@@KeenRunnerHonestly, that isn't always the right answer. No point staying and starving to death and if the forest isn't that big or you know in which direction it's located relative to the compass, you can use the sun (rises in east, sets in west) to figure out where to go.
Generally speaking there's very few places you want to actually stand still. At least in the US. Generally speaking unless you're in a very remote part of the us you want to walk in one consistent direction then hit a road. You need to know how to travel in the same direction but generally unless you're in areas where this doesn't work you're not likely to be lost. And if you're lost you just needed to not travel in a bad direction or go hike in a desert after being black out drunk. Usually within 3 days is 90% of us that people are actually in. And you can find water in most of these areas. Once you hit a road you can pick left or right and generally speaking be very near to people. Almost always you'll hit a house that you can ask for help or a driver you can flag down. But this assumes you've got somewhat knowledgeable information about where you're lost. So you'd want to do this hiking... Or, IDK you've looked at a map somewhat recently. It's not for for "stuck on an island" or ", lost in northern Canada" lvls of stuck.
@@TheBcoolGuy I have worked for search and rescue for my county’s sheriffs office in Oregon, always stay put, as long as at least someone knows where you are, and when you left we can find you, but in general keep a charged electronic device on you, if you can, that way if something does happen we can find you faster using your S.O.S feature or using calling your emergency service line. If you are lost or injured, don’t try to find your own way out, this is one of the fastest ways to get even more injured or even more lost, you have a much higher likelihood of getting recovered if you stay put, hopefully near a trail, if you know where you are and are not injured, that is a different situation. Currently SAR runs in all 50 states and most territories, with the exception of parts of alaska, we have the ability to make it to any location within the United Sates within 48 hours. Next time you go hiking: 1. Tell someone where you’re going, how long you will be gone, and what trails you will be taking. 2. Bring an electronic emergency device. 3. Unless you are a professional stay close to the given trail you were expecting to take. In any case you will be fine if you follow these steps.
@@weibrot6683 My brother in christ, I already use sponsorblock. That doesn't mean you can't salute creators who actually don't make it necessary to enjoy their videos.
It's all just taken straight from the corresponding Wikipedia articles. He doesn't cite them but if you look at the articles you'll see that a lot of the wording is just taken straight from there, but changed slightly. He's done far less work than it seems, and he doesn't credit the original authors at all. Straight up plagarism.
@@Bobo-ox7fj What do you have against Wikipedia? And besides, it's still true that the creator of these videos is profiting off of the work of others. It doesn't matter if you hate the people he's stolen from, that's still dishonest and wrong, right? If an academic paper plagiarised Wikipedia would you have a problem with that?
"What is the maximum packing density of worms of length = 1 in a bounded forest of arbitrary length and width, if worms cannot overlap each other or the edge of the forest?" - the next unsolved problem
Having looked through the paper cited in the wikipedia article on Bellman’s lost in a forest problem, it appears that the case of the triangle is actually solved. Like, wikipedia says the equilateral triangle case is still open, but the paper they cite presents a solution. (Edit: The wikipedia article has been fixed)
@@GarryDumblowski Simply because a better solution is known (and was proven to be optimal). It’s a zig-zag with three line segments at very specific lengths and angles that ends up being a bit shorter than a side length of the triangle, and hence better than a straight line.
@@T.THobbies Solving the problem means to prove that certain solution is right and I don't think that has been done. What has been done is finding a solution that mathematicians think is right. That doesn't count as a mathematical proof.
The fact we dont know square packing for either 11 or 17 is crazy to me. I was gonna guess 111 when some small number; and watch someone state n = 111 is actually known in the comments
In orienteering, lost in a forest is a common thing. That is why you are given a safety bearing which is a direction you are to walk to reach a known feature such as a road where rescuers will eventually find you. The only variation is when you are too injured to walk and you sit still and blow your whistle
Love how the video just ends, as if it was released mid-production and we can expect an update when the scholars get around to finding more impossible conundrums
when you show something interesting, like 1:37 or some equation, i think you should show it for more than a couple seconds.. had to get back and pause the video
Lmfao. I was watching this, and i got really confused because i kept rewinding to figure out where the first one started. Then i realized that my brain literally stops listening when you said "square packing" because i immediately thought it was a square space ad.
7:28 This section confused me because you said the upper bound was "improved" (implying "reduced") to a bigger number. It turns out you had the initial number correct: 0.845 The other two however, you have erred by replacing the 4 with a second 8. "0.88414" should be 0.84414, and "0.88409" should be 0.84409. We also have a lower bound on this problem of 0.832, so we have quite a good answer even if it's not perfect/proven.
If you don't know your position or orientation, heading in a straight line towards a corner takes longer than a straight line towards the middle of a side
The packing conjecture has real world applications! Assuming roughly spherical atoms, the tightest that atoms in a crystal lattice can pack is also 74%. There are two ways to do this: Face-Centred Cubic and Hexagonal Close-Packed (best to look at images, but basically each layer of spheres sits in the divots of the spheres beneath them). A direct application of this is something called a "slip plane". This is the plane through which a line of atoms can easily slip past each other, which is one of the ways materials can deform. As such, some metals like magnesium are stronger in one direction than another - all as a direct result of the packing of spheres!
What am I misunderstanding about Lesbegue’s problem? A circle with diameter 1 is a valid cover for all shapes with diameter 1, and for a cover to be a valid cover for all shapes of diameter 1, it has to be able to cover a circle with diameter 1, therefore the smallest possible cover is a circle with diameter 1…
It’s like they saw a transliteration of the Russian name or saw the Czech name „Stanislav” but then did some weirdness with the caught-cot merger so they spelt it with that o but then…didn’t bother to check? Idk 😂
11:25 Correction: As of December 21, 2007, the Kobon Triangle problem actually is solved at k=10, with the upper bound and best solution agreeing at 25 (unlike how the video states the upper bound is 26). k=11 is still unsolved though!
Someone a couple thousand years ago accidentally dropped ten wooden dowels down a flight of stairs and picked them up without realizing they accidentally had a really interesting thing
The last one is a blast, you can play with it pretty easy on a piece of paper: draw 10 lines and see if you can hit 26 triangles, the supposed limit. It's really hard to even hit 20, let alone the 25 of the best known figure (at exactly 11:26).
For the equilateral triangle, it's walk the amount of the shortest distance from the center to the edge, then go 45° (either left or right, doesn't matter) in a straight line until out.
I would not use the word approximate for the current best estimate for the square packing problem. I would say "less than or equal to" as in this is an upper bound on the solution, but it's unproven that a better packing does not exist.
The Kobon triangle problem is very interesting because it seems computationally approachable since the exactly angles of the lines don't matter as long as the topology of the graph (the order in which other lines intersect a line) they create doesn't change. I 'conjecture' that I am wrong in this assertion.
I feel like there’s an interesting discussion to be had about whether Ulam’s packing conjecture can be disproven using a non-convex solid which is constructed out of convex solids. (Picture the silhouette of a venn diagram or an hourglass)
Kepler's conjecture was proven in 1998 by Thomas Hales (well we can't be too sure since the proof was too complicated), getting officially published in 2005. Nobody was really sure about whether or not it contained an error, so in 2017, a team directed by Thomas Hales managed (with an insane amount of work) to get an entirely computer verified proof. So it has been proven, no doubt about it
@@avonbarksdale5821 What makes you say this? I can find no definition that says this. The Wikipedia page "List of mathematical conjectures" has a section called "conjectures now proven", leading me to believe they can be proven. Are you sure you aren't thinking of axoims?
To solve Lesbegue's problem, I think it's easier to think of it as "create the shape with the largest possible area, in which any 2 points are separated by no more than 1 unit length", as this will cover any shape with a diameter of 1 unit, rather than thinking about covering shapes with other shapes. The solution is either a reuleaux triangle or something very similar to it, but its exact parameters are very difficult to work out.
Do you have a proof that that would work, or is this just your intuition? My intuition tells me that this wouldn't actually result in the right answer. The Reuleaux triangle is the curve of constant width that encloses the largest possible area, so I'm guessing that you're correct that it's the shape you're looking for. However, a Reuleaux triangle of diameter 1 is not a valid cover. This is visually shown by an illustration on this Wikipedia article: en.wikipedia.org/wiki/Lebesgue%27s_universal_covering_problem As you can see, a Reuleaux triangle is included as one of the covered shapes, without being a cover itself (and indeed being significantly smaller).
The reason you should not go anywhere on a path is because it's likely that other people also will use the same path at some point, and they will run into you, especially if it's on a federally protected land, where there's more likely to be a park ranger who will do that trail every once in a while. This is incidentally why you should not leave trails. Unless you're extremely experienced or with someone who is or know how to navigate in the wilderness, regardless of if you know it well in this particular case or not. but as an aside gps devices are so much cheaper than they used to be, like to a ridiculous degree. you can have a little thingy that will tell you literally exactly where you are, and it probably has a function to call for help, and it sends out like a red alert to the nearest emergency service people who are trained to get people who are using gps because they have to. Even ten years ago, something like that cost five hundred dollars, unless you've had a lot of diy skills. now you can get one easily for under 100.
I like convex solids a lot. More than that sentence accurately communicates on its own. But in a way that I think is, at most, vaguely relevant in the context of ulam's conjecture.
LiaF reminded me of a similar problem. I saw a problem about a spaceship. I didn't solve it, but I thought of a similar problem in 2D. In LiaF terms, the forest is a half of a plane, but you know you're at the distance a from the boundary. I then changes to a different problem, found a solution for it and translated it to the 2D version of the original problem. Then, someone else came up with another solution. I successfully translated it to the different problem. It could prove that my solution was more optimal, but the calculations said otherwise
each point on the surface of a sphere is equidistant from the center of the sphere the ratio of a sphere with diameter x to a cube with side length x is π:6 in order for ulam's packing conjecture to be disproven, there would have to be some convex 3d shape A with a ratio of its own volume (or the volume of whatever pattern of multiple As is closest to 1:1 with a rectangular prism) to the volume of a rectangular prism equivalent in length, width, and depth less than π:6, which seems obviously impossible considering the equidistance of points on the outside of a sphere
Slightly upset your packing of balls was one of the more inefficient packings. Move the top row over 1/2 a ball width and you can push them closer together
for packing balls in space it doesn't matter whether it's a square or a triangle pattern. Both result in the same packing density, just different perspective.
@@Fassleexcept in both arrangements, layers need to be offset from each other. the goal was to visualize the concept of 3D packing with 2D circles, so they should have chosen the arrangement that shows up in both 2D and 3D.
Think of it this way, things are more efficiently packed (in 3D) when they have a larger percentage of their faces touching. Spherical objects only touch eachother slightly on the faces, because they have so much curvature.
What's funny to me about these videos is that I learned a lot of this from the video game 4D Golf from the little trivia crystals. Definitely recommend if you like mini golf and geometry
I"m confused by the cover thing. Isn't the smallest possible cover just the largest possible diameter 1 shape, which would be the circle, as any growth to the circle would make it larger, and any cover that could not cover the circle would not be a cover?
No, the cover has to have every possible shape of diameter 1 fit inside, not just take up at least as much space as every shape of diameter 1. An average broom takes up less volume than there is inside of an average backpack, but you can't fit a broom in a backpack. A disk of diameter 1 is not a valid cover; for example, an equilateral triangle with side lengths of 1 has a diameter of 1, and it cannot fit inside.
Great video. I think you could have left some of the example pictures on the screen for a few more seconds to give us time to see them. I had to rewind and pause a few times.
What’s really interesting (or frustrating, depending on your point of view) about math is that there are technically true statements in math which are nonetheless unprovable. So if something doesn’t have a proof it could just be because no one has found one yet, it’s true but no proof exists so any attempt to find a proof would be fruitless, OR you can’t prove it true because it’s false and you haven’t thought of the false proof yet. So there will always be more math to do
i am definitely missing something with the Lesbegue's covering problem. I dont see how it can be smaller than a circle, when it should be able to cover the circle?
@@isavenewspapers8890 oh. Mb. How about a pot of wings? Strap rope on birds and fly up. Jack sparrow did something similar with turtles, why not birds?
@@isavenewspapers8890heh, I thought I commented a few days ago saying “I’ll try it and let you know how it went” I went today (a few days later to seem as if I ‘actually did it’ and would tell you “it did work but the birds are a little hard to catch and wobbly. Set a few traps and caught a few. Tricked them with bird seed on a stick (like the carrot on a stick from MC) and they flew. After being able to see everywhere and be able to be taken anywhere, I let go of them and took the quickest path out”
For Lesbegue's Universal Covering Problem, would the solution with the smallest area not be 0.785? The question is what is the smallest area a shape can have that also covers every shape with the same diameter (a diameter being defined as the longest line you can fit within a solid that passes through the center). A circle of diameter 1 would cover every other possible shape with a diameter of one no matter the rotation, and the area of a circle with diameter 1 (3.14*(0.5^2)) is 0.785. Is there a rule I am misunderstanding?
I think I'm missing something about Lesbegue's universal covering problem: wouldn't the shape with a diameter of 1 that has the smallest area simply be a line with a length of one and no width?
@@IsmailLev hold on, translating your name so it's incorrect in English but makes more sense for Polish speakers. It's baffling that English still does it. We don't write "Stephen" "Stiwen".
@@DeuxisWasTakenThis is common in most natural languages. In Polish, transliteration of people's names isn't that common anymore, but we still do it - consider "Królowa Elżbieta II" (Queen Elisabeth II from Britain), or even "Chrystus" (Christ) for a biblical example. For everyday people, we also do this all the time for non-latin alphabet names, such as names in Cyrilic. Not to mention exonyms for country names - Łotwa, Holandia, Węgry, etc. Just because it's common doesn't mean it's a good thing of course, but I digress.
The video doesn't even show the case for 7 squares. Do you mean better than the currently known best case? Erich Friedman proved that this is impossible in 1999, so that cannot be the case.
Man I loved this video, but I’ll be honest the brutal inaccuracy and inconsistency of the lines in the square packing problem was very unsatisfying. The lines didn’t touch in places the should have, uneven gaps, different thicknesses etc
When you are lost in a forest stay put. Exept when you are in central Europe. Go 500m in any direction. There will be a path or even village. If you went for an hour without encountering a village you are not in central Europe anymore. 😂 Seriously. As a child I wondered how anybody could get lost in the woods until I learned that not every place is like that
For unpacking squares... do you not add 1/2 of the area squared overall each time? So you add 1/2 of a square. Or a triangle. Meaning, tou can rotate in 15° segments as you did to get 45° to asd 1/root[2]. So... 11²=121 121-5² solved=96 96/3=32 (since when adding 1/4 of a 2x2 3 are empty, and the middle 5 is rotated to take root[2] more in its corners) 32/2=16 (l+w of unit square 1) Root[16]=4 3 from above and 1/root[2] omitted for simplicity are 3*(1/root[2]) A=4+(3/root[2]) 2/root[2] more than 10 squares. Then just use Penrose tiling?
Visual from 10 squares at 1:08, if moves 2/root[2] more in white spaces it has enough that we can move the top row all the way over, turn the (2,3) on an angle, and fit 1 more ar ~(3,3)
60 degree rotation, first row 3 last spot requires 2/root(3) as 90-60=30 for length, and 1/root(3) for height as opposite tan(30°|60°) 3+(2/root[3]), 3+(1/root[3]) Error in original comment as 4, at 15° it requires 5.121, at 45° 4.1414, at 60° it is 3.434 or 3+0.434=3+(root[2]/3) Inverted sign in my head. 90°*3=270°,90°*3.5=315°; -60°=210°,255°;360°-=120°,105°; 90°-=30°,15° 3 squares and adding half a square twice, 60° difference is 30°,15° as 3/4 of 4*90° missing and accounting for last 1/4 of 4*90°. First square moves 15°, second an additional 15° for 30°, adds 2/root(3) of the 3×3 for length and 1/root(3) of a 3×3 for height following Penrose tiling. I think, not validated in my head and on phone calculator seems to work.
@@jimmypickinsA lot of what you’ve written is not really interpretable as math, but it’s worth noting that just adding 1/sqrt(2) to the side length of the big square every time is far from optimal. Also, I’m sure you could try tiling smaller patterns to get a bound on larger ones, but it’s likely that the larger ones could optimized further than the small patterns that you’re tiling. Also, why specify that the tiling be a Penrose tiling? I don’t see how it being aperiodic makes the problem any simpler.
A little mistake when you talk about Ulam Conjecture and Kepler Conjecture : Kepler's Conjecture HAS BEEN proved ! The demo dates back to 2014 and was accepted in 2017. Stille a really good video thanks
Maybe I've misunderstood Lesbegue's universal covering problem, but why a circle with diameter of 1 (and area of pi/4 which is about 0.78) can't be a cover? I can't think of any shape with diameter of 1, that this circle wouldn't cover
why cant you prove ulam's packing conjecture with something relating to the fact the spheres can only touch each other at points as opposed to other shapes which can have areas of their faces touching?
@@ojfehevery convex shape with nonzero sized flat faces can always at least pair off on single nonzero area faces of contact, yeah? When packing, anyway. There's at least an intuitive sense that at least pairing off some portion of the faces like this when possible would be important in most packing strategies, but i dont have the rigor to prove it personally
A semi-sphere with a 46 (45.smallest usable value) degree angled cone on the back, with a rounded point, should be less efficient at packing than spheres. If enough people care, ill look into drafting it.
I've realised this is just all of your videos. To 99.9% of viewers, it is unclear where this information has come from. At no point in the video do you mention the articles you took this from. You don't even flash them up on screen. This is incredibly dishonest and disrespectful to the people who put hours into writing and editing those articles.
i failed nearly every math class i took in high school but theres something about geometry that always interested me. only math class i got a D in first try.
Let me know if there's a topic you'd like me to cover next. 😊
3x+1
Maybe
I’m lost in the woods and I’m very glad I found this survival video.
Stay put! 😮
i hope it wasn't an equilateral triangle
@@KeenRunnerHonestly, that isn't always the right answer. No point staying and starving to death and if the forest isn't that big or you know in which direction it's located relative to the compass, you can use the sun (rises in east, sets in west) to figure out where to go.
Generally speaking there's very few places you want to actually stand still. At least in the US.
Generally speaking unless you're in a very remote part of the us you want to walk in one consistent direction then hit a road. You need to know how to travel in the same direction but generally unless you're in areas where this doesn't work you're not likely to be lost. And if you're lost you just needed to not travel in a bad direction or go hike in a desert after being black out drunk.
Usually within 3 days is 90% of us that people are actually in. And you can find water in most of these areas.
Once you hit a road you can pick left or right and generally speaking be very near to people. Almost always you'll hit a house that you can ask for help or a driver you can flag down.
But this assumes you've got somewhat knowledgeable information about where you're lost. So you'd want to do this hiking... Or, IDK you've looked at a map somewhat recently.
It's not for for "stuck on an island" or ", lost in northern Canada" lvls of stuck.
@@TheBcoolGuy I have worked for search and rescue for my county’s sheriffs office in Oregon, always stay put, as long as at least someone knows where you are, and when you left we can find you, but in general keep a charged electronic device on you, if you can, that way if something does happen we can find you faster using your S.O.S feature or using calling your emergency service line.
If you are lost or injured, don’t try to find your own way out, this is one of the fastest ways to get even more injured or even more lost, you have a much higher likelihood of getting recovered if you stay put, hopefully near a trail, if you know where you are and are not injured, that is a different situation.
Currently SAR runs in all 50 states and most territories, with the exception of parts of alaska, we have the ability to make it to any location within the United Sates within 48 hours.
Next time you go hiking:
1. Tell someone where you’re going, how long you will be gone, and what trails you will be taking.
2. Bring an electronic emergency device.
3. Unless you are a professional stay close to the given trail you were expecting to take.
In any case you will be fine if you follow these steps.
No intro, no outro. This guy doesn't play around with our time! We love to see it!
How do you not know about sponsor block, intros, outros and sponsor segments haven't been an issue for years....
@@weibrot6683 My brother in christ, I already use sponsorblock. That doesn't mean you can't salute creators who actually don't make it necessary to enjoy their videos.
It's all just taken straight from the corresponding Wikipedia articles. He doesn't cite them but if you look at the articles you'll see that a lot of the wording is just taken straight from there, but changed slightly. He's done far less work than it seems, and he doesn't credit the original authors at all. Straight up plagarism.
@@MatthewJohnson-hi2th oh nyo, lucky wikipedos deserve naught but disdain
@@Bobo-ox7fj What do you have against Wikipedia?
And besides, it's still true that the creator of these videos is profiting off of the work of others. It doesn't matter if you hate the people he's stolen from, that's still dishonest and wrong, right?
If an academic paper plagiarised Wikipedia would you have a problem with that?
GUYS. Watch every section all the way though. Ended up in a forest for several days and did not find a solution.
RIP
3:15 sorry😢
If mathmaticians dont solve them in the next year Ill have to get involved 🐺
As if you could prove any of them.
Also, mathematical conjectures can take eternities for proofs to be published. Just look at how long it took for someone to solve the FLT.
the chosen one 🙏
@@Gordy-io8sb Anyone can do math, it's only a matter of what tools they have available to them
@@Gordy-io8sb clearly you haven't met ben
2:51
Correction: the equilateral triangle is the only regular polygon that isn’t fat. its solution is known, but it isn’t a straight line path.
well what is the solution
@@drilledbean7434zigzag line composed of 3 straight line segments of equal length
If anyone else was wondering, the solution for equilateral triangles is a zig-zagging line with three segments of equal length.
huh apparently can get notified despite my reply here having been deleted by something
Nerd
We are even further from solving the "Worms lost in a forest" problem
"What is the maximum packing density of worms of length = 1 in a bounded forest of arbitrary length and width, if worms cannot overlap each other or the edge of the forest?" - the next unsolved problem
@@Roset595how wide are the worms
@@Roset595European or African worms?
@@Roset595 Sounds like the setup to an XKCD comic
My toxic trait is believing each question will take me 5 minutes to solve
please do. we would all be grateful
i believe in you
Take 5 years, for real
Same
Dunn-Kruger Effect
Me too.
Having looked through the paper cited in the wikipedia article on Bellman’s lost in a forest problem, it appears that the case of the triangle is actually solved. Like, wikipedia says the equilateral triangle case is still open, but the paper they cite presents a solution. (Edit: The wikipedia article has been fixed)
Huh, that's odd. I was actually just about to ask in the comments why moving in a straight line isn't the optimal solution for a triangle.
@@GarryDumblowski Simply because a better solution is known (and was proven to be optimal). It’s a zig-zag with three line segments at very specific lengths and angles that ends up being a bit shorter than a side length of the triangle, and hence better than a straight line.
@@saschabaer3327but not worth the effort in practice. Academia moment.
Obviously not meant for people in a forest.
Maybe it can be used in certain search/optimisation algorithms
This is a math video lol, we don't do not worth it in practice here!
Moving Sofa problem is also an unsolved geometry problem
Well, only if you don't know how to PIVOT!!!!!!!!
I think the moving sofa problem was solved already
@@T.THobbiesWe have a solution that mathematicians *think* is optimal, but proving that it is is a considerably harder task.
@@T.THobbies Solving the problem means to prove that certain solution is right and I don't think that has been done. What has been done is finding a solution that mathematicians think is right. That doesn't count as a mathematical proof.
well that's just because we haven't invented time travel yet
The fact we dont know square packing for either 11 or 17 is crazy to me. I was gonna guess 111 when some small number; and watch someone state n = 111 is actually known in the comments
In orienteering, lost in a forest is a common thing. That is why you are given a safety bearing which is a direction you are to walk to reach a known feature such as a road where rescuers will eventually find you. The only variation is when you are too injured to walk and you sit still and blow your whistle
what if the forest is triangular? /s
Love how the video just ends, as if it was released mid-production and we can expect an update when the scholars get around to finding more impossible conundrums
when you show something interesting, like 1:37 or some equation, i think you should show it for more than a couple seconds.. had to get back and pause the video
Why, I mean I'm high but I understood it good enough
5:42 japanese flag jumpscare
7:01 😮
3:51 😱
Lmfaooo I was about to comment this but yt has comments with time stamps where you are in the video
So you're telling me I gotta move 11 box tomorrow and mathematicians can't even tell me how to arrange them ?! What are we paying them for ?
3:18 instructions unclear: lost in a forest for 3 years and nobody came yet
I like tiling and packing problems, yet I like neither tiling nor packing.
Could you make a video about problems which went unsolved for a very long time, similar to these, but then were solved?
Lmfao.
I was watching this, and i got really confused because i kept rewinding to figure out where the first one started.
Then i realized that my brain literally stops listening when you said "square packing" because i immediately thought it was a square space ad.
My favorite 3D print I've made is a version of the current best 17 square packing as a puzzle. It is deeply unsatisfying and I love it for that.
7:28
This section confused me because you said the upper bound was "improved" (implying "reduced") to a bigger number.
It turns out you had the initial number correct: 0.845
The other two however, you have erred by replacing the 4 with a second 8.
"0.88414" should be 0.84414, and "0.88409" should be 0.84409.
We also have a lower bound on this problem of 0.832, so we have quite a good answer even if it's not perfect/proven.
It's kind of interesting that getting to the edge of an equalateral triangle fastest from a random point does not encourage going in a straight line.
If you don't know your position or orientation, heading in a straight line towards a corner takes longer than a straight line towards the middle of a side
6:06 the line is not on the corner
THE LINE IS NOT ON THE CORNER
I have a solution to these problems but the proof is too large to fit in the comments
Write a paper about It you might receive money on it
Slow down there, mister Pierre de Fermat
Super underrated comment
What is the minimum size of comment needed to fit your proof?
pop off hashpram
The packing conjecture has real world applications! Assuming roughly spherical atoms, the tightest that atoms in a crystal lattice can pack is also 74%. There are two ways to do this: Face-Centred Cubic and Hexagonal Close-Packed (best to look at images, but basically each layer of spheres sits in the divots of the spheres beneath them). A direct application of this is something called a "slip plane". This is the plane through which a line of atoms can easily slip past each other, which is one of the ways materials can deform. As such, some metals like magnesium are stronger in one direction than another - all as a direct result of the packing of spheres!
5:06 I thought he was starting a limerick there
What am I misunderstanding about Lesbegue’s problem?
A circle with diameter 1 is a valid cover for all shapes with diameter 1, and for a cover to be a valid cover for all shapes of diameter 1, it has to be able to cover a circle with diameter 1, therefore the smallest possible cover is a circle with diameter 1…
Okay, I think I found the flaw in my thinking. An equilateral triangle with diameter 1, for example, does not fit in the circle with diameter 1.
Thank god I found this comment 😅 I've been going back and forth with chatgpt about this and I felt like I was going crazy lol.
@ericfox7021 Thanks for your comment, I was initially thinking the same thing
4:55 his name is misspelled - it should be Stanisław
Yeah, it is so extraordinarely misspelled that I can't imagine how could someone even come up with the writing in the video
They didn't say it was an _unbelievable_ misspelling
It’s like they saw a transliteration of the Russian name or saw the Czech name „Stanislav” but then did some weirdness with the caught-cot merger so they spelt it with that o but then…didn’t bother to check? Idk 😂
Sometimes I think mathematicians come up with these things so their field of study stays relevant.
11:25
Correction: As of December 21, 2007, the Kobon Triangle problem actually is solved at k=10, with the upper bound and best solution agreeing at 25 (unlike how the video states the upper bound is 26). k=11 is still unsolved though!
Someone a couple thousand years ago accidentally dropped ten wooden dowels down a flight of stairs and picked them up without realizing they accidentally had a really interesting thing
The content of the exam the teacher said wasnt going to be too hard:
i like the usage of proper math terms along with giving a brief definition
I know nothing about geometry or even math tbh but I watched this with full interest
The last one is a blast, you can play with it pretty easy on a piece of paper: draw 10 lines and see if you can hit 26 triangles, the supposed limit. It's really hard to even hit 20, let alone the 25 of the best known figure (at exactly 11:26).
Lebesgue
Reuleaux
He also missed an 8 after the comma for the original upper bound on the Lebesgue problem
9:58 visual error, should be 0.2618
For the equilateral triangle, it's walk the amount of the shortest distance from the center to the edge, then go 45° (either left or right, doesn't matter) in a straight line until out.
This video just starts. That's it: straight to the point; no chit chat
I found a truly marvelous solution to all of these problems, which this comment is to narrow to contain
feels like each one can have a 6-15min video of their own too
5:17 optimal packing density for balls
I would not use the word approximate for the current best estimate for the square packing problem. I would say "less than or equal to" as in this is an upper bound on the solution, but it's unproven that a better packing does not exist.
2:32 Never in my life have I seen someone call a forest fat.
Wow that last one is tantalizing
The Kobon triangle problem is very interesting because it seems computationally approachable since the exactly angles of the lines don't matter as long as the topology of the graph (the order in which other lines intersect a line) they create doesn't change.
I 'conjecture' that I am wrong in this assertion.
I feel like there’s an interesting discussion to be had about whether Ulam’s packing conjecture can be disproven using a non-convex solid which is constructed out of convex solids. (Picture the silhouette of a venn diagram or an hourglass)
Kepler's conjecture is proved.
there is a difference between a formal proof and an algorithmic justification.
Common misconception. In fact, a conjecture can never be proven.
Kepler's conjecture was proven in 1998 by Thomas Hales (well we can't be too sure since the proof was too complicated), getting officially published in 2005.
Nobody was really sure about whether or not it contained an error, so in 2017, a team directed by Thomas Hales managed (with an insane amount of work) to get an entirely computer verified proof.
So it has been proven, no doubt about it
@@avonbarksdale5821 What makes you say this? I can find no definition that says this. The Wikipedia page "List of mathematical conjectures" has a section called "conjectures now proven", leading me to believe they can be proven. Are you sure you aren't thinking of axoims?
@@_arie_s 1976 called, they want their computer proof controversy back
In the process of generations of mathematicians working on these problems many useful discoveries are made.
11:25
“The upper bound is 26 tringle”
To solve Lesbegue's problem, I think it's easier to think of it as "create the shape with the largest possible area, in which any 2 points are separated by no more than 1 unit length", as this will cover any shape with a diameter of 1 unit, rather than thinking about covering shapes with other shapes.
The solution is either a reuleaux triangle or something very similar to it, but its exact parameters are very difficult to work out.
Do you have a proof that that would work, or is this just your intuition? My intuition tells me that this wouldn't actually result in the right answer. The Reuleaux triangle is the curve of constant width that encloses the largest possible area, so I'm guessing that you're correct that it's the shape you're looking for.
However, a Reuleaux triangle of diameter 1 is not a valid cover. This is visually shown by an illustration on this Wikipedia article:
en.wikipedia.org/wiki/Lebesgue%27s_universal_covering_problem
As you can see, a Reuleaux triangle is included as one of the covered shapes, without being a cover itself (and indeed being significantly smaller).
The reason you should not go anywhere on a path is because it's likely that other people also will use the same path at some point, and they will run into you, especially if it's on a federally protected land, where there's more likely to be a park ranger who will do that trail every once in a while.
This is incidentally why you should not leave trails. Unless you're extremely experienced or with someone who is or know how to navigate in the wilderness, regardless of if you know it well in this particular case or not.
but as an aside gps devices are so much cheaper than they used to be, like to a ridiculous degree. you can have a little thingy that will tell you literally exactly where you are, and it probably has a function to call for help, and it sends out like a red alert to the nearest emergency service people who are trained to get people who are using gps because they have to.
Even ten years ago, something like that cost five hundred dollars, unless you've had a lot of diy skills. now you can get one easily for under 100.
I like convex solids a lot. More than that sentence accurately communicates on its own.
But in a way that I think is, at most, vaguely relevant in the context of ulam's conjecture.
LiaF reminded me of a similar problem. I saw a problem about a spaceship. I didn't solve it, but I thought of a similar problem in 2D. In LiaF terms, the forest is a half of a plane, but you know you're at the distance a from the boundary. I then changes to a different problem, found a solution for it and translated it to the 2D version of the original problem. Then, someone else came up with another solution. I successfully translated it to the different problem. It could prove that my solution was more optimal, but the calculations said otherwise
each point on the surface of a sphere is equidistant from the center of the sphere
the ratio of a sphere with diameter x to a cube with side length x is π:6
in order for ulam's packing conjecture to be disproven, there would have to be some convex 3d shape A with a ratio of its own volume (or the volume of whatever pattern of multiple As is closest to 1:1 with a rectangular prism) to the volume of a rectangular prism equivalent in length, width, and depth less than π:6, which seems obviously impossible considering the equidistance of points on the outside of a sphere
The Kepler conjecture was proven true back in 1998 by Dr. Thomas Hales; it's a theorem now.
I feel like these are very similar to NP hard problems. The solution requires searching thru all possible reasonable solutions.
Slightly upset your packing of balls was one of the more inefficient packings. Move the top row over 1/2 a ball width and you can push them closer together
for packing balls in space it doesn't matter whether it's a square or a triangle pattern. Both result in the same packing density, just different perspective.
@@Fassleexcept in both arrangements, layers need to be offset from each other. the goal was to visualize the concept of 3D packing with 2D circles, so they should have chosen the arrangement that shows up in both 2D and 3D.
Think of it this way, things are more efficiently packed (in 3D) when they have a larger percentage of their faces touching. Spherical objects only touch eachother slightly on the faces, because they have so much curvature.
What's funny to me about these videos is that I learned a lot of this from the video game 4D Golf from the little trivia crystals. Definitely recommend if you like mini golf and geometry
11:24 Tringle? Is that a kind of pringle?
7:26 the second figure is a worse upper bound, not an improvement
“We must define convex solids.”
Me, looking at the pictures: A convex solid is a D&D dice
Thank god, i was wondering how I would find my way out the forest.
I"m confused by the cover thing. Isn't the smallest possible cover just the largest possible diameter 1 shape, which would be the circle, as any growth to the circle would make it larger, and any cover that could not cover the circle would not be a cover?
No, the cover has to have every possible shape of diameter 1 fit inside, not just take up at least as much space as every shape of diameter 1. An average broom takes up less volume than there is inside of an average backpack, but you can't fit a broom in a backpack. A disk of diameter 1 is not a valid cover; for example, an equilateral triangle with side lengths of 1 has a diameter of 1, and it cannot fit inside.
Me: Sees video is on square packing problem.
"I AM READY FOR 17 (a true classic)."
Great video.
I think you could have left some of the example pictures on the screen for a few more seconds to give us time to see them. I had to rewind and pause a few times.
What’s really interesting (or frustrating, depending on your point of view) about math is that there are technically true statements in math which are nonetheless unprovable. So if something doesn’t have a proof it could just be because no one has found one yet, it’s true but no proof exists so any attempt to find a proof would be fruitless, OR you can’t prove it true because it’s false and you haven’t thought of the false proof yet. So there will always be more math to do
6:33 You spelled Lebesgue wrong. There's no lesbian in him.
and the pronunciation was even worse
Wouldnt a circle have less area for the covering problem? Or will it not cover everything
I think one of Ulam's friends wanted his help packing for a trip or for moving and Ulam tried to make that friend regret asking
i am definitely missing something with the Lesbegue's covering problem.
I dont see how it can be smaller than a circle, when it should be able to cover the circle?
For the first problem, a link to Erich's packing center is missing here.
Ulam seems like he’s packin‘
What about the sofa problem
That was covered in another video.
ua-cam.com/video/6HGEaZ8ROeg/v-deo.html
For the lost in a forest one, can’t you try to scale a tree or something like that? Doing so would help you see your surroundings
The trees are composed entirely of molten rock, so touching them makes your hands hurt.
@@isavenewspapers8890 oh. Mb. How about a pot of wings? Strap rope on birds and fly up. Jack sparrow did something similar with turtles, why not birds?
@@phillipholland7521 I don't see why not.
@@isavenewspapers8890 got it. I’ll try it and get back to you
@@isavenewspapers8890heh, I thought I commented a few days ago saying “I’ll try it and let you know how it went” I went today (a few days later to seem as if I ‘actually did it’ and would tell you “it did work but the birds are a little hard to catch and wobbly. Set a few traps and caught a few. Tricked them with bird seed on a stick (like the carrot on a stick from MC) and they flew. After being able to see everywhere and be able to be taken anywhere, I let go of them and took the quickest path out”
For Lesbegue's Universal Covering Problem, would the solution with the smallest area not be 0.785? The question is what is the smallest area a shape can have that also covers every shape with the same diameter (a diameter being defined as the longest line you can fit within a solid that passes through the center). A circle of diameter 1 would cover every other possible shape with a diameter of one no matter the rotation, and the area of a circle with diameter 1 (3.14*(0.5^2)) is 0.785. Is there a rule I am misunderstanding?
I think I'm missing something about Lesbegue's universal covering problem: wouldn't the shape with a diameter of 1 that has the smallest area simply be a line with a length of one and no width?
But that's not what we're looking for. We're looking for an object that can have any shape of diameter 1 fit inside.
It's Stanisław Ulam, not Stanislov
It is translated, so incorrect in polish but makes more sense for English speakers.
@@IsmailLev hold on, translating your name so it's incorrect in English but makes more sense for Polish speakers.
It's baffling that English still does it. We don't write "Stephen" "Stiwen".
@@DeuxisWasTakenThis is common in most natural languages.
In Polish, transliteration of people's names isn't that common anymore, but we still do it - consider "Królowa Elżbieta II" (Queen Elisabeth II from Britain), or even "Chrystus" (Christ) for a biblical example. For everyday people, we also do this all the time for non-latin alphabet names, such as names in Cyrilic.
Not to mention exonyms for country names - Łotwa, Holandia, Węgry, etc.
Just because it's common doesn't mean it's a good thing of course, but I digress.
@@alliumchocolateTo be fair, Christ isn't a name. It's a title. It comes from the greek khristos meaning Messiah or anointed.
Im trying to figure out how many brownies to put into this pan - how should i square fudge pack?
it took me not joking 5 minutes to find a better packing area for 7 boxes.
The video doesn't even show the case for 7 squares. Do you mean better than the currently known best case? Erich Friedman proved that this is impossible in 1999, so that cannot be the case.
Drawing a line through a donut won't always have it pass through empty space. The line would be smaller though
Man I loved this video, but I’ll be honest the brutal inaccuracy and inconsistency of the lines in the square packing problem was very unsatisfying. The lines didn’t touch in places the should have, uneven gaps, different thicknesses etc
When you are lost in a forest stay put. Exept when you are in central Europe. Go 500m in any direction. There will be a path or even village. If you went for an hour without encountering a village you are not in central Europe anymore. 😂
Seriously. As a child I wondered how anybody could get lost in the woods until I learned that not every place is like that
For unpacking squares... do you not add 1/2 of the area squared overall each time?
So you add 1/2 of a square. Or a triangle. Meaning, tou can rotate in 15° segments as you did to get 45° to asd 1/root[2].
So...
11²=121
121-5² solved=96
96/3=32 (since when adding 1/4 of a 2x2 3 are empty, and the middle 5 is rotated to take root[2] more in its corners)
32/2=16 (l+w of unit square 1)
Root[16]=4
3 from above and 1/root[2] omitted for simplicity are 3*(1/root[2])
A=4+(3/root[2])
2/root[2] more than 10 squares.
Then just use Penrose tiling?
Visual from 10 squares at 1:08, if moves 2/root[2] more in white spaces it has enough that we can move the top row all the way over, turn the (2,3) on an angle, and fit 1 more ar ~(3,3)
60 degree rotation, first row 3 last spot requires 2/root(3) as 90-60=30 for length, and 1/root(3) for height as opposite tan(30°|60°)
3+(2/root[3]), 3+(1/root[3])
Error in original comment as 4, at 15° it requires 5.121, at 45° 4.1414, at 60° it is 3.434 or 3+0.434=3+(root[2]/3)
Inverted sign in my head. 90°*3=270°,90°*3.5=315°; -60°=210°,255°;360°-=120°,105°; 90°-=30°,15°
3 squares and adding half a square twice, 60° difference is 30°,15° as 3/4 of 4*90° missing and accounting for last 1/4 of 4*90°.
First square moves 15°, second an additional 15° for 30°, adds 2/root(3) of the 3×3 for length and 1/root(3) of a 3×3 for height following Penrose tiling.
I think, not validated in my head and on phone calculator seems to work.
This is the first comment I didn't understand after reading three times
@@jimmypickinsA lot of what you’ve written is not really interpretable as math, but it’s worth noting that just adding 1/sqrt(2) to the side length of the big square every time is far from optimal.
Also, I’m sure you could try tiling smaller patterns to get a bound on larger ones, but it’s likely that the larger ones could optimized further than the small patterns that you’re tiling.
Also, why specify that the tiling be a Penrose tiling? I don’t see how it being aperiodic makes the problem any simpler.
Yes on replacing pi with tau!
5:33 what about a variant of the conjecture where we find the best possible packing shape?
unless it's dependent on the shape of the space
Cubes. You can fill 100% of the space using cubes.
Am I the only one who heard each one and thought, yeah I can totally believe that’s unsolved?
A little mistake when you talk about Ulam Conjecture and Kepler Conjecture : Kepler's Conjecture HAS BEEN proved !
The demo dates back to 2014 and was accepted in 2017.
Stille a really good video thanks
the first time i heard about the square packing problem i got a stroke
Maybe I've misunderstood Lesbegue's universal covering problem, but why a circle with diameter of 1 (and area of pi/4 which is about 0.78) can't be a cover? I can't think of any shape with diameter of 1, that this circle wouldn't cover
Oh nevermind, it literally talks about shapes of constant width afterwards! Makes sense now
Yeah a triangle of edge length 1 is an example
For a simple counterexample, you can't cover an equilateral triangle of diameter 1 with a circle of diameter 1
why cant you prove ulam's packing conjecture with something relating to the fact the spheres can only touch each other at points as opposed to other shapes which can have areas of their faces touching?
I mean… why would you be able to prove it using that? Also, there are many other shapes for which this happens, not just the sphere.
@@ojfehevery convex shape with nonzero sized flat faces can always at least pair off on single nonzero area faces of contact, yeah? When packing, anyway. There's at least an intuitive sense that at least pairing off some portion of the faces like this when possible would be important in most packing strategies, but i dont have the rigor to prove it personally
I would love if some random ass kid from like Timbuktu sees this video and solves it
If no one knows you're there find running water and follow it down stream.
If you hit a lake or the ocean pick a direction
A semi-sphere with a 46 (45.smallest usable value) degree angled cone on the back, with a rounded point, should be less efficient at packing than spheres. If enough people care, ill look into drafting it.
John Wetzel’s Pretzel Conjecture
Isn't like... all of this video just taken straight from the corresponding Wikipedia articles? Why don't you cite any sources?
I've realised this is just all of your videos. To 99.9% of viewers, it is unclear where this information has come from. At no point in the video do you mention the articles you took this from. You don't even flash them up on screen. This is incredibly dishonest and disrespectful to the people who put hours into writing and editing those articles.
@@MatthewJohnson-hi2th thank you for pointing this out.
Man these red squares seem so familiar
JUSTICE FOR THE OPAQUE SET PROBLEM
i failed nearly every math class i took in high school but theres something about geometry that always interested me. only math class i got a D in first try.