# Talk:Face (geometry)

WikiProject Mathematics (Rated C-class, Low-priority)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 C Class
 Low Priority
Field:  Geometry
WikiProject Uniform Polytopes
This article is within the scope of WikiProject Uniform Polytopes, a collaborative effort to improve the coverage of uniform polytopes on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.

## Old comment

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)

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)

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

## 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)

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)

## Face, n-face, and Facet?

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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
More reworking, not to say great, but closer to being helpful? Tom Ruen (talk) 02:30, 20 May 2013 (UTC)
I appreciate that you're trying to contribute but I think many of your edits are incoherent and wrong.
1. 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?
2. 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.
3. You have removed the formal definition of a face (either as an intersection with a halfspace or with a hyperplane). Why?
4. You have removed the fact that a polytope is a face of itself. This is again incorrect.
5. 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.
6. 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)
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)
(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)
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)
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)
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)

───────────────────────────────── I'd hate to defend historical usage, facts don't need defending and this particular issue needs to be worked around not defended. Usage has been changing fast over the last couple of decades, as higher-dimensional and abstract theories have developed. While Coxeter's Regular Polytopes remains a classic reference, its terminology is somewhat dated. Even Grünbaum's Convex Polytopes is beginning to follow the same path. In three-dimensional geometry we obviously have works such as Cromwell's Polyhedra to refer to, but the situation with regard to advanced material is far less tractable. The main unifying theme seems to be abstract polytope theory, which is why I suggest adopting the terminology used in that article. It in turn adopts the more recent terminology developed by McMullen and Schulte, who have co-authored the leading reference works in that field. The latest work of theirs which I have to hand is unfortunately only dated 1997, so if anybody has a more recent work please feel free to update me. I quote:

"The elements of rank j are called the j-faces .... For j = 0,1,n − 2,n − 1, we also call j-faces vertices, edges, ridges, and facets, respectively."

and

"When F and G are two faces of a polytope with F ≤ [i.e. of lower or equal dimension to] G ..."

So unless this is outdated, we should be talking in general of "faces", with "facets" as an alternative term for (n − 1)-faces but with no alternative term for 2-faces. Any other usage found necessary at some point in our more advanced content should be noted and referenced in that article. Does anybody disagree with this suggestion? — Cheers, Steelpillow (Talk) 10:34, 20 May 2013 (UTC)

With regard to Facet (mathematics) I'd suggest correcting the howlers and merging the improved content into Facet (disambiguation). I may just get on and do that, you can always revert me. — Cheers, Steelpillow (Talk) 10:44, 20 May 2013 (UTC)
I merged and redirected the small cell (geometry) and 4-face (hypercell) articles here as well, added as new sections at the bottom. Sections might be renamed 3-face and 4-face, but redirect anchors need to be corrected then! Tom Ruen (talk) 20:33, 20 May 2013 (UTC)

### Some redirects that may need attention

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)

I added a new first sentence to the lead to indicate the most common, non-technical meaning of the term; I also moved the k-face section down so as to bring the section on the non-technical polygonal meaning up to the top, all as per WP:MoS. Hope no one minds! Bryanrutherford0 (talk) 00:43, 25 October 2013 (UTC)

Quite the contrary, thank you for doing that. I have cleaned up the elementary discussion accordingly. — Cheers, Steelpillow (Talk) 09:39, 25 October 2013 (UTC)
I have no argument with the cleanup, but curious if solid geometry is sufficiently comprehensive, and notice there's nothing about star polygon faces of star polyhedra which are not "solid", and not abstract either. I remember originally Kepler-Poinsot polyhedra were called Kepler-Poinsot solids until renamed... in 2007. Tom Ruen (talk) 14:55, 25 October 2013 (UTC)
I also agree that your rearrangement is a good idea, particularly with respect to WP:TECHNICAL. Thanks. —David Eppstein (talk) 15:34, 25 October 2013 (UTC)
Re "solid geometry", the whole situation is a mess with authors bending words to mean what they want at the time. For example in a single work, a respected author such as Coxeter will happily describe polyhedra as solids while defining them as surfaces or even skeleta ("a polygon is a closed chain of points and lines, a polyhedron is a closed assembly of polygons"). Reference to polygons and polychora means that we cannot talk of "three-dimensional geometry" either. I have used the term "elementary geometry", which (according to Tarski [1]) is geometry derived from Euclid's approach in his Elements, most notably without the use of set theory. In practice it often also seems to have connotations of an elementary learning level.
With respect to polyhedra, Euclid famously constructed the five Platonic solids and his methods allow the construction of many other polyhedra. Even Poinsot's description of his star polyhedra is purely geometric and not set-based and therefore they qualify. But modern treatments of higher dimensions are almost universally set-based and so need to be left out: there remains perhaps a residual elementary introduction to 4-polytopes and their higher-dimensional analogues.
Now that I think about it, would a distinction between elementary and set-theoretic treatments be a good way to divide the article? — Cheers, Steelpillow (Talk) 16:22, 25 October 2013 (UTC)

## N.W. Johnson: Geometries and Transformations, (2015)

Could somebody tell me, how is this work available? 89.135.8.194 (talk) 05:31, 12 October 2015 (UTC)

I have a preprint copy. It is being published by Cambridge University Press. Tom Ruen (talk) 05:39, 12 October 2015 (UTC)
The book has missed many publication dates and has not yet been published, nor has any peer review. Until then its claimed content remains subject to change and its claimed encyclopedic significance remains unverifiable. — Cheers, Steelpillow (Talk) 12:27, 12 October 2015 (UTC)
When Johnson died, someone (Tom?) said the book is still going forward … —Tamfang (talk) 07:00, 16 January 2018 (UTC)
CUP now says February [2]. But until it is actually published I don't think we should use it as a source. —David Eppstein (talk) 07:08, 16 January 2018 (UTC)