I was helping my daughter with her homework today, and came across the question of How Many Edges does a Cylinder have, she automatically put down 0, but thought that I had better look on the net and ask the question, finding this has been a great help. Thankyou.
On Mon, 1 Apr 2002 12:48:47 -0500 (EST), Walter Whiteley wrote: >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 to many plausible responses. > >One is almost certainly starting with the know 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). Pehaps higher dimensions. > >If the context is topology, then one works down the topological >features of the convext 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 polydron >pieces, so that the faces are still discs, then you have a new form >of Euler's formula |V| - |E| + |F| =0. >Similar foruma exist for covering other surfaces with discs, edges, vertices. > >However, to make those forumla 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. I 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, with out 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. 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). >Still does not seem to be a mathematically interesting description. > >Still does not really help with calculating surface area >- 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 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 base 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 is 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 > >[ps I write this in part because I am trying to have this >converstation with some text book 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?] > > > >> >> I was amazed t the previous email chain. >> >> I too am trying to assist my grader research an incorrect test answer >> that he thinks was correct. >> >> How many edges are there on a solid cylinder? >> >> He put "2" he got it wrong and asked his dear old dad to research >> onthe net and find if it was really wrong. >> >> IS THERE A DEFINIIVE ANSWER TO THE QUESTION: "HOW MANY EDES ARE THERE >> ON A SOLI CYLINDER?" >> >> Thanks! >>