# Talk:Dual polyhedron

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## 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.—Tetracube 22:14, 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?

Tom Ruen 23:10, 17 February 2006 (UTC)

I prefer to put them together, since otherwise there will be a lot of needless repetition. So option 1 sounds good to me.—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! <shudder!>.
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) — 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)

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.—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. :-)—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. — 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. — 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. — 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. — Cheers, Steelpillow (Talk) 20:49, 12 February 2013 (UTC)

## r12 . r22 = 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#Circle inversion?

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

Yes indeed. Now fixed. — Cheers, Steelpillow (Talk) 17:46, 2 December 2014 (UTC)

## Assessment comment

The comment(s) below were originally left at Talk:Dual polyhedron/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

 Would benefit from: a diagram showing polar reciprocity, a full section on Wenninger's 'infinite prism' duals, perhaps a table of some common or interesting dual pairs (e.g. the regular pairs and the Csaszar and Szilassi polyhedra).

Last edited at 21:44, 4 June 2007 (UTC). 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.— 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. — 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.) — 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. — 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. — 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. — 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:
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.) — 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. — 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"?
— 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? — 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. [1] 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. — 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. — 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. — 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#Duality. 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! — Cheers, Steelpillow (Talk) 15:56, 13 February 2017 (UTC)