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Your Thoughts on Quiz Question...
Some time ago in a 'Pub Quiz' this question was asked.....
"What 3 dimensional object has only one side and one edge?"
The answer given as being correct was the infamous 'Mobius Band'.
Now, I answered a 'Solid Sphere', and argued the fact that for a 'Mobius Band' to actually exist it must have some thickness....this implies therefore that it must have more than one side and hence more than one edge.....no matter how 'thin' it was.
A 'Solid Sphere', on the other hand, can only have one side due to it's very nature......as to how many 'edges' it had kind of got lost in the persuing 'How thick is a Mobius Band?' debate which had now spilled out onto the car-park.
So, what would be your answer to that question? Is it a 'Mobius Band'....a 'Solid Sphere'.....or some other object?
Just to add, the guy who set the question was incredibly inteligent so to prove him wrong was more of a victory than winning the quiz!
We didn't win the quiz either.
Regards.
"What 3 dimensional object has only one side and one edge?"
The answer given as being correct was the infamous 'Mobius Band'.
Now, I answered a 'Solid Sphere', and argued the fact that for a 'Mobius Band' to actually exist it must have some thickness....this implies therefore that it must have more than one side and hence more than one edge.....no matter how 'thin' it was.
A 'Solid Sphere', on the other hand, can only have one side due to it's very nature......as to how many 'edges' it had kind of got lost in the persuing 'How thick is a Mobius Band?' debate which had now spilled out onto the car-park.
So, what would be your answer to that question? Is it a 'Mobius Band'....a 'Solid Sphere'.....or some other object?
Just to add, the guy who set the question was incredibly inteligent so to prove him wrong was more of a victory than winning the quiz!
We didn't win the quiz either.
Regards.
Answers
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I think you are being over pedantic re the edges. If a similar quiz question asked how many edges are there on a sheet of A4 paper, would you double it just because it has a finite thickness and is therefore three dimensional? No, you would consider the paper to be a plane surface of 2 dimensions.
Similarly, the M�bius band is considered a 2D surface distorted in 3D space.
As for your solid sphere, there is no way to argue that it has a single edge since it has zero edges.
Finally, the setter could have used the more topologically correct term 'Boundary Component' but since this was a pub quiz (and not one for mathematicians) he was right to use the more widely understood term of 'Edge'.
Similarly, the M�bius band is considered a 2D surface distorted in 3D space.
As for your solid sphere, there is no way to argue that it has a single edge since it has zero edges.
Finally, the setter could have used the more topologically correct term 'Boundary Component' but since this was a pub quiz (and not one for mathematicians) he was right to use the more widely understood term of 'Edge'.
Nope, you're on to a loser there.
You could as easily try to argue that a line must have some width because you have to draw it or that a point has dimensions to be drawn.
But we are talking about mathematical objects here and a line drawn on a piece of paper is not a line but rather a representation of a line.
You could attempt to argue that topologically the mobius strip is a 2 dimensional object and not a 3 dimensional one (the 3D analogue being a klein bottle). If he argues that it has a physical representation and is not a mathematical object then your physical width argument holds water again. (You can't have it both ways)
It's a bit moot though because your sphere doesn't have an edge.
You could as easily try to argue that a line must have some width because you have to draw it or that a point has dimensions to be drawn.
But we are talking about mathematical objects here and a line drawn on a piece of paper is not a line but rather a representation of a line.
You could attempt to argue that topologically the mobius strip is a 2 dimensional object and not a 3 dimensional one (the 3D analogue being a klein bottle). If he argues that it has a physical representation and is not a mathematical object then your physical width argument holds water again. (You can't have it both ways)
It's a bit moot though because your sphere doesn't have an edge.
In my day our geometry teacher would define an edge as being where two surfaces meet. He hadn't heard of Mobius!
But there's nothing to stop a Mobius strip being 3d and having thickness. Imagine it being made from, say, a thick, flat metal bar, where the surface has been filed or otherwise shaped to produce sharp edges, (so that its cross-section looks like the area the red line passes through here). It's unquestionably 3D, having length, width, and thickness.
The real problem is that there are exceptions to many of our so-called ''definitions''.
But there's nothing to stop a Mobius strip being 3d and having thickness. Imagine it being made from, say, a thick, flat metal bar, where the surface has been filed or otherwise shaped to produce sharp edges, (so that its cross-section looks like the area the red line passes through here). It's unquestionably 3D, having length, width, and thickness.
The real problem is that there are exceptions to many of our so-called ''definitions''.
I realise that the edge of a solid must have a thickness of at least one atom, Geezer, but I was looking at the question from a more practical point of view. If you like, a drawing of the figure I suggested might suffice to demonstrate the 3d-ness of the shape, since, as Jake says, ''a line has no thickness''.
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