“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

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Joined 2 years ago
Cake day: July 25th, 2024
  • Lemmy Shitpost@lemmy.world•The cylinder must remain unharmedbyTheTechnician27@lemmy.worldEnglish
    17 days

    Okay, if a torus with a single patch of empty space on its surface is two holes, then this is three holes. But now I’m confused, because I don’t know if that’s true topologically. Otherwise, it’s just a two-torus with a perforation on its 2-surface.

    Edit: I’ve been informed by a topologist friend that this has two holes and that the lip of the mug is just a boundary.

    Edit 3: Ignore the case below if the lip is a smooth curve. I had a brainfart when thinking through the loop contraction.

    Edit 2: So to clarify, that’s a boundary if the lip represents an abrupt transition to either side (i.e. if there’s no “curve” into and out of the mug).

    Since that’s less realistic, let’s assume that you can smoothly walk from the interior surface of the mug to the exterior and show that it isn’t topologically a hole for that case either.

    A picture of three rings around two holes and the purported "lip hole"

    I’ve created a continuous loop around the two holes and the candidate “lip hole”. These loops are embedded in the surface of the mug. You can plainly see that you can’t contract the loop to a single point around the handle hole or the donut hole without crossing over the hole.

    But now let’s take our blue loop (the lip loop) and trivially contract it down to a point.

    Four steps to contract the loop to a point

    1. Our loop is on the outside around the lip.

    2. Take the loop inside of the lip since this is still a smooth, continous surface. The loop is now on the outer wall of the inner mug.

    3. Now bring the loop down toward the bottom along the wall.

    4. Contract the loop to a single point at the bottom.

    In this case, it isn’t even a boundary anymore, and it provably isn’t a hole.