How to find centroid of a hemisphere using Pappus's centroid theorem?

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Try it with a sheet of paper. Draw a line across the width on one side and along the length on the other.
You can turn it into a cylinder by bringing together two opposite edges and gluing them together, then into a torus by doing the same with the opposite ends of the cylinder.
You can do it either so that the first fold leaves the line inside as a circle and the outside line straight, or so that the outside line is a circle around the cylinder and the inside line is straight.
Check linkage of the lines in the finished torus in each case.
Well first I learnt that making a cylinder is relatively easier but making a toroid is way too difficult, the paper doesn't want to change its surface's curvature and IIRC there is a theorem like that by Gauss (??).

Anyway, to the conclusion, the blue circle are red circle are not linked like a chain in one but they are linked in folding the other way around as I expected. 100% their linkages are different, they have to be different!
 
  • #27
haruspex
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Well first I learnt that making a cylinder is relatively easier but making a toroid is way too difficult, the paper doesn't want to change its surface's curvature and IIRC there is a theorem like that by Gauss (??).

Anyway, to the conclusion, the blue circle are red circle are not linked like a chain in one but they are linked in folding the other way around as I expected. 100% their linkages are different, they have to be different!
Right, but for each of those two ways of folding the paper there were two ways you could have done it. If the lines are red and blue, your first fold could have have formed a circle on the inside with either the red line or the blue, or formed a circle on the outside with either the red or the blue.
So the reconstructed torus has any of four configurations:
Linked rings, red inside, blue outside
Linked rings, blue inside, red outside
Unlinked rings, red inside, blue outside
Etc.
The question becomes, which of the possible transitions could pulling the torus inside out through a hole produce? If it started with red outside, blue inside then obviously it ends with red in blue out, but they could be linked or unlinked.. and it should be obvious the linkage status cannot change.
 
Last edited:
  • #28
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27
Right, but for each of those two ways of folding the paper there were two ways you could have done it. If the lines are red and blue, your first fold could have have formed a circle on the inside with either the red line or the blue, or formed a circle on the outside with either the red or the blue.
So the reconstructed torus has any of four configurations:
Linked rings, red inside, blue outside
Linked rings, blue inside, red outside
Unlinked rings, red inside, blue outside
Etc.
The question becomes, which of the possible transitions could pulling the torus inside out through a hole produce? If it started with red outside, blue inside then obviously it ends with red in blue out, but they could be linked or unlinked.. and it should be obvious says the linkage status cannot change.
Oh yes, I see it now,
002.png

If we did the first fold horizontally (and the horizontal fold should be in opposite direction to the unfolding otherwise we will end up with what we started with) instead of vertically then the circles wont be linked and thus that is not what is happening in inverting a tube with linked circles!

Also I noticed that the tube was narrower and longer before inverting it inside out.
 

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