http://www.smbc-comics.com/comic/dress
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If you ask a topologist what their house is like they start counting windows, doors, and chimneys.
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She just needs a dress with a strange number of button holes
Wait, where’s the fourth hole?
Arm, neck, other arm, bottom of dress
Nope.
Consider a sphere with a single hole through it, like a bead. If this were scaled up appropriately, it could almost be worn as a dress, but one would have to keep their arms against their body. Two additional holes would need to be made on opposing sides, each connecting the exterior of the sphere to the hold on the inside. Thus, three holes.
4 holes works if its a spherical surface rather than a volume.
You can flatten a sphere like you’re thinking of into a disc with three holes, so there are really only three holes in it, just like this dress.
I’m starting to think topologists don’t get invited to parties in the first place
We get invited to plenty of parties with actually interesting people.
Aren’t the arms just one hole too? If you take the bead and drill another hole perpendicular to the first one you now have space for both arms
Nope, two holes. By drilling into the solid part of the remaining sphere and breaking through to the centre shaft, which is now “outside” the sphere, you’ve created one hole; continuing the drilling into the other side and out the side of the sphere is yet another hole. You have fully pierced the sphere thrice.
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