Here's a confounding topological curiosity

boing
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Originally published at: Here's a confounding topological curiosity | Boing Boing

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That’s just playing fast and loose with the definition of “hole.”

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Video link for the BBS

https://www.youtube.com/watch?v=S5fPwE7GQOA

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The claymation video makes it seem like you can.

We need more claymation if you axe me.

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Gumby and Pokey? :pleading_face:

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QuickVibrantBichonfrise-size_restricted

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He had fun with poles, too!

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“Can you mold a clay pretzel around a rod? Signs point to YES.”

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You can replicate it with a pencil and your fingers. Just remember that topology and geometry are not the same thing.

Describing how to do it with fingers is a bit harder than doing it, so bear with me.

  1. Make a circle with thumb and index finger. Put the pencil (pen, chopstick, whatever) through the circle.
  2. Take the index finger and thumb of the other hand and touch the tips of both to the first circle. Topologically, you now have a two hole object. (we’re ignoring arms, etc.)
  3. As long as you maintain contact of the tips of those fingers with the first circle, you can slide them around and still have a two hole object.
  4. Now slide them around so they also form a circle. First with the pencil going through it. Then without the pencil going through it.

Prepare to be unamazed and unimpressed.

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What they ask is deceiving because it’s not clear from the beginning that this is soft clay and that you are allowed to re-mold the form. I assumed from the beginning of the video that it was perhaps rubbery like silicone but not that it was completely malleable like clay.

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That’s kinda like my comfort zone these days. Sort of.

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2315b90a263610f2ac866341c54d90b83ab9f864_2_549x500
Swiped from fellow boinger, but I forget who, sorry.

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It’s like using a blow torch to solve one of those bar puzzles with the chains and rings. What do you mean I can’t just expand the ring large enough to get it off?

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Best claymation ever:

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What did you think topological meant? And if you had known it was soft clay, would this result really have been instantly obvious to you? I thought it was a cute illustration of how topological deformation creates equivalences that one might not find intuitive.

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Confounding for anyone who can’t count (topological) holes. Somehow satisfying when you can.

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It’s most interesting when contrasted with what things are left invariant by as a liberal set of equivalences as topology gives you.
For example, can you give a good accounting why this stick-crossing number is not an invariant, but the winding number is? This feels interesting.

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I still can’t tell if this is topologically sound and there is no cheating. It “feels” like it shouldn’t be topologically possible to envelope an object. If he can make it go from 1 → 2, he should also be able to make it go from 1 → 0? [Edit: no he cannot!] When he bends it on itself at 0:24, it seems he’s hiding that he’s doing an illegal break/join? [Edit: no he is not!]

Albeit, the guy who made the video is a math professor, so I am loathe to think he’d play some cheap trick like that. (I guess what’s annoying is he does demonstrate it “like a trick”; that is, he doesn’t slow it down, he repeats it the same way when he is claiming to elucidate the phenomenon. And why turn off comments: that’s just obnoxious.)

I’ve never more needed Playdoh than I do now.

LATE-BREAKING EDIT:
This is the explanation that worked for me (after staring at it for too long to make sure there was no “cheating”):

Problem
image

Solution
image

So basically, 2 objects (of however many holes, each) are either Attached or they are Not Attached. Any attachment is equivalent (ie, can be converted) to any other attachment. (I think!)

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It’s topologically sound with no cheating. Basically, there are three types of loops you can follow that might potentially go around the stick:

  1. you can loop around the left hole, leaving the right handle alone
  2. you can loop around the right hole, leaving the left handle alone
  3. you can loop around the outside, leaving the middle bridge alone

All three are actually topologically equivalent…what’s happening is turning one into another by pulling the middle bridge out to become the right handle. So it goes from the left and outside loops circling the stick, to the left and right loops circling the stick. Nothing disconnects though.

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So if an object has N holes, and a [infinitely long] pole starts being through 1 of them, you can deform the object to go through all of the holes?

But you cannot topologically deform the object to free it from the pole (ie, going through 0 holes)?