on the polyhedron. You are probably thinking of some specialised definition that is not part of elementary geometry. Sorry, there is nothing weird about star polyhedra - the same historical comments apply. Perhaps you have not read the right sources. Cromwell gives a careful explanation in his book on Polyhedra, which I suggest you check out. &mdash; Cheers, Steelpillow (Talk) 21:38, 8 February 2017 (UTC)
 * For an even worse example, look at the rightmost polyhedron in Figure 1 of my paper Steinitz theorems for simple orthogonal polyhedra, a cube with a smaller cubical divot taken out of one edge. All faces are simple polygons, but it has a pair of faces that share two edges with each other. The dual would have to have two different edges connecting the corresponding two dual vertices, but that is not possible geometrically because there do not exist two different line segments with the same two endpoints. I have no idea what you mean by "in elementary theory". The definition of nonconvex polyhedra has not been standardized, as the Grünbaum 2007 reference that I added makes clear. (And Lakatos' book Proofs and Refutations makes even more clear.) In particular I reject the implicit assumption that there is a standard definition that allows the Kepler–Poinsot polyhedra but does not allow these examples. This is not an area of mathematics where it works well to just naively assume that definitions that are valid for convex polyhedra continue to be valid more generally. For that matter, would you assert that the Szilassi polyhedron and Császár polyhedron are not polyhedra? They also have simple polygon faces, manifold boundaries, and enclose a volume (unlike Kepler–Poinsot) but do not obey Euler's formula. —David Eppstein (talk) 21:42, 8 February 2017 (UTC)
 * There is no requirement for a geometric dual to be a valid geometric polyhedron - the Wenninger duals of the "hemi" uniform polyhedra are well known. Just derive the Hasse diagram of your figure and read it upside down. Either these are both valid abstract polytopes or they are not, you cannot have one without the other. If you want to pick nits over definitions, there is no standard definition for "polyhedron", whether convex or otherwise. "Elementary geometry" is that branch of geometry which derives from the ideas expressed in Euclid's Elements. It happens in Euclidean space and is the most widely accepted basis for the geometry of things like polyhedra, unless another model is invoked (which we are not doing here). And it does have a very clear definition of non-convex polyhedra - I refer you again to Cromwell. Lakatos' whole thrust was that our understanding evolves, and what was once a satisfactory definition may cease to be so. Abstract theory, developed since his time, is the modern touchstone and we have to run with that. (As for the dual pair you mention, they are topologically toroids and they do obey the appropriate modification of Euler's formula. Seriously, if you are that far out of touch you should not be making these sorts of edits but reading up on the material I have suggested - please add Richeson; Euler's Gem: The Polyhedron Formula and the Birth of Modern Topology, Princeton (2008) to that.) &mdash; Cheers, Steelpillow (Talk) 22:13, 8 February 2017 (UTC)
 * I am familiar with Euler's formula and its generalizations. But when you say "there is no requirement for a geometric dual to be a valid geometric polyhedron" you appear to be allowing sentences like "all non-convex polyhedra have dual polyhedra" that use the same word "polyhedron" with two different meanings in the same sentence. The first polyhedra must be geometric, for otherwise what would it mean to be convex or non-convex? But now you're allowing the second polyhedra in the same sentence to be abstract lattices without even any topology, let alone geometry. That's just bad mathematics. —David Eppstein (talk) 23:42, 8 February 2017 (UTC)
 * Good, then since you are familiar with the application of Euler's formula, you will understand why it applies to globally toroidal polyhedra but not to locally toroidal faces. I am disappointed that I had to remind you of this. Similarly, may I remind you that the article currently divides the discussion into (geometric) polar reciprocity and abstract polyhedra. If I did not make differing statements about the two differing classes of understanding, now that would be bad mathematics! Furthermore, to fail selectively to distinguish between specifically convex polyhedra and polyhedra in general really is bad mathematics, and your ongoing edits while we discuss this are compounding that error. &mdash; Cheers, Steelpillow (Talk) 11:36, 9 February 2017 (UTC)
 * I'm disappointed that you mistook, and are still mistaking, my simplification for purposes of discussion of Euler as referring only to spherical surfaces, for ignorance. It makes you look condescending and ignorant yourself. It is not very helpful to say that of course any polyhedron obeys Euler's formula for a different genus, because then we can just define the genus as being what it obeys and the generalized formula doesn't actually provide any constraint on what a polyhedron is. Your later comments essentially asserting that computational geometry is not mainstream and can be ignored are equally condescending. Grow up. —David Eppstein (talk) 16:49, 9 February 2017 (UTC)


 * I trust you feel better for having got that off your chest. One day we will sit down, enjoy a cup of tea, and laugh this off. &mdash; Cheers, Steelpillow (Talk) 17:29, 9 February 2017 (UTC)
 * It seems to me that any polyhedra with simple polygonal faces that the dual polyhedron exists topologically, but the practical problem comes down to how to represent that geometrically. And the toroidal (genus-1) Szilassi polyhedron and Császár polyhedron are relatable in this topological sense. An in general I'd say you can map a genus-k polyhedron onto the surface of a k-torus, and draw the vertices, edges and faces there, and then its easier to see the topological dual as a tiled surface. The "star" forms get messy, like when faces pass through the center, the dual won't exist, like the hemipolyhedron. Tom Ruen (talk) 22:09, 8 February 2017 (UTC)
 * More correctly, a concentric reciprocal will not be a polyhedron. The dual does exist, and if you move the sphere off-center it becomes finite. &mdash; Cheers, Steelpillow (Talk) 22:16, 8 February 2017 (UTC)
 * Thanks, yes, hemipolyhedron dual topology still exists, still just ugly geometry. This does remind me on dual polytopes that you could have toroidal cells in a 4-polytope, but I've never explored what exists there! (Like Higher Toroidal Regular Polytopes) Tom Ruen (talk) 22:21, 8 February 2017 (UTC)
 * The point is that "topological polyhedra with simple faces", "geometric polyhedra with self-intersecting polygon faces that form a manifold", "geometric polyhedra with non-intersecting polygon-with-holes faces that enclose a volume", "geometric polyhedra with simple faces and at most one edge per pair of faces that form a manifold and enclose a volume", "geometric polyhedra with simple faces and at most one edge per pair of faces that form a topological sphere and enclose a volume", "abstract Eulerian lattice", etc, are all different but valid definitions, and only some of them have duals in the same class. We should not say that non-convex polyhedra have duals, without qualification, because without an explicit definition of what a non-convex polyhedron is, that statement is so far from meaningful that it is not even wrong. —David Eppstein (talk) 23:11, 8 February 2017 (UTC)
 * I should point out that the same definitional issues apply even to polygons in the Euclidean plane. In computational geometry, it is much more standard to allow polygons to have holes (separate boundary cycles enclosed within the outer boundary cycle) than to allow them to have self-crossings. And in either case, do you allow repeated vertices? If you wrap around the same convex polygon more than once, is the result still a valid non-convex but regular polygon, and if so why doesn't this give us many more regular polyhedra in which we replace the faces of a Platonic solid with these wrapped polygons? If you glue two squares vertex-to-vertex, is the result still a polygon, or are there two different polygons with the same edges and vertices, one in which you go around both squares clockwise and another in which you go around one clockwise and the other counterclockwise? Does every cyclic sequence of points define a polygon, or is it required that consecutive points in the sequence be unequal to each other? Everything is much simpler if you stick to simple polygons, but then half the Kepler–Poinsot polyhedra would be disallowed. —David Eppstein (talk) 00:06, 9 February 2017 (UTC)
 * The Kepler–Poinsot polyhedra have topologically simple pentagram faces or vertex figures. The great disnub dirhombidodecahedron is the only uniform star polyhedron with ambiguity and can be considered degenerate, and need to be seen as coinciding edges to be topologically manifold.
 * Vertex-transitive polygons can have ambiguity of coinciding vertices, like this sequence, slightly offset: Regular_truncation_6_-1.5.svgRegular_truncation_6_-1.0.svgRegular truncation 6 -0.75.svg
 * Tom Ruen (talk) 03:38, 9 February 2017 (UTC)
 * You appear to have misunderstood my intent. I am not seeking enlightenment from you. Those were rhetorical questions. I was not asking you to provide the one true definition, because I don't believe that such a thing exists. You can answer them in different ways. Lots of people would think, quite reasonably, that a pentagram is not a polygon. And I was not asking about "slightly offset" positions of vertices, but exactly coinciding ones. The point is that defining non-convex polygons and non-convex polyhedra requires care. You and Steelpillow do not appear to have been exercising the appropriate level of care. —David Eppstein (talk) 03:48, 9 February 2017 (UTC)
 * We are now getting to the nub of why this article discusses just two definitions. Computational geometry uses a definition of "polygon" which differs from those in mainstream geometry textbooks, and so on. All that is dealt with in the articles on the polygon and the polyhedron. They follow the same pattern of focusing on elementary and abstract theories. This article needs to do the same, specifically focusing on duality within these two domains. I think David may be conflating the old Victorian conceptual agonies over "What is a polyhedron?" with the modern radiation of specialist definitions. This is why we need to take answers to that question from popular and reputable sources such as Cromwell and Richeson. WP:RS requires that "Articles should rely on secondary sources whenever possible" and not contradict what they tell us by invoking primary sources. So I think the only solution to David's and my differences is going to be a citation fest. Properly-cited content cannot then be deleted by an edit warrior, but only by the consensus which is so sorely lacking here. (By the way, I forgot to say last night that Cromwell is a good source for the "what is a polyhedron?" debate, he does not address duality. I might also add that interiors are irrelevant to the duality issues, significant sources only ever consider the bounding manifold.) &mdash; Cheers, Steelpillow (Talk) 11:36, 9 February 2017 (UTC)
 * As a starter for ten, let's see if we can get an article lead to stick. I cited a primary source for the idea that the dual of every abstract polyhedron is also an abstract polyhedron, there is probably a better one but it needs to be found before this one can be removed. &mdash; Cheers, Steelpillow (Talk) 12:15, 9 February 2017 (UTC)

An analogy
I thought this was settled but today's edits convince me otherwise.

Suppose we had an article stating that "every number has a square root", and an editor objected to this as being too sloppy and replaced it with "every positive real number has a real square root, and every complex number has a complex square root, but some other number systems don't always have square roots". You would agree, I hope, that it would be unreasonable to revert this with an explanation that all numbers are complex, or that any exceptions only belong to such esoteric branches of mathematics that they can be safely ignored. So why do you think it acceptable to take the position that all polyhedra are abstract polyhedra?

It's not true, both because abstracting a polyhedron loses a lot of information and also because there are reasonable and intuitive definitions of polyhedra that do not fit the abstract polyhedron requirements. In particular the requirement in an abstract polyhedron that every 1-section be a line segment disallows some perfectly-reasonable flat-sided solids that many would call polyhedra. WP:NPOV requires that our article "represent all significant viewpoints", which in this context means that we must discuss widespread alternative definitions rather than pretending that abstract polyhedra are the only possible definition. In its current state, I think the article is ok in that respect, but I am dismayed at recent attempts to downplay this generality and move back to a view of the world in which all polyhedra are abstract polyhedra. —David Eppstein (talk) 00:11, 11 February 2017 (UTC)
 * An article lead is intended to summarise the main text. Whether I chose to revert to "every number has a square root" would depend on the extent to which the main text explained these complexities. The present article confines its main text to polar reciprocity and abstract or combinatorial duality. The lead needs to reflect this.
 * Similarly, the article deals with the duals of polyhedra as defined in the polyhedron article. That deals primarily with elementary geometry and, therefore, so should this article. Any other kinds of "polyhedron" with significant literature on their duality may be given a subsection of their own. There are several good secondary and tertiary sources for elementary polyhedra and some of them deal with duality as well. These must be our main sources for this article and no contradictions to these sources from, say, some learned primary source should be allowed without building a consensus as to why. I am not aware of examples from such secondary and tertiary sources which support the claims you are making, for example where are the "reasonable and intuitive definitions of polyhedra that do not fit the abstract polyhedron requirements"?
 * &mdash; Cheers, Steelpillow (Talk) 11:06, 11 February 2017 (UTC)
 * The most obvious way that a shape bounded by flat sides would fail to be elementary would be for it to violate the "every 1-section is a segment" constraint. For instance, take a polycube with six cubes, four of which form a square and the other two of which rise above diagonally-opposite cubes of the bottom four, so that they touch along an edge. The section from this touching edge to the top of the Hasse diagram is not a segment — the edge touches four squares, not two. But you can find plenty of literature saying that polycubes are polyhedra. And certainly they would fall into the naive "shape bounded by flat sides" conception of non-convex polyhedra, regardless of whether that conception can be formalized as a proper definition. —David Eppstein (talk) 18:03, 11 February 2017 (UTC)

Non-convexity
An editor cannot summarily remove say a clarification or citation tag placed in good faith, without first dealing with it either in the article or on the talk page. I tagged "Grünbaum (2007) argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron." because the remark stands alone, without explanation, as the conclusion to a subsection and I fail to see any encyclopedic value in it. Is there any? &mdash; Cheers, Steelpillow (Talk) 10:32, 11 February 2017 (UTC)
 * Here's the original edit in the self-dual section and part of a longer sentence. Tom Ruen (talk) 10:50, 11 February 2017 (UTC)
 * I had removed the bit about the excavated dodecahedron because any discussion of a definitional muddle (long resolved), between it and Brückner's icosahedron of the same outward appearance, belongs in that or the polyhedron article and not here. This leaves the Grünbaum quote with nothing to amplify. Hence my tagging it to see if anything worth amplifying came up. My tag was reverted without discussion, which breaches our behavioural guidelines (You are letting your tea go cold, David), so I opened a discussion. &mdash; Cheers, Steelpillow (Talk) 11:26, 11 February 2017 (UTC)
 * I don't have any strong feelings about the inclusion or not of the excavated dodecahedron, but if there is a reasonable definition of geometric duality for non-convex polyhedra (that we have not yet presented in the article, because we only define abstract duality and have a sentence about polarity running into difficulties) then it might make a reasonable example for that. It at least has a well-defined vertex figure (as the link of each vertex lies entirely within the halfspace perpendicular to the center) whose dual polygon could be used as the face shape of a geometric dual. —David Eppstein (talk) 18:10, 11 February 2017 (UTC)

Pause?
I have reported David at WP:ANI. Until that is resolved, I would be grateful if folks could regard this discussion as on hold. &mdash; Cheers, Steelpillow (Talk) 19:29, 11 February 2017 (UTC)
 * "On hold" means that you get your way — the article stays in its misleading-to-readers state. Why should that be the default outcome of your drama-mongering? —David Eppstein (talk) 20:05, 11 February 2017 (UTC)
 * In case other editors are worried by that, you are of course welcome to keep editing the article as usual. I am only suggesting that this discussion be put on hold for a bit. &mdash; Cheers, Steelpillow (Talk) 20:40, 11 February 2017 (UTC)
 * Now that the ANI issue is basically settled, I will return to these discussions once I know where they are taking place: both the duality and the "what is a polyhedron?" discussions have been forked to Talk:Polyhedron. May I suggest that the "what is a polyhedron?" definitional issue should be addressed there, but that the subsequent duality issue is best addressed here, where it began? It is not helpful to have parallel discussions on the same issue. If folks have a problem with that, please reply at the higher-level article discussion rather than fork the discussion about the fork! &mdash; Cheers, Steelpillow (Talk) 15:56, 13 February 2017 (UTC)

Unclear formula in Polar reciprocation?
I received this message on my talk page and copy it here:

Hi!

I'm asking you just because you are the last user who edited the WikiVisually article entitled "Dual polyhedron", which i can not edit...

In its 1.1 paragraph, entitled "Polar reciprocation", is a very unclear mathematical writing; just after this equation of sphere:

x^2 + y^2 + z^2 = r^2,

the dual of a polyhedron P is expressed as:

P° = {y € R^3 / y.x =< r^2 for all x € P}

where y.x denotes the dot product of y and x.

Please, could you improve this writing into something like:

P° = {N € R^3, N // M, & 0 =< N.M =< r^2}(M € P)

?

In advance, thank you very much! :-)

I'm so sorry for bothering you... Entschuldigung! :-P

RavBol
 * N and M are not good choices of variables for 3d vectors. —David Eppstein (talk) 17:17, 31 August 2020 (UTC)
 * Having just defined x, y and z coordinates, the current x and y are probably even worse. &mdash; Cheers, Steelpillow (Talk) 17:55, 31 August 2020 (UTC)
 * True. But other than in changing the variable names, I don't think the proposed formula is an improvement. —David Eppstein (talk) 18:09, 31 August 2020 (UTC)

David Eppstein: I agree: i wrote "u" and "v" instead of "M" and "N". :-)

(But with my mobile, i could not find the "parallel" symbol, so i wrote "//"; of course, i could not reduce the space between the two "/"...)

WatchDuck: Thank you very much for copying my message in the appropriate talk page, and so, making me understand that yes, i could edit this article: in the right Wiki. :-)

RavBol (talk) 18:23, 31 August 2020 (UTC)


 * There's no point in specifying that the point in P should be the parallel one with positive dot product, because the same inequality is valid for all the other points in P as well. The added specification just complicates the formula for no good reason. —David Eppstein (talk) 18:33, 31 August 2020 (UTC)

When i edited the writing of the definition, i had not realized that David Eppstein had already answered "no" (there is very little space left on my mobile's screen when it displays its virtual keyboard, i receive only a few notifications on the mobile version of Wikipedia, & i thought nobody would care about my message)...

Sorry for making you guys waste your time... :-/

RavBol (talk) 21:23, 31 August 2020 (UTC)


 * It wasn't a waste of time. The resulting change in variable names was a clear improvement. —David Eppstein (talk) 21:45, 31 August 2020 (UTC)

Is it possible to cite a mathematics webpage written in French, to source a statement in a mathematics English Wikipedia article, please?

In advance, thank you very much for your answer!

RavBol (talk) 18:51, 8 October 2020 (UTC)


 * Sources written in languages other than English are allowed. However, they must be reliable sources (usually meaning, published by reputable journals or book publishers). Web pages are usually not reliable in this sense. —David Eppstein (talk) 19:12, 8 October 2020 (UTC)

Thank you again for your answer; don't worry: i won't create my own mathematics website & cite it... & even if i did, you would recognize my presentation/writing style! ;-)

RavBol (talk) 00:17, 9 October 2020 (UTC)

Compounds
I just removed a short (though recently expanded) section called "Self-dual compound polyhedra". I feel that it consisted entirely of material that was mathematically dubious, not supported by sources, or off-topic in an article on duality. To be precise: the first sentence was true only if "dual" means "polar dual" (and it is not clear what sense to make of applying other notions of duality to compounds of polyhedra); the second sentence was tautologous; the third sentence was incomprehensible (what is the "this property" in question?) and also not supported by the cited source (the Hart reference makes no claim of uniqueness of anything, and the statement was uncited before that but with a different claim of uniqueness); the fourth (parenthetical) sentence was completely off-topic; and the fifth sentence is not really sourced to Hart and not really on-topic. To the extent that this is mathematically rescuable and can be supported with references, its natural home would be as a discussion of duality in the context of compounds of polyhedra, rather than as a discussion of compounds of polyhedra in the context of a discussion of duality. --JBL (talk) 22:51, 2 September 2020 (UTC) - Why removing before discussing...?

In the third sentence: in a (short) section entitled "Self-dual compound polyhedra", "this property" in question was obviously self-duality. Anyway: why removing the whole section, instead of improving the writing?

The fourth (parenthetical) sentence was a proof that the regular compound of ten tetrahedra is self-dual.

The fifth sentence was on-topic: one can legitimately wonder why the regular compound of two tetrahedra & the regular compound of ten tetrahedra are self-dual, whereas a regular compound of five tetrahedra is not.

"The dual of a compound of polyhedra is the compound of the duals of the polyhedra"... such basic properties must be treated in a context of duality. In a context of compounds of polyhedra: duality must be used to construct dual compounds of polyhedra, & to derive other compounds of congruent polyhedra; but convex hulls, common cores, different symmetry groups... must be treated.

The ref i had added was not sufficient, but it sourced some elements; why removing the whole section, instead of improving the sourcing?

RavBol (talk) 01:18, 3 September 2020 (UTC)
 * Do you really have a published reliable source about self-duality of compound polyhedra? My searches for such sources found only Gailiunas and Sharp "Duality of polyhedra" (which has an informative example of two non-regular compounds that were previously claimed to be dual to each other but actually both self-dual) and Coxeter "Regular compound tessellations of the hyperbolic plane" and Rigby "Some new regular compound tessellations" (both very sloppy, appearing to define self-duality merely by reversal of Schläfli symbols, and not containing the special claims included here).
 * In general our polyhedron articles have often been overrun by original research, and the Gailiunas and Sharp example shows that it's also an area where it's easy to make big mistakes and where the value of having high-quality rigorous sources is high. The example here where we said for some time that the stella octangula is the only regular self-dual (which appears to be incorrect), based on no sources, should similarly serve as a warning about what happens when we don't use good sources. So I think that rather than keeping the original research in the article and hoping to turn up sources that sort of say something resembling it later, it's important to base the content here on reading the sources first and only adding material that comes from good sources.
 * I also think that regular polyhedra are a somewhat specialized subtopic of the theory of polyhedra more generally, and regular compound polyhedra more so. Additionally, because there are many ways of defining polyhedra and their duals, only some of which are compatible with compound polyhedra, pushing compound polyhedra into this article acts as a way of forcing one particular definition to the front to the exclusion of others, which is not necessarily appropriate. So I agree with the opinion that if this material can be written from the start from reliable sources (rather than doing original research and sourcing later), it still may be better to include it in the article about compound polyhedra (where compatibility with other definitions isn't problematic) than here. —David Eppstein (talk) 04:57, 3 September 2020 (UTC)


 * Having checked the cited source, I have to agree on Joel and David's conclusion. Some of the criticisms voiced may be overly opinion-based, while others seem a bit over-stated; if we were to apply their professed standards of rigour across all our polyhedron-related articles, the result would be absolute mayhem. But a slightly more balanced approach would still reap a very large harvest and that would certainly be an improvement. Might some of the deleted remarks find a better home in the article on polytope compounds? Perhaps, if better sources can be found, but I know of none. &mdash; Cheers, Steelpillow (Talk) 07:56, 3 September 2020 (UTC)


 * RavBol, with respect to "obviously", I do not think it was obvious that this is what was meant -- the claim before you edited it appeared to be that the only regular compound that consists of a single self-dual polyhedron and its dual is the stella octangula. This is very close to what Hart says in the first paragraph of the reference you added, incidentally. I am happy with Hart as a reference for the existence of compounds of polyhedra with their duals, and that some of these compounds have other properties.  But I think they are interesting examples of nice polytopal compounds, not nice properties of duality.  If someone wants to add material about duality together with some nice examples (properly sourced) to polytope compound, I certainly would not object. --JBL (talk) 11:54, 3 September 2020 (UTC)

- Still: why removing before discussing? :-/

RavBol (talk) 14:08, 3 September 2020 (UTC)
 * It appears in your latest efforts that you have ignored the main reason for the removal, now discussed extensively above, the lack of adequate reliable sources, needed to prevent this from being original research. Please do not continue adding poorly-sourced material back to this article. Find the sources FIRST, and then base your additions on what they actually say, not on what you would like them to say. —David Eppstein (talk) 18:59, 5 September 2020 (UTC)


 * David Eppstein, it looks to me like the Hart claim is fine (if trivial) and is how I read the claim in the article before RavBol changed it, namely, that the only compound that is both a compound of a single [uniform] polyhedron with its dual and a compound of multiple congruent polyhedra must consist of a compound of a single self-dual uniform polyhedron with its dual. --JBL (talk) 19:30, 5 September 2020 (UTC)
 * The Hart claim is technically correct, but I seem to recall it was wrongly presented here. The Stella Octangula is indeed the only self-dual compound of a single regular polyhedron. What we do not have is a source for the claim that the regular compound of ten tetrahedra is the only self-dual compound of a regular compound. &mdash; Cheers, Steelpillow (Talk) 20:16, 5 September 2020 (UTC)


 * There is no obligation to discuss before making a bold edit, and indeed I began discussion earlier in the WP:BRD cycle than is typically called for. (You will note that everyone else who has weighed in has agreed with my assessment of the problems in the material!) In comparison, making essentially the same edit while ignoring the substantive objections of others is a much more serious breach of protocol, as is adding unsourced content after an objection. --JBL (talk) 19:30, 5 September 2020 (UTC)

- I thought the poverty of source was about the regular compound of two tetrahedra & the regular compound of ten tetrahedra being THE ONLY regular self-dual compounds of polyhedra; so i thought that stating just existence instead of stating existence & "unicity" really solved the problem of source.

PS: sorry for the "Rewrited" that i wrote in my Edit summary...

RavBol (talk) 23:39, 5 September 2020 (UTC)
 * Everything needs a source, not just the part that was dubious and disputed. Getting rid of that part does not eliminate the need for a source. —David Eppstein (talk) 00:48, 6 September 2020 (UTC)

Dorman Luke example needs better explanation
The Dorman Luke example of going from the cuboctahedron to its dual, the rhombic dodecahedron, needs a better explanation — especially of why the vertex figure is drawn as a 1:√2 rectangle.216.161.117.162 (talk) 01:41, 26 September 2020 (UTC)
 * if the reader cannot see that from the drawing of the cut apex, is there any point in explaining it? &mdash; Cheers, Steelpillow (Talk) 08:51, 26 September 2020 (UTC)

ALWAYS avoid quantifier symbols ∀ and ∃?
The MOS says: "Articles should avoid common blackboard abbreviations such as wrt (with respect to), wlog (without loss of generality), and iff (if and only if), as well as quantifier symbols ∀ and ∃ instead of for all and there exists."

Within a sentence in English: OK.

But within a math formula, like in Dual polyhedron, replacing "∀" with "for all" results in mixing math symbols and English words...

Moreover, it would be consistent to replace "|" with "such that", and "{...}" with "the set of ...", too...

Suggestion: in a math formula, using the necessary math symbols, but bluelinking each of them to an explanation article.


 * what do you think of this suggestion, please? --JavBol (talk) 01:29, 27 June 2021 (UTC)


 * In general, it is better to spell out quantifiers in English words rather than using symbols, even within set-builder formulas as in this case. There are occasional exceptions where one is specifically discussing quantified formulas (see the first example displayed formula in true quantified Boolean formula) but they are rare and do not apply in this article. For the same reason, I would also strongly consider using the English word "in" in place of the set membership operator $$\in$$ in the same formula. We need to keep in mind WP:TECHNICAL and the fact that many readers of this article are likely to be non-mathematicians; words are easier to read than unfamiliar notation. —David Eppstein (talk) 05:50, 27 June 2021 (UTC)


 * Right. In other words: a fake consistency does not serve readers well, and serving readers is our job.  Links in formulas are confusing and unclear, as well, and in 95% of cases should be removed, usually by rewriting to give an in-text gloss. —JBL (talk) 11:47, 27 June 2021 (UTC)

In the same formula, why not replace $$q \in \mathbb{R}^3$$ with $$q$$ in $$\mathbb{R}^3$$, too? --JavBol (talk) 14:04, 27 June 2021 (UTC)


 * I think maybe it's better without the $$\in\mathbb{R}^3$$ altogether. It's unambiguous which set $$q$$ comes from, so I think it just makes the formula unnecessarily complicated. —David Eppstein (talk) 16:28, 27 June 2021 (UTC)

Actually, i appreciated the specification "$$q$$ in $$\mathbb{R}^3$$"; indeed: on Wikipedia, polyhedra are defined as "3-dimensional" figures, but there are other 3-dimensional spaces than $$\mathbb{R}^3$$.

However, i guess that the formula in question is not the right place for such a specification. --JavBol (talk) 20:17, 27 June 2021 (UTC)

Anyway, some non-mathematician readers may assume that "q|q·p" means: "q|q", denoting another operation, possibly some kind of division, followed by the dot product with p. --JavBol (talk) 21:46, 28 June 2021 (UTC)

Then, what about writing:
 * $$P^\circ = \{q$$ in Euclidean space $$\mid~q \cdot p \leq r^2,$$ for all $$p$$ in $$P\}$$ ?

--JavBol (talk) 23:41, 30 June 2021 (UTC)

False statement being restored
It is wholly incorrect to say that, as one editor currently insists, "for some types of non-convex geometric polyhedra, the dual polyhedra may not be realizable geometrically." In fact any figure may be realized unfaithfully. Any attempt I make to improve matters gets reverted without discussion.
 * The latest revert was justified with this claim that I had changed it to "the dual is a polyhedron, but the dual geometric figure might not be a polyhedron". That again is incorrect. I had changed it to, "Every geometric polyhedron corresponds to an abstract polyhedron in this way, and has an abstract dual polyhedron. However, for some types of non-convex geometric polyhedra, the dual geometric figure may not be a polyhedron", which for illustrative purposes we may precis as "the dual is an abstract polyhedron, but the dual geometric figure might not be a geometric polyhedron." That is a perfectly correct statement and wholly consistent with the article lead, which states that, "In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra." So the claim of nonsense is itself patent nonsense; it disses what I did not say, while reverting what I did.
 * The reverting editor also rejects insertion of the term "faithful" as it has not been defined in the article, while insisting on "realized", which also has not been defined in the article. Both are established terms in abstract theory. If we are to be rational and consistent about this, we should allow either both or neither here.

Does anybody have any rational objection to restoring this, hopefully slightly clearer, version; "Every geometric polyhedron corresponds to an abstract polyhedron in this way, and has an abstract dual polyhedron. However, for some types of non-convex polyhedra, the dual figure may not be a valid geometric polyhedron"? &mdash; Cheers, Steelpillow (Talk) 08:04, 1 July 2021 (UTC)


 * This latest version (right above) seems clearer to me.
 * Besides, it also seems to me that certain editors make an excessive use of the "Revert" part of the BRD cycle, & tend to become harder if the cycle goes on after they've exposed their first arguments.
 * The BRD page says:
 * "Revert an edit if it is not an improvement, and it cannot be immediately fixed by refinement. Consider reverting only when necessary. BRD does not encourage reverting, but recognizes that reversions happen."

--JavBol (talk) 15:30, 1 July 2021 (UTC)

Anyway, I have deleted the false statement until all involved can agree on a correct version. &mdash; Cheers, Steelpillow (Talk) 16:25, 1 July 2021 (UTC)
 * Grünbaum (a reference to a similar statement in polyhedron) states that
 * Convex polyhedra have duals (as we know)
 * For polyhedra in general (defined by him as sets of simple flat polygons meeting two per edge and with a simple cycle of polygons per vertex, allowing self-intersections) "the only consistent approach to duality is via polarity". He doesn't state that polarity works for these general polyhedra but it appears to, as long as you don't insist on the origin of polarity being at a particular point that might be on one of the polygons. If you want to maintain symmetry, this is a problem, because sometimes the center of symmetry (the only reasonable choice for the origin) does lie on a polygon.
 * For acoptic (non-self-intersecting) polyhedra it doesn't work and there exist polyhedra without duals.
 * —David Eppstein (talk) 16:53, 1 July 2021 (UTC)
 * That paper is decades old. It discusses realizations only in the context of acoptic polyhedra (all of which are orientable). By 2009 (Graphs of polyhedra; polyhedra as graphs, Discrete mathematics 307) he was filling out his treatment by allowing the "realization of a nonorientable abstract polyhedron" and referring to planar nets as realizations. Egon Schulte defines faithfulness in Chapter 18 of the Handbook of Discrete and Computational Geometry, 3rd Edition, 2017. I have only his draft, which states, "The realization is faithful if each βj is a bijection; otherwise, it is degenerate." (His emphasis). So basing anything here on that paper alone is unwise. &mdash; Cheers, Steelpillow (Talk) 20:25, 1 July 2021 (UTC)

Ignoring the over-wrought and irrelevant stuff above, I personally am perfectly happy with "the dual figures may not be valid geometric polyhedra" instead of "the dual polyhedra may not be realizable geometrically"; also I think "the abstract dual" could be substituted for "the dual figure". It seems that this is also consistent with Grunbaum. --JBL (talk) 17:16, 1 July 2021 (UTC)
 * The abstract dual is very much not the dual figure; there is a sharp distinction between the abstract and the geometric. "the dual geometric figures may not be valid geometric polyhedra" would be more correct, but it seems unnecessary to say "geometric" twice. I would have no problem with "the dual geometric figures may not be valid polyhedra". &mdash; Cheers, Steelpillow (Talk) 20:25, 1 July 2021 (UTC)
 * Well, ok, you have successfully convinced me again that you are wrong (or, at least, that you prefer to endow words with your own peculiar definitions that do not appear to agree with those in common use). So I am back to supporting the preservation of the correct statement currently in the article. --JBL (talk) 20:44, 1 July 2021 (UTC)
 * Eh? What did I get wrong there? What words have I messed with? If you don't like my suggestion, I'd be as happy as you to go with the one you are perfectly happy with. &mdash; Cheers, Steelpillow (Talk) 04:15, 2 July 2021 (UTC)

Indeed: i didn't comment JBL's latest 2 replies, because i don't understand them.

Anyway, the Dual polyhedron subsection is totally abstract... It should show the diagrams of a small poset & of its dual, a dual which ideally could not have a faithful geometric realization; shouldn't it?

This way, the terminology in question would seem slightly less crucial... ;-) --JavBol (talk) 05:25, 2 July 2021 (UTC)
 * I think Steelpillow is...not wrong, but reverting to the "there can only be one definition of a polyhedron and its geometric realization, the abstract polyhedron + mapping from vertices to points one that I use" attitude that caused so much difficulty on Talk:Polyhedron in 2017. The correct resolution now, as then, is to make the article recognize and address the multiplicity of definitions rather than being dogmatic about which definition is the only one the article can use. —David Eppstein (talk) 05:56, 2 July 2021 (UTC)
 * Right; I'm not sure I articulated this well in my last comment, but I agree it's the same phenomenon. --JBL (talk) 11:19, 2 July 2021 (UTC)
 * I would agree with that. Pragmatically, we all seem happy to change "the dual polyhedra may not be realizable geometrically" to "the dual figures may not be valid geometric polyhedra". Is there anything still preventing that change? &mdash; Cheers, Steelpillow (Talk) 12:36, 2 July 2021 (UTC)
 * Well, I prefer the current version, and I infer from DE's comments & actions that he does as well (please correct if this is wrong). So that's a thing preventing the change. --JBL (talk) 14:14, 2 July 2021 (UTC)
 * OK, so you have changed your mind (apparently because you dislike my approach, but perhaps we could ignore that hyperbole). If you do want to keep one particular definition of a polyhedron and its geometric realization, then do you agree that the others should also be represented, as David says - all properly cited, of course? &mdash; Cheers, Steelpillow (Talk) 15:05, 2 July 2021 (UTC)
 * Perhaps you should also give your definition of "hyperbole", because what actually happened was that I came to believe something you said was reasonable, and then you argued with me about it, and (as a consequence of the content of your argument) I no longer believe what you were saying was reasonable; there is nothing hyperbolic about any of that. The sentence in question does not require an enumeration of different possible definitions, what it needs is to be flexible enough to encompass the reality of various choices of definition of "geometric polyhedron". The version that currently exists does that, correctly. Maybe it will be helpful if I spell out briefly what I think the paragraph in question actually says.  The first half of the first sentence (implicitly) defines the notion of realizability: for a given class of geometric polyhedra, an abstract polyhedron A is realizable if there is a geometric polyhedron in the class whose associated abstract polyhedron is A.  The second sentence in the present version (correctly) observes that it may or may not be true (depending on what class of geometric polyhedra we're talking about) that the dual of a realizable abstract polyhedron is again realizable.  This reading has the following virtues: it assigns the paragraph a true meaning, via reasonable and widespread conventions for assigning meanings to words in the context of mathematics; it correctly accounts for the variation in definition of the word "polyhedron"; and under this reading the statement is clearly worth making in an encyclopedic article about duality of polyhedra. --JBL (talk) 02:58, 3 July 2021 (UTC)
 * My apologies if reasoned discussion came across as arguing with you. You used the term "hyperbole" with reference to some of my remarks, It did not occur to me that you would be unhappy to have it used in the reverse direction - we all assume good faith here. Back on topic, my criticism of the putative sentence is not to do with duality per se but with realizability. I have cited two impeccable RS to show this. What you are describing as realizable is what Schulte defines as "faithfully" realizable. Examples of his "degenerate" realization are seen in Gr&uuml;nbaum (2009), where any injection of the abstract into a real space is a realization of some kind or other. One can discuss usages of the term interminably, which is precisely why it is better to excise it from an entry-level discussion of duality than to try and accommodate those varied usages. &mdash; Cheers, Steelpillow (Talk) 07:00, 3 July 2021 (UTC)
 * You used the term "hyperbole" with reference to some of my remarks If you mean in the discussion on this page, that is definitely false. —JBL (talk) 14:34, 3 July 2021 (UTC)
 * OK, I have amended my comment, I trust it is now correct. If you feel that "hyperbole" is more perjorative than "over-wrought and irrelevant stuff", I am happy to offer my apologies once again and to withdraw it. Do you yet have any response relevant to the points I brought forward on the article content? &mdash; Cheers, Steelpillow (Talk) 15:44, 3 July 2021 (UTC)
 * "hyperbole" ... pejorative I don't object to "hyperbole" as pejorative, I object to it as a clearly false description. Like most people, I sometimes use hyperbole, and have no objection to that being pointed out when relevant, but there is no use of hyperbole whatsoever in the comment you called hyperbolic.  I have amended my comment That ... also is definitely false.  I think you will have to forgive me if, at this point, I abandon this conversation. --JBL (talk) 20:52, 3 July 2021 (UTC)

"The correct resolution now, as in 2017, is to" add something like: "Some recent publications prefer to say that the realizations of the abstract duals may not be valid geometric polyhedra.", isn't it? --JavBol (talk) 15:31, 3 July 2021 (UTC)
 * No. This sort of mealy-mouthed bothesidesist WP:WEASEL is no good. Also, it is not a matter of what the publications prefer (as if a publication is a thing that could have a preference) but rather what follows from what definitions. We need to be much more specific, something like: "The existence of a dual for all polyhedra depends on the chosen definition of a polyhedron. If a polyhedron is defined as [definition X], some polyhedra do not have duals. According to [definition Y], a dual can always be constructed by [construction], but it may not be possible to preserve the symmetries of the polyhedron. However, according to [definition Z], a dual may always be constructed but may not always be faithful." All with proper reliable sources, if there are more to be found on this issue than Grünbaum. If Grünbaum is the only source we can find, we should follow what he says. —David Eppstein (talk) 15:43, 3 July 2021 (UTC)
 * Would that be Grünbaum (1999) or Grünbaum (2009)? Is the Egon Schulte chapter quoted above not a suitable RS? I am sure I have some more somewhere. But seriously, the problem sentence is not so much leaving the definition of "polyhedron" woolly, but that of "realizable". May I suggest that if the article simply avoids the term in this sentence, then the problems go away? &mdash; Cheers, Steelpillow (Talk) 16:05, 3 July 2021 (UTC)
 * I wrote "something like"; besides, i used the term "publications" instead of "authors", because there seems to be an evolution in Grünbaum's approach & terminology; & anyway, i'll leave this addition to you guys... --JavBol (talk) 18:45, 3 July 2021 (UTC)

--JavBol (talk) 23:02, 3 July 2021 (UTC)
 * What's the point in restoring a typo ("perjorative" > "pejorative" > "perjorative")?
 * What's the point in restoring an indentation error ("::" followed by "::") that glues comments by different users?
 * I placed my latest reply above Steelpillow's latest reply, because mine is not important & requires no reply, whereas his is important & requires a reply.
 * Maybe you should address your questions to the editor who made those edits, rather than to me? Also you should read WP:THREAD, which clearly explains why others have been repeatedly correcting your indentation and sequencing. --JBL (talk) 23:12, 3 July 2021 (UTC)

Twice now my posts have been corrupted by a problem with my login session being lost. It is a known bug with the current version of MediaWiki and has been going on for months. This has led to various confusions and upsets in this discussion. It was delicate enough anyway and has now descended into an unpleasant farce, so I am pulling out to avoid further upsets. I will just say that my use of "hyperbole" should be considered withdrawn with sincere apologies (others may redact instances if you so wish), and that "perjorative" is the correct spelling of the adjective - "pejorative" is the noun. Etiquette on indentations may be found at WP:INDENT. I just hope this post arrives with my sig attached and not my IP address again. &mdash; Cheers, Steelpillow (Talk) 07:56, 4 July 2021 (UTC)

--JavBol (talk) 15:49, 4 July 2021 (UTC)
 * @: Sorry for my confusion (the particular context mislead me). Still, horizontal lines separating comments can be added without breaking threads & indentations. Such a line should be automatically added right below each signature! :-V
 * @: I can find "perjorative" neither on Reverso, nor on Linguee, nor in my English/French dictionary, nor in my English dictionary...

Dorman Luke construction: over-specified?
What is it actually supposed to be? The first step has always been under-specified; 's changes seem like they are trying to fix it, but they leave the text too garbled for me to understand. --JBL (talk) 13:40, 10 July 2021 (UTC)
 * @: Starting with a polyhedron having an intersphere s, the points A, B, C, D of tangency to s of the tangent lines connected to a vertex V are equidistant from V. --JavBol (talk) 17:52, 10 July 2021 (UTC)
 * The text of the article says the midpoints of the edges. Now you say the points of tangency of the midsphere. Which is it? —David Eppstein (talk) 19:07, 10 July 2021 (UTC)
 * Starting with a convex uniform polyhedron, the midpoint of each edge connected to V is also a point of tangency to s; we aim at this point. But starting with e.g. the rhombic dodecahedron, it is another point on each connected edge than its midpoint that is also a point of tangency to s; we still aim at this point (see the "Canonical dual" (subsubsection) compound figure); don't we? --JavBol (talk) 19:57, 10 July 2021 (UTC)
 * Is that what reliable sources say about this construction, or are you trying to figure it out for yourself? —David Eppstein (talk) 20:12, 10 July 2021 (UTC)
 * I don't have reliable sources (according to Wikipedia standards) about the DL construction. But if the claims about starting with a polyhedron having a midsphere, at the end of the "Dorman Luke construction" section, are to be kept, then the specification "VA = VB = VC = VD" is also to be kept; isn't it? --JavBol (talk) 21:22, 10 July 2021 (UTC)
 * So I looked up both of our supposed sources for this, Cundy&Rollett and Wenninger. (1) The method they describe is only for Catalan solids, the duals of Archimedean solids. They do not mention its applicability to more general solids with midspheres, for which this is anyway just a messy way of describing the polar dual. (2) As they describe it, it recovers the shape of a single dual face (congruent to all other faces), but it is not described as constructing "each face" in a way that links together to form the whole dual polyhedron. (3) Their construction does not use edge midpoints or midspheres. For a given vertex v, it chooses an arbitrary distance (less than the length of an edge), finds points on each incident edge at that distance, constructs a circle through these points, and forms a polygon from tangent lines to this circle. My feeling is that this material should be removed from this article to Catalan solid (because its sources only describe it as applying to construct the face shapes of Catalan solids) and cut down to what the sources describe (without mention of midpoints or midspheres). —David Eppstein (talk) 23:32, 10 July 2021 (UTC)
 * I thought you were talking about sources for "VA = VB = VC = VD" only, so i didn't mention: Dual Polyhedron -- from Wolfram MathWorld

https://mathworld.wolfram.com/DualPolyhedron.html ;
 * but this entry mentions uniform polyhedra, midpoints, & midspheres several times each. --JavBol (talk) 00:52, 11 July 2021 (UTC)
 * MathWorld is not usually a high-quality source, and its single short paragraph about DL is very vague about whether it applies to anything beyond the Catalan solids (it says that certain conditions are necessary but does not say they are sufficient). If you pretend that it is not vague then it is outright incorrect (it says to use midpoints of edges in all cases but then also says that it might work for polyhedra with midspheres). —David Eppstein (talk) 01:15, 11 July 2021 (UTC)
 * I've never said this entry was clear. But IF the DL construction works when starting with any convex uniform polyhedron, then it is a pity that reliable sources don't state it, as the proof is probably the same (probably only the number of edges connected to each vertex varies)... --JavBol (talk) 16:00, 12 July 2021 (UTC)
 * Still missing the point. We have a separate article about duals of convex uniform polyhedra. The construction belongs there, not in this article about more general duals. —David Eppstein (talk) 16:31, 12 July 2021 (UTC)
 * I am inclined to agree. --JBL (talk) 19:09, 13 July 2021 (UTC)
 * I've never said the DL construction should stay in the general "Dual polyhedron" article. I just said it would be a pity to apply DL only to the Archimedean solids (to construct the shape of the faces of any Catalan solid), when MathWorld claims DL can construct the faces of any convex uniform polyhedron (& is probably right on this point)... --JavBol (talk) 14:45, 14 July 2021 (UTC)

You reverted my move of this material to Catalan solid, citing WP:BRD. The discussion has already happened; it is here; there was little or no opposition to the move. Please explain your revert or restore the move. —David Eppstein (talk) 18:31, 23 August 2021 (UTC)
 * I am not convinced that this section should be moved to the article on the Catalan solids, the duals of the convex Archimedeans. (In reply to David's concern that "the discussion has already happened", I was not a part of it and, had I been following it, I would certainly have interjected. I do so now.) Firstly, while the points made above by are broadly valid, they do not take into account the fact that both Cundy & Rollett and Wenninger discuss this construction in the context of duality. Cundy & Rollett describe it in Section 3.8 on "Dual solids", while Wenninger's book is titled Dual Models, a bit of a giveaway really. So there is a strong prima facie case for describing the construction under a similar heading here, viz. the present article. Secondly, Cundy & Rollett accept the regular-faced pyramids as among the Archimedeans, and therefore the Dorman Luke construction also applies to them. On the other hand Wenninger's discussion of the Dorman Luke construction is described in his chapter on "The Thirteen Semi-Regular Convex Polyhedra and their Duals", i.e. excluding the regular-faced pyramids, while our article on the catalan solids also omits these pyramids. So DL ought to be described in an article which embraces both the Catalans and the regular-faced pyramids. No better alternative than the present article has been offered. &mdash; Cheers, Steelpillow (Talk) 18:55, 23 August 2021 (UTC)
 * Catalan solid is exactly about the duality of uniform polyhedra, the case relevant for the Dorman Luke construction. This article is not; it is about more general constructions valid for polyhedra more generally. It is as if, in an article about real number division, you went on and on about simplification of rational numbers to simplest terms; yes, it's sort of relevant, but only to a special case, not the general topic of the article. Citing popularizations whose coverage of polyhedra concentrates primarily on the uniform polyhedra rather than the general case is unconvincing as evidence for anything, really. —David Eppstein (talk) 19:04, 23 August 2021 (UTC)
 * David Eppstein, could you respond to the point about C&R including the duals of the regular-faced pyramids in the construction, but our article on the Catalan solids failing to do so? Had we no article recognising the rationals as a whole, but did have multiple sources discussing the division of numbers in general, where else could one discuss their division? &mdash; Cheers, Steelpillow (Talk) 19:14, 23 August 2021 (UTC)
 * Just a naive (ridiculous?) suggestion: if David Eppstein (or Steelpillow) directly contacted Eric Weisstein (if E.W. knows Steelpillow), & "simply" asked him to clarify his MathWorld entry "Dual Polyhedron", then perhaps E.W. would improve this, & so the decision / the most appropriate Wikipedia article for the Dorman Luke construction would become easier to make...? :-P --JavBol (talk) 22:32, 23 August 2021 (UTC)
 * [edit conflict] Ok, then put the construction for uniform duals in uniform polyhedron, where we talk about all uniform polyhedra and their duals, rather than separating out the uniform polyhedra that are not prisms, pyramids, or Platonic solids (a bad way of classifying things by what they are not, but whatever). The answer to your question is "not here". It's a specialized subtopic and should not clutter this article. [added after edit conflict]: Yes, that is a ridiculous suggestion. We do not and should not follow MathWorld in what they do, and we should not advocate that sources change what they do as a way of making those changes here. MathWorld is full of mistakes and original research. We should follow better scholarly sources. —David Eppstein (talk) 22:34, 23 August 2021 (UTC)
 * Of course, I meant prisms not pyramids (which are not isogonal). Wenninger devotes several pages at the beginning to polar reciprocation, setting a clear context for DL as the "the easiest way" to make the construction. So it should really be dealt with in that context. Also, one can equally see it as a specialised subtopic of the uniform polyhedra, to not go there either. Wenninger spends several pages discussing polar reciprocation at the beginning, which sets a clear context for DL as "the easiest way" to make that construction for the Archimedeans. Given that our views can only be editorial opinion, I feel that we need better reason before diverging from the approach of such established sources. I have moved DL in as a subtopic, to mirror that treatment and see how it looks. &mdash; Cheers, Steelpillow (Talk) 05:01, 24 August 2021 (UTC)
 * You're arguing that because it might be seen as a specialized subtopic of a specialized subtopic, it should not go into the subtopic article, but into the general-topic article? Really? In what world is there any logic behind such an argument? And you have not addressed my point, that you are relying on sources that are primarily about symmetric polyhedra; of course in that context, they're going to discuss concepts that relate to that class of polyhedra, but that doesn't make such concepts, which are inapplicable to almost all polyhedra, relevant for the general article here. Should we also have a separate multi-paragraph illustrated section about how the dual polyhedron of a disphenoid is another disphenoid, filled with detailed calculations of its vertex coordinates and face normals? Do you think that sort of cruft would be on-topic here? No wonder our polyhedron articles are such a disaster, if even this small attempt at de-crufting them and trying to avoid saying everything about all polyhedra in every polyhedron article runs into so much opposition. —David Eppstein (talk) 05:37, 24 August 2021 (UTC)
 * If your arguments were supported by treatments in RS, I would take more cognizance of them. Your remarks illustrate very clearly how the whole "specialized subtopic" issue needs reference to RS, and that is what I do. Let us now consider an example of omission. In his Polyhedra, Cromwell eschews duality entirely. He discusses the Archimedeans and, on pp.367-8, the Catalans. Because he avoids duality, he therefore avoids any mention of DL. So here is an RS which discusses the uniforms and Catalans without mentioning DL. Back with DL, Gailiunas and Sharp's paper on the Duality of Polyhedra (Int. J. Math. Ed. in Sci. & Tech., 2006) touches on it without specifically mentioning the Archimedeans (p.625, Section 5). So your arguments are not borne out by RS, whereas mine are. &mdash; Cheers, Steelpillow (Talk) 06:47, 24 August 2021 (UTC)
 * I really can not make my own (humble) opinion on this difficult technical point, as long as every occurence of "pyramids" that should be replaced with "prisms" is not; but a user is not allowed to change other users' comments... :-P --JavBol (talk)

Besides, is there a rigorous mathematical proof anywhere (online) that, when applied to an Archimedean solid (or perhaps more generally to a uniform polyhedron), the DL construction necessarily provides the face shape of the dual polyhedron, please? --JavBol (talk) 15:47, 24 August 2021 (UTC)
 * I am not aware of any rigorous treatment online. The proof is simple enough. It is a theorem of projective geometry that polar reciprocation of any polyhedron about any quadric curve will yield a dual figure. Given a centrally-symmetric convex polyhedron reciprocated about a concentric sphere, the dual will be convex and finite. We define a vertex figure, for this purpose (there are others, see for example my own web page on the subject) as the intersection, of the polyhedron containing the vertex (pole), with its dual face plane (polar). For such a polyhedron which also has an intersphere (tangent to all edges), the Dorman Luke construction follows. The Archimedeans provide examples. I would assume naively that the construction applies to any canonical dual pair and indeed to any polyhedron having an intersphere, but I have never seen any proof of that. &mdash; Cheers, Steelpillow (Talk) 11:59, 26 August 2021 (UTC)
 * Thank you very much for your detailed answer. :-O (However, if anyone ever sees a proof (of the DL construction) at a lower math level (more direct, not using a general theorem), i would also be interested.) ;-) So, does your sentence: "Given a centrally-symmetric convex polyhedron reciprocated about a concentric sphere, the dual will be convex and finite." work even if the centrally-symmetric convex polyhedron is not bounded, please? :-P --JavBol (talk) 18:28, 26 August 2021 (UTC)
 * I am not sure what you mean. Both "finite" and "bounded" can mean different things in different contexts. Can you give an example of a centrally-symmetric finite polyhedron which is not bounded? &mdash; Cheers, Steelpillow (Talk) 19:35, 26 August 2021 (UTC)
 * A "regular tetrahedron" ("centered" on the origin) with all its vertices, edges, & faces at infinity...? :-P Didn't you use "finite" as a synonym for "bounded", in the sense of "included in a ball having a finite radius"? With this definition, a centrally-symmetric convex polyhedron that would not be bounded, would probably have all its vertices, edges, & faces at infinity; then, its dual "polyhedron" would be reduced to the (trivially convex & bounded) center of the sphere... This is probably why the condition that the polyhedron be bounded is not specified in your sentence in question: "Given a centrally-symmetric convex polyhedron reciprocated about a concentric sphere, the dual will be convex and finite."; isn't it? :-P --JavBol (talk) 21:12, 26 August 2021 (UTC)
 * You change my question, to align with your unjustifiable assumptions about my own usage for terms which I have just said vary in meaning! You really must stop changing other editors' comments, or someone will lose patience with you and seek sanctions. In Euclidean geometry "infinity" does not exist. In extended Euclidean geometry, the polyhedron you describe becomes a tiling of the plane at infinity and is flat not convex. With the meanings you state, there is no such thing as your "centrally-symmetric convex polyhedron which is not bounded". I think this discussion can end here, as it has lost relevance to duality. &mdash; Cheers, Steelpillow (Talk) 08:50, 27 August 2021 (UTC)
 * Thank you for your answer. I knew my ``"regular tetrahedron" ("centered" on the origin) with all its vertices, edges, & faces at infinity`` is a tiling of a plane, so i thought it could be convex.
 * Excuse me for my (good faith) replacing of your "finite" with "convex". I owe you an explanation for this blunder: i thought you had unintentionally written "finite" instead of "convex", because:
 * For a polyhedron, the only meanings of "finite" that i can imagine are: "included in a ball having a finite radius", or "with a finite number of faces". To me: the former is a definition of "bounded", & the latter doesn't seem to fit in your sentence in question (does it?).
 * Previously, you had unintentionally written "pyramids" instead of "prisms", "quadric curve" instead of "quadric surface" (hadn't you?), & in your question: "Can you give an example of a centrally-symmetric finite polyhedron which is not bounded?", you had forgotten to write "convex" on top of "centrally-symmetric finite" (hadn't you?).
 * I thought you were quoting the terms of my question: ``does your sentence: "Given a centrally-symmetric convex polyhedron reciprocated about a concentric sphere, the dual will be convex and finite." work even if the centrally-symmetric convex polyhedron is not bounded?``, so that i would clarify these terms by an example.
 * --JavBol (talk) 19:42, 27 August 2021 (UTC)
 * If you think someone has made a mistake in their comment then you ask, you do not change. That is fundamental point of etiquette, see WP:AVOIDABUSE. Why you did it is irrelevant. &mdash; Cheers, Steelpillow (Talk) 05:28, 28 August 2021 (UTC)

With an origin O & 3 orthonormal axes placed as on the following figure, the vertices & edge midpoints of a regular faced square pyramid j1 & the vertices of its dual have the coordinates specified in this file page's Description part. (With the original vertex coordinates, & by their symmetries, one checks that: all original edge lengths are the same (2); all original base angles are right.) So an "extended" DL construction applies to j1, even though: j1 & its dual are not uniform, since they are not isogonal; j1's base lies on the center O of s (as the 3rd coordinate of every base vertex is 0). One similarly checks (but with most point coordinates (specified in the following file page's Description part) & corresponding plane equations being less simple) that the right square pyramid with side/base edge length ratio: √2, also has s as intersphere, & has the same properties regarding its vertex figures, except that its side edge points of tangency to s are not also their midpoints. So the DL construction also applies to this pyramid, even though this & its dual are not uniform, since they are not isogonal & their side faces are not regular. So, in my opinion, the DL construction part should not be moved to the Uniform polyhedron article. --JavBol (talk) 02:47, 3 September 2021 (UTC)
 * The coordinates of the midpoint e.g. B(0,1/√2,1/√2), resp. E(1/√2,1/√2,0), of the edge SU, resp. TU, satisfy the equation x²+y²+z² = 1 of the unit sphere s; so B, resp. E, lies on s. The dot product of the vectors OB & SU(0,√2,-√2), resp. OE & TU(-√2,√2,0), is 0; so these are orthogonal. So the edge SU, resp. TU, is tangent to s at B, resp. E. So, by symmetry, every edge of j1 is tangent to s at its midpoint. So: s is the intersphere of j1; for every vertex V of j1, & for every edge of j1 lying on V, by denoting I its point of tangency to s, the length VI is the same (1).
 * The 3rd coordinate of A, B, C, D, K, L, M, & N is 1/√2, so their coordinates satisfy the equation (√2)z = 1 of the polar plane pS of the apex S(0,0,√2), so A, B, C, D, K, L, M, & N lie on pS. A(1/√2,0,1/√2), B(0,1/√2,1/√2), C(-1/√2,0,1/√2), & D(0,-1/√2,1/√2), so PA = PB = PC = PD where P(0,0,1/√2): P is the center of the circumcircle cS of the apex figure ABCD. E.g.: B lies on cS; B(0,1/√2,1/√2) is the midpoint of also K(1/√2,1/√2,1/√2) & L(-1/√2,1/√2,1/√2). The dot product of the vectors PB(0,1/√2,0) & KL(-√2,0,0) is 0, so these are orthogonal. So KL is the line tangent to cS at B. So, by symmetry, LM, resp. MN, & NK, is the line tangent to cS at C, resp. D, & A. So KLMN is the tangent polygon of ABCD.
 * Similarly: the 2nd coordinate of B, E, F, K, & L is 1/√2, so their coordinates satisfy the equation (√2)y = 1 of the polar plane pU of the vertex U(0,√2,0), so B, E, F, K, & L lie on pU. B(0,1/√2,1/√2), E(1/√2,1/√2,0), & F(-1/√2,1/√2,0), so QB = QE = QF where Q(0,1/√2,0): Q is the center of the circumcircle cU of the vertex figure BEF. B lies on cU & the line KL. The dot product of the vectors QB(0,0,1/√2) & KL(-√2,0,0) is 0, so these are orthogonal. So KL is the line tangent to cU at B. "Similarly": e.g. E lies on cU & the line "KJ", where J("0,0,∞"). The line QE is horizontal & the line "KJ" is vertical, so they are orthogonal. So "KJ" is the line tangent to cU at E. So, by symmetry, "LJ" is the line tangent to cU at F. So JKL is the tangent "polygon" of BEF.
 * E.g. the side face plane STU has the equation x/(√2)+y/(√2)+z/(√2) = 1, so K(1/√2,1/√2,1/√2) is the corresponding vertex of the dual. So, by symmetry, L, resp. M, & N, is the vertex of the dual corresponding to the side face plane SUV, resp. SVW, & SWA.
 * But for its apex figure: by symmetry, SA = SB = SC = SD. And for its base vertex figures: e.g., by symmetry, UE = UF. UE & UB both = (√3)(1+2√2)/7. So UE = UF = UB. So, by symmetry, VF = VG = VC, WG = WH = WD, & AH = AE = AD.
 * Because of your original research? That's not how Wikipedia works. You need to provide published sources, not calculations. —David Eppstein (talk) 21:39, 18 September 2021 (UTC)

Center of mass taken to be center of sphere?
In Dual polyhedron, the following passage: "For bounded polyhedra having multiple symmetry axes, these will necessarily intersect at a single point" should be more specific, such as: "For a bounded polyhedron having multiple symmetry axes, these will necessarily intersect at a single point: its center of mass"; shouldn't it? --JavBol (talk) 02:21, 31 July 2021 (UTC)

@: Do you have an opinion on this point, please? ;-) --JavBol (talk) 04:32, 2 August 2021 (UTC)
 * I'm not Steelpillow, but I have an opinion. It is: this point is the center of symmetry (already stated), the center of mass of the vertices, the center of mass of the edges, the center of mass of the faces, the center of mass of the solid, the center of the bounding ball, and probably a lot of centers besides. Why is it important to say more than that it's the center of symmetry, when it is symmetry that we are already talking about? I don't see how mentioning that it happens to also be a different center makes it any more specific. —David Eppstein (talk) 06:43, 2 August 2021 (UTC)
 * Thank you for your detailed answer. But precisely:
 * In the passage in question, the center of symmetry is rightly not mentioned, because this does not always exist. (Example: the regular tetrahedron is bounded & has multiple symmetry axes; these intersect at a single point: the center of mass (of the vertices, of the edges, of the faces, of the solid); but this is not a center of symmetry.)
 * Mentioning the center of mass can prevent from making this tempting error.
 * Mentioning the center of mass efficiently explains why the symmetry axes necessarily intersect at a single point (& why in this case the centers of mass (of the vertices, of the edges, of the faces, of the solid) are all equal).
 * --JavBol (talk) 14:01, 2 August 2021 (UTC)
 * Of course the regular tetrahedron has a center of symmetry. I agree with David here. &mdash; Cheers, Steelpillow (Talk) 21:48, 2 August 2021 (UTC)
 * BTW see Center of symmetry which makes clear that, according to that definition, it is indeed a center of symmetry. It is not the center of a central symmetry, because that's a different phrase with a different meaning. Maybe the similarity of wording is the cause of confusion here? —David Eppstein (talk) 21:55, 2 August 2021 (UTC)
 * Sorry: i hastily thought that David Eppstein had hastily rejected my suggestion (as he sometimes does), & that he had meant "the center of a central symmetry"... :-P
 * Still, the whole sentence:
 * "For bounded polyhedra having multiple symmetry axes, these will necessarily intersect at a single point, and this is usually taken to be the centroid."
 * should be rephrased, such as:
 * "For a bounded polyhedron having multiple symmetry axes, these will necessarily intersect at a single point: its centroid, and this is usually taken to be the center of the sphere.";
 * shouldn't it?
 * --JavBol (talk) 22:42, 3 August 2021 (UTC)
 * "taken to be the centroid" is wrong or misleading. It cannot be taken to be the centroid of the polyhedron; it *is* the centroid, but that's incidental. What it is usually taken to be is the center of the sphere that we are using to define the polarity. I have rewritten to make this clearer. —David Eppstein (talk) 22:55, 3 August 2021 (UTC)
 * Thank you for your answer & for your edit. The whole passage is now:
 * "For symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. For bounded polyhedra having multiple symmetry axes, these will necessarily intersect at a single point, and this is usually taken to be the center of the sphere."
 * But some symmetrical polyhedra also having an obvious centroid have only 1 or even 0 symmetry axis. So, the following changes should be made:
 * "For a symmetrical polyhedron having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. In particular, for a bounded polyhedron having multiple symmetry axes, these will necessarily intersect at its centroid, and this is usually taken to be the center of the sphere.",
 * shouldn't they?
 * --JavBol (talk) 22:34, 5 August 2021 (UTC)
 * Just some observations. The centroid is well defined. As used here the term "obvious" appears to be an assertion that the centre of symmetry is the intuitively "obvious" place to put the centroid, i.e. it is anticipating what is about to be explained. Also, terms such as centre of mass are loaded; only if we weight all points equally is the barycenter (CG) of a figure coincident with the centroid, so it might be simpler to avoid talk of mass. &mdash; Cheers, Steelpillow (Talk) 07:27, 6 August 2021 (UTC)
 * Thank you for your answer. I agree with it, & i think it supports my latest suggestions of changes; indeed:
 * If "In particular" is not added, then some readers may deduce that an obvious centroid is always the intersection point of multiple symmetry axes.
 * But some symmetrical polyhedra having only 1 or even 0 symmetry axis also have an obvious centroid. Examples:
 * a parallelepiped with (parallelogram) faces of 3 different shapes & 0 right angle is symmetrical about the common midpoint of its body diagonals & of its body medians, has 0 symmetry axis, but has an obvious centroid;
 * an oblique prism with isosceles triangle bases & 1 rectangle side face is symmetrical through the plane lying on the opposite side edge & perpendicular to the rectangle side face;
 * a right pyramid with a symmetrical base polygon (except the regular tetrahedron) has only 1 symmetry axis (& several symmetry planes containing it); the centroid of the solid pyramid is 1/4 the distance from the base centroid to the apex;
 * ... --JavBol (talk) 21:31, 6 August 2021 (UTC)
 * The parallelepiped has a center of symmetry, so there is no reason to talk about centroids for it. The prism has a center of symmetry for equilateral triangle bases, and for other bases I don't see a good reason to choose the centroid rather than say the center of the circumscribed sphere. Again, for the prism, I don't see a good reason to choose the centroid over the center of the circumscribed sphere or inscribed sphere (it has both, not necessarily with the same center). Why are you so fixated on centroids over other centers? The only case where the choice of a good center is unambiguous is when there is a center of symmetry, because that is exactly the case where all centers (defined in a symmetric way) coincide. —David Eppstein (talk) 21:37, 12 August 2021 (UTC)
 * Thank you for your answer. But i had specified: "an oblique prism", so that even with equilateral triangle bases, it has no circumsphere, & generally no insphere.
 * I'm not fixated on centroids over other centers: it's the passage in question & the sentence just after it that were, before you changed the former & surprisingly (*) removed the latter:
 * "Failing that, a circumscribed sphere, inscribed sphere, or midsphere (one with all edges as tangents) is commonly used."
 * Thank you also for mentioning in the article the case with a center of symmetry. (*) Though, some polyhedra have a center of circumsphere or insphere, but no center of symmetry. Also, my specified parallelepiped example has a center of symmetry, but no circumsphere (since its faces have no circumcircles), no intersphere (since its parallel faces, even in the case where they are rhombi, are not perpendicular to the line joining their centers (of incircles)), & generally no insphere. Finally, as your edit summary says, you simplified the text, because: "we don't need to attempt to classify symmetries here". Then, we don't need to attempt to classify symmetries for polar reciprocation efficiency in any Wikipedia article; do we?
 * --JavBol (talk) 21:48, 13 August 2021 (UTC)

@: Then, what about replacing "this can be used" with "this is sometimes used"? Because the sentence just after that claims "it is possible to reciprocate a polyhedron about any sphere". --JavBol (talk) 22:05, 22 August 2021 (UTC)
 * Do you have any published evidence that it "is sometimes used"? Who has used it, in circumstances where a center of symmetry did not exist, and what did they write about using it? Why are you continuing to try to add material to this article without basing it on published sources? —David Eppstein (talk) 01:39, 23 August 2021 (UTC)
 * No, but this sentence had been in the article for a long time, & everyone used to be happy with it... --JavBol (talk) 03:58, 23 August 2021 (UTC)
 * So what? That is not addressing the point and not a valid argument for including it. —David Eppstein (talk) 03:59, 23 August 2021 (UTC)
 * Many mistakes reside unnoticed for years on Wikipedia. That is neither proof they are not mistakes, nor reason to leave them there once someone does notice them! &mdash; Cheers, Steelpillow (Talk) 09:54, 23 August 2021 (UTC)
 * (But this sentence had no mistake.) Is there any example of a polyhedron having a circumsphere, insphere, or intersphere, where reciprocating it about this (by using the tangency points of it & this, or of its dual & this, or of both & this) is not more convenient than about a random sphere, please? :-/ --JavBol (talk) 16:48, 31 August 2021 (UTC)
 * You are still asking the wrong question. We should not be basing article content on your intuition of convenience. We should be basing it on what sources say. I ask again, do you have any published evidence that these choices are used? —David Eppstein (talk) 17:30, 31 August 2021 (UTC)

Dorman Luke construction for all uniform polyhedra?
Hello,

I tried Dorman Luke construction for many uniform polyhedra, It doesn't seem to work when the vertex figure is self-intersecting. Are you sure it doesn't mean the face of a face-transitive polyhedron is dual to it's dual polyhedron vertex-figure? And are you sure it's written in the source «Uniform polyhedron» not «Archimedean solid»?

Thanks, Hirbod Foudazi2 (talk) 15:26, 15 September 2021 (UTC)


 * You will find many, if not all, of the constructions in Wenninger's book Dual Models (Cambridge University Press, 1983). It worked for him. &mdash; Cheers, Steelpillow (Talk) 16:17, 15 September 2021 (UTC)


 * No, It worked for convex examples for him. I saw page 30, it is written:"The easiest way to find the faces of convex duals (i.e., The Archimedean or semi-regular duals) is to use the so-called Dorman Luke construction."  Thank you, Hirbod Foudazi2 (talk) 16:46, 15 September 2021 (UTC)
 * See also paragraph on Pages 32-5 and the whole of Chapter IV. Indeed, I strongly recommend studying his entire text and accompanying drawings before jumping to conclusions. &mdash; Cheers, Steelpillow (Talk) 17:21, 15 September 2021 (UTC)


 * Hello, I don't have access to the pages you mentioned (I found p.30 in Google books). So as you say it works for any uniform dual. Could you please show me how it works to find the faces of great triambic icosahedron for example? I might have understood it wrongly. Thanks for your attention, Hirbod Foudazi2 (talk) 07:48, 16 September 2021 (UTC)
 * The great triambic icosahedron is the dual of the great ditrigonal icosidodecahedron (W87). Its Dorman Luke construction is shown on Page 49. If you enable user email in your preferences and send me an off-wiki message, I can send you a photo of the page. &mdash; Cheers, Steelpillow (Talk) 08:22, 16 September 2021 (UTC)

Proposed section move
Wenninger uses the Dorman Luke construction for many if not all of the nonconvex uniform duals. While it would appear to work for all polyhedra with a midsphere, including canonical dual pairs, I know of no reliable source stating this. So, while Wenninger's use also clearly makes the article on the convex Catalan solids an inappropriate home, I do not think that any argument can be sustained against moving it to that on the Uniform dual polyhedra. I personally dislike my own suggestion, as I am sure that one day we will have to move it again, but for now it does seem to be the home that best conforms to Wikipedia's policies. &mdash; Cheers, Steelpillow (Talk) 15:52, 19 September 2021 (UTC)
 * Support per my earlier discussion here. —David Eppstein (talk) 18:51, 19 September 2021 (UTC)
 * Ditto. --JBL (talk) 23:02, 19 September 2021 (UTC)


 * I have moved it across and left behind a small subsection, which others of you may or may not wish to rearrange. &mdash; Cheers, Steelpillow (Talk) 08:28, 20 September 2021 (UTC)

Hello and sorry for delay. I found page 32 and read it and I found out what is going on, thank you. About the thing you said, maybe page 117 in this reliable source is useful. Thank you, Hirbod Foudazi2 (talk) 06:00, 23 September 2021 (UTC)


 * See two last paragraphs of the page I mentioned in particular. Hirbod Foudazi2 (talk) 06:05, 23 September 2021 (UTC)
 * I would assume that Steelpillow is already familiar with Cundy & Rollett's coverage of this topic. —David Eppstein (talk) 06:50, 23 September 2021 (UTC)


 * I did not say he/she doesn't. Hirbod Foudazi2 (talk) 08:37, 23 September 2021 (UTC)
 * Indeed I am. It is the original published source of the Dorman Luke construction. But I am not sure what point is being made here. Nor am I clear that I am worth arguing over. &mdash; Cheers, Steelpillow (Talk) 08:41, 23 September 2021 (UTC)

missing link
In the passage "For example, the dual of a regular tetrahedron", there should be a link from 'tetrahedron' to the Wikipedia article on that topic. Kontribuanto (talk) 00:34, 2 March 2024 (UTC)