Talk:Face (geometry)

The Formal Definition here really SUCKS, and without references it is hard to evaluate for cleanup. spacetime??? intersection of any supporting hyperplane of P and P? Nonsense and errors for all I know! Tom Ruen 06:50, 14 October 2006 (UTC)[reply]

I am trying to understand this definition so I can help my child do his homework. Nowhere does it say whether a face is flat, or planar. Please write something that I can understand. Thank you.

68.6.69.34 03:37, 23 February 2007 (UTC)[reply]

this does not make a dot of sense —Preceding unsigned comment added by 79.64.78.217 (talk) 17:07, 12 September 2007 (UTC)[reply]

Holes

Can a surface still be a face if it has a hole in it? 24.85.161.72 (talk) 21:02, 27 March 2013 (UTC)[reply]

A good question, doesn't seem covered here. There are other special cases of general polygons that cause problems - nonplanarity, concavity, self-intersection, and self-contacted boundaries. In all cases triangulation (geometry) would seem to offer an workable solution. That is, if you take a polyhedron and triangulate its faces with new edges that represent the hole, then you can work from those simple polygon faces. Afterwards, you could consider the collection of triangles glued together back and consider what restrictions you want to allow. Tom Ruen (talk) 22:54, 27 March 2013 (UTC)[reply]

Face, n-face, and Facet?

There is discussion on these three confusing usages of face at: Template_talk:Infobox_polychoron#Face. I think this article is the problem. Tom Ruen (talk) 05:57, 19 May 2013 (UTC)[reply]

The formal part of the article looks to be completely in agreement with standard research usage to me. However, the lead section somewhat contradicts it. The problem, as I see it, is that some of our articles (and the template in question) use "face" in a way that does not match the formal definition here, to mean only the 2-faces. It is the other articles (and the lead of this article) that need to be corrected. —David Eppstein (talk) 06:30, 19 May 2013 (UTC)[reply]

ETA: I've edited the article to fix some technical mistakes (the formal definition needs to intersect the polytope with a halfspace, not a hyperplane) and to remove some strange notions about space-time. I also rewrote the lead to more accurately reflect the contradiction between different meanings. I also added some much higher quality sources than the links previously included with the article (which are still present as they were before in the external links section). —David Eppstein (talk) 06:43, 19 May 2013 (UTC)[reply]

The sad truth is that different sub-disciplines and different authors have used such terms as "face" and "facet" differently over the years, each perverting previous meanings to their own sometimes over-specialised or even unwise ends. For example elementary polyhedron theory and abstract polytope theory both mean very different things by "face" and "facet" - and even within each of these relatively well-defined areas, usage still differs. For example I have seen the terms "face" and "facet" used by different authors to refer to the same thing, with edit battles over "j-face" vs. "j-facet", both well enough attested in the literature. This is not a simple problem to unravel.

Clearly each article needs to ensure that the term is adequately defined - either in that article or a more foundational one which it links to - and then ensure that the article is both self-consistent and as consistent as possible with related articles (with any discrepancies noted). Where topic areas collide in an article or say a template such as Template:Infobox polychoron is re-used, conflicts can emerge.

Personally I would like to see us Wikipedians develop and record a consensus approach and then stick to it, using phrases like "Authority X uses the term 'facet' to mean a 'face' as defined here." My own view again is that the elementary (schoolkid) and more advanced (undergraduate) levels will still be a bit inconsistent, but with care in addressing the appropriate audience for the topic, that should be manageable. For tha advanced topics, I'd suggest the abstract polytope article as a starting point. It has been fairly thoroughly fought over and what is there now seems reasonably stable.

In the current case of Template:Infobox polychoron, I think the key question to ask is, is this elementary or advanced information? If school kids may be reading it and we are stopping at 4 dimensions then "faces" is probably fine. But if it is really for the more mathematically advanced editor, who may for example be contemplating a Template:Infobox 5-polytope and beyond, then we should get these things right and several of those labels need to change. Looking at some of the articles which use it, I am slightly inclined to the latter view but open to persuasion.

Sorry about the long rant and lack of a firm PoV, but I hope there is at least some sense in there. — Cheers, Steelpillow (Talk) 12:45, 19 May 2013 (UTC)[reply]

Whatever else we decide, (and I think the only answer is to split definitions here and cite usage in every context of interest) I totally disapprove of saying polygon for 2-face elements of a polyhedron. A polyhedron can have cyclic subsets of edges that makes polygons that are NOT faces, like a cuboctahedron has central hexagons and squares that are NOT 2-faces. Petrie polygon are another example of polygons in polytopes that are NOT faces. Tom Ruen (talk) 22:04, 19 May 2013 (UTC)[reply]

Ok, but what alternative do you propose? I don't think "face" is acceptable to refer to the 2-dimensional things in articles about polytopes of dimension greater than three. —David Eppstein (talk) 22:27, 19 May 2013 (UTC)[reply]

Opening a book from Coxeter, I find he describes the 24-cell with 16 vertices, 32 edges, 24 square faces, and 8 cubic cells. So polygonal face would seem to be a generic term for 2-face, and context free face would seem to imply 2-face. So you can generalize k-polytope face='k-face for k>=2, but what do we mean with no k? Coxeter might also say polyhedral face generally or cubic face (instead of cell) for 3-faces elsewhere, but I have no immediate examples. Tom Ruen (talk) 23:12, 19 May 2013 (UTC)[reply]

In Coxeter's Regular Polytopes (book), page 264, says The vertex figure of {5/2,5,3} is, of course, a dodecahedron, whose edges and faces are vertex figures of pentagrams and of {5/2,5}'s [cell]s. He doesn't qualify as polygonal faces, so again he is implying face means 2-face in 4D figures. Tom Ruen (talk) 23:29, 19 May 2013 (UTC)[reply]

Cubic_crystal_system#Cubic_space_groups like face-centered cubic lattice, is another example where common usage face means 2-face. A cubic honeycomb is topologically identical to 4-polytopes, space filled by cells, cells separated by common faces, faces and cells sharing common edges, and all sharing common vertices. Tom Ruen (talk) 23:45, 19 May 2013 (UTC)[reply]

The context-free "face" certainly does not imply 2-face in more recent work in polyhedral combinatorics. See the three major textbooks on the subject that I added as sources to this article yesterday, all of which agree that a "face" without additional context means the things of all dimensions. —David Eppstein (talk) 23:20, 19 May 2013 (UTC)[reply]

So what we've determined is context counts, and this article should more clearly express the 2-face original definition that has been swallowed in polytope theory. So if we search books and find TWO DIFFERENT usages, then they BOTH deserve proportional expression on this wikipedia article. Tom Ruen (talk) 23:29, 19 May 2013 (UTC)[reply]

And they are both represented here. But you are still missing my original point which is that in articles about higher dimensional polytopes we need different terms for 2-faces and arbitrary-dimensional faces, because we need to talk about both kinds of things. Your stubborn refusal to use any word other than the unadorned "face" for the 2-faces is making it impossible to include information about these shapes that refers to the faces of all dimensions, because it preempts the only word that can be used to talk about those things. —David Eppstein (talk) 00:06, 20 May 2013 (UTC)[reply]

I don't think I'm refusing anything except calling a 2-face element a polygon, but I made my FIRST change on the example for 4-polytope: Tom Ruen (talk) 01:35, 20 May 2013 (UTC)[reply]

3-faces – 3-dimensional cells (polyhedral faces)
2-faces – 2-dimensional faces (polygonal faces)

Okay, I did a bit of reworking, mainly added a section polygonal face and redirect 2-face to Face (geometry)#Polygonal faces. Tom Ruen (talk) 02:03, 20 May 2013 (UTC)[reply]

More reworking, not to say great, but closer to being helpful? Tom Ruen (talk) 02:30, 20 May 2013 (UTC)[reply]

I appreciate that you're trying to contribute but I think many of your edits are incoherent and wrong.

You have put (n-1)-face (facet) as a primary meaning of "face", before the meaning that applies to faces of all dimensions Do you even have a source for the meaning of "face" being restricted to the n-1-dimensional ones?
In the lead, you have modified a sentence that said there were two meanings to say there are three, but left in place a source for that sentence that says there are two meanings. This makes the source incorrect for that sentence.
You have removed the formal definition of a face (either as an intersection with a halfspace or with a hyperplane). Why?
You have removed the fact that a polytope is a face of itself. This is again incorrect.
You have entitled the section about what I would consider the main meaning of faces (the faces of all dimensions) "k-faces". These are not k-faces, they are just faces. k-faces is a variation of this terminology used only when you want to restrict attention to faces of a specific dimension.
If you don't want to make that restriction, you call them faces. Given what I see as a big pile of mistakes, I'm going to revert your edits, but we can continue to discuss these issues here if you think there is some value that is lost by this reversion. —David Eppstein (talk) 03:36, 20 May 2013 (UTC)[reply]

Feel free to rework what I've done, but reverting is unfair. I can undo some of your complaints piecewise. Tom Ruen (talk) 03:48, 20 May 2013 (UTC)[reply]

(1) - I'll move facet last. (2) I see 3 meanings, change as you like, but I'll put it back to 2. (3) Formal definition is still there. (4) I'll restore n-face of n-polytope as you like, but it doesn't fit the hyperplane definition. (5) I'll remove my attempted intro addition. Feel free to improve what's there. My primary defense is for the polyhedral face section. Tom Ruen (talk) 03:54, 20 May 2013 (UTC)[reply]

p.s. The intro two related but inconsistent meanings clearly is referring to 2-face versus (n-1)-face (see also Facet (geometry), NOT a general face. So someone ELSE can write the intro to make sense! Tom Ruen (talk) 04:02, 20 May 2013 (UTC)[reply]

Re your p.s. I believe you are misreading the source. He is calling the two-dimensional faces of a 3-dimensional polyhedron "facets", in keeping with the higher-dimensional terminology; he explains that 2-dimensional faces of a three-dimensional polyhedron are more often called "faces" but that he is using "faces" (as defined not far below in the same source) to mean faces of all dimensions. There is no implication that "faces" should ever be taken to mean the (n − 1)-dimensional faces of an n-dimensional polytope for n ≠ 3.

Re "that does not fit the hyperplane definition": that's why the definition is not the intersection with a supporting hyperplane. Ziegler uses the intersection with a zero-set of a valid inequality, a slightly more general notion that includes supporting hyperplanes, disjoint hyperplanes (giving the empty set), and the whole space (giving the whole polytope). Other authors use intersection with a halfspace, the definition we have here. Although it's a matter of definition, there are multiple reasons why the mathematics is cleaner using a definition of faces that includes the empty set and the whole polytope: for instance, Euler's formula is simpler (it's always zero rather than alternating between 0 and 2 depending on dimension), and also because it's needed to make the face lattice be a complete lattice.

By the way, before Michael Hardy comes here and starts yelling at you about the way you have formatted your formulas: please read MOS:MATH#Typesetting of mathematical formulae. E.g. you have written things like "n-1", with a roman variable name, a hyphen in place of a minus sign, and no spacing around the minus sign; it should be n − 1. —David Eppstein (talk) 04:16, 20 May 2013 (UTC)[reply]

I'll let Steelpillow defend the inconsistent historical usages of face vs facet if he likes. I'm also not against Facet (geometry) being copied and redirected here if that helps anything. We're still stuck with 100+ polyhedron/polytope articles linking face (geometry) for 2-face, but with the prominent first section, I'm satisfied. 2-face directs to Face (geometry)#Polygonal faces, and easier to write if someone wanted to replace face in all the specific polytope articles. Tom Ruen (talk) 05:21, 20 May 2013 (UTC)[reply]

Incidentally current redirect here are: Polytope face, Polyhedron face, Face (mathematics), Hedra, Faces (geometry), 2-face 5-face 6-face 7-face 8-face, although I didn't look which are used in articles. Tom Ruen (talk) 23:41, 19 May 2013 (UTC)[reply]