Edges and Faces: A Matter of Definition

(An archive question of the week)

Having looked at the matter of faces, edges, and vertices from several different perspectives, I want to look at one more question and answer, to tie it all together.

Definitions

The question is from 2008:

Definitions of Edge and Face in 2D and 3D

Different resources define "edge" in different ways.  What is the "official" definition of "edge", specifically is an edge restricted to the intersection of two non-coplanar faces or do two dimensional shapes have edges?  I also have a similar question about "faces"?  How many faces does a two-dimensional shape have?

The use of edges and faces arose in a problem about Euler's Formula relating numbers of faces, edges and vertices in a three-dimensional shape.  The same terminology was applied to the two-dimensional net for the three-dimensional shape.  In exploring the definitions some Geometry books use the terms without defining them well and many internet sites have conflicting definitions and uses of these terms.  Is it that there is no consistent definition?  What source would be definitive for future definition questions?

I could really use a mathematical dictionary or encyclopedia at home.  The internet sites brought up by a search are not guaranteed to be scholarly or correct.

An entry on this site defines edges for two-dimensional shapes.  Other sites restricted edges to intersections of two faces on three dimensional shapes.  I realize that I need to refine my own conceptualization of edge and face so I can use them correctly.  I teach high school and coach high school math teachers so the correct use and understanding of these terms is something important to me. I respect the scholarship of this site.  I had no way of evaluating the information on other sites.

Definitions, and their variability, have been one of the things that fascinate me as a Math Doctor. Since we get questions from all over the map (both in the global sense, and in terms of the many fields of math), we get to see how each field, and each country, can have its own use of words. Context is everything. There are no single official definitions; and having them would in a sense actually be bad for math.

Sue here is troubled by conflicting definitions; part of this is due to poor sources, as she suspects, but some is also due to different contexts. I answered:

Just as in ordinary usage, mathematical definitions depend on context. One word can have several slightly different definitions centered around a common idea, depending on how it is being used.  One problem we find is that elementary texts often try to use words in ways that mathematicians don't, wrongly assuming they have a uniform meaning that can be applied in all cases.  Another problem is trying to use a general definition without considering the special restrictions imposed by a specific context.

You can see the variation of definitions by looking in any dictionary. Choosing an arbitrary word, consider “paper”. On the surface (no pun intended), this seems straightforward; but consider these variations in its use:

• A sheet of paper
• The substance that sheet is made from
• A document
• A specific type of document: a scholarly paper
• A newspaper
• Financial documents
• Identification documents

These are all, as I said above, “centered around a common idea” (not unrelated words that happen to look alike, which also happens); but each usage is adapted from that for a particular use. Depending on context, you would know which I meant if I just said, “Here are my papers”, though the definitions are different.

Geometry

There are two quite different contexts in which “face” and “edge” are used, namely geometry and topology; and within geometry, we have both plane and solid geometry. In each context, we use these words with different nuances. What makes this confusing is that these fields are all interrelated, and it can be hard to be sure which we are really in!

I first looked at geometry:

In solid geometry, the basic definition of "edge" is "the intersection of two faces of a polyhedron".  It also applies to the segments comprising a polygon in plane geometry.

Similarly, a face is one of the (flat) polygons comprising the surface of a polyhedron.  The term is not really relevant in plane geometry, though it could be applied without trouble to a polygon that is part of a plane figure.  It is also irrelevant to discussions of curved surfaces, such as cylinders, cones, and spheres, within solid geometry.

In these contexts, an edge is a line segment and a face is a polygon. Why? Because those are the elements we are interested in there. (Faces as part of a plane figure typically arise in the context of topology, to be discussed below, though one often doesn’t realize that.)

Outside of the context of polygons and polyhedra, there are no standard definitions for "edge" and "face".  If one wants to talk about the "edges" or "faces" of a cylinder, it is necessary to either extend the definitions in a way that fits this new context, or to keep the restricted definitions and use some new terms for "curved edges and faces".  This would be done on a case by case basis--if someone has a reason to make such a modified definition (in order to be able to state certain theorems efficiently, say), he will state his definitions at the top of his paper.  Unfortunately, many elementary texts evidently make up a variety of solutions to this issue, so that kids who learn one thing from their text but see something different on the web get very confused.  The right thing would be not to use these terms at all except for polyhedra.

The reality is that mathematicians don’t usually have a reason to talk about “faces” of a cylinder, so there is no clearly defined term for either the circular or tubular portions of its surface. When we do, as I said, we can just state how we are using terms in our own local context. Therefore, as I have said previously, curriculum writers are doing their students a disservice when they teach some particular usage that those students will find conflicts with other sources when they do research (which includes just trying to cheat by looking on the Internet!).

Topology

Now I turned to topology, the study of geometrical entities with regard only to their connectedness, dropping all considerations of length or direction.

But there's more.  The same words are used with related but different meanings in topology.  Here, straightness and flatness are irrelevant, but connections matter: an edge must be a curve with two endpoints (which, for some purposes, must be distinct), and a face must be a simply connected region bounded by edges.  It is here that Euler's formula arises, so straightness doesn't matter, but the theorem imposes other restrictions--which are too often ignored or oversimplified in elementary treatments--namely the connectedness issues I just mentioned.  The formula can't be blindly applied to any solid (or plane) figure.  For a discussion of the restrictions, and how to relate the geometrical and the topological aspects, see this page:

  Faces, Vertices, and Edges of Cylinders, Cones, and Spheres
  http://mathforum.org/library/drmath/view/64540.html

In order to apply the formula, the vertices, edges, and faces have to meet certain connectedness criteria, and the entire surface must be equivalent to a sphere--its interior must be simply connected.  For example, as the link at the bottom of that page discusses, a torus needs a more general formula called the Euler characteristic.

A major source of the confusion over these issues is that Euler’s formula is not really about geometry proper, but about topology. In its original form, applied to polyhedra, it was thought of geometrically; but it was eventually realized that it was far more general, and relied only on how parts of a figure are connected. As a result, the conditions for it to hold are not those we think of in geometry; and the way it is typically stated at an elementary level can be misleading.

For our purposes the important thing is that the definitions we apply in talking about the formula are not those we use in geometry. The apparent inconsistency Sue found in definitions comes from the easily-overlooked difference in context.