Talk:Face (geometry)

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?

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)

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)

• 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)

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)

Non-technical lead

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 ) 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 . 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)

It was published in 2018 by Cambridge University Press. And my name is on the back cover! —Tamfang (talk) 22:03, 19 June 2023 (UTC)

Metric spaces

Is there an established definition of "face" for metric spaces? I am thinking of a definition similar to that for subsets of vector spaces, but where we instead say that p is strictly between x and y if dist(x, p) + dist(p, y) = dist(x, y) and all three distances are strictly positive. So the definition for a metric space M would be that F ⊆ M is a face if together p ∈ F and p is strictly between x and y implies that x, y ∈ F. —Quantling (talk | contribs) 14:27, 20 December 2024 (UTC)

Proposed merge of Extreme set into Face (geometry)

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

To not merge; topics are sufficiently distinct to warrant separate discussion; different audiences. Klbrain (talk) 19:29, 1 May 2025 (UTC)

---

Duplication of Face of a convex set. fgnievinski (talk) 19:13, 9 April 2025 (UTC)

Oppose. The topics are too different. Specifically, this article is primarily focused on polyhedra and polytopes (things that have a finite and discrete set of faces). The extreme set article is focused on more general convex objects like spheres and ellipsoids that can have an infinite or continuous family of faces. I think it would be too confusing to readers to try to cover both topics in a single article, and lead to an article that is too WP:TECHNICAL in that one would have to have significant background in both polyhedral combinatorics and convex geometry, a combination of expertises that is uncommon among even professional mathematicians. —David Eppstein (talk) 20:33, 9 April 2025 (UTC)

Oppose. What David Eppstein says. Also, Extreme set promises to eventually talk about topological vector spaces, which is a radically different topic than polyhedra or polytopes. (TVS's are not "just" infinite-dimensional, but come with a whole blob of properties that make them suitable for quantum field theory, quantum gravity and homology and what-not in such spaces, for which there is no analog in the finite-dimensional case.) 67.198.37.16 (talk) 20:20, 25 April 2025 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Tangent plane

@Fgnievinski thank you for your recent edits. With regards to how a face is defined in terms of a tangent plane, I am worried about the case where a plane includes only a single vertex of a polyhedron. By some definitions of tangent plane, that plane is a tangent plane, and yet a single vertex generally isn't considered a face. Any thoughts on how we can avoid misleading the reader who is wondering whether this single-vertex case would be considered a face? (Similarly, there is a case where a plane includes only a single edge of a polyhedron ... but edges are not faces either.) Thank you —Quantling (talk | contribs) 18:53, 14 November 2025 (UTC)

Why don't we sidestep pathological cases with a caveat like "(non-degenerate) tangent plane"? fgnievinski (talk) 21:04, 14 November 2025 (UTC)

The phrase "non-degenerate tangent plane" doesn't mean anything to me. I assume it is to mean a plane for which none of the polyhedron intersects strictly on one of the sides of the plane (but the polyhedron does intersect on the plane and, usually, on the other side), and the plane intersects the polyhedron with a positive area (which a vertex or edge would not have). However, wanting it to mean that, isn't the same as it having that meaning. Unless it has that meaning in other contexts, it isn't going to be useful to the reader to know that it also applies in this context. —Quantling (talk | contribs) 02:21, 15 November 2025 (UTC)

@Quantling the common case is a tangent plane spanned by at least three vertices; I've now made a caveat in the article. I guess the more exotic exceptions would apply to non-convex polyhedra? they are definitely harder to define in general, not only their faces. the other article (polyhedron) gives the following definition: "The faces of such a [general] polyhedron can be defined as the connected components of the parts of the boundary within each of the planes that cover it." I'd propose to include it here. fgnievinski (talk) 05:39, 15 November 2025 (UTC)

@Fgnievinski Ack, the example in my edit comment is not good, but I don't know how to amend an edit comment. Nonetheless, I think we need to figure out how we are going to write this sentence before we insert it into the article. That we find ourselves dealing with single vertex, single edge, and non-convex polyhedra is a sign that this isn't a clean way to define a face. Perhaps the textbook you are citing has language we could lean on (though not outright infringing on the copyright). —Quantling (talk | contribs) 02:04, 16 November 2025 (UTC)

I still fail to see the importance of tangent planes defined on a single vertex or a single edge; it seems an exception that proves the rule. fgnievinski (talk) 03:28, 18 November 2025 (UTC)

I suppose it depends upon context. Tangent planes in Euclidean space are typically used to separate an object such as a polyhedron from empty space -- the tangent plane intersects the object but every point of the polyhedron that isn't on the tangent plane is on the same side of the tangent plane as every other such point. For example, by considering the intersection of every half space that includes the whole object, one gets the convex hull of the object.

In this scenario we don't care whether the tangent plane intersects a single vertex or a single edge, it all still works. Also, when the object isn't convex, we are exploiting that the plane that includes a face that isn't on the boundary of the convex hull is not on a tangent plane.

I consider this a typical use of tangent planes. That we have to shoe horn tangent plane to make it work in the current article is a warning sign for me. Maybe it can be done anyway; if this is something you'd like to see happen, I encourage you to keep thinking about it and propose language here. —Quantling (talk | contribs) 14:45, 18 November 2025 (UTC)

originally the article had only a non-definition: "a face is a polygon on the boundary of a polyhedron"; not all polygons on the boundary of a polyhedron are a face (for example, a triangle connecting any three vertices on the square face of a cube). the inserted text is an actual definition: "the face of a convex polyhedron is the plane segment resulting from the intersection of the polyhedron with a tangent plane" (including a source citation). I'm not sure which definition you have in mind for tangent planes, but it's not just a touching plane, it's also perpendicular to the polyhedron normal direction. you're insisting on tangent planes at a vertex or on an edge, where the normal is not well defined; such an exotic case would require an ad-hoc definition of vertex normal. fgnievinski (talk) 02:53, 19 November 2025 (UTC)

Putting on my WP:CALC hat, I think that we're getting somewhere. I can phrase it terms of unit normal vectors, but there may be a way in terms of tangent planes as well. A face is a maximal, connected region, of the surface of a polyhedron, having a unique, shared unit normal vector. Okay, the wording sucks, but it does cover the single-vertex, single-edge, and non-convex-polyhedron "exceptions" in a way that isn't merely a laundry list. I suspect that "tangent plane" is more accessible to the average reader than is "normal vector", so I'd want to swap that in somehow ... but the definition of tangent plane that I am used to doesn't require that its intersection with the polyhedron be more than vertex or edge, nor that it be connected.

Regardless, we are pretty much forced to use a citation from a noteworthy source, and mimic the technical language from it. If you have it handy... what language does your source actually use? —Quantling (talk | contribs) 03:11, 19 November 2025 (UTC)

I like your definition, originally I wanted to use "maximal region" but couldn't find a source using it (related to area maximization). I had cited a source for my proposed definition, it could be simply complemented with a definition of tangent plane in parentheses: A face of a convex polyhedron is the plane segment resulting from the intersection of the polyhedron with a tangent plane (the infinite plane spanned by the face vertices). Your proposal could appear next: More generally, the face of a polyhedron is a maximal region on the surface of a polyhedron having a constant normal direction. ("region" already takes care of connectedness). fgnievinski (talk) 06:26, 22 November 2025 (UTC)

I don't think it's possible to give a clear and precise definition, because there is no single accepted clear and precise definition for a polyhedron (see polyhedron). More, for many of the definitions of a polyhedron the faces are something that is used to define the polyhedron rather than something that can be extracted from a polyhedron that is defined in some other way. And saying that it's a maximal polygonal subset of the boundary, or a polygon within the boundary and bounded by the edges of the polyhedron, also doesn't work: not all definitions give you a solid subset of space for which the polyhedron is a boundary of anything, and the edges have the same definitional problem as the faces.

It is easy to define the 2-faces of a convex polyhedron but this section is not restricted to convex polyhedra. See Polyhedron § Definition; all the same definitional problems detailed in that section apply here to faces. A face is one of the polygons used to define a polyhedron. I think that's the best we can say without getting far into the weeds of how all the different definitions work. —David Eppstein (talk) 08:46, 26 November 2025 (UTC)

The complications of general polyhedra should not preclude a useful definition of faces of convex polyhedra. The narrower scope could be emphasized by saying something like: "Convex polyhedra are more easily defined and so are their faces: the face of a convex polyhedron is a plane segment resulting from the intersection with a tangent plane (the infinite plane spanned by coplanar adjacent vertices)." fgnievinski (talk) 03:53, 10 December 2025 (UTC)

That definition is already circular, or how do you define adjacent? But that's an avoidable problem. The bigger problem is that this article purports to be about polyhedra, not just about convex polyhedra, so defining faces only for convex polyhedra is far too specialized. —David Eppstein (talk) 07:16, 10 December 2025 (UTC)

The article is about faces in geometry generally, so it could easily accommodate a paragraph about faces of convex polyhedra.

Here's a simpler and non-circular definition:

A face of a convex polyhedron P is a positive-area polygonal region on the boundary of P resulting from the intersection of P with a plane.

If needed, an explanatory footnote could be added:

A region is a non-empty, connected, and open set. The restriction of non-zero area excludes edges and vertices, as well as cross sections of the convex polyhedron.

The definition is adapted from the following source:

Croft, H.T., Falconer, K.J., Guy, R.K. (1991). Polygons, Polyhedra, and Polytopes. In: Unsolved Problems in Geometry. Problem Books in Mathematics, vol 2. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0963-8_3

The full quotation is as follows:

"A boundary point x is a vertex if there is some plane that intersects the [convex] polyhedron P in the single point x. A line segment L in the boundary is an edge of P if there is a plane that intersects P [only] in the segment L, and a region in the boundary P is a face if it is the intersection of a plane with P and has positive area."

fgnievinski (talk) 02:39, 11 December 2025 (UTC)

That looks good to me. —Quantling (talk | contribs) 15:50, 11 December 2025 (UTC)