Smale's inside out paradox
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- Опубликовано: 4 ноя 2016
- This week’s video is about the beautiful mathematics you encounter when you try to turn ghostlike closed surfaces inside out. Learn about the mighty double Klein bottle trick, be one of the first to find out about a fantastic new way to turn a sphere inside out and have another go at earning the Mathologer seal of approval by accepting the Mathologer inside out challenge.
Latest news (November 7, 2016): Arnaud Chéritat just finished an absolutely stunning animation of the deformation of the outer dome that I talk about in this video. Check it out! www.math.univ-toulouse.fr/~ch...
Make sure you explore what the sliders can do and rotate the model around with your mouse.
More latest news (November 10, 2016): Arnaud just rerendered his torus eversion in HD using a colourblind friendly colour scheme. Here are links to the versions showing the full torus and the half torus • Torus eversion: colorb...
• Torus eversion: colorb...
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Mathologer inside out challenge
1. Marco Souza de Joode, 2. Max Brain, 3. Cory Williams, 4. Stefan Linden, 5. Saelben Noa, 6. Mehmed Adzemonic, 7. Lachie Miles, 8. Rory McAllister, 9. Alejandro Robles, 10. Marcin Szyniszewski, 11. Sam Jones, 12. Jack Leightcap, 13. Christian Callau, 14. Richard Schank, 15. Daniel Feuerstein, 16. Irene Meunier, 17. Sinom, 18.Denny Eggroll, 19. Joshua Pirie, 20. Grillet Lucien, 21. Lea Werle, 22. Dominic Birkwood, 23. Andrei Maria, 24. Marco Rozendaal, 25. Khalis Totorkulov, 26. Kevin Tsang, 27. Thiasam, 28. Batonkal, 29. Grillet Lucien, 30. Arnaud Cheritat, 31. Cichy Wodór, 32. Manex Vallejo, 33. Matthew Giallourakis, 34. Eric K., 35. Kai Wolder, 36. Mei Li, 37. Mad Cuber, 38. Nelly Lin, 39. Sam Amber, 40. Devansh Sehta, 41. Samuraiwarm Tsunayoshi, 42. Joris van Duijneveldt, 43. Craig Montgomery, 44. Warren Brodsky, 45. Jonathan Fowler, 46. Nathan Petrangelo, 47. 정재윤 , 48. George Milis, 49. TrianguloY, 50. Potii92 (Daniel), 51. Ha Quang Trung, 52. Jerry Stoops, 53. Conall Kavenagh, 54. Dean Reichel, 55. Pavel Klimov, 56. Griffin Keeter, 57. Tian Chen, 58. frobeniusfg (Andrew), 59. Nathaniel Gofourth, 60. Benjamin Seidel, 61. Miloš Stojanović
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Check out the following animations of different ways to turn a sphere and a torus inside out
• Sphere Eversion : teaser Arnaud Chéritat’s sphere eversion (bottom right among the four eversions I show). The animation is joint work between Arnaud Chéritat and Jos Leys (make sure to also check out Jos Leys' channel and website/in my list of recommended channels).
• Torus eversion: turnin... Arnaud Chéritat’s torus eversion (the half torus version • Torus eversion: turing... ). Also check out Arnaud’s website for other mathematicial treasures www.math.univ-toulouse.fr/~che...
• how to theoretically t...
• Sphere Inside out Part... the video “Outside in” split into two parts (Thurston’s eversion, top right among the four eversions I show). An absolute must-see !! I think Outside in and what I talk about in this video complement each other very nicely. The clip at 3:00 is also part of Outside in.
• The Optiverse the automatic “Optiverse” eversion (bottom left among the four eversions I show). Also check out this really nice write-up by John Sullivan torus.math.uiuc.edu/jms/Papers...
• Mathematics visualizat... the “Holiverse” eversion by Iain Aitchison another (just like me) mathematician from Melbourne, Australia. Read about his eversion here www.ms.unimelb.edu.au/~iain/to...
• Turning the Sphere Ins...
• Turning the Sphere Ins... “Morin’s eversion” (top left among the four eversions I show). This first animation of an eversion was produced by Nelson Max.
• A simple sphere eversion “deNeve/Hills eversion” Also check out these pages for more details about this eversion: www.usefuldreams.org/sphereev.htm and www.chrishills.org.uk/ChrisHil...,
For a very nice history of sphere eversions visit this page torus.math.uiuc.edu/jms/Papers...
Here is a link to Derek Hacon’s notes on his eversion hosted on his son’s Christopher Hacon’s website www.math.utah.edu/~hacon/spher... (Christopher Hacon is also a mathematician). Here is my adaptation using level curves: www.qedcat.com/misc/deformatio...
(and, of course, now there also Arnaud's animation that I mention earlier on.)
And here is another writeup of Derek Hacon’s eversion by his PhD supervisor E. Christopher Zeemann zakuski.utsa.edu/~gokhman/ecz/...
Thank you very much to Arnaud Chéritat, Christopher Hacon and Cliff Stoll for their help with this video.
Enjoy,
Burkard
One more video credit: The nice clip of the punctured torus turning inside out at is based on a video by Greg Mcshane: • torus eversion
“Is this it, Is this a sphere turning inside out!”
You bet! That wasn't easy to follow, was it?
*?
A sharp bend!
@@gigaprofisi check "outside in" on youtube, an old video
literally just watched that yesterday
"Watch out!"
*"THATS A SHARP CORNER."*
*LIKE A BASKETBALL*
YoU mUsTn'T tEaR oR cReAsE iT
I remember watching that video on how to turn a sphere inside out years ago :D
That's Outside In (link in the description). I'd really recommend it to everybody. Actually that's the video that got me interested in sphere eversions in the first place :)
I remember it being a recommended video for like 10 years, then finally watching it.
+martinshoosterman the same happened to me and then it never showed up again.
Tsskyx that was basically how I learned English. I remember trying to guess what they meant by "frown" and "saddle". Best teaching video ever.
William Pereira Gomes same
that specific video from 94 is every youtube addict's introduction to topology. a very slippery slope to the math side of youtube also known as the klein kult
So im not alone, I just watcheed that video and now I'm here
Mathologer: *Explains an interesting spacial "paradox"*
Me: "Wait, why does his arm appear behind the transparent images, while his shirt appears in front?"
Not sure about that, they don't seem layered.
His shirt isn't 000000 though
@@Marhathor I mean. Download the video and look at it.
58 67 68
#3a4344
It's definitely not black. Its luminance value is 59, almost exactly 1/4 of white.
his skin is green but it was colour corrected to be skin colour
THANK YOU! I have always wanted to turn my donuts inside out!
:)
So your donuts have no mass and can pass through each other? That doesn't sound very filling.
Key_Of_Destiny ba dum tiss
😭😂
@@key_of_destiny4712 the donut eats him
Is it just me or is everyone that speaks about Klein bottles online completely obssessed with them?
No, I can speak about klein bottles just fine. I mean, they're pretty cooland all, but no- _Oh hey look! Kein Bottles!_ *Klein Bottles!*
I cannot talk about Klein bottles normally.
Also after playing FNV I cannot talk about them or Mobius strips without pretending I'm one of the Think Tanks shitposting about each other
@@TheSentientCloud yeah, i am too mobias like some are too klein...
Klein bottles are funny
I recognized cliff from numberphile! and recognized your inside out sphere video.. I love this channel!
Cliff is awsome ^^
Inside out is awesome ^^
Arthur Reitz yup c:
Cliff for president!
He's seriously so awesome though.
Cliff is my spirit animal
He's the adorable old grandpa I aspire to be like. Super *true* passion. It's so hard to find people like that anymore. He gets so excited to talk about Klein bottles and is all hand flappy and happy about them and it's adorable as fuck.
when i saw the first part of the sphere inversion, my first thought was "hey that kinda looks like a sphere with a slightly larger torus around it, is that how its gonna work?"
A Mathologer and a 3blue1brown video within 24 hours, both on topology. :D
Coincidence?
I think not!
Great news: Arnaud Chéritat just finished an absolutely stunning animation of the deformation of the outer dome that I talk about in this video. Check it out!
www.math.univ-toulouse.fr/~cheritat/eversion/Hacon/
Make sure you explore what the sliders can do and rotate the model around with your mouse.
Question. I would have expected the green "outside" and purple "inside" switch places at the end of the deformation. But at the end, the outside is still green and inside is purple. What am I missing here?
woowooNeedsFaith The outer dome becomes the inner dome, so at the end it's inside out.
Have another look at the part where I explain the Hacon eversion. The two domes just exchange places and in this way contribute to the overall surface turning inside out :)
Perhaps a version with stripes, perhaps semi-transparent, would be more helpful for visualizing the whole shape?
Mathologer How dare you not link to the "how to turn a sphere inside out video"!!!!!
More latest news (November 10, 2016): Arnaud just rerendered his torus eversion in HD using a colourblind friendly colour scheme. Here are links to the versions showing the full torus and the half torus ruclips.net/video/INdOWVFb8fk/видео.html
ruclips.net/video/Cw4aTVi8ndQ/видео.html
Still having a hard time wrapping my head around the fact that this does not count as breaking the surface.
Think ghosts :)
Because it can pass through itself
I agree. Most of these inversions involved self-occlusion. If that's allowed then why is the circle impossible -- just make one side ghost through the other, and you're done.
@@MijinLaw That’s really the same, as pushing the North and South Poles of the sphere through each other. What made it work, for the sphere, was that pumpkin-looking intermediate phase; but, in a 2D-plane, there’s just not enough freedom of movement to do that in. If you embedded the circle into our 3D-space, it really wouldn’t have inside and outside, anymore, would it, now?
Two weeks ago at the university where I work people gutted the room I usually record my videos in and so today's video had to be done in a different location using a very different setup and probably has a little bit of a different feel to it than usual. How does this work for you?
Mathologer it works terribly for me :(
Seriously?
It's good, just too much backlight. Great content as always though 👍
I didn't notice to be honest
+TomRaj Good to hear. I am now experimenting with a setup that is similar to what the weather people on TV use. Still needs a bit of tweaking but may eventually be a blessing in disguise that they destroyed my home for making these videos :)
Mathologer: Half is green and half is red.
The 5% of men who are red-green colorblind: If you say so.
“Remember, you mustn’t crease or tear it”
ok, you can turn a sphere and a torus inside out, but what about a Klein bottle?
Just you wait for the Christmas video. You are in for a surprise :)
A Klein bottle can't be turned inside out because it doesn't have an inside and an outside, it's a one-sided surface.
It is inside out and inside in at the same time.
@@midiplay A double Klein bottle has 2 surfaces!
*Look out, that's a sharp bend*
Outside In has enjoyed quite a renaissance on RUclips lately, so it's no surprise the algorithm directed me here.
Just ordered a small glasses case like that from grand illusions in Great Britain, so laughed when you showed your case at the very end.
If you are interested in inside out toys maybe also check out the Switch Pitch. I show that at the end of one of my early videos: ruclips.net/video/gWgXlvIlpzM/видео.html :)
I remember watching the sphere eversion and it absolutely blowing my mind :D
Thanks Doc. Didn't expect to get my Daily Dose here on youtube.
And here I am, barely able to turn my socks inside out.
What a stunning way to turn a sphere inside out. I have seen the Outside In video many times and it used to be the "easiest" method to visualize. But this method is so elegantly simple. Tsk tsk to the magazine for deeming it too trivial - I was always taught that the shorter the solution or proof, the better it was. Thank you for sharing this with the world.
Actually, Outside In was what got me interested in this subject. Definitely one of the best math videos ever produced. Still, even after watching it 10+ times and reading up on the background of the particular eversion shown in that video and pretty much all others out there I was never completely satisfied until I stumbled across Derek Hacon's notes.
Are you sure Hacon is dead, though?
Probably one of the most interesting videos on your channel. Immediately I imagined a Mobius strip and two points on it at a distance A (there is only one dimension - length). Then I made a cut between points A / 2 and got two measurements: length and width. By a simple action, he increased the number of dimensions in space. Naturally, if you glue the Mobius strip again, the number of measurements will decrease. Wonderful math. How interestingly space and time are arranged.
Very nice Mathologer! As always, thanks for your excellent and clear explanations. Also, it was cool to hear you use the particle/antiparticle analogy in the video :D
Yes, I really liked that analogy when you mentioned it the other day. Actually I also remembered where I had seen this Rubik's cube particle antiparticle business before. It was not in one of Hofstedter's articles/books. It was in an article by Solomon Golomb. I must have it somewhere but is probably also easy to find if you are interested :)
Thanks for the suggestion, I'll look it up!
The nice thing is that I think you could also perform the torus inversion on solids if you consider 4D space, because the place where the klein bottles would intersect could be separated then
This was quite a fun challenge to work out!
YEAH! The sphere inside out video! I saw that video for the first time about 10 or so years ago, and to see it appear in a video like this is really cool.
(after watching the video a bit more, the specific video I mean is the one that he shows a clip of a 3 minutes)
Yes, that's Outside In (link in the description). That's also the video that got me interested in this subject. Still one of the best maths videos ever produced in my opinion :)
Beautiful
The sphere vid went viral a few years back to the point it was a meme in and of itself how viral it was. I think I was one of the few who actually loved it lol.
Here's a question. Using the same set of rules, imagine two concentric spheres (or toroids). Your goal is to invert this so the inner and outer swap places, but here's the challenge: the inner sphere cannot touch the outer sphere.
Is it possible for them to swap places?
Same here. That video is gold.
As for your question, it doesn’t seem possible, at first glance; but there just might be some obscure trick you probably wouldn’t think of. I mean; I was pretty tempted to say turning a sphere inside out is impossible, but I was wrong. I’m thinking: Turn both of those spheres/tori inside out (I’m assuming you can change their sizes, because it’s topology; so, you can make sure (at least, in the spheres’ case) that they don’t touch, in the process); and then, just define the inside (the volume around the center of the spheres/tori) to be the new outside, and the outside (where the observer is probably situated) to be the new inside; but that might be kind of a mathematical/semantic cheat. I don’t know. You decide. 🤷🏼♂️
@@PC_Simo Aw great, now this thing that I was thinking about 4 years ago is back in my head, but I'm actually currently holding space for trying to calculate the time complexity of those water sort puzzles. I asked an old professor to not tell me the answer but tell me if I was on the right track, and well, I ended up with O(2n^2-3n+1) which is O(n^2) which he ended up saying was "close but no cigar." I'm still trying to figure out what I did wrong, because when I worked it out the apparent growth was 3-10-21... which of course are the even triangulars which the formula I wrote describes... but I think I may be missing a step? Maybe I need to go deeper in the recurrence... I may only have *half* the solution... and I need to finish solving the recurrence out. Maybe what I have is not O(2n^2-3n+1) but instead T(2n^2-3n+1)... IDK my dude you could reply to me 7 years from now and I'll be telling you my PhD thesis for some new problem in graph/network theory
But good to know a "new" problem to occupy my mind once I finish solving this one... but I guess half a math and half a CS degree ain't enough for this one lol.
Klein bottles are the answer the life, the universe, and everything it seems...
Yes, once you notice them, they really seem to pop up everywhere :)
Yes, small bottles are the answer
"but can you turn a sphere inside out?"
This gave me flashbacks
Can you turn a hyper sphere inside out? and is there a general rule for an N ball?
I seem to remember reading that apart from the 2d sphere only the 6d sphere can be everted :)
Nice question. I checked, and it is covered here for an n-sphere (not n-ball). math.stackexchange.com/questions/479383/turning-higher-spheres-inside-out
Mathologer's memory was right on. The answer also mentions that you can also evert S^0 (i.e. a pair of pointa).
Roice Nelson Whats the differance between an n ball and an n sphere?
martinshoosterman, an n-ball includes the volume inside an (n-1)-sphere. So a 2-sphere looks like a 3-ball from the outside, but the 2-sphere is just the surface.
en.wikipedia.org/wiki/Ball_(mathematics)
Roice Nelson thanks.
This feels like turning a sphere inside out at the start...
God Bless you & I thank you for your teaching.. i just can't shake off the feeling you are a bit Math Magician..
Glad you enjoy all this and, yes, every once in a while I indulge in a bit of math magic :)
klein bottle guy PogChamp
UbererSK SeemsGood lul
Be careful not to make sharp bends though
Mathologer is my favorite multidimensional RUclipsr
Excellent video. So another 4D trick
love your stuff
That's great and thank you for saying so :)
Wow. I'm a french student in Toulouse, and, next monday, Arnaud Chéritat will introduce the sphere eversion to us. Now, thanks to your video, I'm really excited about this subject :D
That's great :) I had a bit of an e-mail exchange with him over the last couple of weeks while I was working on this video. His sphere eversion is also very nice and actually is a little bit similar to the one by Hacon that I focus on in this video. Anyway, a really nice guy with very good taste in mathematics. So, that talk should be great fun :)
The glass box, I've got one before. Amazing designing!.
5:18 Also; there’s not enough freedom, in the 2D-plane (you can’t do the ”pumpkin-trick”, for example); and, if you lift the circle outside of the plane, then it doesn’t really have an outside and an inside, anymore, does it?
Double möbius strips form a klein bottle, and double klein bottles form a torus. brilliant
Thanks Doc
Torus: *turns inside out*
Brain: *turns inside out*
Great stuff!
Welcome back Steve :)
Thank you, Sir. I've got most of my stuff moved into the new shop now. It's in utter chaos still, but at least I can start using my tools again. I hope to be back into the full swing of things pretty soon.
thanks doc
Could we turin people inside out?
Would not make a good disney movie
It was also part of a Goosebumps book that I forget the name of. Back in the day...
Put your finger in their A$$Hole and pull out. That's it...person inside out...
topologically three-hole torus (2 nostrils, mouth and anus, all connected). Affix nose tip, shorten palate so that you have just the nose between two nostrils and mouth, over a large throat cavity. Then stretch anus to some 2 meters, stretching the bowels as needed. You end up with a disk - skin on one side, intestine inner surface on the other, with face and 3 small holes in the center. Then proceed with the double Klein bottle trick to flip that inside out...
We can turn ghost people inside out. So, sure, definitely a good movie there.
Easy, reach down their throats, grab them by the ankle, and yank. Boom! Inside out.
👁👁👂🏼👂🏼👁👁👂🏼👂🏼very interesting video. First time hearing about the Kline bottle. Regards✋🏼
Thanks Mathologer, im now in the description!
Yes, you are. Glad you enjoyed the challenge :)
Your deformation level image reminds me of some work Scott Carter did in knot theory, showing the relationship between S^2 surfaces imbedded in R4 and and slices of them as S^1 in R3, i.e. knots. Cheers.
For the challenge, please just send me an e-mail with a link to your video (burkard.polster@monash.edu). Some people sent me private RUclips messages, however, for some reason they don't show up :(
Mathologer Here's an easy way to make the ring: make a cylindrical ring out of a 4x1 rectangle and flatten it, then fold it in half. You should end up with 4 square faces. Next fold the square on the diagonals.
Where can I find a file to download this challenge?
AdrenalineL1fe No, I don't check my spam folder :)
Absolutely amazing (as usual) both the incredibly beautiful ideas and your excellent explanations. It is a shame most people wont ever be curious about mathematics just because they only see numbers and not the superb beauty in structure, pattern and ingenuity. Congratulations for creating such a wonder
Glad you enjoy my videos :)
Is it possible to evert a projective plane? If so, is it possible to evert all 3D objects (since they're always a connected sum of projective planes/spheres/tori)..??
this is actually very similar to one of the videos that you showed an example of, but in the video they do it at multiple points throughout the sphere instead of just one
Your channel is awesome full of great material, I want to ask you
Is it possible to have your slides?
Thanks
Thank you!!!
Awesome T-shirt!
Very interesting !
Since you enjoyed this, I'd say definitely also check out the video Outside In. One of the best maths videos ever made which complements what I talk about here very nicely (or the other way around, my video complements Outside In :)
This video is a reasonably good example of how "time" animates, as when running as a Computer Clock, the shaping/emulation of the interpenetrating images of reality, by the projection of QM-Time (tensor-vector) fields, mathematically, in the Universal Quantum Computation.
Fun to imagine.
This was neat. I never heard about this problem before.
Make sure to also watch that other video I refer to: Outside in. Link is in the description. One of the best maths videos ever made and the main reason for me getting interested in all this :)
Mathologer Thanks for suggesting that. I watched it and it was really good.
When the surface can pass through itself "like a ghost" is that because it's a 3d object in 4 dimensional space?
No, in this case it's really just a matter on defining it this way to get into territory where things get more interesting than the solid case. Having said that that whole self-intersecting business is pretty much forced on us when we try to visualise objects like Klein bottles in 3d :)
is it because 'topologically', the only problems are really when you have 'kinks', everything else, along with self intersection, can still be thought of in terms of 'smooth' curves/functions?
It's just a hypothetical object not a real thing. We just allow it to pass through itself to turn it inside out .
@@Mathologer but if we were to want to build these things as solid but maleable objects, instead of passing through themselves maybe some extra dimensions could make it achievable
what would it look like if you made a klein bottle (with the missing disc that your glass ones have) out of a flexible material like rubber and pulled the “handle” part upwards? i’m trying to imagine it folding in on itself but can’t.
Another way to put my question is: how could I perform the step shown in second 50 of the video with the inner tube of a tyre without actually punching two cuts in the inner tube?
Your video is mirror-inverted. One could say, you're turned inside-out....
Well played sir, well played... :D
(at least from 3:53 - 5:37)
Well spotted ! I actually messed up when I recorded the video in the part where I describe the twisting and the only way to fix this without redoing everything was to simply mirror-reverse the clip for this part of the video :)
3:00 Can't you do the normal torus inversion method there when you push in the 2 sides and have that ring since that ring is just a torus and then push them back out when the ring is is inverted
the donut style eversion is number 1 in my book
Can you make a tutorial about a DIY Klein Bottle please?
“Ah so it’s like undoing to loop of a belt!”
This is such the manifold. The double Klein. Or the Klein. Depends if you feel better having antimatter universe part of our manifold but twisted? Or if you want to make that twist a delineating factor. Is there a version that’s lowest energy?
I know it doesn't affect the math in the least but my brain screams "no, no, wrong!" at the racecar's port side being green and starboard red.
Can you transform the circle into something like a rectangle and then close it, i mean, put the bases together? Making something like a 2D torus, and then, do the same like in the torus
I'd better just stay in this dimension.
Firstly great video. Secondly that is a fantastic set of videos in the description. Thurston's one is something I'd seen previously. You make it easy to follow. Would you consider doing something on packing problems?
Second this, I think I might have requested this before. I think 2d packing problems in particular are really fascinating, because they look so elementary and make you think that mathematicians would have figured them out long ago, but in reality many aren't solved at all
Definitely a great topic and a really tricky one. Just to give a definition of an optimal infinite packing is not trivial :)
The very first visualization of how to turn a sphere inside out (using a Boy surface) was actually made by Bernard Morin and Jean-Pierre Petit in a _Pour la Science_ article in 1979. Check it ! www.jp-petit.org/science/maths_f/Retournement_sphere/Retournement_sphere4.htm
Topology and mindfuck are synonymous to me.
My 8 year old boy is telling me, but wait dad, you cant just make the skin pass through itself and make that klein bottle thing with a real torus. Son, that overlap turn move was a slight of hand that the great magicians would appreciate !
Just picked up my double Klein bottle from the post office and it's beautiful! www.qedcat.com/misc/double-klein.jpg
What would you call 1^-1 = 1 & 1^1 = 1 so that log(1)1 = -1 & log(1)1 = 1 so log(1)1 /= log(1)1? It's a paradox right?
Nevermind, why is log base 1 not defined?
because 1^x=1 for all x
So why is it important to avoid creases in topology? Is it similar to avoiding discontinuities in calculus? When you integrate a two dimensional function, you want there to be no discontinuities so that you don't have to split your function into multiple functions in order to integrate.
That's pretty much it except that topology dealing with objects like surfaces usually does not worry about creases, just about not cutting things. It's really a special class of surfaces that we are considering in this exploration.
What is the definition of the "ghostlike" property that allows you to carry out what you call the "double Klein bottle trick"? Without this I am stuck because when you twist the torus/half-torus it looks like you would have to puncture the surface, otherwise how would we ever see any of the inside? I'm imagining trying to evert a bicycle inner tube, which before this video I would have called a closed surface, but I am clearly missing something. If I got it correctly, you're not suggesting I could take a football or an innertube and actually evert it this way. So what is it about a "ghostlike" closed surface that makes this possible?
can you pour 2 different liquids in the double klein bottle? Will they mix? Will they fall out of the double-bottle?
Remember, the double klein bottle is still the torus in disguise, so it still has a single "inside" volume and a single "outside" volume. It would be interesting to think deeper about how to extend the ghosting rules to these two volumes, rather than just the surface (and if that can even makes sense). Assuming you can think about them in some consistent way, I would say you could get the liquids to mix or not, depending on which side of the torus you poured the two liquids.
could you show the full sphere deforming in the way you described smoothly and with shading
Arnaud Chéritat finished an absolutely stunning animation of the deformation of the outer dome that I talk about in this video. Check it out.
www.math.univ-toulouse.fr/~cheritat/eversion/Hacon/
Make sure you explore what the sliders can do and rotate the model around with your mouse.
This is awesome :) thank you .. but it doesnt tun inside out like how you say (green is still where it starts) but is still pretty cool
The green dome does not turn inside out, its position swaps with that of the red dome, which then results in the sphere turning inside out, have another close look :)
Whoa finally something intellectually puzzling for me!
Hey Mathologer, its really fun to watch your stuff. I got a question or more like a request if you are interested. Could you make a video about why you cant divide by zero? Lol that would be interesting to see what you have to say about it!
Funny enough this is sort of the topic of the upcoming video (well I'll show you how you do it anyway although it's forbidden :)
8:54
Look at your right arm. I think YOU are the real ghost here.
Yes, I definitely exhibit ghostlike features in this video :)
Alpha value up!
One interesting bit about everting a torus is that longitudinal circles are turned into latitudinal circles, and vice-versa. Perhaps you could edit that into one of the illustrations of the torus eversion? I believe I first saw this in a Martin Gardner article in Scientific American many years ago. P. S. Your videos remind me of his articles in many ways.
But the Klein Bottle requires 4 dimensions. Similarly, if you can use a higher dimension than your shape, inverting a circle is easy. Any child can flip a rubber band around. I don't know much about topology. What am I missing here? Thanks.
A rubber band is not two-dimensional. It is a torus rather than a circle.
If you try the same with a circle, you have to invert the orientation. You end up with a mirror of your original circle. The torus stays the same torus even when you turn it inside out.
But if we allow motion in 3d, then we can just flip the whole circle over to fix the inversion.
But that's not an inversion, flipping the whole circle over will not switch the inside and outside diameters.
Until I almost finished the video I didn't even realized this was 6 years old
I know i'm a bit late to the show, but where can i get / make a glasses case like that?
Suddenly i'm getting recomendations how to turn things inside out and it started of with outside in what?
What kind of sorcery you did at the end of the video!!! :'D
Same principle as the "Jacob's ladder" :) en.wikipedia.org/wiki/Jacob%27s_ladder_(toy)
Now I want a donut
Just make sure you don't go for the hollow version :)
I could give you a cup of cofee instead, they are not diffirent
Hmm. I wondered how the polar of each dome twisted? wouldn't it creates creases of some kind?
Torus Aversion - great name for a math rock band
i am bad at thinking about topology. but, why would there necessarily be a crease when you turn a sphere inside out?
Hmmm if we consider oriented surface (e.g. stripes on the surface of the torus, either going through the hole or around the circle), then there seem to be two ways to turn the torus inside out - one that preserves the orientation (the one you demonstrate) and one that doesn't (as if you "pull" the torus through a "hole" in its surface, the way you demonstrate with the double torus). The second seemingly ends with pinching but we can get rid of that if we apply the sphere inversion method on it.
Wait there, in the video he isn't pulling the torus through any hole in the surface; he's pulling it through the second torus. You can't just pinch a hole into the torus to turn it. That changes the topology of the torus and makes the problem trivial. For example, if you were allowed to do this, you could also turn a circle inside out by just ripping it apart, straightening it and then bending it the other way.
Or maybe I misunderstood your comment.
SpaghettiToaster there would be a hole where the two torus meet, in the video you can see there is a hole if you look close enough. The blue animation on the bottom is only showing the small torus
Red green color-blindness is very common. In general, different colors should be used so that more people can see, like blue and orange.