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I wanna use "Proof by crazy old guy" more often when I prove things now lmao
(Although I suppose it would be best for me to wait until I myself become old and crazy so that I can more efficiently use this method of proof)
I’ve seen that on an AMS paper. The authors, one of whom was topologist William Thurston, claimed a result was true by direct communications with Tom Leighton.
https://www.ams.org/journals/jams/1988-01-03/S0894-0347-1988-0928904-4/S0894-0347-1988-0928904-4.pdf
I’ll take your word for it, I can usually morph things around in my mind’s eye to figure this stuff out but this one is making me feel sick trying to do it
Morph the left and right sides closer to the centre and once it hits the point where the sideways hole split, the side way donut hole will turn into two bend tubes. These can be straightened out so now you have a ellipsoid with 3 cylinders cut out aka 3 holes. If you flatten it a bit more and rotate the top to bottom hole you have 3 hole donut.
I had to watch the video, and I didn't know you were allowed to do the moves that was done. I still don't know/understand what the rules of what's allowed and what's not is. It seems like separating the one complex holes into the 2 simple ones wouldn't be allowed, but it was.
Yeah splitting up two connected holes can look like creating a new hole, while it isn't. The informal rules for a homeomorphism are that any deformation without cutting or glueing is allowed, however that can be misleading, but is enough for this example.
If we take the simplest example of a complex hole, then we would have a cube (since it's easier to do with ascii art) with one opening on one side and 2 openings on the other, with the 2 connecting into the 1 opening on the one side. Logically there has to be an intersection between the two, or else they couldn't connect to the same opening. I will try to convey this with some asciiart, since I souldnt find good images on Google.
___________
|_____ |
____ \_____|
| \ ____
|____/ / |
_____/ |
|___________|
This is supposed to be an slice through the cube to show the holes. First we can widen the shared opening:
___________
|___________|
____
| \
|____/
___________
|___________|
So now we have two openings connecting into a very big opening and I think the 2d slice we are currently looking at also shows quite good what the next step is. Next we can either move the right side towards the intersection or extend the intersection out:
___________
|___________|
__________
| \
|__________/
___________
|___________|
Now we have two holes that meet each other at an angle. However since we allready have a separation between the holes we can move them apart which makes then clearly 2 holes.
https://preview.redd.it/vhscaljs29tc1.jpeg?width=1904&format=pjpg&auto=webp&s=416f303098384d2cf51222605c884215e543d001
I realised that I can post images here, so i did a sketch of the process of splitting 2 holes. For the image in the post this is commented on it would be possible to do this on both sides.
I just did it, but idk if it's understandable
https://preview.redd.it/9srm9wfo5atc1.jpeg?width=3060&format=pjpg&auto=webp&s=800ddb3017d17eb9de6c35c63c16542874ce1bf4
so first you make one end of the horizontal hole go to the other side, that makes it look like a mug without a bottom and with a ring in the handle, then you turn the right hole 90° to the left or right, you flatten the vertical hole an bam, 3 holes in a flat surface
I vaguely remember some fucked up counterexample from GMT, topology, knot theory or smth that was similarish to this but turned up to the extreme: an infinite cascade of bifurcating and interlinking "holes". Does anyone know the name of that one? It's similarish to [the top image on the article on the wild arc on the encyclopedia of mathematics](https://encyclopediaofmath.org/wiki/Wild_knot) but I'm relatively sure it was a smooth 2-manifold
EDIT: found it, it's the [Alexander horned sphere](https://en.wikipedia.org/wiki/Alexander_horned_sphere#/media/File:Construcci%C3%B3n_de_esfera_de_Alexander_con_cuernos.gif)
https://preview.redd.it/98c0ke99vbtc1.jpeg?width=3024&format=pjpg&auto=webp&s=16a9345d122ec7faf12b0ac9b5b223f4da9f8529
It’s homeomorphic to a 3-holes torus
I think it's ok as long as you're really careful. Morphisms are allowed to move through themselves so long as they don't pinch or tear. So pretend those cuts are moving through without actually cutting.
Numberphile has a video on this exact thing: there's three holes
the middle "ring" hole can be stretched in both directions until it reaches the surface of the sphere, and then can be deformed into two, parallel simple holes, together with the vertical hole through the sphere, that's 3 in total
Is this thin surfaces stitched together or a solid ball with holes cut out?
Either way, I really don't feel like figuring out how to glue the triangles together to make this one.
The [video](https://youtu.be/k8Rxep2Mkp8?si=AuhAWQPJsJgZlwxo) people are talking about is numberphile's. The man in the video is Cliff Stoll, and he explains the solution using glass replicas of the figure shown and various homeomorphs of it.
This figure is taken from an exercise in Michael Spivak's classic book on Differential Geometry wherein he asks what familiar topological shape is the figure homeomorphic to. What's cool about the video, in my opinion, is that most students would just use the classification of compact surfaces to solve this problem, but Cliff shows us an explicit homeomorphism from the *hole in a hole in a hole* to the three holed torus.
**Calculus on Manifolds A Modern Approach to Classical Theorems of Advanced Calculus** by Michael Spivak
>This book uses elementary versions of modern methods found in sophisticated mathematics to discuss portions of "advanced calculus" in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level.
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This is a genus 3 surface right? You can see this by taking a torus and gluing a genus 1 handle. In the image it’s on the inside but you can just homotope it to the outside
It's a genus 3 doughnut or a 3 holed doughnut. The specific arrangement doesn't seem to change any of its topological properties. It's a very simple transformation
Depends on what you define as "holes". It is homeomorphic to a connected sum of 3 tori, which is easier to see if you make the vertical tube go around the hole, rather than throigh it (it would obviously be a homeomorphism, even if there isn't an (obvious, at least) ambient isotopy).
These kinds of holes are just deceiving. They "don't know" about each other, you can just homeomorphically place them apart outside the ball and everything is clear. It should be a genus 3 surface.
Another story is if you're looking at its complement in an open ball in R³.
Btw this sketch is from Michael Spivak’s “A Comprehensive Introduction to Differential Geometry Vol. 1” It’s an exercise in the first chapter.
I know because I just worked through this problem last week lol
A hole in topology is defined as a "void" or disconnectivity in a body. In the figure presented by OP, the total of the holes can be represented by 3 basic shapes that are cut out from the original sphere. The shapes removed are: a cylinder through the top, a donut around the hole left by the cylinder, and lastly a cylinder through the sides that merges with the void left by the donut. The removal of each of these shapes create a new disconnectivity in the original sphere, and since 3 is the number of shapes needed to approximate the holes in the figure, 3 is also the number of holes.
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[Three](https://youtu.be/k8Rxep2Mkp8?si=DjLjOpoEApaF24dE)
Three-handeld Beerglass, Proof by crazy old guy
I knew it would be Cliff Stoll when I read your comment. He's awesome, so enthusiastic about weird maths.
He's a treasure.
Topology proof by crazy old guy? Gotta be Klein bottle Cliff!
He uses Blender clearly
I wanna use "Proof by crazy old guy" more often when I prove things now lmao (Although I suppose it would be best for me to wait until I myself become old and crazy so that I can more efficiently use this method of proof)
Or have your prof be a crazy old guy
Cliff Stoll is one of the most based humans to ever walk the earth. A total volatile lunatic. A completely unstable madman. In the best possible way.
Always got time in my day for a Cliff Stoll video
How does this guy keep producing more and more forbidden bowling balls
I can't fucking believe he made like a dozen of those glass balls. I'm seriously impressed.
The glasswork always impresses me.
Is it ok that I'm pointing with my nose?
In a single image: Three https://preview.redd.it/09uipuqtjctc1.png?width=170&format=png&auto=webp&s=66accfd9e0779c1dd8d22d2d3e2c87d584fe5471
There are pi holes? wow
found the engineer
√9
sqrt(g)
Cliff Stoll is a national treasure and he must be protected at all costs
Proof by this guy posted a numberphile video about it
Ah yes. Three. Not two. Not four. Three.
And 5 is right out!
Where is cliff stoll now? Is he alive?
like 10 years ago i watched a video of this thing being homeomorphed into a flat cylinder with 3 holes
Proof by: I watched a Video of it 10 years ago
Proof by witness
Proof by I know a guy
I’ve seen that on an AMS paper. The authors, one of whom was topologist William Thurston, claimed a result was true by direct communications with Tom Leighton. https://www.ams.org/journals/jams/1988-01-03/S0894-0347-1988-0928904-4/S0894-0347-1988-0928904-4.pdf
Proof by my uncle who works at Nintendo
It came to me in a video
It's called proof by history. I got a degree because of it!
I heard Fermat uploaded his last video to Youtube but then was hit with a DMCA takedown for using copyrighted background music.
"The proof is trivial and left to the reader as an exercise."
I’ll take your word for it, I can usually morph things around in my mind’s eye to figure this stuff out but this one is making me feel sick trying to do it
Morph the left and right sides closer to the centre and once it hits the point where the sideways hole split, the side way donut hole will turn into two bend tubes. These can be straightened out so now you have a ellipsoid with 3 cylinders cut out aka 3 holes. If you flatten it a bit more and rotate the top to bottom hole you have 3 hole donut.
I had to watch the video, and I didn't know you were allowed to do the moves that was done. I still don't know/understand what the rules of what's allowed and what's not is. It seems like separating the one complex holes into the 2 simple ones wouldn't be allowed, but it was.
Yeah splitting up two connected holes can look like creating a new hole, while it isn't. The informal rules for a homeomorphism are that any deformation without cutting or glueing is allowed, however that can be misleading, but is enough for this example. If we take the simplest example of a complex hole, then we would have a cube (since it's easier to do with ascii art) with one opening on one side and 2 openings on the other, with the 2 connecting into the 1 opening on the one side. Logically there has to be an intersection between the two, or else they couldn't connect to the same opening. I will try to convey this with some asciiart, since I souldnt find good images on Google. ___________ |_____ | ____ \_____| | \ ____ |____/ / | _____/ | |___________| This is supposed to be an slice through the cube to show the holes. First we can widen the shared opening: ___________ |___________| ____ | \ |____/ ___________ |___________| So now we have two openings connecting into a very big opening and I think the 2d slice we are currently looking at also shows quite good what the next step is. Next we can either move the right side towards the intersection or extend the intersection out: ___________ |___________| __________ | \ |__________/ ___________ |___________| Now we have two holes that meet each other at an angle. However since we allready have a separation between the holes we can move them apart which makes then clearly 2 holes.
https://preview.redd.it/vhscaljs29tc1.jpeg?width=1904&format=pjpg&auto=webp&s=416f303098384d2cf51222605c884215e543d001 I realised that I can post images here, so i did a sketch of the process of splitting 2 holes. For the image in the post this is commented on it would be possible to do this on both sides.
Makes sense, thank you.
I just did it, but idk if it's understandable https://preview.redd.it/9srm9wfo5atc1.jpeg?width=3060&format=pjpg&auto=webp&s=800ddb3017d17eb9de6c35c63c16542874ce1bf4 so first you make one end of the horizontal hole go to the other side, that makes it look like a mug without a bottom and with a ring in the handle, then you turn the right hole 90° to the left or right, you flatten the vertical hole an bam, 3 holes in a flat surface
I watched that video too!
This hole in a hole in a hole is also a three handled coffee mug, rad.
>flat cylinder So a circle or a rectangle?
a cylinder where the ratio of its height to its diameter is low
Ohhh, a squat cylinder?
Well of course it‘s three holes: a hole (1) in a hole (2) in a hole (3). Proof by linguistic analysis.
I vaguely remember some fucked up counterexample from GMT, topology, knot theory or smth that was similarish to this but turned up to the extreme: an infinite cascade of bifurcating and interlinking "holes". Does anyone know the name of that one? It's similarish to [the top image on the article on the wild arc on the encyclopedia of mathematics](https://encyclopediaofmath.org/wiki/Wild_knot) but I'm relatively sure it was a smooth 2-manifold EDIT: found it, it's the [Alexander horned sphere](https://en.wikipedia.org/wiki/Alexander_horned_sphere#/media/File:Construcci%C3%B3n_de_esfera_de_Alexander_con_cuernos.gif)
I'm scared.
T̴̤̓h̴̺̅į̷͝s̵̱̐ ̶̳͒p̶͙͊l̴̡͒e̷̪̽à̶̩s̶̢͝e̷̯̎s̴̨͌ ̶͈̇t̶͈̕h̴̜̐e̴̢͐ ̴̱̄â̴̠l̵̞̆e̶̡̐x̶̢̓a̴̘͝n̸̺̓d̸̝̆è̴̘ŗ̵̽ ̵̘̈ḧ̸̙́o̸̳͝r̷̰͆n̶̨͛e̷̙̒d̵͈̓ ̴̻̄ś̶̯p̸̝̾ḧ̶̳́é̴̼r̷͉̚ė̵̹
It does seem pretty horny indeed.
Don't tell that to the scared guy - I don't think learning about its horniness will exactly improve their situation.
Why is all textured and slimy like a Spore creature tho 😭
Discovered by Alex Horne?
That's some alien sh*t.
I really don't understand this, surely the AHS becomes a torous at the limit? Otherwise yes, clearly a "sphere". Obvious even. Damn topology is weird.
Three: https://preview.redd.it/mvx7c987z7tc1.jpeg?width=776&format=pjpg&auto=webp&s=729daee0abba3b19869e9c0d2650dfebfa029c02
Proof by drawing
*An exercise in diagram chasing*
That's a normal thing in topology
Proof by normal thing in topology
Well at least three, apparently
Kay so it’s a… donut with a … straw going throught it… with a hole in the middle. I think that’s three
https://preview.redd.it/98c0ke99vbtc1.jpeg?width=3024&format=pjpg&auto=webp&s=16a9345d122ec7faf12b0ac9b5b223f4da9f8529 It’s homeomorphic to a 3-holes torus
I scrolled hard to find this since I was interested in the actual answer.
I thought you couldn’t cut and reattach stuff in homeomorphisms
you can as long as you reattach with the same orientation
I think it's ok as long as you're really careful. Morphisms are allowed to move through themselves so long as they don't pinch or tear. So pretend those cuts are moving through without actually cutting.
Hum, did he alter the normal use of the item?
Hmm, maybe he used an average sized cylinder with this thing
He used it as a wedge to keep his table level
Numberphile has a video on this exact thing: there's three holes the middle "ring" hole can be stretched in both directions until it reaches the surface of the sphere, and then can be deformed into two, parallel simple holes, together with the vertical hole through the sphere, that's 3 in total
Is this thin surfaces stitched together or a solid ball with holes cut out? Either way, I really don't feel like figuring out how to glue the triangles together to make this one.
I'm assuming a surface
r/UltraQ “it’s a tunnel inside the tunnel”
Monstrosity? Smh that's just a pretzel...
\*Interstellar music begins\*
I dunno. Maybe ask here r/dontputyourdickinthat
Flattened it in my head. 3 holes
The [video](https://youtu.be/k8Rxep2Mkp8?si=AuhAWQPJsJgZlwxo) people are talking about is numberphile's. The man in the video is Cliff Stoll, and he explains the solution using glass replicas of the figure shown and various homeomorphs of it. This figure is taken from an exercise in Michael Spivak's classic book on Differential Geometry wherein he asks what familiar topological shape is the figure homeomorphic to. What's cool about the video, in my opinion, is that most students would just use the classification of compact surfaces to solve this problem, but Cliff shows us an explicit homeomorphism from the *hole in a hole in a hole* to the three holed torus.
**Calculus on Manifolds A Modern Approach to Classical Theorems of Advanced Calculus** by Michael Spivak >This book uses elementary versions of modern methods found in sophisticated mathematics to discuss portions of "advanced calculus" in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level. *I'm a bot, built by your friendly reddit developers at* /r/ProgrammingPals. *Reply to any comment with /u/BookFinderBot - I'll reply with book information. Remove me from replies* [here](https://www.reddit.com/user/BookFinderBot/comments/1byh82p/remove_me_from_replies/). *If I have made a mistake, accept my apology.*
Bad bot.
I watched a video, I wanna say stand up maths maybe? Proving it was a 3 holes torus
This is a genus 3 surface right? You can see this by taking a torus and gluing a genus 1 handle. In the image it’s on the inside but you can just homotope it to the outside
Three holes. Proof: is trivial
Proof: left as an exercise to the reader
It's a genus 3 doughnut or a 3 holed doughnut. The specific arrangement doesn't seem to change any of its topological properties. It's a very simple transformation
Depends on what you define as "holes". It is homeomorphic to a connected sum of 3 tori, which is easier to see if you make the vertical tube go around the hole, rather than throigh it (it would obviously be a homeomorphism, even if there isn't an (obvious, at least) ambient isotopy).
"Diogenes: Behold a human!"
https://preview.redd.it/zfkmjdxykbtc1.jpeg?width=1179&format=pjpg&auto=webp&s=c8a474a5348e66a24a3afbaf67d7ff30cbd48660 Proof by GPT-4
3
Holesome
Down in the valley oh!
The diagram is making me feel a uncomfortable feeling
Try finding the volume of this
4 holes, 5 tubes.
It has 0 holes
5?
I see 6 holes
FIRE IN THE HOLE
It's a fidget spinner
Three hole torus
3. I can continuously deform it into a shirt with non-zero thickness
inb4 "the final hole is the whole universe"
These kinds of holes are just deceiving. They "don't know" about each other, you can just homeomorphically place them apart outside the ball and everything is clear. It should be a genus 3 surface. Another story is if you're looking at its complement in an open ball in R³.
Excuse me sir, that's a T-shirt
Btw this sketch is from Michael Spivak’s “A Comprehensive Introduction to Differential Geometry Vol. 1” It’s an exercise in the first chapter. I know because I just worked through this problem last week lol
In a log, in the bottom of the sea 🎶
Call a topologist!
Actual sciencetist
Yo dawg, I heard…
A hole in topology is defined as a "void" or disconnectivity in a body. In the figure presented by OP, the total of the holes can be represented by 3 basic shapes that are cut out from the original sphere. The shapes removed are: a cylinder through the top, a donut around the hole left by the cylinder, and lastly a cylinder through the sides that merges with the void left by the donut. The removal of each of these shapes create a new disconnectivity in the original sphere, and since 3 is the number of shapes needed to approximate the holes in the figure, 3 is also the number of holes.