Smale's inside out paradox
ฝัง
- เผยแพร่เมื่อ 4 พ.ย. 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
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“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
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
"Watch out!"
*"THATS A SHARP CORNER."*
*LIKE A BASKETBALL*
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
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
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
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.
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
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 :)
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 th-cam.com/video/INdOWVFb8fk/w-d-xo.html
th-cam.com/video/Cw4aTVi8ndQ/w-d-xo.html
A Mathologer and a 3blue1brown video within 24 hours, both on topology. :D
Coincidence?
I think not!
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?
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?"
Mathologer: Half is green and half is red.
The 5% of men who are red-green colorblind: If you say so.
Beautiful
*Look out, that's a sharp bend*
Thanks Doc. Didn't expect to get my Daily Dose here on youtube.
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!
Torus: *turns inside out*
Brain: *turns inside out*
“Remember, you mustn’t crease or tear it”
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.
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
I remember watching the sphere eversion and it absolutely blowing my mind :D
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: th-cam.com/video/gWgXlvIlpzM/w-d-xo.html :)
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
This was quite a fun challenge to work out!
Outside In has enjoyed quite a renaissance on TH-cam lately, so it's no surprise the algorithm directed me here.
Thanks Doc
love your stuff
That's great and thank you for saying so :)
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 TH-cam 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 :)
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 :)
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?
Thank you!!!
thanks doc
klein bottle guy PogChamp
UbererSK SeemsGood lul
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!
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.
The glass box, I've got one before. Amazing designing!.
Mathologer is my favorite multidimensional TH-camr
Excellent video. So another 4D trick
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.
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 :)
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.
This feels like turning a sphere inside out at the start...
"but can you turn a sphere inside out?"
This gave me flashbacks
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 :)
Awesome T-shirt!
And here I am, barely able to turn my socks inside out.
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 :)
Be careful not to make sharp bends though
Double möbius strips form a klein bottle, and double klein bottles form a torus. brilliant
Thanks Mathologer, im now in the description!
Yes, you are. Glad you enjoyed the challenge :)
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.
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.
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 :)
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.
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?
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
thank you :)
👁👁👂🏼👂🏼👁👁👂🏼👂🏼very interesting video. First time hearing about the Kline bottle. Regards✋🏼
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.
I'd better just stay in this dimension.
the donut style eversion is number 1 in my book
Topology and mindfuck are synonymous to me.
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.
Red green color-blindness is very common. In general, different colors should be used so that more people can see, like blue and orange.
Wow.
Torus Aversion - great name for a math rock band
be even more impressed at a circle turning inside out
No can do, as I show in the video :)
Can you make a tutorial about a DIY Klein Bottle please?
“Ah so it’s like undoing to loop of a belt!”
This only works in theory
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
Until I almost finished the video I didn't even realized this was 6 years old
We all know why you're here. Smiles and frowns, right?
should've brought cliff stoll as a guest on this one
A tricky one since he is in California and I am in Australia. Also, Numberphile did such a great job with him ... :)
Whoa finally something intellectually puzzling for me!
"I hope you were as impressed as I was" yeah sure
One of those bottles would be the bottle of Klein , i mean that the curves have not to vary a lot , it would have a global minimum of variations of the derives for very high
mathematicians succeeding to draw it . If you see it and why.
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!
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
sweet
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 :)
Your channel is awesome full of great material, I want to ask you
Is it possible to have your slides?
Thanks
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 !
I thought Klein bottles could only have an odd number of openings.
how about that ring on the equator
I got lost from the donut.
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
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.
Turning that double torus inside out kind of looked like a 4 D sphere being rotated, just based off an animation that i saw
This is a nice follow up after seeing 3blue1brown's new video about topology.
Yes, that's a great one, isn't it :)
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.
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
3:00 Woah, I remember that video from like 8 years ago!
Yes, a real classic :)
"Just reach into the hole and pull out"
*THAT'S AGAINST THE RULES*
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 :)