Euclid's Amazing Orchard

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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):

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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 (:

joaocandeias

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.

xyz

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.

Djorgal

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!

LeiosLabs

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.

kenhaley

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.

liweicai

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 YouTube itself.

danielsteel

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)

AlbySilly

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.

michaelmanning

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

mihailazar

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)

xxcheckerxx

What happens if you start at a different point such as (2, 0)?

RD-zcjw

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).

dohduhdah

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.

danielsteel

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.

zforzero