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#1




Does a cylinder have any corners?
I say it has 2 corners because the definition of a corner is "the position at which two lines, surfaces, or edges meet and form an angle". Therefore, it seems to me that each edge where the top and bottom circles meet the plane curve can be described as corners. However, my third grade daughter's math teacher told her that a cylinder has no corners. Any opinions? Thanks!

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Last edited by KneadToKnow; 02062008 at 10:24 AM. 
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#4




I'm having trouble finding the definition of a "corner".
Is a corner the intersection of 3 planes or just 2? Wouldn't the intersection of 2 planes be an "edge"? 2 intersecting surfaces = edge 3 intersecting surfaces = corner 


#5




Yes. In a solid, I'd say a corner is the intersection of three planes (or possibly just three faces  they needn't be plane faces)
(in a plane figure, a corner is the intersection of two lines) So a cylinder has no corners. 
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Well, the definition I used above is straight from the dictionary and it specifies "two" lines, surfaces, etc. I see what you are saying, though, and I guess I'm wondering if an edge and a corner can sometimes be the same thing? (and, no, I don't know why I'm wasting time thinking about this...) Last edited by Morrison; 02062008 at 10:39 AM. 
#7




I'd think maybe a corner can be defined as an intersection of 2 lines if it's a 2dimensional figure like a square.
But once you jump to 3 dimensional spaces and objects you'd need 3 lines or surfaces. 
#8




A sphere has no corners.

#9




The intuitive idea of a "corner" on a surface would seem to me to include the vertex of a cone, where only one surface is involved, but there is a singularity in the tangent planes of the surface. So a definition might need to be in terms of singularities of tangent planes, rather that in terms of number of intersecting surfaces. In the case of the circles at each end of a cylinder, you have a set of singularities forming a continuous curve, and so yoiu might define as "edge" as a set of singularities in the tangent planes that form a continuous curve, and and a corner as either:
(1) an isolated singularity (e.g., the vertex of a cone); or (2) a point where three or more edges meet; or (3) a point at the end of an edge; or (4) a point in the middle of an edge where there is a discontinuity in the tangent lines to the edge. (I leave dreaming up cases of (3) and (4) as exercises for the reader). Last edited by Giles; 02062008 at 10:57 AM. Reason: typo 


#10




Let's say an ant is walking along the length of a cylindar and comes to the end. If he then goes around and starts to walk across one of the caps, wouldn't you say that the ant had turned a corner?

#11




In math, most everything is a matter of definitions, and definitions are generally only valid insofar as they are useful.
In talking in a strict, mathematical sense about the nature of solid shapes, it's useful to describe three special places on the surface of the solid: 1) Sides. A side is made up of points between which there's a smooth continuity. Sometimes planar, sometimes not, but in any case there's no sharp discontinuity between points on the same side. The shape of a side is a plane area, or a nonplane surface. You can basically (if not very rigorously) say it's a region of the surface over which the first derivative of the surface exists in every direction. 2) Edges. An edge is made at the intersection of two sides, usually, and in this case it's the set of contiguous points which are all members of both side 1 and side 2. Sometimes an edge is linear, sometimes not, but in any case the edge is the discontinuity at the boundary of a side. The shape of an edge is either a line, or a nonlinear curve. You could say it's a region of the surface where the first derivative exists, but only in the direction along the edge. 3) Corners. A corner is different than an edge  its' a set of points made up of exactly one point. Often it's the intersection of multiple edges, but it doesn't have to be  take the tip of a cone. This is a region of the surface where the first derivative of the surface curvature doesn't exist, along any direction. So you've got places where the surface doesn't bend sharply, places where it bends sharply in some directions but is smooth in others, and places where it bends sharply in every direction. It's relevant to distinguish between them. and in learning math, it's useful to learn to work with definitions. Using the definitions I've given above, a cylinder has no corners, but it has two edges. It's also useful to be a creative thinker, and the kid who gets the most out of math class recognizes that the definitions are only definitions, useful as a framework for classifying things, and not gospel from the Math Doctor. But take this comment: Quote:
On preview: hajario basically makes the point more eloquently. 
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#15




There's no hard and fats mathematical defitnion of 'corner' and Morrison's defitnion is not an uncommon usuage of the word. So cyclinders have two corners if that is how you choose to define 'corner'.

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My suggestion: Mathematically  yes, it has an infinite number of corners at each end. Literally  No. 
#18




What would you call the intersection of two edges, like in the roof of this picture?
It doesn't seem to fit the definition of either an edge or a corner. 
#19




Cylinders have two edges.
If you called them corners, what would you call the part on a cube where three planes meet, since you'd be calling the edges (where two planes meet) of the cube "corners"? 


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Just like a kid looks at a zebra and says, "Stripey horse!", even though a zebra isn't a horse. Even the poster Zebra isn't a horse. *yes, I am actively campaigning for the title of worst analogist (anologizer?) ever  why do you ask? 
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__________________
Crows. Keeping our highways clear of roadkill for over 80 years 
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Well, also according to OED...
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This seems a more accurate description of the are being described. 
#29




"the position at which two lines, surfaces, or edges meet and form an angle". Therefore, it seems to me that each edge where the top and bottom circles meet the plane curve can be described as corners.
Wrong. You're thinking of a "cylinder" as if it had some thickness. A mathematical cylinder has none. 


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So, to quote wolfstu, who got it so right that it's a wonder this thread needs to go on: Quote:
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Last edited by Indistinguishable; 02062008 at 07:16 PM. 
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#33




Seems to me Wolfstu nailed it, and with superb clarity. Bravo, Sir.
IMO, the LSLGuy Linguistic Uncertainty Principal applies here & quoting duelling dictionary entries attempting to define technical terms is pointless. After that, all we can bicker about is the colloquial use of the terms "edge" and "corner" when spoken by a probablynotverybright 3rd grade teacher speaking to typical 8 yearolds. YMMV. 
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#35




And all of the preceeding posts just prove why the Oval Office is a pain in the ass to vacuum.

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