Definitions of Edge and Face in 2D and 3D

Date: 10/10/2008 at 23:35:25
From: Sue
Subject: Looking for definitions of edge, and face for 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.

Date: 10/11/2008 at 22:46:42
From: Doctor Peterson
Subject: Re: Looking for definitions of edge, and face for 2D and 3D

Hi, Sue.

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.

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. 

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.

See the following page for some discussions of these issues:

  Do Cones, Cylinders Have Edges?
    http://mathforum.org/library/drmath/sets/select/dm_cone_edge.html  

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. 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/