Talk:Polyhedron/Archive 4
Latest comment: 3 years ago by JayBeeEll in topic On varying definitions
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Lack of definition?
My definition of a polyhedron would be a finite number of polygonal faces connected edge to edge, with exactly 2 faces sharing each edge. But if I read or skim through the article none of these facts seem immediately apparent. Tom Ruen (talk) 17:29, 21 April 2019 (UTC)
- We've been through this in excruciating detail before, but: That is not a definition. What do you mean by face? What space do these things live in? How are they allowed to intersect? What are you using as the published reliable source for these vaguely-specified notions that might with care and nurturing become a definition? —David Eppstein (talk) 18:08, 21 April 2019 (UTC)
- In other words, mathematicians have no interest in communicating simple facts simply on encyclopedias. Tom Ruen (talk) 18:30, 21 April 2019 (UTC)
- The first sentence communicates the rough idea in plain words: "solid in three dimensions with flat polygonal faces". The definition section clearly states that "the formal mathematical definition of polyhedra that are not required to be convex has been problematic" and goes through several different approaches. It is not constructive to state that your vague notions outlined in your initial comment of this thread are "facts" nor to pretend that there is one simple definition that works for everything. See also Wikipedia_talk:WikiProject Mathematics#"Where triangle's area is triangle's area": "many (non-mathematicians) believe that all mathematical notations are an eternal global indestructible convention". We do not need to feed this misconception. —David Eppstein (talk) 18:39, 21 April 2019 (UTC)
- In other words, mathematicians have no interest in communicating simple facts simply on encyclopedias. Tom Ruen (talk) 18:30, 21 April 2019 (UTC)
- Compare to 4-polytope: 4-polytope#Definition
- A 4-polytope is a closed four-dimensional figure. It comprises vertices (corner points), edges, faces and cells. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Each face must join exactly two cells, analogous to the way in which each edge of a polyhedron joins just two faces. Like any polytope, the elements of a 4-polytope cannot be subdivided into two or more sets which are also 4-polytopes, i.e. it is not a compound.
- Why can't we say the equivalent?
- A polyhedron is a closed 3-dimensional figure. It comprises vertices (corner points), edges and faces. A face is a polygon. Each edge must join exactly two faces, analogous to the way in which each vertex of a polygon joins just two edges. The elements of a polyhedron cannot be subdivided into two or more sets which are also polyhedra, i.e. it is not a compound.
- Tom Ruen (talk) 18:50, 21 April 2019 (UTC)
- Because that doesn't describe a lot of things that we want to call polyhedra. See early in the definitions section, "Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid polyhedra (such as solids whose boundaries are not manifolds)." For instance, consider the two different polycubes formed by removing two non-adjacent cubes from a 2x2x2 block of cubes. One of these has two cubes meeting edge-to-edge but not face-to-face; the second one has a non-manifold vertex at the center of the block but no non-manifold edges. Your definition would exclude the first (there are four faces along the meeting edge of the cubes) but not the second, despite the fact that both have non-manifold boundary. Why? Can you find any instance in the published literature rather than in your own imagination where an author disallows one kind of non-manifold boundary but allows another, except by mistake? I'm not convinced that it's a good definition in the 4-polytope article, either, but this is not the place to argue that. —David Eppstein (talk) 19:03, 21 April 2019 (UTC)
- It would seem useful to have a qualifier like topological polyhedron for this definition, or I suppose 3-polytope as the topologically connected definition. 3-polytope could even be its own article for this definition. Things like Euler characteristic, duality and orientability are undefined outside of such definitions. Tom Ruen (talk) 19:15, 21 April 2019 (UTC)
- The Euler characteristic is well defined as long as you know how to count vertices, edges, and faces. It just doesn't have the value you expect it to have when the polyhedron surface is not a manifold. —David Eppstein (talk) 19:23, 21 April 2019 (UTC)
- What does it even mean to have an Euler characteristic for two cube volumes attached by a common internal face? It would make more sense to call that structure a sort of 4-polytope, where you can count cells and call the exterior the last cell, and use V+F=E+C. Tom Ruen (talk) 19:33, 21 April 2019 (UTC)
- What I mean is that the union of the cubes forms a volume, whose boundary consists of the edges, vertices, and square faces that touch both interior (the inside of one of the chosen cubes) and exterior (the space where we have not chosen any cubes), and that the Euler characteristic is the number that you get from the formula V+E-F. It has a topological meaning: if you shrink the interior and exterior away from the boundary, you get two manifold-bounded shapes (possibly disconnected) and V+E-F is just the average of the Euler characteristics of these two shapes. When all the internal and external cells are topological balls then V+E-F is the number of them (maybe what you are calling C?) but when the cells have more complicated topologies (such as the toroidal interior cell that you get from the 2x2x2 block minus two opposite cubes) then even that is not true. In any case all this requires a proof and I don't know of a published source for it, so it should not go into our articles. —David Eppstein (talk) 21:19, 21 April 2019 (UTC)
- I think a simple way to express that is to say that the two joined cubes comprise a decomposition of a bounded manifold. The Euler number is an important characteristic of bounded manifolds, as well as unbounded ones. — Cheers, Steelpillow (Talk) 09:52, 22 April 2019 (UTC)
- This seems hopelessly confusing. We might as well considered two squares sharing a common internal edge as a type of polygon that fails the polygonal version of Euler's formula, V=E.— Preceding unsigned comment added by Tomruen (talk • contribs) 12:20, 22 April 2019 (UTC)
- I have a new blog post at https://11011110.github.io/blog/2019/04/23/euler-characteristics-nonmanifold.html explaining what I mean in more detail. Part of the point of it for this discussion is that one of the examples that it mentions, the 2x2 cube with two opposite 1x1 cubes removed, meets Tom's intuitive definition of a polyhedron (it is a collection of 24 square faces meeting edge-to-edge with two faces per edge) but, because it is not a manifold, it acts strangely: its Euler characteristic is odd. Incidentally I checked with a colleague today what his intuitive definition of a polyhedron was, and his definition turned out to be that it should be a boundary of a solid, with flat faces, forming a manifold. So this example would not meet his definition of a polyhedron, but intuitions differ. —David Eppstein (talk) 00:41, 24 April 2019 (UTC)
- This seems hopelessly confusing. We might as well considered two squares sharing a common internal edge as a type of polygon that fails the polygonal version of Euler's formula, V=E.— Preceding unsigned comment added by Tomruen (talk • contribs) 12:20, 22 April 2019 (UTC)
- I think a simple way to express that is to say that the two joined cubes comprise a decomposition of a bounded manifold. The Euler number is an important characteristic of bounded manifolds, as well as unbounded ones. — Cheers, Steelpillow (Talk) 09:52, 22 April 2019 (UTC)
- What I mean is that the union of the cubes forms a volume, whose boundary consists of the edges, vertices, and square faces that touch both interior (the inside of one of the chosen cubes) and exterior (the space where we have not chosen any cubes), and that the Euler characteristic is the number that you get from the formula V+E-F. It has a topological meaning: if you shrink the interior and exterior away from the boundary, you get two manifold-bounded shapes (possibly disconnected) and V+E-F is just the average of the Euler characteristics of these two shapes. When all the internal and external cells are topological balls then V+E-F is the number of them (maybe what you are calling C?) but when the cells have more complicated topologies (such as the toroidal interior cell that you get from the 2x2x2 block minus two opposite cubes) then even that is not true. In any case all this requires a proof and I don't know of a published source for it, so it should not go into our articles. —David Eppstein (talk) 21:19, 21 April 2019 (UTC)
- What does it even mean to have an Euler characteristic for two cube volumes attached by a common internal face? It would make more sense to call that structure a sort of 4-polytope, where you can count cells and call the exterior the last cell, and use V+F=E+C. Tom Ruen (talk) 19:33, 21 April 2019 (UTC)
- The Euler characteristic is well defined as long as you know how to count vertices, edges, and faces. It just doesn't have the value you expect it to have when the polyhedron surface is not a manifold. —David Eppstein (talk) 19:23, 21 April 2019 (UTC)
- It would seem useful to have a qualifier like topological polyhedron for this definition, or I suppose 3-polytope as the topologically connected definition. 3-polytope could even be its own article for this definition. Things like Euler characteristic, duality and orientability are undefined outside of such definitions. Tom Ruen (talk) 19:15, 21 April 2019 (UTC)
- David is quite right. Many kinds of thing have been called polyhedra. The definition that we currently have is a consensus-driven least worst option. I seem to recall that the opening phrase used to read, "In elementary geometry". I am not sure how or why the "elementary" got removed, as even in geometry many of these other definitions rear their heads. But I would not contest anything else after all that got us here. I certainly would not open the topological Pandora's box. — Cheers, Steelpillow (Talk) 19:31, 21 April 2019 (UTC)
- If we can't handle definitions, perhaps we can offer examples failures with interpretations of each. Is a cube with the top face removed a polyhedron? Are two cubes that share a common internal face a polyhedron? Are two cubes attached by a common edge a polyhedron? (or if the common edge is two coinciding edges?) Are two cubes attached by a single common vertex a polyhedron? Are two separated cubes in space a polyhedron? Is a small cube inside of a big cube a polyhedron? Tom Ruen (talk) 19:59, 21 April 2019 (UTC)
- That level of complexity is not suited to an article lead. You should already find an approach to it in the main body of the text below. A discussion on historical examples of what might or might not be classed as a polyhedron and why might be useful. There are many examples in 19th century literature, some re-examined in the twentieth (e.g. Cromwell). It would need some thought to place that sensibly in the current article structure. Otherwise, I am finding it difficult to understand the problem you are having with the lead as it stands, beyond "WP:I don't like it". — Cheers, Steelpillow (Talk) 09:52, 22 April 2019 (UTC)
- I think my goal would be to find a way to isolate portions of this article that satisfy topological polyhedra, like my definition, so readers don't have to be hopelessly confused what applies to what definitions. Or pretty much EVERYTHING outside of the "definition" section is about my definition, so having a large confusing definition section that has nothing to do with the rest of the article seems simply existing to purposely confuse every reader who doesn't already know what a polyhedron might be. Tom Ruen (talk) 12:20, 22 April 2019 (UTC)
pretty much EVERYTHING outside of the "definition" section is about my definition
is very not true: just glancing quickly at sections that have drawn attention on this talk page, neither the concepts of duality nor volume are best put in the context of your pseudo-definition. On the other hand, I agree that some basic facts (or even the simpler one that an edge joins two vertices) that are common to many (if not all) definitions could be laid out more clearly somewhere. Possibly the section "characteristics" should begin with a collection of more basic properties. --JBL (talk) 15:04, 22 April 2019 (UTC)
- I think my goal would be to find a way to isolate portions of this article that satisfy topological polyhedra, like my definition, so readers don't have to be hopelessly confused what applies to what definitions. Or pretty much EVERYTHING outside of the "definition" section is about my definition, so having a large confusing definition section that has nothing to do with the rest of the article seems simply existing to purposely confuse every reader who doesn't already know what a polyhedron might be. Tom Ruen (talk) 12:20, 22 April 2019 (UTC)
- That level of complexity is not suited to an article lead. You should already find an approach to it in the main body of the text below. A discussion on historical examples of what might or might not be classed as a polyhedron and why might be useful. There are many examples in 19th century literature, some re-examined in the twentieth (e.g. Cromwell). It would need some thought to place that sensibly in the current article structure. Otherwise, I am finding it difficult to understand the problem you are having with the lead as it stands, beyond "WP:I don't like it". — Cheers, Steelpillow (Talk) 09:52, 22 April 2019 (UTC)
- If we can't handle definitions, perhaps we can offer examples failures with interpretations of each. Is a cube with the top face removed a polyhedron? Are two cubes that share a common internal face a polyhedron? Are two cubes attached by a common edge a polyhedron? (or if the common edge is two coinciding edges?) Are two cubes attached by a single common vertex a polyhedron? Are two separated cubes in space a polyhedron? Is a small cube inside of a big cube a polyhedron? Tom Ruen (talk) 19:59, 21 April 2019 (UTC)
- Because that doesn't describe a lot of things that we want to call polyhedra. See early in the definitions section, "Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid polyhedra (such as solids whose boundaries are not manifolds)." For instance, consider the two different polycubes formed by removing two non-adjacent cubes from a 2x2x2 block of cubes. One of these has two cubes meeting edge-to-edge but not face-to-face; the second one has a non-manifold vertex at the center of the block but no non-manifold edges. Your definition would exclude the first (there are four faces along the meeting edge of the cubes) but not the second, despite the fact that both have non-manifold boundary. Why? Can you find any instance in the published literature rather than in your own imagination where an author disallows one kind of non-manifold boundary but allows another, except by mistake? I'm not convinced that it's a good definition in the 4-polytope article, either, but this is not the place to argue that. —David Eppstein (talk) 19:03, 21 April 2019 (UTC)
On varying definitions
The following paper in Synthese may be of interest to editors of this page: [1] . --JBL (talk) 18:20, 12 July 2021 (UTC)
- Paywalled. Try this link — Cheers, Steelpillow (Talk) 18:49, 12 July 2021 (UTC)
- Wikipedia editors often can get access to paywalled sources via the Wikipedia Library. In this particular case, any editor of sufficient standing (account at least 6 months old, 500+ edits including at least 10 in the last month, not currently blocked) has access to the "library bundle". The bundle includes ProQuest, and this article can be accessed there. --JBL (talk) 19:07, 12 July 2021 (UTC)