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Klein bottle

BarnacleBob

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Winston_Room101

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Is Topology Not Fun ....

Place a finger on your nose ... and then
without lifting your finger from your skins' surface
move your finger to the inside of your mouth ... continue
moving your finger down your esophagus ( a thought experiment ) ...
thur your small intestine ...
thur your large intestine ... and out the other end
with your finger now returned to your skins' exterior surface

and so now it has been demonstrated that
the surface of your skin is the
same surface as the interior of your digestive system

FWIW
 

Fatrat

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Neat.....
 

arminius

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Is Topology Not Fun ....

Place a finger on your nose ... and then
without lifting your finger from your skins' surface
move your finger to the inside of your mouth ... continue
moving your finger down your esophagus ( a thought experiment ) ...
thur your small intestine ...
thur your large intestine ... and out the other end
with your finger now returned to your skins' exterior surface

and so now it has been demonstrated that
the surface of your skin is the
same surface as the interior of your digestive system

FWIW

To do that, that finger needs to be at least 20+ feet long...
 

Bottom Feeder

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another topology gotcha —
You cannot turn a torus inside out (think an inner-tube)
(without destroying it, that is)
 

Unca Walt

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And if you cut a Klein bottle in half the right way, you get two Mobius strips.

If you then cut the Mobius strip lengthwise all the way around...

You get a smaller Mobius strip. But you also get a 3-dimensional strip of twisted paper that is looped through the new Mobius strip.
 

Unca Walt

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One more -- I attended a topological luncheon one time where the napkin holders were Mobius strips, and the coffee was served in Klein bottles. Nothing sinister happened to the coffee.

You can picture how a Klein bottle can be a coffee cup if you hold it "upside down" (eg: skinny, bendy part down, and they had holders on the table to keep the Klein bottles upright.)
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Nowadays, you can go to the Road Runner/Wiley Coyote store: Acme Klein bottle Mug. $93.

You can get a coffee mug Klein bottle -- the photo seems to make no sense, so I will add the diagram of what you are looking at:

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