Tree Gaps and Orchard Problems - Numberphile
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- Опубліковано 21 вер 2024
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Pi, fibonacci, golden ratio, probability, magnitudes of infinity, Riemann zeta function... it's like all these years of watching numberphile has prepared us for this one video lol
I was expecting the probability of 1/e though, Euler's number is sadly missing.
HasekuraIsuna 😄
HasekuraIsuna
*everyone is here*
@@Ulkomaalainen Pi is here and so is 0, so we just need to imagine really hard and... oh, there it is!
false.
This infinite orchard almost solved world hunger, but unfortunately the harvesters couldn't find any trees since they were all points and had a 0% chance of being seen.
Yeah, no.
The tree must be growing in a logarithmic scale. If they expected an infinitesimal amount of time they may only see trees on the field.
But the fruit were all poisonous anyway.
How does a person pick a fruit off a point tree? I'm glad you ask. Here's another case where we want pie but pi shows up. Here's the proof....
Dismantle the food industry and throw your TV out the window, that is the only way to solve "world hunger".
If a tree falls in an infinite forest but you're looking in an irrational direction, does it make any sense?
You, Sir, just made my day xD
Applied math has application, but pure math is completely useless. :)
but fun!
@@mikeguitar9769 It actually isn't; it's just that the uses of it often come 100 - 300 years after the math itself is discovered.
For example, abstract algebra is pure math, and it's used all over cosmology and fundamental physics, e.g. to identify particles in particle collisions.
Your question is irrational
Somewhat unrelated but I was told by a guy who works in forestry that sometimes trees are planted in a fibonacci arrangement to maximise sunlight exposure. In a spiral like that seen in the centre of a sunflower
Spiral yes but single Fibonacci spiral would get too wide to be efficient really quickly leaving big spaces. The only way it could work is multiple interlaced sprials like Roger Penrose examples. Otherwise rows and columns is always more efficient which is why no commercial places use other methods unless it's stacked rows and columns.
more efficient in the respect of harvesting and tending to the crop I guess... getting machinery/equipment around a spiral compared to up and down in rows :)
If the sun stayed still. But the relative motion makes any "most efficient" arrangement only temporary until a different epicenter would need selected to maximize light gain.
But surely there is an arrangement (or set of arrangements) which are on average most efficient.
+Matt McConaha; yeah, by alternating the rows w rows that are offset by half a tree's width.
I love the "mindfuck" aspect of mathematics and I always have. It's stuff like this where reality and intuition are on complete opposite ends of the spectrum that I love the most.
I know words, I have the best words. Nobody respects women more than me. I am the least racist person who you have ever met. Nobody lies better than me. Believe me. Sad!
>where reality and intuition are on complete opposite ends of the spectrum
Funny, that's also the feeling you get when you find an inconsistency. The moment when sh*t blows up because it's a logical fallacy.
In this case the example is on the other side of reality since nowhere in the real world are there point width trees to make this example even realistic. Our intuition is right for realistic examples. But for theory like the e.g. then intuition might not get us to the right answer.
Kevin Potts i
+Dole Pole; i would call anything experienced or imagined reality but there is physical reality that has tactile reification whereas abstract theories dont. We can experience an integer in our mind (or imagination land in southpark).
This was brilliantly presented and really fun. I would never think of this type of problem but I am super glad to have stumbled upon the fact that this kind of thinking exists!
I know words, I have the best words. Nobody respects women more than me. I am the least racist person who you have ever met. Nobody lies better than me. Believe me. Sad!
Here is Douglas Adams using the same maths: “It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is also zero, and that any people you may meet from time to time are merely the products of a deranged imagination.”
This doesn't sound right. Any percentage of infinity is infinity. Therefore, the number of planets that are inhabited is not finite, if the universe is infinite.
Please address all complains to Douglas Adams.
Sure. But, I didn't know he was so poor in maths. :-)
I disagree with his assertion that not every world being inhabited implies that a finite number of worlds are inhabited.
I MEAN UH, COMEDY.
If there are an infinite number of inhabited worlds, there are an infinite number of beings. But in your lifetime you will only meet a finite number of beings. The chances of you meeting a given being are therefore zero, and it follows that any person you think that you've met is the product of a deranged imagination.
The least close line being the golden ratio... wow
What does it mean that the golden ratio is "the least well approximated by a rational number"? I'd like to see a video just about that. It sounds like a very interesting property.
It is indeed! The youtuber +Mathologer has done a video about this.
If there is a « least well approximated by a rational number » , is there a « best well approximated by a rational number » ??
The rational number p/q is a best rational approximation of some real number x if it is closer to x than any other rational number with a smaller denominator (as the denominator gets larger you can get more precise). The golden ration is the least well approximated in the sense that the best rational approximations are the worst possible. Some numbers converge as slowly but none more slowly.
There are countably infinite numbers that are best approximated by rational numbers. We call them the rational numbers!
Alexander F what about 2.618 etc ? Aka 1 + phi. Does this have the similar propert?
this was surprisingly interesting, well done. For those wondering why the golden ration is the "most irrational", it's because of its continued fraction. The golden ratio can be expressed as such:
phi=1+1/(1+1/(1+1/1+1/(1+1/(............ extending to infinity.
If you stop somewhere (say after n steps and ignore the rest), you get a rational approximation of the golden ratio. The fact is that the smaller the numbers you have in the continued fraction, the worse the approximation. Because there are only ones, this is the most irrational number. Hope I made myself clear.
I think if you’re allergic to apple trees and find yourself in the middle of an infinite apple forest you’ve made some wrong choices in life!
Or the DMT has kicked-in not hard enough
[Record scratch]
[Freeze frame]
See that guy right there? Looks like he made some wrong choices in life. Well that guy is actually me.
Doctor King?
At least you don't see a single tree if you do not look at one on purpose.
Maybe we are all in a infinite apple tree forest but we never notice it since no one ever saw a tree?
Every direction you look you won't see a tree sounds like something out of the Hitchhiker's guide.
"There is unrest in the forest, there is trouble with the trees, for the maples want more sunlight, and the oaks ignore their pleas."
I am amazed how a simple problem had so many underlying principles involved. One of the best videos on Numberphile.
Totally mindbending, fantastic, wonderful! Also, I feel like an idiot - until today (age 41), I thought pi = 22/7, then watched this video, paused it halfway through, did some Googling, and mind blown, 22/7 is only a lucky approximation! But I thought that was how you actually calculated pi, just do the math for 22/7 for many many decimal places. Nope. WRONG. Calculating pi is actually MUCH harder than that, something no one ever taught me in school or in the years inbetween. So just that much more love for this particular video for opening my eyes!
Cast: Riemann's Zetafunction, the Golden Ratio, the Fibonacci Sequence, Pi, Primes
So many shout outs to Dr. James Grime. It's like he knows he's the best Numberphiler.
He and Matt are the best
What about Matt Parker?
Future Astronaut He is a Parker Square of a Numberphiler, he's almost the best, but not quite.
I think you'd just say he's the best Numberphile.
That was pretty cool actually. Weird maths turning up in places you don't expect is always great fun
Awesome video. I feel like this is the math equivalent of a crossover episode. A lot of our favorite recurring characters are back: Reimann zeta function, pi, golden ratio, Fibonacci sequence....
Moral of the story:
Don't plant infinitely many infinitely thin trees in a square pattern, or there will be a huge number of people walking into them, because all they see is gaps and they don't know the exact gradient they have to walk or just miss it.
Invasion of the invisible trees incoming...
and because they're infinitely thin, the strain applied upon walking into one will be infinite, slicing anyone who would be so unfortunate clean in half.
But because they're infinitly thin, their stiffness in bending is infinitly small, so might just not feel anything after all!
Fortunately the people sliced in half by an infinitely thin tree get better, since zero cells were harmed by the slice.
@Dan Powell Actually you're right... So you can walk through something without even realizing it? That would be hillarious! But I think it comes down to the question of "what's the smallest thing that makes everything up?" or "what are the smallest things that make everything up?". And with that: What happens if you cut those things in half? If it would cause something like a error in the matrix of the universe it could get really ugly... But I think you only would have two halfs noone cares about :D
If there is a smallest thing that makes everything up, what does it mean to cut it in half?
I can't get enough of this guy.
me neither
the intertextuality these problems display blows my mind. from irrational numbers to riemann to the golden ratio.
And Fibonacci Sequence.
@@Bartooc Which is pretty closely linked to the golden ratio. Not that mindblowing, if you ask me.
There is a German saying: "Den Wald for lauter Bäumen nicht sehen" / "To not see the forest because of too many trees" .. This suddenly makes sense.
Daily Drum Lesson that’s also a saying in English - usually phrased as “can’t see the forest for the trees”
Didn't know that, thanks for the info!
Or in Dutch: ik zie door de bomen het bos niet meer - I don’t see the trees through the forest.
"C'est l'arbre qui cache la forêt", French expression; litterally "this is the tree that hides the forest"
vor*
Can't be more grateful for your videos. You made me feel like a child again with this very well edited videos.
You guys, make UA-cam great!!
Thanks a lot for contribute to this little nerd community
So if I'm understanding this correctly you would see no trees since in-order to see something that is a point (infinitely thin trunk) you would need to have them in every direction you look so every point is blocked out and you see a "solid" wall but since there are infinitely more gaps due to irrational fractions than there are blocked points you see no trees. That is wild isn't it!?
I think that any video that has an appearance of the Riemann Zeta function, phi, or "least rational numbers" is guaranteed to have those weird relationships. It's one of the more exciting areas of math, honestly.
I think it's better to think of it like an atmosphere of trees....you see a similar effect with a large number of atoms occupying far less space than is otherwise empty.
Yet we still see the atmosphere emerge.
Granted tho. Even an atom is infinitely bigger than a point
Sorry, but you've misunderstood the video. You can see any tree... if you look directly at them. Problem is, we didn't say were looking directly at them and, in fact, were directly looking at random points. Think about it like this: take a step a meter forward. What is the chance that you walked exactly one meter? Yes, it's possible, but it's completely unrealistic to imagine that ever happening because it's a single point and we have no way to connect our steps with exact distances. When you look in a random direction, you're mimicking that inability to precisely pick a point, though it is still always possible.
@@SilverLining1 You bring up an important question: can any event with an infinitesimal chance ever succeed? It is more than 0 by definition but 1 over infinity is ridiculously small. I don't think such a thing is actually possible, even in a universe where infinity makes sense. As was said in the video, the chance of seeing a tree is 0%. So there seems to be no misinterpretation.
The chance of seeing a tree is 0%. But humans have a CHOICE. We can choose to look at the tree regardless what the probability is. And that is why humans have such a problem understanding probability. Because the universe isnt random but it is chosen and ordered according to intelligence.
So let me see if I have this... he has managed to clearly and understandably present, in 14 minutes, a topic that includes geometry, trigonometry, orders of infinity, pi, the golden ratio, the Fibonacci sequence, and last but certainly not least, the Riemann Zeta function. Absolutely brilliant.
In german we say "Man sieht den Wald vor lauter Bäumen nicht mehr" which roughly translated means "you can't see the forrest through/because of all those trees" - finally i get something that validates that saying xd
We have the same in english- "Can't see the forest for the trees"
This might be one of the best videos I've seen in terms of tiny mind-blowing factoids.
I'm going to look along a gradient of TREE(3)
Please be patient whilst I just calculate how to angle myself..
!remindme AA(187196) seconds
Hey, it's Alan Key from that pi video years ago! How are you doing?
Booskop - Hello! I am alive and well!
RIP
Simple, just look at the tree on the point (1, TREE(3)), and start walking.
Presumably the breakdown in intuition here is because when we look in a direction, we actually look in a spread of directions. We would have to be able to look in an infinitely thin line for this to make sense to us.
Many times when traveling by train in Northern California have I watched the hypnotic patterns of passing vineyard rows and thought, "now there's some fascinating mathematics waiting to be written up". But I also suspected somebody had to have already explored this matter in detail. Thanks for pointing it out. It's one of those things that's hard to google.
Clickbait title: "Mathematician proves that you can't see forest for the trees"
:D
:D
the mathematical significance of being T H I C C
This absolutely blew my mind. Like, after the golden ratio and fibonacci sequence came into play I literally had to pause and set my head on my desk for a second to gather the bits of my exploded brain back together. Well done, maths.
If a mathematician walks into an orchard
On a trajectory based on the golden ratio
How is he going to pick apples for his
π?
His arms have a length which is rational. That way he will eventually be close enough to pick some apples.
use his hands :P
His arms don't have to have a rational length, they just have to be longer than some epsilon.
He can get as close to the nearest tree as he wants if he walks far enough, so eventually he can reach out to pick the apples.
You picked the low hanging fruit with that pun.
This video started out cool, then the stuff about the golden ratio, Fibonacci, and zeta function were mind blowing.
The Fibonacci transition - mind blown!
It is 10 mins away from 3 am while I watched this video and right as he was about to mention the golden ratio I figured that is what he would say, my toes clenched with excitement and I clapped with joy. I love patterns. I miss Math classes and hope to keep growing in my understanding of Math 💗
Sweet. I really enjoyed this one, reminds me of some of the older numberphile videos
The number of in-jokes between different mathematical sitautions and equations is astounding. It's like they all give each other cameos.
Best episode yet
I have been watching for many years, and this might be my favorite video. I really loved this one
This was such a absolutely lovely video!
Just fantastic. When I lie in my hammock in my forest, I can always see a tree. Thank goodness.
Interesting, this makes me think of the useful irrationals we haven't discovered yet.
The sudden appearence of Pi and the Golden Ratio is just beautiful!
very nice video which presented MANY math concepts - well done!
Mathematicians: "These trees are infinitely thin"
Also mathematicians: "Woah you can't see any of them :o"
I think it’s important to stress that not only are the trees single points, but your field of view is infinitely thin. No peripheral vision is allowed in your scenario.
He talked about the vision as a laser beam in fact.
Perhaps. I didn't notice him saying it until the end.
I absolutely love this guy, he's such a clear explainer, and seems like a top bloke to have a beer with. Really down to earth.
Sometimes I am astounded how many different mathematical concepts converge into each other...
Nym Alous true !!!
nym alous so true !!!
of all the different mathy theory-y things that I've casually observed, this video definitely has been blowing my mind the most of all of them
I feel like the claim that the line at the golden ratio (φ) gradient "avoids" trees "the most" depends on whether your definition of "most" is exclusive or not-surely it is at least tied by 1/φ, since that's basically the same line just reflected across the diagonal
One of the most fun days I had as a kid was when the family was camping beside a cultivated grid of mature pine trees. We played tag there. Because of the thickness of the trees it was easy to "disappear" whenever you angled away from the person who was "it". 😋
Matt Parker made a video once where he approximated pi by rolling two dice and using the probability of 6/pi² for two co-primes
matt parker and appoximating pi with increasingly insane methods make the best pait
The laser beam analogy makes more sense to me; since the human field of vision obviously has width; and thus, we’d be bound to see the trees; no matter, how thin they are (including points) 🌳.
Golfers will be thrilled to hear trees are now 100% air.
Videos like this are why I love Numberphile. Taking math concepts that we are already somewhat familiar with and using them in new ways or finding them in unusual places
If the golden ratio is the "least near" any trees, isn't there also another line that is equally "least near" any trees if you reflect the golden ratio along the line y=x (So the line would be first going between 1,1 and 2,1 instead of 1,1 and 1,2)? Is there a special name for that line too, like the silver ratio or something like that?
Wouldn't the slope of that line be simply the inverse of the golden ratio? And I don't think we need to invent another name for that number.
Isn't that just the conjugate of the golden ratio. phi-1 or aka the magnitude of the other solution to the definition of the golden ratio
The golden ratio has already "captured" that solution, like Hemant mentiones: If we denote the golden ratio by φ then interestingly 1/φ = φ-1. If we ignore signs then φ is the only number with this property.
Yes, (1-sqrt(5))/2, the conjugate of the golden ratio.
What would be the least near line in a 3D lattice? What coordinates would the elastic lines catch on?
This was one of the first Numberphile videos where I actually figured out most of the answers ahead of time... albeit mostly from watching earlier Numberphile videos as well as ViHart videos.
I can't see any trees. There's an Orchard in the way.
Can't see the trees for the orchard.
I love the videos like this where numbers like pi and the golden ratio just appear out of nowhere!
Seems like a Parker forrest to me. It's there, but not really.
I love the videos where they're totally unintuitive but once you hear the explaination it makes total sense.
A mathematician stands inside a forest with infinitely many trees and has no chance of seeing any of them. Brilliant!
This may be my favorite numberphile video
Me too. Covered so many number problems in one go. I didn't expect that at the start!
The only issue that I have with this problem (the first one), is that the question then turns into "Can you see something that is truly one-dimensional (or two-dimensional)?”, and the answer is obviously "no”.
Fausto Inomata it's not literally meant to be about whether you can see trees. It's an analogy to demonstrate a mathematical concept.
This is the best Numberphile video, IMHO: builds and references other videos that make the topic more enjoyable if you have seen those or plants curiosity on them, is interesting on its on right and easy to grasp although without subtracting complexity to the topic. And i like Ben's subdued but ever present enthusiasm. Even though "positive" answers to a video sometimes are just "background noise" to call them something, I was really excited to see such encapsulation of the Numberphile experience.
ViHart fans know that the problem is mostly "how are you going to give me a random real number?" If there's infinite digit positions, how likely is it that for a random number in R, suddenly the digit sequence becomes all 0's forever?
Yes, that's where the mathematical reality fails on the infinite nature of the decimal (and also binary) representation of irrational numbers. With limited RAM, even if you don't limit yourself to any small number of bits for a float number construct you always work with rational numbers.
It's funny, in school, I was a "pure math" major. When conceiving of a plane with infinitely-small points, I feel I have an easy intuition for how it works. But, start introducing "realistic" elements like the thickness of your line or sight, or thickness of the tree and oop now i have no idea. I'm glad there are practical-minded people out there because if everyone was like me humanity would be screwed.
The way to get a continuum of points on a unit interval is to have n points each with width 1/n . Then let n go to infinity. No matter what "n" is, n*(1/n) = 1 , even inside a limit, since limits are linear.
Does this sort of explain why crystals, which are organized in lattices, are see-through?
Ben Sparks is awesome to listen to! Please get him in another video :)
Mind blown.
Absolutely brilliantly presented. If this person ever decided to teach math in high school he’d produce an entire new generation of mathematicians
The golden ratio strikes again.
i love seeing twisty puzzles in the background of numberphile scenes because my desk is littered with them and it's nice to see most of these guys have similar interests
I used to think of it as a child and promised to figure a solution out. What a shame somebody had already done
Most things you think about and try to figure out have already been discussed or solved. I kinda find it cool when I see a problem or a question I've been asking myself be already out there
It always makes me happy when I see a problem where at every step my intuitive answer turns out right.
Happens very rarely, so it's nice when it does. :D
Intuitively, the only slope at which you can't see trees is at an irrational slope.
Hey there! Here is one perfectly average person here, repeatedly awed and entertained by math videos. Thanks for the great work you all are putting into!
I wonder what happens if the trees' radii are not constant, but instead r(x,y), some function depending on the tree's coordinate.
Well, if the radii either shrink, or grow no more than linearly by distance, then the problem is essentially the same
Wow! You guys are amazing. Makes me humble to see how things are connected at fundamental level.
If an infinitly thin tree falls in the forest and you're standing at the edge, does it make a sound, and do you see it?
It cuts through the floor.
My friend Russel said " Yes but Only when its windy ."
The title didn't intrigue me, but having watched the video I was immensely impressed. Very well done.
An infinite orchard means infinite apples, which sounds cool to me.
But they're infinitely small.
Don't tell Princess Peach!
But I don't like apples.
+K.o.R How? Have you ever eaten one?
Yes. I'm more of a orange person.
Fascinating as usual. Glad you mentioned thickness of the 'laser' as our field of view is much more complicated than a single thin line and of course we catch more than a single photon in one fix AND light is bent by air and even the trees themselves to some degree. And we have two eyes both with wide fields of view. In which case of course it's trees from every angle.
line with gradient *2/(1+sqrt(5))* is also the farthest...
Cheers for recognizing the reflective symmetry on either side of the 45 degree angle!
Any number with the same non-integer part as phi will have said property, including 1 / phi and phi^2.
This video is chock full of mathematical concepts.. amazing!It seems that it could be a starting point for maths altogether.
If you're allergic to those trees and wanna dodge them, just go between two rows, and walk along them, you'll have most distance from the trees.
All this infinity stuff blows my mind especially that when the trees are points you see out but when they have any thickness whatsoever you can’t see anything. I feel like I understand why Plato loves maths - it’s this tiny world of perfection that we can only imagine
10/10 ad transition
This has my vote for best Numberphile video. Surprising, exciting, intuitive results, and who doesn't love a guest appearance from pi!
I don't think the way he describes "you will not see a tree" is intuitive. In fact, you'd see an infinite number of trees as long as your field of vision is wider than 0 (i.e., between two real numbers instead of at an exact real number).
If you wanted to stay away as far as possible from all trees just walk 0,5 units up and then go right lol
There is a board game called Photosynthesis which is liked by most and you plant seeds, grow your trees, try to gain sunlight, and chop trees down for points. Give it a go.
I want James back, but this guy is pretty good too.
What's all this talk about James Grime? He happens to not be in this video but people seem pretty fixated on that. Did I miss something?
Interesting timing. A problem regarding this occured in a math programming contest just yesterday.
Alcesmire interesting.. where it happened?
If a man is standing in an orchard minding his own business, does he still run into riemann zeta, fibonacci, pi, and the golden ratio?
I really enjoy the way that the guys explain the concepts. Easy to follow
How do we know that the golden ratio is the least well approxated by ratios?
Yeah I feel like it’s just the best example we’re aware of
It can be proven that the golden ratio is the most difficult number to approximate by rational numbers (meaning it takes large denominators to approximate the golden ratio well). This is related to Euclid's algorithm for computing the greatest common denominator.
Watch numberphile's video "The Golden Ratio (why is it so irrational)"
I love the fact that the brown paper from the Graham's number video got framed and is hanging on the wall! Would be great to have a photocopy of that on a poster!