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Topic: How Many Edges On SOLID CYLINDER
Replies: 3   Last Post: Oct 26, 2004 9:16 AM

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 Mrs Deryn Bosch Posts: 1 Registered: 1/25/05
Re: How Many Edges On SOLID CYLINDER
Posted: Oct 26, 2004 9:16 AM

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:
&gt;This is a common issue among elementary teachers, and some
&gt;elementary text book writers. Basically different sources
&gt;
&gt;The underlying issue is: what is the context? What is the
&gt;larger mathematics one wants to engage with? Without
&gt;this, there are to many plausible responses.
&gt;
&gt;One is almost certainly starting with the know definitions and
&gt;answers for convex polyhedra. The faces are flat plane regions.
&gt;The edges are where two faces meet (and lie along lines). The
&gt;vertices are where three or more faces meet (three or more edges
&gt;as well) and are points. The overall pattern has a nice mathematical
&gt;structure given by Euler's formula |V| - |E| + |F| = 2.
&gt;
&gt;Now one wants to extend this to other creatures. Perhaps non-convex
&gt;but spherical polyhedra. Perhaps general topological surfaces
&gt;(the counts do not change if the polyhedron is made of rubber and
&gt;deformed without cutting or gluing). Pehaps higher dimensions.
&gt;
&gt;If the context is topology, then one works down the topological
&gt;features of the convext polyhedron.
&gt;(a) A face is a topological polygon - a disc with boundary of edges
&gt;and vertices.
&gt;(b) An edge is a closed curve, with two boundary edges - vertices.
&gt;(c) A vertex is a point.
&gt;Together the vertices and edges form a connected graph.
&gt;(Whether you allow two edges between a fixed pair of vertices,
&gt;or a loop which has the same vertex at both ends is not critical
&gt;for the topology.)
&gt;
&gt;With this in mind, you still can talk about spherical topology.
&gt;Basically, a connected graph drawn on a sphere without crossings,
&gt;with the regions cut out forming the faces. This still satisfies
&gt;Euler's formula: |V| - |E| + |F| = 2.
&gt;
&gt;If you draw a graph on, say, a torus, or make a torus out of polydron
&gt;pieces, so that the faces are still discs, then you have a new form
&gt;of Euler's formula |V| - |E| + |F| =0.
&gt;Similar foruma exist for covering other surfaces with discs, edges,
vertices.
&gt;
&gt;However, to make those forumla work in that simple context, you need
to
&gt;ensure the faces have a single polygon as the boundary, and the
&gt;edges do have vertices at their ends.
&gt;
&gt;How might this relate to a cylinder? Well, in many elementary texts,
&gt;one studies the 'net' - the flat paper pieces which one might
&gt;use to fold up to the surface. I standard net for a cylinder
&gt;has two circular discs on the ends, and a rectangle which is to be
&gt;taped together along two opposite sides to form the sides of
&gt;the cylinder. If you study the taping of the net, you can tape it
&gt;up with three pieces of tape (one at each end, and one along the
sides).
&gt;You have two points at which a couple of pieces of tape meet - the
ends
&gt;of the side slit. You have three faces which are discs.
&gt;In this image, you have |V|=2, |E|=3, |F|=3 and |V|-|E| + |F| = 2.
&gt;So this image makes good sense from the point of view of topology
&gt;and counting with Euler.
&gt;
&gt;Note that we had to slit the tube of the cylinder, creating an extra
&gt;edge, in order to make that face a disc, and to restore the formula.
&gt;It is, however, a sensible process.
&gt;
&gt;In the same spirit, one would have to cut up a sphere in order to
&gt;make it work in the topology. E.g. Put down an equator, with a
vertex
&gt;where the two ends of the equator meet. This would give
&gt;|V| - |E| + |F| = 1 - 1 + 2 = 2.
&gt;
&gt;However, some elementary texts and test writers decide they
&gt;know best and give distinct definitions of 'faces' 'edges'
&gt;and 'vertices'. When doing so, there should be some good
&gt;mathematical reason for doing that. Some set of situations one
&gt;is trying to make sense of. Simple extrapolation on one basis
&gt;or another, with out investigating the good and bad patterns
&gt;is a source of trouble. That, unfortunately, routinely happens
&gt;in elementary (and some high school) materials.
&gt;
&gt;If faces are 'flat regions' and 'edges' are straight lines,
&gt;then a cylinder has two faces, no edges, and there is not real
&gt;surface area!
&gt;
&gt;IF faces are regions, and edges are where two faces meet, then
&gt;a cylinder has three faces and two edges (no vertices).
&gt;Still does not seem to be a mathematically interesting description.
&gt;
&gt;Still does not really help with calculating surface area
&gt;- need to cut it open as a net and make the tube into a
&gt;rectangle. Then you have formulae for the areas of faces,
&gt;and also, happen to have the three edges, three faces and two
&gt;vertices needed for the pattern of Euler's formula.
&gt;You still see, in the simple count, what the overall topology is
&gt;(the 2 tells you it is spherical and could be redrawn, topologically,
&gt;on a sphere).
&gt;
&gt;I suspect that whatever answer this particular test expected,
&gt;it is base on a particular discussion in a particular text.
&gt;I can show you different materials with different answers,
&gt;but seldom is there a mathematical discussion. Some people
&gt;have concluded that, as a result, it is simply a bad idea
&gt;(distracting without learning) to use the words faces, vertices,
&gt;edges for such objects. I do not quite agree - but the only
&gt;really useful context I know is the larger topology, and you
&gt;can see that this takes a larger understanding, something
&gt;I only learned at graduate school, and only teach is some
&gt;upper level undergraduate courses (courses most teachers
&gt;have not taken).
&gt;
&gt;Odds are this discussion in the source text or materials
&gt;did NOT give enough context to explain why one would bother
&gt;with these words for this object.
&gt;
&gt;What is the MATHEMATICS one is trying to do!
&gt;That is where one needs to start.
&gt;
&gt;Walter Whiteley
&gt;
&gt;[ps I write this in part because I am trying to have this
&gt;converstation with some text book publishers for elementary
&gt;materials, and with some other curriculum sources for our schools.
&gt;It IS an important topic to make sense out of, and I look forward
&gt;to other contributions on the topic. Are there other
&gt;ways to make mathematical sense out of the choices?]
&gt;
&gt;
&gt;
&gt;&gt;
&gt;&gt; I was amazed t the previous email chain.
&gt;&gt;
&gt;&gt; I too am trying to assist my grader research an incorrect test
&gt;&gt; that he thinks was correct.
&gt;&gt;
&gt;&gt; How many edges are there on a solid cylinder?
&gt;&gt;
&gt;&gt; He put "2" he got it wrong and asked his dear old dad to research
&gt;&gt; onthe net and find if it was really wrong.
&gt;&gt;
&gt;&gt; IS THERE A DEFINIIVE ANSWER TO THE QUESTION: "HOW MANY EDES ARE
THERE
&gt;&gt; ON A SOLI CYLINDER?"
&gt;&gt;
&gt;&gt; Thanks!
&gt;&gt;

Date Subject Author
4/1/02 David Saray
4/1/02 Walter Whiteley
10/26/04 Mrs Deryn Bosch
4/1/02 Guy Brandenburg