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)

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, 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)