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Re: How Many Edges On SOLID CYLINDER
Posted:
Oct 26, 2004 9:16 AM
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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!
>>
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