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#1
Old 02-06-2008, 11:08 AM
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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!
#2
Old 02-06-2008, 11:24 AM
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Quote:
Originally Posted by Morrison
Any opinions? Thanks!
The teacher is only thinking of corners by thinking of the corners of a room. You're thinking more analytically and I encourage you to teach your child to do so as well while also helping her to understand that some people won't be able to see these things.

Last edited by KneadToKnow; 02-06-2008 at 11:24 AM.
#3
Old 02-06-2008, 11:26 AM
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Quote:
Originally Posted by Morrison
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!
#4
Old 02-06-2008, 11:33 AM
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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
Old 02-06-2008, 11:37 AM
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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.
#6
Old 02-06-2008, 11:38 AM
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Quote:
Originally Posted by Hampshire
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

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; 02-06-2008 at 11:39 AM.
#7
Old 02-06-2008, 11:48 AM
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I'd think maybe a corner can be defined as an intersection of 2 lines if it's a 2-dimensional figure like a square.
But once you jump to 3 dimensional spaces and objects you'd need 3 lines or surfaces.
#8
Old 02-06-2008, 11:50 AM
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A sphere has no corners.
#9
Old 02-06-2008, 11:56 AM
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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; 02-06-2008 at 11:57 AM. Reason: typo
#10
Old 02-06-2008, 12:07 PM
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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
Old 02-06-2008, 12:22 PM
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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:
Originally Posted by Morrison
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?
Sure they can. The room I'm in is nearly cubic, so it has 4 vertical edges between the walls, and a bunch more edges where the walls meet either the ceiling of the floor. And yet, I might casually talk about 'the four corners of the room', or be send to go 'stand in the corner' for drawing polygons on the wall. Here we're using corner in a different sense, influenced by our habit of seeing the world a a two-dimensional place we have to navigate. Just like out front of my workplace, where there's an intersection 'at the corner of' Kepler Lane and Black Way.

On preview: hajario basically makes the point more eloquently.
#12
Old 02-06-2008, 12:25 PM
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Quote:
Originally Posted by hajario
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?
That's a corner in the path of the ant. I'd also say the ant was turning corners if its path was a square inside one of the circles forming the ends of the cylinder. So a path on a surface can have a corner if it passes through an edge or not. (And I suspect that in addition it's possible for a path on a surface to be smooth even though it passes through an edge).
#13
Old 02-06-2008, 12:27 PM
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Quote:
Originally Posted by gonzomax
A sphere has no corners.
Or an infinite number of them.
#14
Old 02-06-2008, 12:49 PM
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Quote:
Originally Posted by Q.E.D.
Or an infinite number of them.
Not for the usual definitions of "corner". However, there are surface with an infinite number of corners shaped like the corners of a cube. (Example: take a cube, then form a surface by sticking a smaller cube in the centre of each face, and removing the square in common. Keep on doing that by adding smaller cubes to each face. Clearly, the limiting surface has an infinite number of corners that are formed from the corners of a cube, and it has in infinite number of corners with other shapes as well.)
#15
Old 02-06-2008, 12:58 PM
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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'.
#16
Old 02-06-2008, 01:19 PM
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Quote:
Originally Posted by Giles
Not for the usual definitions of "corner". However, there are surface with an infinite number of corners shaped like the corners of a cube. (Example: take a cube, then form a surface by sticking a smaller cube in the centre of each face, and removing the square in common. Keep on doing that by adding smaller cubes to each face. Clearly, the limiting surface has an infinite number of corners that are formed from the corners of a cube, and it has in infinite number of corners with other shapes as well.)
...and, of course, a finite volume.
#17
Old 02-06-2008, 01:19 PM
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Quote:
Originally Posted by Mangetout
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)
But the idea of planes becomes a bit wishy-washy when you talk about a three-dimensional figure. After all, one of the whole points of a plane is it only has two-dimensional properties.

My suggestion:
Mathematically - yes, it has an infinite number of corners at each end.
Literally - No.
#18
Old 02-06-2008, 01:29 PM
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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
Old 02-06-2008, 01:46 PM
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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"?
#20
Old 02-06-2008, 01:53 PM
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Quote:
Originally Posted by Santo Rugger
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"?
The main face of a cylinder isn't a plane, though.
#21
Old 02-06-2008, 03:57 PM
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Quote:
Originally Posted by Shalmanese
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.
The centre of that roof would seem to me to be the intersection of 6 edges (2 edges being the lines at the tops of the ridges, which continue through the point, and 4 edges being the lines at the bottom of the valleys going up to that centre point). I'd call it a corner.
#22
Old 02-06-2008, 04:07 PM
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Quote:
Originally Posted by Really Not All That Bright
The main face of a cylinder isn't a plane, though.
Well, maybe not exactly. But how is that relevant? If you took a cube and inflated it a bit so that the faces all curved slightly outward, thus failing to be exact planes, I think the natural propensity would still be to distinguish between 8 corners and 12 edges.

Last edited by Indistinguishable; 02-06-2008 at 04:10 PM.
#23
Old 02-06-2008, 04:13 PM
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Quote:
Originally Posted by Indistinguishable
Well, maybe not exactly. But how is that relevant? If you took a cube and inflated it a bit so that the faces all curved slightly outward, thus failing to be exact planes, I think the natural propensity would still be to distinguish between 8 corners and 12 edges.
Good point. If we went by the OP's definition then a cube would have 12 corners.
#24
Old 02-06-2008, 04:58 PM
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Quote:
Originally Posted by Really Not All That Bright
The main face of a cylinder isn't a plane, though.
Substitute "surface" for "plane", as the definition in the OP includes "surface".
#25
Old 02-06-2008, 05:02 PM
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Quote:
Originally Posted by Indistinguishable
Well, maybe not exactly. But how is that relevant? If you took a cube and inflated it a bit so that the faces all curved slightly outward, thus failing to be exact planes, I think the natural propensity would still be to distinguish between 8 corners and 12 edges.
Right, but mathematically they'd no longer technically be corners as they wouldn't be the confluences of 3+ planes. In reality, they're still basically pointy bits that we'd look at and say, "Corners!" (new expletive!), but they wouldn't really be corners.

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?
#26
Old 02-06-2008, 05:08 PM
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Quote:
Originally Posted by Santo Rugger
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"?
Well, according to OED...
Quote:
Originally Posted by Concise OED
corner

noun 1 a place or angle where two or more sides or edges meet.
...the edges, or places where two planes meet, are, in fact, corners.
#27
Old 02-06-2008, 05:26 PM
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Quote:
Originally Posted by Hampshire
Good point. If we went by the OP's definition then a cube would have 12 corners.
If the planes of a cube were considered two at a time, there are 18 corners.
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#28
Old 02-06-2008, 05:30 PM
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Well, also according to OED...
Quote:
edge

noun 1 the outside limit of an object, area, or surface. 2 the line along which two surfaces of a solid meet.
bolding mine
This seems a more accurate description of the are being described.
#29
Old 02-06-2008, 05:43 PM
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"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.
#30
Old 02-06-2008, 08:14 PM
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Quote:
Originally Posted by Really Not All That Bright
Right, but mathematically they'd no longer technically be corners as they wouldn't be the confluences of 3+ planes.
Only if you adopt a definition of a corner as the confluence of 3+ planes (with this meaning exact planes, not mere surfaces). But my whole point is that this is a silly definition to adopt, at least if our goal is to characterize the intuitive concept of a corner. Nothing in math compels this definition; math is flexible, it can handle all kinds of definitions.

So, to quote wolfstu, who got it so right that it's a wonder this thread needs to go on:

Quote:
Originally Posted by wolfstu
In math, most everything is a matter of definitions, and definitions are generally only valid insofar as they are useful.
Quote:
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.

Last edited by Indistinguishable; 02-06-2008 at 08:16 PM.
#31
Old 02-06-2008, 08:43 PM
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Quote:
Originally Posted by maroci
"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.
I'm not even sure what you mean by this. What did he say that invoked thickness?
#32
Old 02-06-2008, 10:39 PM
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Quote:
Originally Posted by hajario
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?
Only if he admitted he had become powerless over alcohol.
#33
Old 02-07-2008, 08:13 AM
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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 probably-not-very-bright 3rd grade teacher speaking to typical 8 year-olds.

YMMV.
#34
Old 02-07-2008, 01:06 PM
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Quote:
Originally Posted by maroci
"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.
I'm not sure what you mean by this either. But note that it is correct to refer as a "cylinder" to the solid, as opposed to the surface (of no thickness) bounding that solid, which would be a cylindrical surface. cite
#35
Old 02-07-2008, 05:06 PM
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And all of the preceeding posts just prove why the Oval Office is a pain in the ass to vacuum.
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