“Falsehood flies, and truth comes limping after it, so that when men come to be undeceived, it is too late; the jest is over, and the tale hath had its effect: […] like a physician, who hath found out an infallible medicine, after the patient is dead.” —Jonathan Swift

  • 114 Posts
  • 1.27K Comments
Joined 2 years ago
cake
Cake day: July 25th, 2024

help-circle


  • So you’ve moved the goalposts from “that’s not just participation, it’s [things that participation entails]” to “well this doesn’t apply unless you need it to exist”*. GN needs a major platform like YouTube to actually reach an audience.

    * By the way, the comic is illustrating a man with a car that’s seemingly from the 1920s or 30s, so there goes that tenuous argument.














  • The hole through the center makes a hole around it inside the mug.

    Yeah, the hole through the center is one hole (the “donut hole”). The handle is the other (the “handle hole”).

    Topologically, the lip of the mug is a boundary, not a hole. This object is of genus 2 given it’s a 2-manifold.


    Edit: The explanation below is the product of a brainfart.

    But we can show this assuming the lip isn’t an abrupt boundary and that instead that the outer wall of the inner loop is part of the same side as the outside of the mug (i.e. that you can continuously walk on the surface from the outside past the lip without crossing an abrupt boundary, which more closely matches the physical reality of this mug).~

    Watch this: imagine a loop around the hole of a donut. Now try to contract that loop down. Without cutting it, you can’t.

    Now take a loop around the lip of this coffee mug. You can contract it to a point by bringing it all over the lip of the mug, then inside, along the outer walls, and then finally down to the bottom of the mug. It all meets up there. No hole. There’s the donut hole and the handle hole, but the colloquial “hole” is either a boundary if the lip is flat or not even a boundary if it continuously curves inward.




  • Topologically, that part is not a hole. The liquid basin not being a hole is where the joke in the OP originates, even: a standard coffee mug with a basin for coffee and a handle is homeomorphic (topologically equivalent) to a torus (a donut). And that’s because the handle is the hole, not the basin.

    In the case of the OP, ignore the handle for a second just to simplify things. That’s a hollowed-out donut (torus) except that you’ve taken a part of that donut’s surface and cut it out. That area that’s been cut out isn’t topologically a hole; instead, you’ve just created a boundary on the 2-manifold (read: flexible surface).

    Topology has rigorous, algebraic definitions under the hood, but that’s what’s going on in this picture topologically: you’ve taken a donut, glued it to another donut, and cut a “hole” (colloquial usage) in the double-donut’s surface, creating a boundary on the surface.


    Edit: To hopefully justify this a bit intuitively for the standard coffee mug (not this abomination in the OP), the utilities problem is a classic toy problem in topology that

    Tap for spoiler

    is unsolvable on a 2D plane

    but

    Tap for spoiler

    can be solved when you embed it into the surface of a coffee mug.