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/
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