Euclid's Amazing Orchard
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- Опубліковано 14 чер 2024
- Guys, this is amazing! Let me know if you agree in the comment section!
Seriously, though. Euclid's Orchard somehow snuck up on me while browsing wikipedia and I fell in love with the concept! I hope you can see why.
Anyway, thanks so much for watching and I'll see you next time!
The music came from Josh Woodward (sped up 1.5 times):
www.joshwoodward.com/
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Very interesting video!! One other neat fact about this infinite orchard is that the percentage of trees that are visible to you is exactly 6/(pi^2), which completely blows my mind! Keep up the great work (:
Everything about this completely blows my mind!
Broke: Every direction I look at is trees. Literally infinite trees for every direction!
Woke: Only when looking in a rational direction, which is statistically impossible given the comparative size of the reals. You see no trees at all.
haha, you are right!
Man, I thought this was amazing! I hope you guys do too!
Let me know your thoughts or if you have any questions or comments. I love hearing from you guys!
LeiosOS Is it x/(x+y) or just x/y ie the slope or gradient.
Hmm... But what if the "tree" had some sort of thickness, so that ratios that are too close to each other will be blocked, (for example 10,7 blocking 13,9)
Yeah, a friend of mine looked into that, it creates a sphere of what can be seen.
Does it do that for all sizes of the circle? What about triangles?
You missed the most amazing thing about the orchard. If you point in a random direction the odds of pointing at a tree, even though the orchard is infinite, is zero! That's because any random direction will have zero chance of correspoing to a rational number. Ironically, it will always be irrational.
Yeah, it's super cool!
From this problem is borne the question *"Why is the sky dark at night?"* (since either there are infinitely many stars, or there aren't; either space is an infinite expanse, or it isn't; either the universe is infinitely old, or it isn't; et cetera).
All of our most fundamental inquiry in Astronomy (up to and including modern Cosmology) could have been (and in some cases, most definitely was) motivated by this "orchard" idea.
Daniel Steel , star light power decrease with distance. Also star light go to red when is far, not because it's far but because universe is expanding, and then go out of visible light.
@@chichikb The question wasn't answered until Hubble (ca. 1920s). Most of what people know about Astronomy (e.g., luminosity, parallax, "standard candles" etc) was being worked out between 1850 and 1950 (thanks in large part to Maxwell and Einstein).
For millenia, astronomers _didn't know that stuff._ Euclid certainly didn't. Hence, the question "Why is the sky dark at night?" was a stubborn one.
Thanks to the work of Perlmutter et al., it's now widely believed that there will be a point in the distant future where the entire night sky will be pitch black (And it won't even occur to future sentient life-forms to be puzzled by starry nights.)
Distant light redshifts out of our perceptual range.
Cool fact: the "density" of this pattern is 6/pi^2, the reciprocal of 1/1^2+1/2^2+1/3^2+... This is also the probability that two random integers are coprime.
they do have a direct correlation.
x/y is in lowest terms *if and only if* x is coprime with y
On Jan 26, 2018, the channel Numberphile published the video (watch?v=p-xa-3V5KO8) "Tree Gaps and Orchard Problems" which currently has 436,415 views.
There's officially such a thing as "mainstream media" within UA-cam itself.
OMFG i actually thought of something like this and wondered if there'd be some way of knowing wether or not the trees would get obscured or not
when i saw it it made me spontaineousely burst into laughter, as i realised it's brilliancy ... MY GOD, you made my day
It's so cool!
Everything in our existence is truly a mathematical equation.
I love prime numbers!
in this case they're coprime.
And because of irrational slopes, there is always a window of no trees in the infinite orchard since no tree would be on an irrational line: i.e. x,y existing in the natural numbers and let m be an irrational slope. Then, y=m*x. Thus, y/x=m, which is false since m is irrational. So, no rational number would be on an irrational line (replace m with pi and you'll see what I mean).
So, look at the slope of pi, and there will be no trees, just blankness forever.
The math's so easily understandable, and still amazing!
Right? It's so cool!
If you sort the visible trees in the orchard on size, you get the Farey sequence (except 0=0/1 and 1=1/1 at the start and end).
You didn't mention that you can look all the way through the forest, as long as the direction is irrational.
Another way to prove that is that there is a discrete number of trees in the orchard, but there is a continuous number of directions toward which you can look at. Thus, there are many more possible directions than trees.
is this the first video with 60fps ? looks great, also awesome video
My camera always films at ~60fps, but my animations are always at 30; however, I had to speed up all the animations this time, so that effectively quadrupled the fps. If you noticed a difference, I'll make future animations 60fps instead of 30.
Also: I'm glad you liked the video! =)
super cool, but it took me a while to figure out that the observer is placed at (0,0) and the first tree in the bottom left corner on (1,1)
awesome
What happens if you start at a different point such as (2, 0)?
I'm writing a paper and would like to feature this. Is there some literature about this orchard question you could recommend? I'm having difficulties finding hardly any.
It's interesting. I am wondering how the finite size of tree (not infinitesimally thin) will change the visibility in the real world.
So, a friend of mine ran this simulation. We actually see a circle of visibility emerge. It's pretty cool!
So when we finally find the last prime number we will have to stop planting new trees? What will happen with the forests then?!
Haha, not sure!
there's a last prime? i thought it was proven there are infinite.
Cool!
Haha, I'm glad you thought so too! =)
I thought he was kidding when he finished the video 😶
Love the enthusiasm. It is cool.
Euclid's orchard is still amazing!
Doesn't he remind you of Cody's lab?
Huh. I had not seen that channel before. Thanks for introducing me to it!
No problem, btw I love the way you speak
Trivial xD
But still amazing!
The video is amazing but I seriously didn't understood even a bit of it.
Euclid's super cool!