Number of Cylinder EdgesDate: 04/01/2002 at 12:16:20 From: David Saray Subject: Number of edges in a solid cylinder Dear Sir: My 8-year-old son was asked "how many edges are there on a solid cylinder?" on a recent math examination. His answer was "2" and it was marked as incorrect. He truly believes in his answer and has asked for my assistance in researching. Can you assist? Thank you. Date: 04/01/2002 at 12:33:11 From: Doctor Peterson Subject: Re: Number of edges in a solid cylinder Hi, David. It depends on how "edge" was defined in his class, which may not agree with his intuitive definition. Often, an edge is required to be straight, in which case a cylinder has no edges. Unfortunately, elementary texts are not always very careful about definitions, and they can ask questions like this that are really worthless. The only definition of "edge" that would make sense in this context would be the one your son is naturally using (a boundary between smooth surfaces making up an object), which would allow a cylinder to have two edges. Asking this question with the other definition only invites confusion, so I wish they wouldn't ask it. I'd like to hear how they did define the word. You can read some other discussions of related issues here: Does a Cone have an Edge? A Vertex? http://mathforum.org/dr.math/problems/mark.03.12.02.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 04/01/2002 at 13:03:33 From: Doctor Sarah Subject: Re: Number of edges in a solid cylinder David asked the same question in the discussion group geometry-pre- college, where it received these responses: http://mathforum.org/kb/message.jspa?messageID=1077941 Subject: Re: How Many Edges On SOLID CYLINDER Author: Walter Whiteley <whiteley@mathstat.yorku.ca> Date: Mon, 1 Apr 2002 12:48:47 -0500 (EST) This is a common issue among elementary teachers, and some elementary text book writers. Basically different sources put down different answers. The underlying issue is: what is the context? What is the larger mathematics one wants to engage with? Without this, there are too many plausible responses. One is almost certainly starting with the known definitions and answers for convex polyhedra. The faces are flat plane regions. The edges are where two faces meet (and lie along lines). The vertices are where three or more faces meet (three or more edges as well) and are points. The overall pattern has a nice mathematical structure given by Euler's formula |V| - |E| + |F| = 2. Now one wants to extend this to other creatures. Perhaps non-convex but spherical polyhedra. Perhaps general topological surfaces (the counts do not change if the polyhedron is made of rubber and deformed without cutting or gluing). Perhaps higher dimensions. If the context is topology, then one works down the topological features of the convex polyhedron. (a) A face is a topological polygon - a disc with boundary of edges and vertices. (b) An edge is a closed curve, with two boundary edges - vertices. (c) A vertex is a point. Together the vertices and edges form a connected graph. (Whether you allow two edges between a fixed pair of vertices, or a loop which has the same vertex at both ends is not critical for the topology.) With this in mind, you still can talk about spherical topology. Basically, a connected graph drawn on a sphere without crossings, with the regions cut out forming the faces. This still satisfies Euler's formula: |V| - |E| + |F| = 2. If you draw a graph on, say, a torus, or make a torus out of polyhedron pieces, so that the faces are still discs, then you have a new form of Euler's formula |V| - |E| + |F| = 0. Similar formulae exist for covering other surfaces with discs, edges, vertices. However, to make those formulae work in that simple context, you need to ensure the faces have a single polygon as the boundary, and the edges do have vertices at their ends. How might this relate to a cylinder? Well, in many elementary texts, one studies the 'net' - the flat paper pieces which one might use to fold up to the surface. The standard net for a cylinder has two circular discs on the ends, and a rectangle which is to be taped together along two opposite sides to form the sides of the cylinder. If you study the taping of the net, you can tape it up with three pieces of tape (one at each end, and one along the sides). You have two points at which a couple of pieces of tape meet - the ends of the side slit. You have three faces which are discs. In this image, you have |V|=2, |E|=3, |F|=3 and |V|-|E| + |F| = 2. So this image makes good sense from the point of view of topology and counting with Euler. Note that we had to slit the tube of the cylinder, creating an extra edge, in order to make that face a disc, and to restore the formula. It is, however, a sensible process. In the same spirit, one would have to cut up a sphere in order to make it work in the topology. E.g. put down an equator, with a vertex where the two ends of the equator meet. This would give |V| - |E| + |F| = 1 - 1 + 2 = 2. However, some elementary texts and test writers decide they know best and give distinct definitions of 'faces', 'edges', and 'vertices'. When doing so, there should be some good mathematical reason for doing that. Some set of situations one is trying to make sense of. Simple extrapolation on one basis or another, without investigating the good and bad patterns, is a source of trouble. That, unfortunately, routinely happens in elementary (and some high school) materials. If faces are 'flat regions' and 'edges' are straight lines, then a cylinder has two faces, no edges, and there is not real purpose in the answer. It does not even help you calculate the surface area! If faces are regions, and edges are where two faces meet, then a cylinder has three faces and two edges (no vertices). This still does not seem to be a mathematically interesting description. And it still does not really help with calculating surface area - you need to cut it open as a net and make the tube into a rectangle. Then you have formulae for the areas of faces, and you also happen to have the three edges, three faces, and two vertices needed for the pattern of Euler's formula. You still see, in the simple count, what the overall topology is (the 2 tells you it is spherical and could be redrawn, topologically, on a sphere). I suspect that whatever answer this particular test expected, it is based on a particular discussion in a particular text. I can show you different materials with different answers, but seldom is there a mathematical discussion. Some people have concluded that, as a result, it is simply a bad idea (distracting without learning) to use the words faces, vertices, edges for such objects. I do not quite agree - but the only really useful context I know is the larger topology, and you can see that this takes a larger understanding, something I only learned at graduate school, and only teach in some upper level undergraduate courses (courses most teachers have not taken). Odds are this discussion in the source text or materials did NOT give enough context to explain why one would bother with these words for this object. What is the MATHEMATICS one is trying to do! That is where one needs to start. Walter Whiteley [P.S. I write this in part because I am trying to have this conversation with some textbook publishers for elementary materials, and with some other curriculum sources for our schools. It IS an important topic to make sense out of, and I look forward to other contributions on the topic. Are there other ways to make mathematical sense out of the choices?] ----------------------------- Subject: Re: How Many Edges On SOLID CYLINDER Author: Guy Brandenburg <gfbranden@earthlink.net> Date: Mon, 01 Apr 2002 17:47:21 -0500 I suppose it depends on what you define as an edge. I believe that most math types would think of it as a straight line segment that is the boundary of two polygonal faces; it runs between two vertices, which are points where 3 polygonal faces meet. Now, since a cylinder has no polygonal faces, then it has no edges at all. That being said, many people not familiar with mathematical terminology might indeed consider the boundaries of the two circles as being edges. But that would not correspond to the normal mathematical definitions of polyhedra, prisms, pyramids, and so on. A cone is not the same thing as a pyramid, for example; a cylinder is not a prism. On the other hand, one could think of the circular bases as being polygons with an infinite number of sides. In that case, there would be an infinite number of edges. But that would make the lateral, curved, sides into no longer curved sides but an infinite number of infinitely thin rectangles. Probably better not take that approach unless you are trying to do calculus-type computations. I am guessing that the expected answer was zero. Guy Brandenburg - Doctor Sarah, The Math Forum http://mathforum.org/dr.math/ |
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