# Talk:Polyhedron

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I have a web page at wordpress.com (URL:http://polyhedron100.wordpress.com) with a variety of Nolidean Polyhedra including some nice Crown Polyhedra in wood. I am hoping I might have a link to them here at the polyhedron page as there is no mention of these types of polyhedra elsewhere at wikipedia. Please advise if this would be acceptable? Thanks and Take Care User:Bertimusminimus 10:15, 29 November 2014 — Preceding unsigned comment added by Bertusminimus (talkcontribs)

Hi again. These are really beautiful, they certainly deserve a wider audience. I will post a link on my facebook page for sure. But they are not normally regarded as polyhedra, because the surfaces are bounded. If they are regarded as toroidal nolids, even then they are not usually understood as polyhedra: for example two (coincident) edges may share the same two vertices, which is not allowed in conventional polyhedron theory, but only if one chooses to extend the theory specially. As a footnote, a "crown polyhedron", sometimes called a stephanoid, is a particular kind of axially-symmetric (pyramid/prism symmetries) polyhedron. In the end, your creations are at heart beautiful symmetrical mathematical sculptures, and best appreciated as such. Not perhaps what you wanted to hear, but I hope it helps clarify things for you. — Cheers, Steelpillow (Talk) 16:47, 29 November 2014 (UTC)

Thank you for responding and your kind words. It's a little disappointing to be sure, but I will of course honor your decision. That said, I do realize that they are not polyhedra in the traditional sense but I thought that they still would fall into that category if one allows for a relaxation of the terms of their definition such as having gaps or holes between polygons. In any event thanks again and Take Care, User:Bertimusminimus 18:15, 29 November 2014 (UTC) — Preceding unsigned comment added by 75.120.178.107 (talk)

## Topological polyhedra

This section currently defines only topological polytopes. Are we to take it that "polyhedron" and "polytope" are synonyms in this context, or that a topological polyhedron is a topological 3-polytope? — Cheers, Steelpillow (Talk) 18:58, 21 December 2014 (UTC)

I don't even know, and it's unsourced. What I would normally do in such situations is try quick Google scholar and Google books searches to determine whether there is in fact a standard meaning for this term; if so, add the sources and clarify the meaning, and if not just delete the section. —David Eppstein (talk) 19:42, 21 December 2014 (UTC)
I found several definitions, each built on more impenetrable buzzwords than the last, so I have no idea even whether they are equivalent or not. Then there is this possibly related remark from Grünbaum & Shephard, 1969:
"A topological polytope P' is the image of a convex polytope P under a homeomorphism Φ : P -> En. The faces of P' are the images of the faces of P under Φ, and the dimension of P' is defined to be the dimension of P. Sometimes we shall use the term geometric polytope for a convex polytope when we wish to emphasise the difference from a topological polytope."[1]
To my poor understanding, En is Euclidean n-space and anything injected into it is perforce a real geometric polytope. But here it is a topological polytope and is being distinguished from the geometric variety. This kind of apparent non sequitur, supported invariably by the most impenetrable of jargon salads, always utterly baffles me. Hence my appeal to someone schooled in such ways. — Cheers, Steelpillow (Talk) 21:33, 21 December 2014 (UTC)
This passage seems clear enough to me, at least. It's describing what you get from a polyhedron when you warp space with a continuous but 1-1 transformation. E.g. you could make a topological cube with six Coons patches whose boundary curves and corners have the same combinatorial structure as the edges and vertices of a geometric cube. —David Eppstein (talk) 22:15, 21 December 2014 (UTC)
Thank you, yes, that makes sense now - as far as polytopes go. The relation to "topological polyhedra" is still undefined. I also begin to wonder as to what extent things like CW complexes are related. — Cheers, Steelpillow (Talk) 23:22, 21 December 2014 (UTC)

## A clear definition is missing, or something is false

In the Polyhedron#Euler_characteristic section we see the folowing statement.

For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected, χ = 2.

This statement isn't true for a solid made by two tetrahedron joining in one of their vertices. Since this is a simply connected solid (bounded by planes) with simply connected faces and χ = 3, either this object isn't a polyhedron, or the above statement is false. I don't see anything in this article that would exclude this wedge-sum object from among polyhedra, so it would be helpful to improve the either the defintion of polyhedrion, or the statement above. 89.135.19.75 (talk) 07:36, 18 May 2015 (UTC)

"A solid made by two tetrahedron joining in one of their vertices" is not a polyhedron as defined in this article. Nor is it a simply-connected manifold as linked to in the article. Now I know some mathematicians squawk in horror at that because they have any of several different specialised definitions of a polyhedron in mind from their favourite standard text on their chosen field. But this is a Wikipedia article introducing polyhedra, and it uses the classical definition unless some variation is expressly stated. I have tidied the article to help clarify one or two aspects. — Cheers, Steelpillow (Talk) 12:04, 18 May 2015 (UTC)
Sorry, but I don't find this statement in Richeson's book. Where is it exactly?
And the other thing: exactly which sentence excludes this object from among the polyhedra in the article? The link points to siply connected space, and this object is also a simply connected space, so the link doesn't. 89.135.19.75 (talk) 17:12, 18 May 2015 (UTC)
p.255 states "We know from the classification of surfaces that the sphere is the only simply connected closed surface", p.182 gives χ for the sphere = 2. I have added a parenthetical comment to separate out the two logical steps for you. Any better? — Cheers, Steelpillow (Talk) 18:25, 18 May 2015 (UTC)
1. I see nowhere in this article, that every polyhedron should be topologically a connected closed surface.
2. It isn't true, that if a polyhedron is topologically a sphere, then its Euler characteristic is 2, see for example a small cube on the middle of the top of a bigger cube (the interior of the contacting face parts are removed). This is topologically a sphere, but its Euler characteristic is 3. The simply connectedness of the faces is an additional necessary condition, so it should not have been removed from the text (see also here) 89.135.19.75 (talk) 20:03, 18 May 2015 (UTC)
[Edit conflict] First, thank you for pointing out the omission re. vertex-connected objects. It is implicit in standard definitions, such as that of an abstract polytope, so I don't think it need be made explicit in the discussion on definition. But I have now added a commentary in the section on the surface characteristics. Richeson illustrates your figure and discusses the problem, so too does Cromwell at a more basic level. Do you think it needs a citation?
The requirement to have a closed surface is expressed in simpler language as a requirement that the surface must not end abruptly, perhaps why you missed it. The cube-on-a-cube is of course excluded by the need for every face to be a polygon: a square with a hole in is not a valid face and the figure is not in fact a polyhedron. So the definition of a face needs clarifying too. I can think of at least one other possible omission. I don't have time to think the changes through now, I'll try and remember to take a proper look tomorrow (assuming nobody beats me to it). — Cheers, Steelpillow (Talk) 20:34, 18 May 2015 (UTC)
So,this concave heptahedron Isn't a polyhedron at all? 89.135.19.75 (talk) 23:04, 18 May 2015 (UTC)
It isn't according to the definition Steelpillow is using. One can find sources that use definitions that would allow it but that may be a minority view of the subject. —David Eppstein (talk) 23:07, 18 May 2015 (UTC)
That is correct. The article takes as its backbone the modern formulation of the definition used by well-known authors such as Euclid, Coxeter, Cromwell and (at an elementary level) Grünbaum and found in every school geometry textbook. From this it classifies "polyhedra" found in the wider literature into families (i.e. sub-species), generalisaztions (broadly compatible) and alternatives (incompatible in some profound way). One may hope that the heptahedron shown comes under one of these alternatives, although its discussion in the linked article is uncited, is not supported by the article sources given, and including it as a "polyhedron" under any sensible definition could just be a lapse of rigour. — Cheers, Steelpillow (Talk) 10:31, 19 May 2015 (UTC)

We shouldn't pretend that there is a single universally-accepted definition, but I don't think our article does that. —David Eppstein (talk) 16:14, 18 May 2015 (UTC)

Thanks! 89.135.19.75 (talk) 20:04, 20 May 2015 (UTC)

## Euler characteristic

The text of this article allows us to regard a polyhedron as a 2-dimensional CW-complex as well as a 3-dimensional one. However the definition of the Euler-characteristic implicitly assumes that it is taken as a 2-dimensional CW-complex (i.e. the definition of the Euler characteristic given here is true for polyhedral surfaces, but it isn't true, if the interior volume is considered also to be part of the polyhedron. Perhaps it would be useful to tell this in the article (see Euler_characteristic#Topological_definition). 89.135.8.194 (talk) 22:40, 25 August 2015 (UTC)

What the text allows and what it should say are very different things. There is no need to discuss the niceties of CW-complexes in an article which does not mention them. In particular, the Euler characteristic as defined for a polyhedron references only vertices, edges and faces: contrary to what you say, whether or not there is an interior has no relevance. A figure decomposed into say tetrahedral cells is no longer just a polyhedron but a more general topological object - and of no relevance to an article on polyhedra. — Cheers, Steelpillow (Talk) 10:05, 26 August 2015 (UTC)

I mean this:

Different approaches - and definitions - may require different realisations. Sometimes the interior volume is considered to be part of the polyhedron, sometimes only the surface is considered, and occasionally only the skeleton of edges or even just the set of vertices.[1]

and this:

For example a convex or indeed any simply connected polyhedron is a topological sphere or ball (depending on whether its body is taken into account).

The problem is, that if I regard a simply connected polyhedron as being topologically a ball, then it is contractible, hence in this case, its Euler-characteristic should be 1 instead of 2. Euler characteristic should be homotopically invariant. 86.101.236.13 (talk) 10:55, 26 August 2015 (UTC)

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

And this is exactly what I mean: "Euler characteristic for a polyhedron is given in the article does NOT invoke the body" I miss this sentence from the article. 86.101.236.13 (talk) 11:18, 26 August 2015 (UTC)

It is there to be read. The section refers to "any simply connected polyhedron (i.e. a topological sphere)". — Cheers, Steelpillow (Talk) 11:33, 26 August 2015 (UTC)
I think the confusion was caused more by the introduction of a ball in this context. Although more complete it brings complexities seldom addressed at this level and best left out. Accordingly, I have edited the general remarks about topological characteristics to confine the discussion to the surface. Any better? — Cheers, Steelpillow (Talk) 11:42, 26 August 2015 (UTC)

This is better already, but still isn't explicit enough in my taste. What about something like this:

From this perspective, a polyhedron is regarded as its surface. Any polyhedral surface may be classed as certain kind of topological manifold. For example a convex or indeed any simply connected polyhedron is a topological sphere.

From this perspective, any polyhedral surface may be classed as certain kind of topological manifold. For example a convex or indeed any simply connected polyhedron is a topological sphere.

86.101.236.13 (talk) 13:08, 26 August 2015 (UTC)

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

Yes, but (according the current text) the Euler characteristic is assigned to the "polyhedron", not to the "polyhedral surface". If you don't like my proposal,we should say at least, that we mean the "Euler characterisic of a polyhedron" the "Euler characterisic of its (polyhedral) surface". Should't we? 86.101.236.13 (talk) 14:24, 26 August 2015 (UTC)

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

I've made an attempt.89.135.8.194 (talk) 06:15, 27 August 2015 (UTC)

## Polygonal faces

Shouldn't we define what does "polygonal face" mean? Shouldn't we require that the polygon must be simple (non self-intersecting)? — Preceding unsigned comment added by 89.135.8.194 (talk) 06:51, 29 September 2015 (UTC)

The faces of star polyhedra are not simple. Precise definitions differ widely and a full treatment would not be useful in the present introductory article. See for example Lakatos, Proofs and Refutations.— Cheers, Steelpillow (Talk) 07:58, 29 September 2015 (UTC)
Even nonplanar or skew polygon faces are not excluded in Polyhedron#Abstract_polyhedra! Simple would seem to be a 2D-space concept?! p.s. I forgot how passionately this article hates pictures. Tom Ruen (talk) 09:55, 29 September 2015 (UTC)

OK, but then what excludes the not simply connected faces? What is a polygonal face at all? For example, what kind of face is defined by the fourth polygon here?

Some polygons of different kinds: open (excluding its boundary), bounding circuit only (ignoring its interior), closed (both), and self-intersecting with varying densities of different regions.

89.135.8.194 (talk) 21:44, 29 September 2015 (UTC)

Could you clarify your question? By "simply connected" do you mean a synonym for "simple" or something more general? The polygon you ask about is not simple, but nor is it excluded. Would you regard it as simply connected? — Cheers, Steelpillow (Talk) 07:51, 30 September 2015 (UTC)
A decagon with nearly coinciding vertices.
A simply-connected polygon faces is a good requirement, excluding disconnected sets (compound forms), and excluding coinciding vertices, edges or faces (degenerate forms). Like this isogonal decagon, left, is approaching a degenerate case if vertices come together. Still, simply connected allows edges to intersect where the interior of polygon is ambiguous. Tom Ruen (talk) 08:18, 30 September 2015 (UTC)
A Petrial cube is a tetrahedron made of 6 skew polygon (hexagon) faces, 8 vertices, and 12 edges.
The case of skew polygon faces is the least referenced generalization of a face, like the Petrial cube. You can see the hexagons on the right as red, orange, blue and green edge-paths around a cube. Every edge has 2 colors (2 skew polygon faces). Tom Ruen (talk) 08:24, 30 September 2015 (UTC)

@Steelpillow: Consider this part of the article:

The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron:

${\displaystyle \chi =V-E+F.\ }$

This is equal to the topological Euler characteristic of its surface. For a convex polyhedron or more generally any simply connected polyhedron (i.e. with surface a topological sphere), χ = 2.

If nothing rules out simply connected faces, then we should explicitly include this condition here as

generally any simply connected polyhedron with simply connected faces (i.e. with surface a topological sphere), χ = 2.

Bul last time you told that a"polygonal face" is always simply connected, so we don't need to specify here the simply connectedness of the faces explicitly.89.135.8.194 (talk) 08:23, 30 September 2015 (UTC)

Yes, the boundary of a real polygon is by definition simply connected (This is not true of complex polygons but that is an obscure side issue, as they are defined very differently and inhabit the complex plane). Polyhedra of course need not be simply connected. The condition arises in abstract polytope theory as a consequence of the rules for the partial ordering of the set, which are explained in that article. In more traditional topology it arises because a piece cut from a contiguous, smooth surface such as a real plane must always have a simply connected boundary. Any polygon may be used as a face of a polyhedron. For example the polygon you ask about can form an end face of a self-intersecting hexagonal prism. — Cheers, Steelpillow (Talk) 08:52, 30 September 2015 (UTC)
Toroidal polyhedron
Guy, going up a dimension, can a "real" 4-polytope exist with toroidal polyhedron cells, not topological spheres? I'd have to work a bit to find a full example.... perhaps like a tesseract's 8 cubes could be merged into 2 sets of 4 cubes by removing 8 square faces, and leaving 2 toroid cells?! Tom Ruen (talk) 09:31, 30 September 2015 (UTC)
Yes. They are called locally toroidal polytopes. Locally projective polytopes also exist, for example having one or more hemicubes as cells. I don't know for sure whether such things necessarily can or can't be faithfully realised in real n-space, but a few minutes' thought suggests to me that some can and some can't. They can make topological analysis difficult as they don't obey the usual Euler formula and its generalisations, for example they cannot be subdivided into simplexes without changing the values of such topological "invariants". — Cheers, Steelpillow (Talk) 10:10, 30 September 2015 (UTC)
This isn't really different from the phenomenon that three-dimensional polyhedra with annular faces also don't obey the Euler formula (even when they have spherical topology), I don't think. —David Eppstein (talk) 06:32, 1 October 2015 (UTC)
The key difference is that a toroidal face does not have a continuous boundary It is therefore not a valid polygon and cannot be used to construct higher polytopes. A three-dimensional toroid is a valid polyhedron and so it can be used. The disruption to topological analysis is the same though. — Cheers, Steelpillow (Talk) 08:49, 1 October 2015 (UTC) [Updated 09:39, 1 October 2015 (UTC)]
4x4 square grid as unfolded flat torus
Cool! Oh, I see my tesseract reconstruction would have two flat tori cells (two sides of a common flat tori surface), as Coxeter's {4,4|4} regular skew polyhedra, a 4D folding of a 4×4 grid from a square tiling! Tom Ruen (talk) 12:11, 30 September 2015 (UTC)
Potential annular faces?
Polyhedra with annular faces (or higher-dimensional shapes with toroidal faces) seem to be perfectly acceptable under our current definition of an abstract polyhedron — they don't violate the 1-section=line segment restriction. The shapes that are disallowed by that restriction are the ones where a face has a hole such that a vertex of the hole coincides with a vertex of the outer boundary of the face. But holes that don't touch the outer face boundary seem to be allowed. They would be disallowed if we made the stronger restriction that every 2-section is an abstract simple polygon. —David Eppstein (talk) 20:40, 23 February 2017 (UTC)
Toroidal cells make more sense to me since they are connected. Here's a set of candidate annular faces, but I'm willing to discount all of them as "degenerate" by some measure. And I only drew interiors as possibilities, and once you go "off plane" even slightly, all bets are off what interior might mean. Tom Ruen (talk) 22:22, 23 February 2017 (UTC)
In your figure, I think the abstract polyhedron model (with the 1-section=segment constraint) would allow A, B, C, and G, and disallow D and E. However, because this model is only about which vertices and edges belong to which faces, it would not make any distinction between A and G (both of which have two four-cycles of vertices and edges) nor between B and C (both of which have three four-cycles). F would be disallowed if the crossing points are listed as vertices, but allowed (and equivalent to A and G) if they are not. —David Eppstein (talk) 23:23, 23 February 2017 (UTC)
Why not the regular small stellated 120-cell {5/2,5,3}, which has genus-4 small stellated dodecahedron cells? Double sharp (talk) 12:53, 30 September 2015 (UTC)
Yes indeed. The "reduced tessaract" is a toroidal 4D equivalent of a dihedron. {5/2,5,3} is a really nice example. — Cheers, Steelpillow (Talk) 13:28, 30 September 2015 (UTC)

@Steelpillow : I see a general misunderstanding between us. I talk abot this sentence:

for any simply connected polyhedron (i.e. with surface a topological sphere), χ = 2

I thought originally, that simply connectedness refers here to the body of the polyhedron, i.e. to the the 3-dimensional domain bounded by its surface. This belief was supported by the fact that this sentence was originally (up to 12:00, 18 May 2015)

For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected, χ = 2

, and that this is in accordance with Lakatos's book:

For a simple polyhedron, with all its faces simply-connected, V-E+F = 2

As far as I see, the referred book of Richeson also doesn't say else.

But now, since you talk about self-intersecting polyhedra, I have doubts about what do you mean. Has at all sense to talk about the body of a self-intersecting polyhedron? What does this sentence mean in your opinion? 89.135.8.194 (talk) 06:23, 1 October 2015 (UTC)

The following applies whether we consider the surface or the interior of a polygon or polyhedron, for if one is simple or simply-connected then the other must be as well: wherever space is locally flat, a topological sphere will always surround a topological ball.
There is a distinction between structural incidence or connectedness on the one hand and geometric coincidence on the other. "Simple" is a geometric property, "simply-connected" is a structural or topological property. In topology, whether a particular geometrical form is simple or self-intersecting has no significance. For example a cross-quadrilateral (a butterfly or hourglass shape) has four vertices each connected to, i.e. incident with, two sides. The crossing point in the centre has no such connection and the two sides are merely coincident at that point. If the quadrilateral is unwound and made convex then it is easy to comment on the fact that it is now a simple polygon, and the fact that it is simply connected is easy to see. When twisted up it is no longer geometrically simple but structurally it is still simply-connected. If we make it the end of a four-sided prism, the prism remains simply connected no matter how we squash or morph it around. Another simply connected polyhedron is the great stellated dodecahedon {5/2, 3}, because (as Cayley noted) it is just such a morph of the convex regular dodecahedron {5, 3}. But of course it is not simple like {5, 3} because it self-intersects. On the other hand the small stellated dodecahedron {5/2, 5} is a toroid of genus around 4 (I can't remember exactly) and is neither simple nor simply-connected.
It is relatively easy to find out whether a certain figure is simple, especially if it is convex. It is much harder to see, by looking or by analysis of things like half-spaces or vertex connectivity, whether a self-intersecting polyhedron is simply-connected or not. The sure way to find out is by discovering its Euler characteristic. This is the analysis which the article tries to explain by first introducing the Euler characteristic of the structural sphere, whether it be geometrically simple or self-intersecting. Does this clarify the situation for you? — Cheers, Steelpillow (Talk) 09:27, 1 October 2015 (UTC)
(Yes, the genus of {5/2, 5} is indeed 4, as its Euler characteristic is −6 and it is orientable.) Double sharp (talk) 08:37, 2 October 2015 (UTC)

Not exactly, but trying to comprehend your words. Of course, we can talk about abstract graphs where doesn't matter if it is planar or not. It is an independent topological space. If it is not planar, then when we draw it on a sheet of paper, then there will appear line crossings that "do not count", i.e, that arent a vertex. Of course we can draw planar graphs also in an entangled form where the lines cross not only at vertices. Still it is planar. But I think, that we can talk about "faces" only when we embed a ${\displaystyle G}$ planar graph in the sphere. If the embedding is the ${\displaystyle f:G\to S^{2}}$ function, then the faces are the connected parts of ${\displaystyle S^{2}\setminus f(G)}$. Generally, the faces depend on ${\displaystyle f}$ too, not only on ${\displaystyle G}$. But how do you define faces in the case of a nonplanar graph? 89.135.8.194 (talk) 05:51, 2 October 2015 (UTC)

You choose some cycles in the graph and call them faces. If you like, you can also associate each chosen cycle with a topological disk and glue the disks together along the edges, but that step is not necessary for understanding the collection of vertices, edges, and cycles as an abstract polyhedron. —David Eppstein (talk) 06:24, 2 October 2015 (UTC)

Is there a difference between your "abstract polyhedron" and CW complexes? 89.135.8.194 (talk) 06:31, 2 October 2015 (UTC)

Is there a difference between your "abstact polyhedron" and CW-complexes? 89.135.8.194 (talk) 06:31, 2 October 2015 (UTC)

Yes. There are abstract polytopes which are not CW-complexes and there are CW-complexes which are not abstract polytopes (although in three dimensions, all abstract polyhedra are CW-complexes). At a foundational level, abstract theory is overtly set-theoretic in nature, so the two theories express themselves rather differently. In terms of the structures allowed, a CW-complex requires all cells (of any dimension) to be topologically simple, while abstract theory is more general in allowing toroidal and other non-simple cells (or j-faces). As it happens, planar faces are always simple so for 3D polyhedra this distinction is trivial. On the other hand CW-complexes are more general in that they do not require cells to be assembled into a higher polytope. I don't know much about CW-complexes but as I understand it, say an n-ball attached to a 0-dimensional CW-complex (0-complex) together comprise a valid n-complex, but this is certainly not a valid abstract polytope. Also, a CW-complex need not "fill in the gaps", for example a graph need not highlight any particular cycles as cells or faces. A skeletal polyhedron is abstractly "unfaithful" or incomplete but, understood as a graph, is a valid CW-complex in its own right. This can be significant, for example consider a skeletal regular icosahedron. Abstractly we may have identified triangular cycles as bounding 2-faces of a convex icosahedron, or we might have identified pentagonal cycles bounding a great dodecahedron: we have to have made the choice. But the CW-complex of the skeleton is sufficient to itself and does not need to choose. (One can of course make the choice anyway and construct a distinct, higher-dimensional CW-complex.) — Cheers, Steelpillow (Talk) 08:52, 2 October 2015 (UTC)

This is very interesting, thank you. I'm starting to understand you. 89.135.8.194 (talk) 06:46, 3 October 2015 (UTC)

## Duality

There has been a bit of editing back-and-forth about duality. I'm inclined to prefer David Eppstein's version (though actually I might even prefer something like this: "For every convex polyhedron there exists a dual polyhedron, having .... (Abstract polyhedra also have abstract polyhedral duals, with the same properties. However, some definitions of non-convex geometric polyhedra may not have duals meeting the same definition.)"). There is one minor thing that concerns me about it: at that stage of the article, the class of convex polyhedra has not been introduced yet. (Possibly this is an indication that convexity should be mentioned earlier.) --JBL (talk) 18:29, 11 February 2017 (UTC)

Oh, and I should mention: the center of this disagreement is the discussion here. --JBL (talk) 18:31, 11 February 2017 (UTC)

I will not enter further discussion with an editor who accuses me of lying and shouts at me in his edit comments. By rights I should be taking this straight to WP:ANI. I will say here that I have cited the case for nonconvex polyhedra in both articles - Wenninger's is a popular and well-regarded text from a reputable academic publisher. — Cheers, Steelpillow (Talk) 18:46, 11 February 2017 (UTC)
If you want to take David Eppstein to ANI, go for it, but it has nothing to do with choosing among a few options for how to word a sentence. --JBL (talk) 18:58, 11 February 2017 (UTC)
Yes, that's exactly the problem. — Cheers, Steelpillow (Talk) 19:01, 11 February 2017 (UTC)
I cannot decipher this comment. --JBL (talk) 19:12, 11 February 2017 (UTC)
[edit conflict] Since this response was not particularly informative, let me describe what I think Steelpillow's position is (he can obviously correct me if he would like to actually participate in the discussion rather than blustering). It is that the theory of abstract polyhedra provides a valid form of duality for almost all instances of what people call polyhedra (true) and therefore that all uses of the word "polyhedra" in our articles (unless otherwise qualified) should be assumed to mean abstract polyhedra. Under this interpretation, the sentence "all polyhedra have duals" is true, because what it really means is "all abtract polyhedra have duals". My own position, on the other hand, is that most readers are likely to come to the article with a naive conception of what it means to be a polyhedron (involving something embedded into Euclidean space with flat sides), and that the sentence "all polyhedra have duals" is likely to seriously mislead these readers into thinking that all non-convex Euclidean things with flat sides have dual Euclidean things with flat sides, something that generally isn't true. If we say things that we can reasonably expect to lead to false beliefs in our readers, we are lying to them. (This, by the way, is what Steelpillow thinks of as incivility: pointing out situations where what we write may cause readers to have false beliefs.) To avoid lying to the readers, I would prefer to qualify the statement by saying which kinds of polyhedra have duals: convex polyhedra have convex duals, and abstract polyhedra have abstract duals, but other kinds of polyhedra may not have duals within those other classes of polyhedra. —David Eppstein (talk) 19:02, 11 February 2017 (UTC)
This summary accords with my sense of things. May I ask specifically about your feelings about the alternate wording I proposed, and the question of whether the location of the discussion of convexity in the article is problematic for it? --JBL (talk) 19:12, 11 February 2017 (UTC)
It's a bit more cumbersome than my attempt but I have no objections to it. You're absolutely right about convexity not having been introduced yet; I think it would make sense to move the convexity section above "Characteristics", since it provides readers with an important class of examples to use in understanding the more technical parts of the Characteristics section. The same problem also happens earlier in Characteristics, in the "Topological characteristics" subsection, which mentions convex polyhedra without their having been introduced yet. So this ordering issue needs to be addressed regardless of what we decide about duality. —David Eppstein (talk) 19:18, 11 February 2017 (UTC)
Well, I made a second attempt but apparently now Steelpillow feels that the source Steelpillow added to support "all polyhedra" in fact supports "uniform polyhedra" but *not* "convex polyhedra." Sigh. --JBL (talk) 16:24, 12 February 2017 (UTC)

I have reported David at WP:ANI. Until that is resolved, I would be grateful if folks could regard this discussion as on hold. I need hardly add that he has grossly misrepresented my position. — Cheers, Steelpillow (Talk) 19:28, 11 February 2017 (UTC)

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

I think it's confusing to say without qualification that all polyhedra have duals. Intuitively, I'm guessing that all “sensible” polyhedra – with simple faces (or at least faces with simply-connected edges) and each edge joining precisely 2 faces – have duals, with the caveat that the polar reciprocal duals of non-convex polyhedra may be self-intersecting. If I'm guessing correctly (not sure), then maybe something like that would be clearer and/or more accurate (assuming it can be phrased better)? Κσυπ Cyp   23:29, 11 February 2017 (UTC)

As we already discussed on Talk:Dual polyhedron, see Figure 1 (right) of my paper Steinitz theorems for simple orthogonal polyhedra, a cube with a smaller cubical divot taken out of one edge. It has simple-polygon faces, each edge joining two faces, and no geometric dual, because there are two faces that share two edges and it is not possible to have two different geometric line segments between the corresponding two dual vertices. Like most of the other examples we're talking about, you can throw away the geometry to get an abstract polyhedron, which has a dual, but the dual is not a geometric polyhedron. —David Eppstein (talk) 01:17, 12 February 2017 (UTC)
Oh, sorry, I had misinterpreted that at a cubic hole in a face instead of in an edge, and hadn't seen the paper/figure itself. Maybe if also adding the constraint that two faces may only share one edge. Don't know, or maybe there are counterexamples to everything I can come up with… Either way, I think having an abstract dual which doesn't make sense geometrically is something that probably ought to be mentioned in some form. Κσυπ Cyp   10:48, 12 February 2017 (UTC)
Some people do indeed add a constraint like that, even for abstract polyhedra. See e.g. Whiteley, who discusses the stronger rule that, when faces share more than one vertex, there can be only two and they must share an edge that's also on both faces. One drawback to such rules is that unless you're careful you end up with only the same topological structures as convex polyhedra, just with different geometries (so as abstract polyhedra there would be no point to calling them non-convex). Anyway, I agree that we should mention the existence of non-convex geometric polyhedra with no non-convex geometric dual. That has been my point all along, and I don't see why it has become so controversial. —David Eppstein (talk) 18:35, 12 February 2017 (UTC)
I rather liked this version -- in particular, I feel like it separates the claim that the most common classes of polyhedra do have duals from the more subtle issues. I would set about finding supporting sources if you two agreed with using it as a base. It still needs convexity to be mentioned earlier, of course. I can't decide if just cutting and pasting the convexity section before this one is the right thing; maybe make convexity an (early) subsection of the big Characteristics section? --JBL (talk) 14:22, 13 February 2017 (UTC)
Double sharp, you've been active on this article and talk page as well; would you care to weigh in? --JBL (talk) 14:42, 13 February 2017 (UTC)
Here is a nice survey of Grunbaum and Shepard from 1969; on page 260 we find the statement that all [convex] polytopes have [convex] duals, as well as the statement that polar reciprocation provides the dual. (Of course Grunbaum's text also would serve as a source.) --JBL (talk) 14:39, 13 February 2017 (UTC)
May I suggest that the "what is a polyhedron?" definitional issue should be addressed here, but that the subsequent duality issue is best addressed where it began, at Talk:Dual polyhedron? It is not helpful to have parallel discussions on the same issue. Do folks have a problem with that or is there a better way to structure the content discussion, say keeping both issues on the same page? Now that the ANI issue is basically settled, I will return to these discussions once I know where they are taking place. — Cheers, Steelpillow (Talk) 16:51, 12 February 2017 (UTC)

### Proper citations

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

## reliable source

A relevant citation of a reliable source cannot be summarily removed based on editorial opinion. It needs to be discussed and consensus established first. If that is to be done then let us revert to an earlier version of this section and stop making unwarranted edits. — Cheers, Steelpillow (Talk) 15:14, 12 February 2017 (UTC)

Demanding that other people discuss while refusing to engage substantively is not constructive. David Eppstein has repeatedly asked you some very simple substantive questions; can you answer them? --JBL (talk) 16:17, 12 February 2017 (UTC)
A content dispute is one thing, but bad behaviour by editors is quite another. I am at present happy to engage on the second - on whether or not a relevant citation of a reliable source can be summarily dismissed, and if not then whether the content it supports may be allowed to contradict it. Are you willing to accede that all editors here should abide by WP:POLICY and edit per reliable sources and not per their personal opinions? Are you willing to acknowledge that a pertinent passage, supported by a reliable citation, should not be summarily deleted or perverted just because an editor disagrees with it but can themself produce no better citation? — Cheers, Steelpillow (Talk) 16:51, 12 February 2017 (UTC)
I will answer David once I know that he can assume good faith and keep a civil tongue in his head, there is no mileage in a slanging match. I will say to you that I think the "what is a polyhedron?" definitional issue should be addressed here, but that the duality issue is best addressed at Talk:Dual polyhedron. It is not helpful to have parallel discussions on the same issue. Do you have a problem with that or do you think there is a better way to structure the content discussion, say keeping both issues on the same page? — Cheers, Steelpillow (Talk) 16:51, 12 February 2017 (UTC)
This attempt at organizing the discussion, while laudable, still seems to miss the point: we should not insist on having only a single definition for a polyhedron and standardize on that one definition throughout our articles on the subject. To do so would violate WP:NPOV. Rather, to the extent that multiple competing definitions have been covered in reliable sources, we should describe them all here, and describe their dualities at the other article. —David Eppstein (talk) 17:29, 13 February 2017 (UTC)
Yes, that would need to be a part of the discussion - which definitions do we present and how? I offer my own view on that in a new discussion below. — Cheers, Steelpillow (Talk) 18:21, 13 February 2017 (UTC)

## Characteristics#Volume

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

Yes, the second section is certainly not trivial, something most readers would ignore. Tom Ruen (talk) 15:16, 13 February 2017 (UTC)
I see Steelpillow restored it on 2009/Dec/9 [2], and original anonymous editor added it on 2009/Nov/19 [3] with talk discussion here Talk:Polyhedron/archive2#section_added. Tom Ruen (talk) 15:23, 13 February 2017 (UTC)
I restored a wholesale delete by an IP editor in part because there was an ongoing discussion about it and deleting material wholesale while it is under discussion is seldom helpful. I have no opinion beyond my comments made back then. — Cheers, Steelpillow (Talk) 15:43, 13 February 2017 (UTC)
I was the one who removed it back then, moving a copy to the talk page. I'm not a deletionist at all and don't delete material wholesale when it is worthy of discussion. Tom Ruen (talk) 16:47, 13 February 2017 (UTC)
No worries, I was just going by my edit diff that you posted. — Cheers, Steelpillow (Talk) 16:59, 13 February 2017 (UTC)

## Definitions

Our differences over duality suffer from different ideas of what a polyhedron is.

The basis for definition, currently given in this article, focuses on two kinds of polyhedron, those arising in Elementary geometry (which are called variously elementary, geometric or traditional polyhedra) and those arising in abstract polytope theory. Any geometric polyhedron is said to be a realization of an associated abstract polyhedron. Some other definitions are mentioned in the sections on generalisations and alternative usages, but are otherwise excluded from the main discussion. David has suggested that more of them need bringing into the main discussion. I would disagree: Wikipedia needs an introductory article which gives no more than a glimpse of all those complexities, and WP:NPOV requires balancing for WP:DUE weight. If a more advanced blow-by-blow treatment is useful, then it should have its own article. Convex polyhedra also currently have their own section, being the only sub-class which has.

No explanation of the realization process or further definition of a geometric polyhedron is given here, save the remark that there are many such definitions bandied around.

The details of any such definition are critical in deriving the nature of a polyhedron's duality with other geometric figures. An obvious constraint here is that any definitions under consideration should arise within the context, of elementary geometry and abstract theory, that has been set for the rest of the article.

These definitions need reliable citations. Wikipedia prefers widespread secondary and tertiary overviews where possible, rather than primary research papers. Since this is an introductory article it is best to turn to such introductory overviews. Among such widespread introductions, Cromwell, Wenninger, Grünbaum and of course Coxeter stand out. We may not agree with everything they say, but that is how Wikipedia works: the sources speak, we editors keep our opinions to the talk pages.

As it happens, few sources - if any - give rigorous accounts of duality. This is the heart of the current problem. In the more detailed discussion at Talk:Dual polyhedron I intend to draw out an approach which can be traced through to Grünbaum's 21st century work and can therefore provide a way forward to acceptable and properly-cited content.

But there is little point in discussing the duality of polyhedra if we do not know what a polyhedron is. Therefore, my aim here is to build a definitional consensus so that we can present the mainstream definiton/s (such as Cromwell's) in this article. This can then provide a basis for a sensible discussion of duality.

Another suggestion I would make is to:

• Shorten the section on the general characteristics, by moving some of its content down
• Expand the section on convexity, which sorely needs it
• Add a section below it on non-convex and star polyhedra, among other things moving here the discussion on topological characteristics.
• Add any further sections which editors feel belong in such a basic introduction

Duality will probably need some mention in several sections, in particular to separate the convex and non-convex discussions.

— Cheers, Steelpillow (Talk) 18:22, 13 February 2017 (UTC)

My position is that pretending that "polyhedron" always means "abstract polyhedron" is false (because many authors actually mean other things), confusing (because readers will come to polyhedron articles with other meanings of polyhedra in mind and Steelpillow objects to even telling them in each instance that certain claims are about abstract polyhedra, e.g. see his insistance on the sentence "all polyhedra have duals"), misleading (because the readers with other conceptions of polyhedra will be led to believe false things such as the idea that non-convex geometric polyhedra always have geometric duals), a violation of WP:NPOV (which tells us to consider all significant viewpoints), and overly dogmatic (leading to declarations of "that's not a polyhedron" for any example that challenges orthodoxy).
As far as I can tell there are three major strains of definition of polyhedra, varying both in their level of abstraction (how much information about an object they convey) and generality (how broad a class of objects they can describe). This is a tradeoff and we should not ignore the loss of information in abstraction when aiming for greater generality. More general is not better, and should not be the sole criterion in situations where we might want to pick one definition instead of listing multiple definitions.
Some of these definitions and their subtypes are:
• Geometric polyhedra: all vertices are represented by points
• Convex Euclidean polyhedra
• Convex hyperbolic polyhedra (combinatorially the same as Euclidean but with different metric properties
• Convex spherical polyhedra (bounded by great-circle arcs on a sphere; differing from the above by including hosohedra)
• Polyhedra embedded in Euclidean space as manifolds (with multiple variations according to whether the faces are simple polygons, weakly simple, or polygons with holes, whether flat dihedrals are allowed, and whether the intersections of faces are restricted to single vertices or edges or whether they are unrestricted)
• Solids with connected interiors and flat sides, or possibly the boundaries of such solids. Probably the closest to the naive conception of a non-convex polyhedron that we might expect readers to come in with. Includes things that are not allowed by our current abstract polyhedron definition (and that some people would declare to be non-polyhedra) such as a polycube with two cubes that share only an edge (connected via other cubes).
• Self-crossing polyhedra (in which the vertices still are geometric points but now the faces can be any cyclic sequence of distinct coplanar vertices), with two faces/edge and possibly also restricted to a single face-edge cycle per vertex so that it forms a manifold
• Metric polyhedra — not as significant as geometric/topological/abstract but important in the context of Alexandrov's uniqueness theorem: polyhedra are specified by the metric space of geodesics on their surface. To be a polyhedron, this metric space should be locally Euclidean except at certain cone points where there is an angular defect. To be a convex polyhedron, all defects should be positive and the defect should sum to 4π. These are almost the same as (Euclidean) geometric convex polyhedra, but they come without a position in space and they necessarily include the doubly covered convex polygons (dihedra) which otherwise might not be considered to be polyhedra.
• Topological polyhedra — polyhedral subdivisions of topological manifolds. Usually here this means that the intersection of two faces can only be a vertex, edge, or empty. Convex polyhedra, geometric polyhedra embedded as manifolds, and self-crossing polyhedra with a single face-edge cycle at each vertex can all be represented topologically, but at the cost of losing the positions of their vertices and of conflating polyhedra that have distinct shapes geometrically into the same topological subdivision.
• Abstract polyhedra — partial orders describing the sub-object relation between vertices, edges, and faces (I won't say subset because I don't want to assume that everything is a set of points — that doesn't work so well for self-crossing polyhedra). The version we already describe make the restriction that every 1-section is a segment, which has the advantage that it allows the order of edges around each face and around each vertex to be recovered but the disadvantage that it cannot represent geometric embedded manifolds with weakly-simple faces. Whiteley suggests a version without this restriction, augmented by supplying the face and vertex ordering information separately. A more restrictive variation (especially for higher dimensions) is the Eulerian posets, but for polyhedra that's the same as the 1-section requirement. Again, some authors (see Whiteley) have considered tighter restrictions on the intersections between faces.
My preference would be to have a list such as this with a clear statement that there is no single universally-agreed-on definition of a polyhedron. Additionally, I would prefer that we use the adjective "non-convex" only to mean non-convex geometric polyhedra; topological and abstract polyhedra are neither convex nor non-convex, because there is no notion of convexity that applies to them. —David Eppstein (talk) 19:06, 13 February 2017 (UTC)
 Convex flat convex concave nonconvex Toroidal nonconvex Self-intersecting nonconvex Abstract nonconvex (Petrial cube with skew hexagonal faces)
I don't think convexity applies to spherical polyhedrons since they are really surface tilings, and so these can be grouped as finite tilings with the infinite euclidean and hyerbolic tilings.
But I'm still confused what non-convex does or should imply besides the obvious not convex. Coxeter used star polyhedron for intersecting geometry (self-crossings) with planar faces. And I'd say concave for non-intersecting flat-faced polyhedra are not identical to the convex hull. Of course a middle (non-concave) case might allow coplanar faces sharing a common edge, and some annoying definitions of polyhedra actually exclude calling these polyhedra at all, even if moving vertices infinitesimally outward in some cases would make them polyhedra. And of course nonplanar faces is another troublesome thing, clearly not useful in a convex polyhedron and again can be called non-convex. Finally we have the cases of polyhedra with simple faces, but topological handles, (F+V-E = χ≠2, like χ=0 for a single hole torus shape). So these are also concave by my definition, but unlike polyhedra that are topological spheres (χ=2), no adjustment to geometry can make them convex.
So almost all of these cases could be considered non-convex, and yet may need to be deal with somewhat differently. A topological-sphere but concave polyhedron's dual can be computed with the ordinary approaches, while I'm not sure what to do with a torus.
I just tested a nice 30x30 square tiling grid torus polyhedron in Stella (software) and its dual is a bit wild, a sort of hyperboloid/cone thing that is clearly not correct! Oops! Topologically, it should have been an offset square tiling grid on the same torus surface! Tom Ruen (talk) 20:37, 13 February 2017 (UTC)
By the way, I don't understand how links to elementary geometry can be used (as they have been above) as a justification for favoring the abstract polytope view. There is nothing in our geometry article (the target of the "elementary geometry" redirect), nor in Euclid's elements (the most natural meaning of the term "elementary") that can reasonably be interpreted as telling us to throw away the geometric positions of polyhedra and treat them purely as abstract incidence structures. And the only article I can find with "elementary geometry" in its actual title, list of formulas in elementary geometry, is about metric properties that do not make sense for abstract polyhedra. One may reasonably read "elementary" as meaning something different, that one should decompose a polyhedron into its elements (vertices, edges, and faces), and study the incidence relations between those elements, but the proper link for that point of view is incidence geometry. —David Eppstein (talk) 02:15, 14 February 2017 (UTC)
There is a Category:Elementary geometry listing some articles which other editors see as relevant. Personally I think it may not be the best term to use here, but it has been in the article a good while. Perhaps Euclidean geometry would be more sensible. The link between geometric and abstract polytopes has been brought out by actual definition of a geometric polyhedron in terms of the realization of the associated abstract poset. This is made clear often enough in works on abstract theory and some of its consequences are explored by Grünbaum in "'New' uniform polyhedra", "Are Your Polyhedra the Same as My Polyhedra?" and "Graphs of Polyhedra: Polyhedra as Graphs" (let me know if anybody needs publication details for this discussion). As Grünbaum remarks in the last of these, "in order to achieve the desiderata mentioned above, nontraditional "polygons" and "polyhedra" need to be admitted. On the other hand, once the initial discomfort wears off, it will be seen that the present point of view provides a very satisfactory solution to various situations and questions." From this perspective, by definition "all polyhedra are [realizations of] abstract polyhedra", even if they might be unfaithfully realized (as some of David's examples are. Again, Grünbaum has long studied and written about such difficult examples. Interestingly, I note that David's paper, which he has cited a couple of times in these discussions for at least one of these examples, does not define the "polyhedra" which it is addressing - a classic example of what Grünbaum called the "original sin". Quite how he can then use this non-existent definition against me remains to be explained). One gets the feeling that David's "initial discomfort" is yet to wear off. David's other concerns are already largely addressed: there is already "a clear statement that there is no single universally-agreed-on definition of a polyhedron", in fact there is even a general discussion of this issue. There are also two lists of such variations. If a treatment of "metric polyhedra" is useful then it can easily be included in the scheme I outlined above. So I am not sure what else David is wanting to change in what is a basic introductory article. Tom brings up the different sub-classes of non-convexity. I think that the depth of the treatment here needs to be appropriate to an introductory article, for example identifying the various classes and their relative importance per WP:DUE. It might be worth mentioning the relationship between concave polyhedra and star domains. The duality issues are best left to brief mention in the section here on duality and expanded on in the dual polyhedron article. — Cheers, Steelpillow (Talk) 08:13, 14 February 2017 (UTC)
My paper is not a useful reference to this topic, merely a convenient source of examples. And your rhetoric about using things against you is best ignored; see WP:BATTLEGROUND. But your claim that the paper does not define the polyhedra it addresses is mistaken. The definition starts at the bottom of page 1: they have the topology of an (embedded) sphere, (flat) simple polygons as faces, and three perpendicular edges at each vertex. So the definition is merely a special case of the embedded-manifold definition that I listed above under geometric polyhedra. The words embedded and flat are omitted from that part of the actual paper, because for the audience of the paper, they can be safely assumed to be understood. Apparently the audience of Wikipedia editors requires a different standard. Anyway, no, our polyhedron article does not clearly state that there is no standard definition of a polyhedron. In fact, it starts with a clear statement that a polyhedron is something specific: a solid with flat sides (the definition I labeled above as "closest to the naive conception"). "Basis for definition" then states that the faces are "polygons – regions of planes", which likewise does not make sense for self-crossing polyhedra (their faces are not regions). And the current article nowhere provides a taxonomy of definitions like the one I laid out above, rather it has a waffly paragraph about how definitions are difficult and then gives primacy of place to abstract polyhedra and their realizations. —David Eppstein (talk) 08:34, 14 February 2017 (UTC)

### A few textbook definitions

I'm not sure how this should relate to the content of the article, but I thought I would leave it here to save others looking it up. There is a formal definition of polyhedra in section 4.1 of the textbook O'Rourke, Joseph (1993), Computational Geometry in C, Cambridge University Press, pp. 113–116. In brief, it is that they are the subsets of Euclidean space that can be represented as the unions of finitely many convex polygons, with each two polygons intersecting in a vertex, edge, or empty set, with each point having a neighborhood topologically equivalent to a disk, and with the whole set connected. (That is, they are piecewise linearly embedded connected manifolds). This is obviously quite restrictive; it doesn't allow the faces to be non-convex simple polygons, let alone allowing self-crossing polyhedra. However, since O'Rourke is more concerned with a polyhedron as a set of points (or as the set of points it encloses) rather than the combinatorial structure of its faces, it's adequate for his purposes. I don't think we can claim that this is the one correct definition of a polyhedron, but it illustrates the diversity of definitions that have been used in this area.

I happen to have in my office a much older textbook, McCormack, Joseph P. (1931), Solid Geometry, D. Appleton-Century Company. Its definition of a polyhedron (p. 416) is much less satisfactorily rigorous: "A solid may be defined as any portion of space completely enclosed. ... A polyhedron is a closed solid bounded by portions of planes." But if we add the reasonable assumption that there should only be finitely many of these planes, it's possible to infer a usable definition from this: that a polyhedron is the union of some of the bounded closed cells of an arrangement of finitely many planes. This doesn't tell us what its vertices, faces and edges are, but one could obtain this by choosing the arrangement to be minimal (that is, only use the planes necessary to define the solid) and then using the vertices, edges and faces of the arrangement that belong to both chosen and unchosen cells. Because this definition is based on solids rather than on boundaries, it doesn't make sense for self-crossing polyhedra. One of the standard computational geometry texts, de Berg, M.; van Kreveld, M.; Overmars, M.; Schwarzkopf, O. (2000), Computational Geometry (2nd ed.), Springer is no better; when it defines polyhedra (in a chapter on a problem taking them as input, p.64) it does so as "a 3-dimensional solid bounded by planar facets".

My copy of Richeson, David S. (2008), Euler's Gem, Princeton University Press (suggested as a source in some earlier comment by Steelpillow) is a review copy but I think it's identical to the published version. On p.28 he writes "there is no single definition of polyhedron that applies to the massive body of literature on these mathematical objects", precisely the point of view I have been advocating here. He then avoids the issue of providing a precise definition by restricting his attention for the earlier parts of the book to convex polyhedra, and then shifting to subdivisions of topological surfaces in the later parts. Kepler–Poinsot polyhedra are mentioned briefly later but with no hint of definitional issues.

In Wenninger, Magnus J. (1971), Polyhedron Models, Cambridge University Press, the only formal definition of a polyhedron is on p.1, which defines a polygon as "a set of line segments enclosing a portion of two-dimensional space" and a polyhedron as "a set of [polygons] enclosing a portion of three-dimensional space". This looks superficially like the definition of McCormack and the four Marks, but because it attempts to describe polyhedra as boundaries rather than solids it is much more problematic. It neglects to specify that there are finitely many line segments or polygons, neglects to specify that they must form a manifold, and neglects to specify that the enclosed portion of the plane or of space must be connected. Wenninger later provides models of self-crossing polyhedra so perhaps "enclosing a portion" should be taken as allowing self-crossings? It's not clear.

I couldn't find an example of a text based on the "realization of an abstract polyhedron" point of view favored by Steelpillow, but I don't have Cromwell's book so maybe that's where one can find it. Still, it doesn't seem to be the view of even a preponderance of recent texts, let alone something universal. I hope also that the difference between O'Rourke and the four Marks lays to rest the idea that computational geometers have somehow settled on their own idiosyncratic definition; I think we're as confused on this issue as everyone else. —David Eppstein (talk) 02:16, 18 February 2017 (UTC)

A few minor remarks: first, thanks for this. Second, of course there are some classes of polyhedra that do have standard universally agreed-upon definitions, notably the convex polyhedra. (And people working in the convex setting often use "polyhedron" to mean "convex polyhedron".) This is one reason I am in favor of making/keeping them central to this article (of course, while still giving fair mention to more general classes). (This is a pattern that seems to reproduce in other related places: for example, there is an accepted definition of a regular complex polytope, even though there is no widely accepted definition of a complex polytope.) Third, "the four Marks" = "de Berg et al."? Finally, given this variety, I feel like it is infeasible to include all well-sourced definitions in this article, but it's great to have these quotes side-by-side for purposes of trying to write a decent summary of the situation. --JBL (talk)
Sorry, yes, inside joke. The four Marks are Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf (now Otfried Cheong). —David Eppstein (talk) 02:46, 18 February 2017 (UTC)
Ok, thanks. (Here in Minnesota we have a paper by the three Dennis's, Dennis White, Dennis Stanton, and Vic Reiner.) --JBL (talk) 02:49, 18 February 2017 (UTC)
On the subject of the content of the article: I would describe the section "Basis of Definition" as making a pretty good stab at addressing this issue (despite its very awkward title). Are there things about the section that make you unhappy? Or is it how the definitional issue flows into the rest of the article? --JBL (talk) 22:47, 18 February 2017 (UTC)
That section has:
• A well-sourced paragraph primarily about how definitions have been difficult and problematic, with very brief mentions of solid-based and surface-based definitions;
• Three paragraphs (counting the bulleted list as one) about the approach based on abstract polyhedra and their realizations, which nevertheless completely fail to adequately define a realization
• Two short paragraphs that could be merged about higher dimensional and topological generalizations.
My preference would be an organization like the following:
• A paragraph much like the existing first one, about how definitions have been problematic and have frequently failed to cover important cases such as star polyhedra. Add a point that is missing from our current paragraph, that there is still no universal agreement on a single definition that covers all important cases.
• A paragraph or bulleted list much like the existing one about how most definitions allow one to identify the polyhedron's features as being vertices, edges, and faces, but noting that the definitions disagree about what kinds of objects these are and about how they can be placed in space with respect to each other. Do not tie this part to the abstract polyhedron part, as it is now, because it is true for all the types of definitions.
• A paragraph about the solid-based definitions: it's an interior-connected and bounded solid whose boundary is a subset of finitely many planes. We can use McCormack and de Berg et al. as sources here. Mention that this fails to cover star polyhedra, and (if we can find adequate sources for this) that this definition may produce faces that some authors would not recognize as polygons (for instance, because they are not topological disks).
• A paragraph about the surface-based definitions: it's a piecewise-linearly embedded connected manifold, perhaps with faces defined from the maximal linear pieces and perhaps with a specified subdivision into faces, and perhaps with some additional restrictions on the face shapes or on how faces can intersect. We can use O'Rourke as a source here. Mention (if we can adequately source this) that this definition is more restrictive than the solid-based definitions, because not all solids have manifold boundaries. Maybe also mention the disagreement over whether two adjacent faces can be on the same plane. We can also include here the purely topological definitions (it's a subdivision of a topological manifold).
• A paragraph about geometric star polyhedra: collections of self-crossing polygons such that each polygon edge is also an edge of another polygon. Mention that, for this type of definition, it does not make sense to think of a polygon or polyhedron as enclosing a set of points or being defined by a set of points that it encloses; instead, it is defined by the lower-dimensional objects (edges or polygons) out of which it is formed.
• A paragraph about how these can all be unified as instances of abstract polyhedra, partial orders describing the inclusion hierarchy of edges, vertices, and faces. Mention the restrictions that each edge have two faces and that each vertex belong to exactly two edges of each face that contains it (these are the 1-section=segment restrictions in more intuitive language) and state explicitly that this restriction disallows some flat-sided solids. Mention the additional restrictions necessary to ensure that the abstract polyhedron is a manifold (namely that each face is incident to a simple cycle of vertices and edges, and each vertex is incident to a simple cycle of faces and edges) and state that with these restrictions abstract polyhedra carry exactly the same information as topological polyhedra. Mention that geometric polyhedra can be defined as realizations of abstract polyhedra but also state that the definition of a realization has been problematic.
I think this would be more well-balanced than the current approach, which essentially focuses on a single style of definition, pretends that realizations are an adequate definition for geometric polyhedra, and treats the other styles only briefly and dismissively. —David Eppstein (talk) 23:28, 18 February 2017 (UTC)
This sounds like a great outline to me (although, as usual, I will say that I would like more mention of convexity). Possibly it is actually more than one section's-worth of material? (But this would depend a lot on how exactly it is written.) --JBL (talk) 00:11, 19 February 2017 (UTC)
Ok, I've made an attempt at a rewrite along these lines. —David Eppstein (talk) 01:01, 21 February 2017 (UTC)
I spent all day yesterday teaching and will spend all day today grading; I looked it over superficially and at a broad scale it looked clear and thorough, and I hope to take a closer look later this week. --JBL (talk) 13:25, 21 February 2017 (UTC)

## Realizations

Branching off from the above discussion, since I hope this can be less controversial: our article mentions realizations of abstract polyhedra without saying what they are (other than that they are geometric polytopes that give rise to the same abstract polytopes, which appears kind of circular if we are taking the position that the meaning of a geometric polytope is that it is a realization of an abstract polytope). Our abstract polytope article, besides spelling realization inconsistently, isn't much better — it says it's "a mapping or injection of the abstract object into a real space", but a mapping of which elements of the abstract object into which kind of things in a real space? Lacking anything better, in a recent blog post http://11011110.livejournal.com/341698.html I chose a more specific meaning that it's a function from the vertices of the abstract polyhedron (or actually a topological polyhedron but it makes no difference) to distinct points in Euclidean space with the property that each face's vertices are mapped to distinct planes. I didn't specify what happens to edges because it wasn't necessary to do so for the polyhedra I was posting about, but probably more generally one would want to add extra restrictions such as that segments on the same line are disjoint. But that was a blog post — we shouldn't be so arbitrary here. Is this material treated properly in a reliable source, and if so what does that source say about how to define realizations? Alternatively (and I hope this is not the case because this seems like a useful topic to include) if it's not properly covered in a reliable source, maybe we should remove it? —David Eppstein (talk) 07:54, 15 February 2017 (UTC) o

This has been a bugbear of abstract theory since its inception. The statement of the basic principle of realization has been made often enough, but (to my knowledge) not yet rigorously pursued. In effect, all the different definitions of "polyhedron" give rise to different classes, each of which has different rules governing their realization. To take one example, are multiple elements such as vertices or faces allowed to coincide? Grünbaum has long argued that they should be, however he concedes that they are then (geometrically) degenerate. Such figures are said to be "unfaithful" realizations. Some definitions of as polyhedron treat a face as a plane region, others as an ordered sequence of vertices, others as a cycle of incident edges. What is degenerate under one definition may not be so under another. Johnson had originally planned some discussion of these issues in his forthcoming book (Due out at last on this coming 30 April [4]), but that later changed its focus from uniform polytopes to Geometries and Transformations so it will be interesting to see if anything of that discussion remains. Johnson also regarded my collage of a newsgroup discussion as accurate and useful, but that is no more acceptable here than your own blog post. (@Tomruen: I think you have a more recent draft than I do, can you say what will be in the book? Now that we have a firm and imminent publication date I am much happier with your citing from it). Meanwhile the reliable sources leave us with the bald statement and the three pieces by Grünbaum that I referred to earlier but, to my knowledge, little or nothing else as yet. An unhappy state of affairs, but I would suggest that the greatest polyhedronist of the latter twentieth century cannot simply be passed by, we have to do him what justice Wikipedia allows itself, so I don't think that plain deletion is the answer. One way ahead might be to move all mention of abstract theory and realization to a subsection of its own (which already exists in embryonic form) and to focus the main line of discussion around the traditional geometric treatments from Euclid to Cromwell. Mathematically this is technically doing it backwards, in going from the particular to the general, but for the novice reader it would be more intelligible. — Cheers, Steelpillow (Talk) 10:28, 15 February 2017 (UTC)
Johnson's book talks more about polytopes in relation to symmetry than the reverse. There are abounding definitions, all matter-of-factly, no real philosphical discussions of the sort Grünbaum enjoys. Tom Ruen (talk) 11:18, 15 February 2017 (UTC)
EXAMPLE: 11.1 Polytopes and Honeycombs: A polygon or a polyhedron is a two- or three-dimensional instance of a polytope, a geometric figure consisting of points, line segments, planar regions, etc., having a particular hierarchical structure. When realized in some Euclidean or non-Euclidean space, a polytope also has certain metric properties, such as edge lengths and (dihedral) angles. A partition of a line or a circle into segments or arcs or a tessellation of a plane or a sphere by polygons joined edge to edge is a one- or two-dimensional example of a honeycomb, a kind of degenerate polytope. ...
So what I'm reading from your responses is that our article is incorrect when it claims that "One modern approach treats a geometric polyhedron as an injection into real space, a realisation, of some abstract polyhedron". Because actually it goes in the other direction — geometric polyhedra are defined in some other way, as collections of points, line segments, and self-crossing polygons with certain connectivity requirements, and then realizations are defined as geometric polyhedra that have the same connectivity as a given abstract polyhedra. Is that accurate? Also, can we agree on whether to spell it realization (US spelling, currently used once in this article) or realisation (UK spelling, currently used twice)? Unfortunately I don't think this topic has enough strong national ties to make the choice of spelling obvious. —David Eppstein (talk) 19:24, 15 February 2017 (UTC)
Your reading is not accurate. I am saying that the claim in the article which you quote is indeed correct but is hard for novices to grasp (especially since realization is so poorly defined). If we present them in the article (which is very different from ordering a hierarchy of definitions) the other way round by focusing mainly on geometric polyhedra (perhaps as defined in the leading introductory text book), then that might be more intelligible. You have argued that abstract theory has been given undue weight here, and I agree to a fair extent. You have wanted to fix that, so this is one suggestion for a way ahead. I'd also suggest that American spelling should be used throughout.— Cheers, Steelpillow (Talk) 20:05, 15 February 2017 (UTC)
Without a non-circular definition of a realization, it is hard for non-novices to grasp also. So, if you are asserting that claim to be accurate, then for consistency I hope you agree that our current article's sentence "Any geometric polyhedron is then said to be a "realization" in real space of the abstract poset" is inadequate and circular? The article also separately says that realizations are injections of the abstract into a geometric space, which looks like the start of a non-circular definition but is still not fully usable without knowing what kinds of objects are being mapped to what kinds of objects, and what the one-to-one requirement of an injection is supposed to mean. —David Eppstein (talk) 20:52, 15 February 2017 (UTC)
It seems to me that "One modern approach treats a geometric polyhedron as an injection into real space, a realisation, of some abstract polyhedron" and "Any geometric polyhedron is then said to be a "realization" in real space of the abstract poset" are saying the same thing: in both cases the geometric is defined in terms of the abstract and its realization. I see no circularity there, so I am not sure where/how you do. I'd agree that your other criticisms are entirely justified. — Cheers, Steelpillow (Talk) 21:09, 15 February 2017 (UTC)
The sentence "Any geometric polyhedron is then said to be a "realization" in real space of the abstract poset" appears to be a definition of a realization (as the correspondence between a geometric polyhedron defined in some other way and its abstract form) but you already said that's not what it means (your reply above that my interpretation was inaccurate). The only remaining meaning I can see is circular: a geometric polyhedron, defined elsewhere as a realization, is said to be a realization. Can you suggest an alternative wording that avoids the appearance that this is either a definition of a realization or a tautology? In particular, if it's intended to be a definition of a geometric polyhedron, it needs to be rephrased to look like one. —David Eppstein (talk) 21:36, 15 February 2017 (UTC)
I see your point, the sentence is ambiguous and can be read with either sense of "is said to be". How about, "Any geometric polyhedron is then defined in terms of some "realization" in real space of the abstract poset"? — Cheers, Steelpillow (Talk) 10:13, 16 February 2017 (UTC)
There are still two problems with that. First, and more importantly, it's not true unless we have an actual definition of a realization, and not just a vague handwaving map from abstract to real. Second, it's too strongly stated, basically asserting that that is the one correct way to define geometric polyhedra. "may be defined" instead of "is defined" would be better in that respect. —David Eppstein (talk) 17:13, 17 February 2017 (UTC)
I don't see "is defined" used in this context now. Within the subsection on abstract polyhedra, "is then said to be" would appear suitably contextualised by the word "then". "May then be said to be" is just a horrible use of language. — Cheers, Steelpillow (Talk) 17:44, 17 February 2017 (UTC) Or, better still, use the same phrase I suggested above (now done). — Cheers, Steelpillow (Talk) 18:03, 17 February 2017 (UTC)

## Duality and citation

Here is some content which has been repeatedly undone:

When applied to polyhedra, the principle of duality states that there exists a dual figure having

• faces in place of the original vertices and
• vertices in place of the original faces.

A polyhedron created in this way is dual to the original.[1]

Dual polyhedra exist in pairs. The dual of a dual is just the original polyhedron again. Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron.[1]

The dual of a convex polyhedron can be obtained by the process of polar reciprocation.[2][3]

The dual of a uniform polyhedron can also be obtained by the process of polar reciprocation in a concentric sphere. However, using this construction, in some cases the reciprocal figure is not a proper polyhedron.[4] In such cases a dual polyhedron may be constructed, at the expense of high symmetry, by moving the sphere appropriately off-centre.[5]:469-470

Abstract polyhedra also have duals, which satisfy in addition that they have the same Euler characteristic and orientability as the initial polyhedron. However, for some definitions of non-convex geometric polyhedra, the dual does not meet the same definition.[citation needed][clarification needed]

Note that this text does not state that "all polyhedra have a dual", a statement which some editors are unhappy with. Specifically, it does not assert that a dual figure is necessarily a polyhedron.

I have said above (and now move here for better visibility) that:

It is not acceptable to delete properly-cited content because it conflicts with an editor's personal point of view, see WP:NPOV. Nor is is acceptable to judge reputable, peer-reviewed, mainstream and widely-referenced textbooks and papers as unreliable for the same reason. See for example WP:NOTTRUTH. By all means add further comment in the interests of WP:NPOV, with appropriate sources, if it is relevant, but keep in mind that this is not the main article. — Cheers, Steelpillow (Talk) 12:54, 17 February 2017 (UTC)

The sources in question are Cundy & Rollett (Oxford University Press), Wenninger's Dual Models (Cambridge University Press) and Grünbaum 2003 (Discrete and Computational Geometry, Springer). These are all reputable peer-reviewed publishing channels: the first two contain much secondary and tertiary material, which is preferable to WP:PRIMARY sources where it is available. They all tell a consistent story, further endorsed by Grünbaum in 2007 (Polyhedra as graphs, graphs as polyhedra) where, on page 452 he explains his motivation in that "Most [geometers] would also wish to be able to associate with each polyhedron a dual polyhedron." All in all, anybody who wishes to discard the work of the leading polyhedronist of the late twentieth century on the basis that it is unreliable, has a great deal of explaining and consensus-building to do here first. — Cheers, Steelpillow (Talk) 13:13, 17 February 2017 (UTC)

(In the above, Grünbaum 2003 is "Are Your Polyhedra the Same as My Polyhedra?")

Can any of the reverting editors, or anybody else, please explain why any of the following is wrong:

1. This is an introductory article to the topic of polyhedra.
2. WP:PSTS requires that we give primacy to secondary and tertiary sources over primary, where possible.
3. WP:NPOV requires that we give WP:DUE weight to each PoV where several exist.
4. Cundy & Rollett and Wenninger are two mainstream introductory texts published under proper peer review and are therefore reliable sources for the present use.
5. Grünbaum's papers give further endorsement and clarification to their PoV, and are also RS.
6. This PoV should therefore be represented in the article.

— Cheers, Steelpillow (Talk) 12:03, 18 February 2017 (UTC)

References

1. ^ a b Cundy & Rollett; Mathematical Models, OUP, 1961, Pages 78-79.
2. ^ B. Grünbaum and G. C. Shepard, Convex Polytopes. Bull. London Math. Soc. 1 (1969). Page 260.
3. ^ B. Grünbaum and G. C. Shepard, Convex Polytopes. Bull. London Math. Soc. 1 (1969). Page 260.
4. ^ Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 0-521-54325-8, MR 0730208
5. ^ Grünbaum 2003
All your linking to policies and guidelines is pure blather and that's why it's getting ignored. To the extent that you have raised a content question (minimal, given your failure to engage constructively with other editors), the answer is already fully contained in this week-old response of David Eppstein -- your preferred phrasing is misleading to readers in exactly the same way that the earlier version was. Adding new undefined words ("figure") does not resolve anything. --JBL (talk) 12:49, 18 February 2017 (UTC)
Also, the phrase "uniform polyhedron" is first mentioned in section 4 of the article -- to include them in this section would require an earlier discussion of the uniform polyhedra, which is a bad idea. --JBL (talk) 12:53, 18 February 2017 (UTC)
Is it really necessary to define a "figure" in a geometry article? Would you prefer say "construction? There is a section on the uniform polyhdra and their duals, it seems more sensible to move related content there than to delete it. Here is Eppstein's post which you link to (shorn of its ad hominem digressions):

Let me describe what I think Steelpillow's position is. It is that the theory of abstract polyhedra provides a valid form of duality for almost all instances of what people call polyhedra (true) and therefore that all uses of the word "polyhedra" in our articles (unless otherwise qualified) should be assumed to mean abstract polyhedra. Under this interpretation, the sentence "all polyhedra have duals" is true, because what it really means is "all abtract polyhedra have duals". My own position, on the other hand, is that most readers are likely to come to the article with a naive conception of what it means to be a polyhedron (involving something embedded into Euclidean space with flat sides), and that the sentence "all polyhedra have duals" is likely to seriously mislead these readers into thinking that all non-convex Euclidean things with flat sides have dual Euclidean things with flat sides, something that generally isn't true. I would prefer to qualify the statement by saying which kinds of polyhedra have duals: convex polyhedra have convex duals, and abstract polyhedra have abstract duals, but other kinds of polyhedra may not have duals within those other classes of polyhedra. —David Eppstein (talk) 19:02, 11 February 2017 (UTC)

This misunderstands my position. I do not assert the supposed inference at all, any more than the cited sources do, and my proposed content makes no such assumption. The phrase "all polyhedra have duals" does not occur in the content under discussion here, so that part of Eppstein's reply is not relevant either.
So, while I thank you for attempting an answer, I am afraid that it is no answer at all and the justification for deletion remains unexplained. It is interesting too that you brush off Wikipedia's content policy and guideline issues here. May I take it that you regard the truth of the matter as more important here? — Cheers, Steelpillow (Talk) 19:51, 18 February 2017 (UTC)
It is completely unnecessary to quote something linked that also appears just above, and it's extremely poor form to post an edited form of someone else's words without their permission. I strongly suggest that you remove the quote.
It is difficult to know how to respond to you substantively, because your answer does not even hint at addressing the key sentence of my previous post: " your preferred phrasing is misleading to readers in exactly the same way that the earlier version was. Adding new undefined words ("figure") does not resolve anything.". Simply declaring things irrelevant does not make them so. --JBL (talk) 22:09, 18 February 2017 (UTC)
To claim that an editor does not reference a comment when they have expressly asked for clarification on it (Is it really necessary to define a "figure" in a geometry article?) seems a tad careless - merely reasserting your claim, as you so acutely observe, does not explain it. Perhaps you could oblige? The point about the reciprocal construction is that it is verifiable from multiple sources and so whether anybody believes it or not is not relevant to Wikipedia. Perhaps you could address that point too? Your focus on your perceived truth instead of verifiability is not in line with policy - I remind you once again of WP:VERIFY and WP:NOTTRUTH. — Cheers, Steelpillow (Talk) 10:52, 21 February 2017 (UTC)
As a result of your refusal to engage in an appropriate, constructive way, I have no further interest in discussing this with you. --JBL (talk) 13:21, 21 February 2017 (UTC)
I have asked you twice for clarification of your comment on the use of "figure" and you have refused twice to give any. I have twice pointed you at our content policies and you have ignored them, too. All you offer is ad hominem wikilawyering. Yet you claim that I am the one being obstructive. OK, perhaps it is best if you keep out of this one. — Cheers, Steelpillow (Talk) 17:50, 21 February 2017 (UTC)

Elsewhere, criticisms of the universality of dual polyhedra have focused around counter-examples, one being a dual with coincident edges and another being a dual which extends to infinity. Both of these are disposed of by Grünbaum. The first arises from his remarks on the metamorphoses of polygons and polyhedra, in which overlapping intermediate forms exist and cannot sensibly be dismissed. In recent years he has amplified the theory using the idea of an unfaithful realization of the associated abstract polyhedron. That is to say, where elements superimpose the geometric polyhedron is still a realization, just not a faithful one and we just have to accept such things if we wish to be consistent - this is his expressed view in the cited source, I should perhaps emphasise, and not just mine. The extension to infinity observed in certain examples of polar reciprocation may be avoided by moving the reciprocating sphere off-centre. He states this explicitly in the cited source, again there is no OR in my paraphrasing of him. Are there any other criticisms which Grünbaum's model does not address? — Cheers, Steelpillow (Talk) 19:25, 23 February 2017 (UTC)

It is unambiguously true (that is, it is a mathematical fact, not an opinion-based criticism) that for some of the sourced definitions in our article, there is no dual that obeys the same definition. In particular this is true for the definition as a bounded solid with connected interior and connected boundary that is a subset of the union of finitely many planes (example: the cube with a dent taken out of one of its edges). It is also true for polarity of convex polyhedra (when the polyhedron is allowed to have the origin as one of its boundary points rather than requiring it to contain the origin). It may or may not also be true for embedded connected closed manifolds in which all faces are simple polygons and the nonempty intersection of two faces can only be a vertex or a single edge (I'm not aware of a counterexample but I'm also not aware of a proof that all of these have duals that meet the same definition). The existence of tweaks to the definition of a polyhedron, unrelated alternative definitions of a polyhedron, or perturbations of the polarity that allow some meaning of a dual to be recovered in some of those cases does not change this truth. —David Eppstein (talk) 20:29, 23 February 2017 (UTC)
It is certainly wrong to claim that in all circumstances every polyhedron has a dual polyhedron. That should not stop us mentioning those circumstances where reliable sources say that it is true. The dual to the nibbled cube has two superimposed edges and in Grünbaum's model there is no problem with that. Skilling's uniform star provides an example with superimposed vertices. So while some definitions do not allow these figures, others do. The point about reciprocity and the sphere centre hinges on whether a claim is exclusive: for any finite polyhedron the sphere may be arbitrarily positioned to yield either a finite or an infinite dual construction, so the emphasis is purely one of custom and presentation. There is also an issue over use of the term "dual". In projective geometry, polarity arises as a consequence of projective duality, which is universal. Few dual constructions in projective geometry are actually dual polyhedra and the term "dual" does not imply that the dual construction is a polyhedron. The reciprocity of polyhedra in Euclidean space differs subtly from this, but it is close enough that if we are to restrict "dual" in this context to mean a dual polyhedron, we need to say so. Otherwise, to claim in a certain situation that there is "no dual" risks being misunderstood as meaning that no reciprocal construction of any kind exists. — Cheers, Steelpillow (Talk) 11:54, 25 February 2017 (UTC)
There is no need to go to Skilling's figure to get that problem; anything with coplanar faces, like the small snub icosicosidodecahedron, will have a geometric dual (constructed the usual way via polar reciprocation) with coincident vertices. I think David Eppstein has already phrased it very well: as I read his comment, he is not claiming that there is "no dual", but that for some definitions of polyhedron it is possible to find a polyhedron such that the dual does not fit that definition. Double sharp (talk) 13:49, 25 February 2017 (UTC)
Specifically, the current article phrasing (after descriptions of the duals for convex and abstract polyhedra) is "For some definitions of non-convex geometric polyhedra, there exist polyhedra whose abstract duals cannot be realized as geometric polyhedra under the same definition." Do you (SP) disagree with this statement or are you just arguing out of inertia? —David Eppstein (talk) 17:00, 25 February 2017 (UTC)
Yes, that is all fine. My point is that for other definitions a dual will always exist and that needs to be in the article too. — Cheers, Steelpillow (Talk) 18:31, 25 February 2017 (UTC)

## Reliability of Cromwell

BTW, since some readers here seem to believe that Cromwell's Polyhedra is the bible of polyhedra, and should be used in preference to all other sources, it may be worth repeating what I found on Peter McMullen's MathSciNet review, MR1458063:

Which is not to say that it's a bad source by any means, just that (like any other source) its author is human, and we should not be uncritical in our reading of it. —David Eppstein (talk) 01:18, 21 February 2017 (UTC)

Thank you. If I meet one of those Cromwell evangelists I will let them know. The key thing about criticism is that it needs to be backed up by a reliable source of that criticism, as you have done here. However, I am unclear as to the relevance of the above criticisms to the present article? WP:PSTS obliges us to take notice of such a widely-read book and the suggestion that we look to Cromwell as a starting point seems a reasonable one. — Cheers, Steelpillow (Talk) 11:01, 21 February 2017 (UTC)

I just ran into one of these myself, sourcing a new article Dehn invariant. On p. 48 Cromwell writes "the key to Dehn's proof is to associate a number (now called the Dehn invariant) to each polyhedron". To call these things numbers is highly misleading and suggests a very shallow understanding of the subject. —David Eppstein (talk) 05:31, 28 February 2017 (UTC)

## Skeletal polyhedra versus polyhedra as graphs

Under "Generalisations of polyhedra" we have a section Polyhedron#Hollow-faced or skeletal polyhedra, while under "Alternative usages" we have a section Polyhedron#Polyhedra as graphs. Both refer to the idea of disregarding to some degree the faces of polyhedra. Is there a reason not to merge these two sections? If not, are there strong feelings about which of the two locations is better? --JBL (talk) 20:31, 4 March 2017 (UTC)

I'd support graphs as a subsection to the first. Tom Ruen (talk) 14:23, 5 March 2017 (UTC)
Ok, I have merged the two. --JBL (talk) 21:17, 7 March 2017 (UTC)

## Merge Polynomial-time algorithm for approximating the volume of convex bodies?

I found this odd-ball article Polynomial-time algorithm for approximating the volume of convex bodies that doesn't seem like it deserves it's own page. Maybe it ought to be merged here? Jason Quinn (talk) 11:03, 1 August 2017 (UTC)

It's a significant topic in geometric algorithms, worthy of its own article. Also, these algorithms work even for smooth convex bodies, so polyhedron is a bad merge target. —David Eppstein (talk) 15:59, 1 August 2017 (UTC)

## History

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

## cube

Why doesn't the illustration at the top of the page have the best-known solid ?5.34.81.117 (talk) 01:41, 26 September 2017 (UTC)

I would think that having one convex regular polyhedron should already be representative of that limited class. I am also not sure if the cube is really significantly more well-known than the tetrahedron, which is after all easily described in layman's terms as a triangular pyramid. Double sharp (talk) 10:47, 26 September 2017 (UTC)