Band brings a woman, a bagel, and, possibly (not my area), a huge inaccuracy: two sides and one surface...
A band, by definition, has two faces: if we flatten it up, then we have upper and lower or over and under, whatever we want to call them, but it is still two sides.
I liked the idea of the bagel being cut like that...
It is a short video.
I think it has a huge inaccuracy.
The knife is not lifted?
Do we lift the knife when spreading cream cheese on a bagel that is opened traditional way?
Why would we invest so much time to open a bagel if we can simply do it in a normal way?
Only female... 1 minute and 47 seconds it says... record time, shortest ever, or something like that.
Apparently, we say face in Mathematics, not side.
In this way, we could stick to the normal definition of side.
This defines side as a surface (American Heritage Dictionary).
According to the same source, the Collins English Dictionary says that side is face.
In this case, she might be right sometimes.
We say that we are measuring surfaces when we calculate area, so that we must think of how to calculate the area of the band.
In this case, it would be only one surface, but then all other figures would have only one surface, since we flatten them all before calculating area.
Surprisingly, Wolfram, which used to be my preferred source for Mathematics in 2001, does accept her definition of side, and then says that the band has one side.
See: Wolfram
"The Möbius strip, also called the twisted cylinder (Henle 1994, p. 110), is a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then reattaching the two ends (right figure; Gray 1997, pp. 322-323)."
A few inaccuracies inhabit Wolfram's pages...
Side cannot be the top or the bottom face, therefore any face but top and bottom, so that the band could also have no sides if we go with what people say: not top or bottom
The best argument I have to state that the Mobius Band has two sides or two faces is that we could use the same reasoning, of the walk, for any common solid, so say a prism.
We start with the finger on one face and we go all over the figure coming back to the same place.
That cannot mean that all those faces became one.
It cannot...
Face is about the aspect: As the dictionary says, it is what we see in front of us.
In this case, we see two faces or two sides in the untouched band: The inner and the outer.
We would have to have at least these two sides or faces to form a three dimensional figure, since only two dimensions could show one-sided things, the flat shape in the Cartesian Plane.
Prism brings a nice collection of prisms. Say we choose this one:
A band, by definition, has two faces: if we flatten it up, then we have upper and lower or over and under, whatever we want to call them, but it is still two sides.
I liked the idea of the bagel being cut like that...
It is a short video.
I think it has a huge inaccuracy.
The knife is not lifted?
Do we lift the knife when spreading cream cheese on a bagel that is opened traditional way?
Why would we invest so much time to open a bagel if we can simply do it in a normal way?
Only female... 1 minute and 47 seconds it says... record time, shortest ever, or something like that.
Apparently, we say face in Mathematics, not side.
In this way, we could stick to the normal definition of side.
This defines side as a surface (American Heritage Dictionary).
According to the same source, the Collins English Dictionary says that side is face.
In this case, she might be right sometimes.
We say that we are measuring surfaces when we calculate area, so that we must think of how to calculate the area of the band.
In this case, it would be only one surface, but then all other figures would have only one surface, since we flatten them all before calculating area.
Surprisingly, Wolfram, which used to be my preferred source for Mathematics in 2001, does accept her definition of side, and then says that the band has one side.
See: Wolfram
"The Möbius strip, also called the twisted cylinder (Henle 1994, p. 110), is a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then reattaching the two ends (right figure; Gray 1997, pp. 322-323)."
A few inaccuracies inhabit Wolfram's pages...
Side cannot be the top or the bottom face, therefore any face but top and bottom, so that the band could also have no sides if we go with what people say: not top or bottom
The best argument I have to state that the Mobius Band has two sides or two faces is that we could use the same reasoning, of the walk, for any common solid, so say a prism.
We start with the finger on one face and we go all over the figure coming back to the same place.
That cannot mean that all those faces became one.
It cannot...
Face is about the aspect: As the dictionary says, it is what we see in front of us.
In this case, we see two faces or two sides in the untouched band: The inner and the outer.
We would have to have at least these two sides or faces to form a three dimensional figure, since only two dimensions could show one-sided things, the flat shape in the Cartesian Plane.
Prism brings a nice collection of prisms. Say we choose this one:
Now, do the same: Run your finger, index finger, over the lateral surface of this prism (and notice that top and bottom is relative: put it standing and what was side according to one of the sources (not top or bottom) will become a non-side).
And now? Do we have only three sides?
You could run your finger over four of the surfaces and come back to the same place...
That is not a good argument...
I found no definition relating finger walk to sides or faces, so that not even the non-mathematical sources support this view.
We must have a mathematical agreement on what a side or face is, however.
We cannot define face based on what we had before connecting elements, before forming the shape, since otherwise all shapes would be the initial flat surface, and therefore would have one face.
Notwithstanding, any three-dimensional shape, perhaps taking away the line, would have to have at least two faces, for otherwise it would fit the Cartesian Plane and it would be in 2D instead.
The definition should be visual, and therefore based on the source that said it is what we see in front of us: if we look at the band from the front, we see one face. If we look at the band from the back, we see another, to the back of the face we just mentioned.
We put a number on what we see and we will have two.
For instance, take the picture that follows (it came from band):
And now? Do we have only three sides?
You could run your finger over four of the surfaces and come back to the same place...
That is not a good argument...
I found no definition relating finger walk to sides or faces, so that not even the non-mathematical sources support this view.
We must have a mathematical agreement on what a side or face is, however.
We cannot define face based on what we had before connecting elements, before forming the shape, since otherwise all shapes would be the initial flat surface, and therefore would have one face.
Notwithstanding, any three-dimensional shape, perhaps taking away the line, would have to have at least two faces, for otherwise it would fit the Cartesian Plane and it would be in 2D instead.
The definition should be visual, and therefore based on the source that said it is what we see in front of us: if we look at the band from the front, we see one face. If we look at the band from the back, we see another, to the back of the face we just mentioned.
We put a number on what we see and we will have two.
For instance, take the picture that follows (it came from band):
Write 1 in all you see when starring at this.
Now put it upside down and write 2.
Any other angle will return the same numbers, so that we would definitely have 2 in the end.
Perhaps we define it as being the largest number we may have when changing angles of sight of a 3D-shape.
In this way, our selected prism would be positioned in a way to show one face at a time to us, so that we get six.
Where we have an edge, we have the encounter of two faces. We can then just count the edges of the band.
We definitely have two: The top of the band and the bottom.
We also have the same faces meeting there, so two.
Got the idea from here (edge):
See:
Now put it upside down and write 2.
Any other angle will return the same numbers, so that we would definitely have 2 in the end.
Perhaps we define it as being the largest number we may have when changing angles of sight of a 3D-shape.
In this way, our selected prism would be positioned in a way to show one face at a time to us, so that we get six.
Where we have an edge, we have the encounter of two faces. We can then just count the edges of the band.
We definitely have two: The top of the band and the bottom.
We also have the same faces meeting there, so two.
Got the idea from here (edge):
See:
