Graham's number is an upper bound to a problem whose actual solution may be as small as 13. While Graham's number is impressive in size, it could very well just be a horribly horribly wrong upper bound to a problem. Whereas TREE(3) has a LOWER bound that is known to be far larger than Graham's number. For this reason, TREE(3) is more fascinating to me. Although I respect Graham's number for being the first 'stupidly large' number to be used in a serious mathematical paper.
The fascinating thing about these numbers to me isn’t that they’re so large, it’s the processes that makes them finite - which is crazy within crazy because it would suggest infinitely itself is easier to understand
This just goes to show that even if something feels infinite, you still have to prove it because there's always a chance that it only holds to an unimaginably large number like TREE(3)
TBH, I find colossal numbers to be much scarier than infinity. If we say the universe is infinite then there's no need to worry about how big it is, (you might even say it doesn't even have a real size in the normal sense) but if it's TREE(3) light years across that's just nuts.
Lunar Delta If scientists were to discover someday that the universe is infinite, it would make me feel less small and insignificant because literally every finite portion is exactly zero percent the whole universe (there is nothing that isn’t so tiny). But if they discovered that the universe is topologically a 3-sphere and has a volume of 10^130 (or so) m^3 that would make me feel insignificant compared to the large structures.
Surely just by going off the rules of this tree game you can assume that tree(3) is not infinite. Is some kind of proof required beyond logical reasoning in this case? If you know you know a tree can't contain previous trees then at some point you're going to run out of iterations.
Tree size isn't a priori bounded, I think the reason why we know that TREE(3) must be finite is because of the graph minor theorem (the whole tree not containing a previous tree thing smacks of the notion of a minor in graph theory, and the graph minor theorem says that every infinite collection of graphs has one that is a minor of some other graph in that collection; there's probably a bit more to the argument because in the TREE game, order matters).
There's actually an even bigger number known as "tree fiddy" which is named after the ammount of times that damn lockness monster will try and deceive you.
What I love about TREE(3) is that unlike other big numbers, they weren't intentionally looking for a huge number. One sprang up out of mathematical inquiry. That makes it more, I guess, legitimate than the likes of Rayo's Number. They had a concept and then out of this curiosity a colossal number emerged.
I've seen attempts to actually show how to "explain" it, but that requires a ton of really weird formulation on how all of the stuff Tony is talking about looks on paper. It can be done, but it's insanely technical.
So Graham's number is G64 iirc. Which G would TREE(3) be? Also, is it known which is the first busy beaver number greater than TREE(3) (or at least greater than the lowerd bound)?
Yes. There is no way to describe how big this number is in layman's terms the way you can explain Graham's number, it requires more advanced mathematical concepts to explain.
Tree(3) is so enourmous since it essentially takes the first tree with 1 seed of Tree(2) which makes you not have any other options that single seed. However, when you still have that seed, it scales up INSANELY
Exactly 🎯! Whichever seed you use, for the 1st tree, is forever out of the picture; you can never use it again, for that, particular forest. Which, in the case of TREE(2), leaves you, with only 1 seed, to work with (0, in the case of TREE(1)). With TREE(3), you still have 2 seeds, to work with; which gives you quite a bit more wiggle-room. 🤔
I'm opening a Discord server dedicated to explaining ordinals and the fast growing hierarchy, which you might be interested in. The end goal will be to reach an understanding of the magnitude of TREE(3) and larger things using only recursion, and lots of it, and you might gain some insight as to how much of a difference one arrow means compared to the difference from TREE(n) to TREE(n+1). discord.gg/5v6ucfN Feel free to join, basic algebra required.
Wait How come that's so small? Surely with four colours you can build the TREE(3) forest without ever using the fourth colour, and then when you've used up all possible trees start using the fourth colour?
@@Dexuz it's not way bigger than TREE(3) if you compare them using the fast growing hierarchy. The difference between TREE and FOREST is literally just adding 1 to a pretty large infinity
I still prefer Graham's Number because you can see the process by which you get there and (to a very limited extent) wrap your head around how absurdly large the number is. TREE(3) is, well, just a really big number. Yes, it's countless magnitudes _larger_ than Graham's Number, but as I like to say, "It's not the size of the pen that matters, but the poetry you write with it." I'm still interested in TREE(3) enough to learn more about out it, and find out why it behaves the way it does, but it still doesn't have that daunting, step-by-step escalation that Graham's Number does.
Rayo(10¹⁰⁰) is probably my favourite big number. It's easy to visualize, and it's reasonable. There are many ways to interpret it. Tree(3) is just like, a number. There's not really another way to visualize it other than it's original meaning, which is kind of boring
@@AbsoluteZero-zg9gj There are specific known lower and upper bounds for TREE(3), though the upper bound is less well-researched. The fact that it's bigger than Graham's number is not remotely "all we know." If you want to know more there is a lot of learning to do to get there, but these number can be parsed more thoroughly than you're aware. Indeed there are estimations comparing the entire TREE(n) function's growth rate as compared to the functions in the Fast Growing Hierarchy (which if you like Graham's number, and don't know about already, I highly suggest looking into).
Maths can really sneak up on you. You think you're ok doing it once, you start with 2+2, maybe someone teaches you some things about real and complex numbers in a dark alley. Next thing you know you're hooked on TREE(3).
Same. I got distracted from my abstract algebra homework to watch a video on graph theory lol. I can't wait to take my graph theory course next semester
I think Ackermann numbers (and Ackermann functions) would make for a really great topic on Numberphile, mainly for people who like stupidly big numbers- like me!
Look up Googology wiki, it's a great resource for this stuff. You can look up the Ackermann function there as well. BTW, don't get too excited, Ackermann function isn't nearly as powerful as TREE() and you're never going to define a number as large as TREE(3) just using the Ackermann function; You CAN easily pass Graham's number with it, though.
I think Tree(3) is the most interesting of these giant numbers because this game of trees looks so simple and all it takes is 3 seeds to produce a number that makes Graham's number look like nothing.
Grahams number is effectively zero compared to TREE(3). It is even bigger than GGGG…G64 with G64 iterations of G. In fact the number of times you would need to iterate the G function to beat it is TREE(3) itself, so basically pointless. You can’t even express TREE(3) using chain arrows. That’s just how big it is
Also TREE(n) has a growth rate between the SVO( Small Veblen Ordinal) and LVO( Large Veblen Ordinal) in fast growing hierarchy. For reference the above ordinals is way beyond gamma zero
@@R3cce " In fact the number of times you would need to iterate the G function to beat it is TREE(3) itself" do you have any reference or explanation to this statement?
@@R3cce It would be fun If someone does a video vizualization Tree(3)'s size. Similar to the videos that visualize the size of the Universe compared to a Plank length.
@@xenky2272 he isn't exactly correct. for example if you iterate G function TREE(3) - 1 times you certainly get a larger number than TREE(3). he's right in the sense that you will be hard pressed to put a number using any meaningful algebra or combination of G functions to reach TREE(3). For example a number such as G(G(G(G(G(G(G(G(G(G(G(...(G64)...))))))))) where you have applied the G function G(64) number of times, is still nothing compared to TREE(3)
'Effectively' zero should mean 'not zero' - or the 'effectively' is redundant?. or is one allowed different sorts of zeroes? Struggling with this one :)
@@tim40gabby25 "effectively zero" refers to the fact that grahams number is so unbelievably small compared to TREE(3) that it might as well be the same as 0 for all intents and purposes when you're on the scale of TREE(3)
Looking at these numbers makes you realize how scary eternity is, for example, when we talk about being immortal, literally immortal, no matter what happens you can't die, you could live Graham's number in years, TREE (3) in years, and still wouldn't have lived even a fraction of your entire life, not even close, you will live literally FOREVER, eternity is scary.
Love both this and the extra footage video! I cannot explain the joy watching these big number videos brings me; I completely empathise with Tony's excitement 😄
Looking at the sample trees for TREE(3), the fact that the function suddenly explodes after n=2 is maybe a little more intuitive than it first appears. Whatever colour you choose for the first tree cannot be used again in the sequence ever, so if you only have one or two to choose from to begin with, you're going to run out of options rapidly. But for n>2, you essentially have a "freebie" disposable seed for the first tree, and then all bets are off after that.
What always fascinates me about large numbers is that they can have very different properties from small ones. Many of the properties of numbers we think about are found in small examples: we have small primes, small perfect numbers, etc. But there are (presumably) types of numbers where there aren’t any small examples, and which potentially exhibit behaviours very unlike any we are used to thinking about. This is kind of incredible thing: usually we conceptualizer large numbers as being like small ones, just bigger, but there may be ones that are very different.
I can actually imagine Tree (3) being mind-bogglingly huge. Because the third and fourth tree that you draw in Tree (3) game only cancels out a fraction of possibility for the fifth tree that you draw. And this fraction gets smaller with each tree in a logarithmic fraction. As the trees become more complex it becomes easier not to have that same arrangement in the next tree. So already without even being told that tree 3 is very huge, I can somehow imagine it being bigger than a trillion if that makes sense.
Ah ah, TREE(3) dwarfs all numbers in common use. "Bigger than a trillion" is an understatement. TREE(3) is so huge that mathematicians in the comments are having trouble explaining it to laypeople. If you were to take the number of atoms in the universe (a big number) and produce a billion-core, terraherz-speed supercomputer for each atom, computers so strong that they can effectively execute any arbitrary exponentiation a billion times every nanosecond, and set them all to work exponentiating 2 and passing their results to the next computer... (in short, if you imagine anything from real life, distorted within reasonable bounds...) they would reach the result of TREE(3) eventually given a stupidly large amount of time, but ONLY because TREE(3) technically isn't infinite. If you imagine that scenario, and then put a time limit on it, any time limit you want, and ask "can they reach or exceed the result of TREE(3)?" The answer would be a resounding "nope!" Crazy big number...
@@ferociousfeind8538 That's a fine explanation but can I ask, what does "reasonable bounds" mean? I mean I know kind of what it means, but how do you define what is reasonable? I've seen it a lot in these comments.
@@SaladDongs as in, as long as your answer isn't "I want to use TREE(3) computers to do it!" The answer will be "it will take an inconceivably long time to calculate the size of TREE(3)
Mathematicians have what is considered an "extremely weak lower bound" for TREE(3). That number is greater than GG1, but less than GG2. In other words, it is greater than G of G1, but less than G of G2.
1. You mean if there is a constant C so that TREE(n) < C for all n? No. 2. Neither. Because of its growth hierarchy, this goes off to infinity too (even though every TREE(n) is finite).
You were being awfully cheeky there lol. Your explanation of TREE(2) and then the graphic of TREE(3) showing a node with 5 coming off THEN 4 coming off THEN 3 three coming off as a way of getting around the common ancestry. I saw that, thought about it for a second, then my head almost exploded. That is crazy!
Graham's Number is alot easier to understand than TREE(3) but TREE(3) is much cooler because it is WAY WAY WAY WAY bigger than Graham's Number and Graham's Number is already unimaginably huge!
I do not know the last digit of TREE(3), the first digit of TREE(3), or how many digits are in TREE(3). But I do know that 2 * arctan(TREE(3)) = 3.141592653589793 rounded to 15 decimal places.
I don’t understand, it was said that TREE(3) could only have 3 colors and 3 “seeds” every iteration, but in the examples on screen at 6:33 show way more seeds
But what if we play that game with Grahams cubes? With 1 color the upper bound is 1. With 2 colkrs it is already g(12) (mich smaller than G(64) but still huge). And with 3 colors?
When you look at the trees you notice they are actually only using two colors, because they need to use one in the first step and then can never use that again. And the second one where they use one color two times is another huge reduction of possibilities...chemistry (I dont know much about molecules) I think actually likes to reuse earlier structures?
I know tree(3) is already so ridicolously huge that cannot be processed but I wonder... do we have an idea on "how quickly this function grows"? I mean, what is the growth rate from tree(3) to tree(4)? Is the difference somehow proportional to the distance we have from tree(2) to tree(3)? Is it growing much faster? Does someone has an idea and does this really matters since tree(3) is already out of every scale?
TREE(4) is even bigger than putting TREE(3) in the repeated G sequence namely GGG…..G(TREE(3)) with TREE(3) number of G’s This shows how insane the function grows! 🤯
The nth tree can only be n nodes (maximum). Albeit your explanation does make sense, the first tree being 2 nodes doesn't work because of the first statement.
In a sense, given a simple and consise calculation of a number, the probability that it will be Tree(3) is tiny because Tree(3) is so big; hence it is difficult to give it any defining features apart from this definition alone.
what a smart trick playing with 3 seeds is to use the 1st type of seed only to start the game and never use it again after that. so, basically, the game goes on only with 2 types of seed, giving us more of a tree(2) than (3), and it still heads somewhere to the infinity... now imagine what crazy horror starts when we ACTUALLY have 3 different types playing the Tree(4)
![Seungmin "그렇게, 천천히, 우리(As we are)" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/kAzmhLHePqU/mqdefault.jpg)
Don't miss the extra footage - Tony says it is better than the main video: ruclips.net/video/IihcNa9YAPk/видео.html
Do a video about tetration, pentation, hexation etc...!
Do a video about extremely big numbers in works of Archimedes!
Is this Bigger
Tree(3)^Tree(3)
Do a video about the number of possible combinations of the Library of Babel!
Lex Viduya,
Yes, it is. But I mean numbers that are used in a mathematical proof.
Continue the logical sequence: 1, 3, ?
Passe-Science really big
Passe-Science or 5
Could be 9, if it's a geometric sequence
"?" is exactly how big TREE(3) is.
Teachers should have this on exams and everyone fails.
The TREE function does have a practical application - the calculation of interest by loan sharks.
Lol
That’s if they are generous
And student loans.
TREE(3) * TRUE!
So the answer is broken kneecaps?
It's like my little sister counting.
"One... three... gazillion billion"
willion
nillion
@vakahsj vnabdbn hillion
@vakahsj vnabdbn oillion
@vakahsj vnabdbn noillion
I laughed really hard when he said "We have a lower limit on it. It's bigger than... well it's certainly bigger than three."
What! i didn't laugh! 😐😐😐.
@@findystonerush9339 ??
Thank you sir
And everybody knows, anything bigger than 3 is just "big".
Ig it'd be smaller than TREE(4)
so, TREE(3) came about because someone gave a mathematician a third colouring pencil?
No, the TREE sequence arose from graph theory.
en.wikipedia.org/wiki/Graph_theory
Rykehuss Annnnd you had to ruin it......
What about a blue pen?
@@Yora21 What have you done
@@rykehuss3435 r/whoosh
Child: I can count to tree.
Me: no I don’t think you can.
I can count to TREE(3 - 1) + 1.
@@no-one-1 you can count to 4
lol
@@roblohub2270 League of leagions lets watch!😂😂😂.
Well, if they are only counting to TREE(1) or TREE(2) it's quite possible for a child.
I prefer grahams number. You can understand its growth even as a non mathematician. Tree3 is just... "Yeah, just believe us, it's big"
That's virtually what Graham's Number is too... "Yeah, it's a bunch of 3s multiplied together..."
You can find full explanations, but they are insanely difficult to understand. I can't wrap my head around them.
Graham's number is an upper bound to a problem whose actual solution may be as small as 13. While Graham's number is impressive in size, it could very well just be a horribly horribly wrong upper bound to a problem. Whereas TREE(3) has a LOWER bound that is known to be far larger than Graham's number. For this reason, TREE(3) is more fascinating to me. Although I respect Graham's number for being the first 'stupidly large' number to be used in a serious mathematical paper.
Retro Game Spacko exactly
Agree
Retro Game Spacko lol yeah
“It’s a big number”
Me: aight
“It puts Graham’s number to shame”
Me: ...aight
lol
The fascinating thing about these numbers to me isn’t that they’re so large, it’s the processes that makes them finite - which is crazy within crazy because it would suggest infinitely itself is easier to understand
This just goes to show that even if something feels infinite, you still have to prove it because there's always a chance that it only holds to an unimaginably large number like TREE(3)
It also really helps hammer home just how big "infinity" is. When we're constructing these colossal numbers that are nothing compared to infinities.
TBH, I find colossal numbers to be much scarier than infinity. If we say the universe is infinite then there's no need to worry about how big it is, (you might even say it doesn't even have a real size in the normal sense) but if it's TREE(3) light years across that's just nuts.
Lunar Delta
If scientists were to discover someday that the universe is infinite, it would make me feel less small and insignificant because literally every finite portion is exactly zero percent the whole universe (there is nothing that isn’t so tiny). But if they discovered that the universe is topologically a 3-sphere and has a volume of 10^130 (or so) m^3 that would make me feel insignificant compared to the large structures.
Surely just by going off the rules of this tree game you can assume that tree(3) is not infinite. Is some kind of proof required beyond logical reasoning in this case? If you know you know a tree can't contain previous trees then at some point you're going to run out of iterations.
Tree size isn't a priori bounded, I think the reason why we know that TREE(3) must be finite is because of the graph minor theorem (the whole tree not containing a previous tree thing smacks of the notion of a minor in graph theory, and the graph minor theorem says that every infinite collection of graphs has one that is a minor of some other graph in that collection; there's probably a bit more to the argument because in the TREE game, order matters).
There's actually an even bigger number known as "tree fiddy" which is named after the ammount of times that damn lockness monster will try and deceive you.
?
"Lockness" lol... read a book.
there's got to be a morning after
*Loch Ness
@@caineblackknife2443 there's a nicer way to do that.
What I love about TREE(3) is that unlike other big numbers, they weren't intentionally looking for a huge number. One sprang up out of mathematical inquiry. That makes it more, I guess, legitimate than the likes of Rayo's Number. They had a concept and then out of this curiosity a colossal number emerged.
Rayos number is boring
Plus its got a funny name, TREE 🔥
Well, there’s Graham’s Number, which wasn’t intentionally meant to be big.
"No physical process you can use to describe it." That's my favorite way to describe truly large numbers.
Me, talking to my sibling after borrowing some money: “how much do I owe you?”
My sibling: 0:00 - 0:14
DAMMMMMMMMMMMMMM
THIS IS SO FUNNY
every time i look at this post i start laughing uncontrollably
😆, 😆, 😜, 😅, 😅!
Or 6:49 to 6:55
More interested in Tree(Fitty).
LOL
Well now I'm startin' to get a little suspicious...
I spit my water out when I read this!
Damn you Loch Ness monster with Tree(Fiddy)
waddabout tree hunnid this is Sparta!!!
I'm waiting for the follow up "The Enormous Tree(3), but everytime they say tree it gets faster"
Fiyaaah I'll get on that
every time they say tree it speeds up by tree(3)%
AnnaIsABanana that's excessive
at that rate, the video would just stop by the first time they say "tree"
AnnaIsABanana no, slows down.
_Well that escalated quickly!_
A four word sentence and you had to edit it?
Interesting.
Also that was already commented here a month ago down the comments. I don't even understand it..
Lucien from “The Originals “
@@3vimages471 I think it was the italics
Imagine having a small number
This post was made by Tree(4) gang
Nico Detalo imagine having a smallER number.
This post was made by the TREE(5) gang. (There is no TREE(6) hahaha)
@@liongames8776 why is there no tree(6)
Shaan Singh no idea but who knows maybe there is but there is a.... TREE(TREE(3))
@@liongames8776 no there is a TREE(6) it’s just not shown here
@@SG2048-meta there could be a TREE(7), TREE(8), TREE(9), TREE(10), and it could just go on forever
But how do you even calculate this? Graham's number could be "grown" via arrow notation, but what about this?
I've seen attempts to actually show how to "explain" it, but that requires a ton of really weird formulation on how all of the stuff Tony is talking about looks on paper. It can be done, but it's insanely technical.
So Graham's number is G64 iirc. Which G would TREE(3) be? Also, is it known which is the first busy beaver number greater than TREE(3) (or at least greater than the lowerd bound)?
You couldn't express it using the "G" system used for Graham's Number, it's just too big.
It's even bigger than G(G(G(....(G(64))...))) for a reasonable number of iterations?
Yes. There is no way to describe how big this number is in layman's terms the way you can explain Graham's number, it requires more advanced mathematical concepts to explain.
Tree(3) is so enourmous since it essentially takes the first tree with 1 seed of Tree(2) which makes you not have any other options that single seed. However, when you still have that seed, it scales up INSANELY
Thank you sir
Huh
Exactly 🎯! Whichever seed you use, for the 1st tree, is forever out of the picture; you can never use it again, for that, particular forest. Which, in the case of TREE(2), leaves you, with only 1 seed, to work with (0, in the case of TREE(1)). With TREE(3), you still have 2 seeds, to work with; which gives you quite a bit more wiggle-room. 🤔
I once thought the difference that one arrow notation makes was big
But then the difference of tree(2) and tree(3) is just colossal
Meh, I've seen crazier. Also, a nitpick, tree(n)
I'm opening a Discord server dedicated to explaining ordinals and the fast growing hierarchy, which you might be interested in. The end goal will be to reach an understanding of the magnitude of TREE(3) and larger things using only recursion, and lots of it, and you might gain some insight as to how much of a difference one arrow means compared to the difference from TREE(n) to TREE(n+1).
discord.gg/5v6ucfN
Feel free to join, basic algebra required.
nguyen eyyy ninja'd also hi from googology discord
@@lamnguyen-uh4tz send invite
Ah, finally a number that can describe the size of my...
love for mathematics, gottem
*GOTTEM*
i thought you were gonna say brain and i was thinking "man, this person is full of themselves"
GOTTEM!!!!!!!!!!!!!!!!
Gottem did not get the hero it deserved, but the one it needed.
Iqbal Mala definitely*
TREE(Graham's number) ?? :D
oh no
aleph null be like hold my beer
TREE(TREE(3))
TREE(G64)
TREE(TREE(TREE(...(TREE(G64))...))) with G64 sets of parentheses
And still, almost all natural numbers are bigger than that.
Infinite natural numbers are larger than that
This is great
Precisely 100% of natural numbers are bigger than that
Yeah, piguy314, and they all contain the digit 3...
Almost all natural numbers are bigger than any natural number anyone cares to name.
Soooooo.... What about TREE(4)?
TREE(4) is actually pretty small, 9 to be exact.
Noooooooooo!!! :D
Wait
How come that's so small? Surely with four colours you can build the TREE(3) forest without ever using the fourth colour, and then when you've used up all possible trees start using the fourth colour?
TREE(TREE)
Poseidon
He was just joking.
Poseidon - What about TREE FIDDY?
And I thought planting 20 million trees was a lot, apparently all we need to is to plant 3
FOREST(3) = TREE(TREE(...TREE(3))...)
Not too much bigger than TREE(3)
SSCG(3) is still way bigger (not even talking about SCG(3) or SCG(13).)
@@metachirality and the Uncomputable functions
@@marketplierr
WAAAAAY bigger than TREE(3)
But also smaller than an infinite number of naturals.
@@Dexuz it's not way bigger than TREE(3) if you compare them using the fast growing hierarchy. The difference between TREE and FOREST is literally just adding 1 to a pretty large infinity
My question is, do we have any way of knowing or determining the first n steps of the optimal sequence of trees for TREE(3)?
For the first step at least, yes.
I still prefer Graham's Number because you can see the process by which you get there and (to a very limited extent) wrap your head around how absurdly large the number is. TREE(3) is, well, just a really big number. Yes, it's countless magnitudes _larger_ than Graham's Number, but as I like to say, "It's not the size of the pen that matters, but the poetry you write with it." I'm still interested in TREE(3) enough to learn more about out it, and find out why it behaves the way it does, but it still doesn't have that daunting, step-by-step escalation that Graham's Number does.
So why don't you like G64!
TREE3 we only know that it's way bigger than Graham Number. We don't know actually how big is it
Rayo(10¹⁰⁰) is probably my favourite big number.
It's easy to visualize, and it's reasonable.
There are many ways to interpret it.
Tree(3) is just like, a number. There's not really another way to visualize it other than it's original meaning, which is kind of boring
@@AbsoluteZero-zg9gjWe sort of do, we know that it's smaller than other massive numbers
@@AbsoluteZero-zg9gj There are specific known lower and upper bounds for TREE(3), though the upper bound is less well-researched.
The fact that it's bigger than Graham's number is not remotely "all we know." If you want to know more there is a lot of learning to do to get there, but these number can be parsed more thoroughly than you're aware.
Indeed there are estimations comparing the entire TREE(n) function's growth rate as compared to the functions in the Fast Growing Hierarchy (which if you like Graham's number, and don't know about already, I highly suggest looking into).
stopped doing my maths to watch maths
Maths can really sneak up on you. You think you're ok doing it once, you start with 2+2, maybe someone teaches you some things about real and complex numbers in a dark alley. Next thing you know you're hooked on TREE(3).
It's truly horrible... I've recently seen a documentation about an addict, he already started doing it in elementary school.
Math*
Connor K a
Same. I got distracted from my abstract algebra homework to watch a video on graph theory lol. I can't wait to take my graph theory course next semester
YES! Finally! I waited for a TREE(3) Numberphile episode for ages!
and next Loader's Number :D
I love how "tree" is a mathematical function. :)
There are actually around 8 tree related functions two of them even faster than tree
We've all been waiting for this since the Graham's number videos
Finally easy video about TREE(3)!!!! Thank you!
MajkG MajkG unexpected factorial
You're right. I shouldn't mix my excitement with mathematic. :D
MajkG MajkG TREE(3)!!!! is a large number indeed
TREE(3) and TREE(3)!!!! are essentially indistinguishable, so they are effectively the same size.
same dude same
TREE(3) is so big it makes short jokes about Graham's Number.
How big are the roots of these trees, and how much wood could a woodchuck chuck from them?
John Michaelson probably allot
ohhh as in plant roots hahahah nice joke
@asd
Spoiler, TREE(3)th root of 1 is small.
@@Dexuz TREE(3)th root of 1 is 1
@@nilesspindrift1934 Honestly, I don't even know why I said root, I should have said 1 divided by TREE(3)
So excited! I've been waiting for this video ever since tree(3) was alluded to in the original Graham's number video.
"To explain what TREE(3) comes from, well it comes from a game of trees."
Well, great, thanks professor.
I think Ackermann numbers (and Ackermann functions) would make for a really great topic on Numberphile, mainly for people who like stupidly big numbers- like me!
Hi Ho Wolverhampton how stupidly big should the numbers get?
I think the Ackermann functions were talked a little bit about in Computerphile
It would be nice to see another look at them in Numberphile
Seconded!
Look up Googology wiki, it's a great resource for this stuff. You can look up the Ackermann function there as well. BTW, don't get too excited, Ackermann function isn't nearly as powerful as TREE() and you're never going to define a number as large as TREE(3) just using the Ackermann function; You CAN easily pass Graham's number with it, though.
Like them? I love them .
"What is it useful for? What does any of this got to do with anything that's important?"
End cut with no answer
They made an extra video with the answer in it
I think Tree(3) is the most interesting of these giant numbers because this game of trees looks so simple and all it takes is 3 seeds to produce a number that makes Graham's number look like nothing.
Grahams number is effectively zero compared to TREE(3). It is even bigger than GGGG…G64 with G64 iterations of G. In fact the number of times you would need to iterate the G function to beat it is TREE(3) itself, so basically pointless. You can’t even express TREE(3) using chain arrows. That’s just how big it is
Also TREE(n) has a growth rate between the SVO( Small Veblen Ordinal) and LVO( Large Veblen Ordinal) in fast growing hierarchy. For reference the above ordinals is way beyond gamma zero
@@R3cce " In fact the number of times you would need to iterate the G function to beat it is TREE(3) itself" do you have any reference or explanation to this statement?
@@R3cce It would be fun If someone does a video vizualization Tree(3)'s size. Similar to the videos that visualize the size of the Universe compared to a Plank length.
@@xenky2272 he isn't exactly correct. for example if you iterate G function TREE(3) - 1 times you certainly get a larger number than TREE(3). he's right in the sense that you will be hard pressed to put a number using any meaningful algebra or combination of G functions to reach TREE(3). For example a number such as G(G(G(G(G(G(G(G(G(G(G(...(G64)...))))))))) where you have applied the G function G(64) number of times, is still nothing compared to TREE(3)
"Grahams nunber is effectively zero compared to tree 3" very funny way to start a vid
'Effectively' zero should mean 'not zero' - or the 'effectively' is redundant?. or is one allowed different sorts of zeroes? Struggling with this one :)
@@tim40gabby25 "effectively zero" refers to the fact that grahams number is so unbelievably small compared to TREE(3) that it might as well be the same as 0 for all intents and purposes when you're on the scale of TREE(3)
@@zenthichutt7071 understood, thanks :)
OMG I've been waiting years for you to cover this! Thank you!
Looking at these numbers makes you realize how scary eternity is, for example, when we talk about being immortal, literally immortal, no matter what happens you can't die, you could live Graham's number in years, TREE (3) in years, and still wouldn't have lived even a fraction of your entire life, not even close, you will live literally FOREVER, eternity is scary.
Love both this and the extra footage video! I cannot explain the joy watching these big number videos brings me; I completely empathise with Tony's excitement 😄
3:30 when scientists discover humans originated in Ethiopia
Looking at the sample trees for TREE(3), the fact that the function suddenly explodes after n=2 is maybe a little more intuitive than it first appears. Whatever colour you choose for the first tree cannot be used again in the sequence ever, so if you only have one or two to choose from to begin with, you're going to run out of options rapidly. But for n>2, you essentially have a "freebie" disposable seed for the first tree, and then all bets are off after that.
What always fascinates me about large numbers is that they can have very different properties from small ones. Many of the properties of numbers we think about are found in small examples: we have small primes, small perfect numbers, etc. But there are (presumably) types of numbers where there aren’t any small examples, and which potentially exhibit behaviours very unlike any we are used to thinking about. This is kind of incredible thing: usually we conceptualizer large numbers as being like small ones, just bigger, but there may be ones that are very different.
I love rewatching this video
We need a digitally playable version of TREE(3)
I can actually imagine Tree (3) being mind-bogglingly huge.
Because the third and fourth tree that you draw in Tree (3) game only cancels out a fraction of possibility for the fifth tree that you draw.
And this fraction gets smaller with each tree in a logarithmic fraction. As the trees become more complex it becomes easier not to have that same arrangement in the next tree. So already without even being told that tree 3 is very huge, I can somehow imagine it being bigger than a trillion if that makes sense.
Ah ah, TREE(3) dwarfs all numbers in common use. "Bigger than a trillion" is an understatement. TREE(3) is so huge that mathematicians in the comments are having trouble explaining it to laypeople. If you were to take the number of atoms in the universe (a big number) and produce a billion-core, terraherz-speed supercomputer for each atom, computers so strong that they can effectively execute any arbitrary exponentiation a billion times every nanosecond, and set them all to work exponentiating 2 and passing their results to the next computer... (in short, if you imagine anything from real life, distorted within reasonable bounds...) they would reach the result of TREE(3) eventually given a stupidly large amount of time, but ONLY because TREE(3) technically isn't infinite. If you imagine that scenario, and then put a time limit on it, any time limit you want, and ask "can they reach or exceed the result of TREE(3)?" The answer would be a resounding "nope!"
Crazy big number...
@@ferociousfeind8538 That's a fine explanation but can I ask, what does "reasonable bounds" mean? I mean I know kind of what it means, but how do you define what is reasonable? I've seen it a lot in these comments.
@@SaladDongs as in, as long as your answer isn't "I want to use TREE(3) computers to do it!" The answer will be "it will take an inconceivably long time to calculate the size of TREE(3)
of course it is bigger than a trillion dummy
Tree(3) isn't just bigger than a trillion, it's bigger than Graham's Number!
I like how he already sounds tired of its bigness as he goes to draw the very first tree of it at 6:15
How to keep a toddler occupied: explain this game and give him 3 coloured crayons.
Mathematicians have what is considered an "extremely weak lower bound" for TREE(3). That number is greater than GG1, but less than GG2. In other words, it is greater than G of G1, but less than G of G2.
I've got an even weaker lower bound of 1
Parker could've gotten TREE(2) up to 10.
He would have used 4 colors though.
Ah yes, a Parker Tree
@@walterrobinson9796 :-)
But if you use 4 colours then you're technically calculating TREE(4)
@PiggyKillerQ Explain the joke then
@PiggyKillerQ Oh, thanks 😂👍
6:20 He starts drawing and knows, he will sit there myraids and myraids of millenia. How many brown sheets will he need?
Other interesting questions I have:
Is TREE(n) bounded?
Is TREE(n)/TREE(n-1) bounded? Or even structured in any way?
TREE(n)/TREE(n-1) can't have n
1. You mean if there is a constant C so that TREE(n) < C for all n? No.
2. Neither. Because of its growth hierarchy, this goes off to infinity too (even though every TREE(n) is finite).
@@magicmulder2 TREE(n) is bounded between the SVO and LVO in fast growing hierarchy
I am confused by the Knuth triple down arrow notation in the description?
I think it means "this way lies madness" as a warning not to try comprehending it.
You have a secret: Tree 1
You tell another person: Tree 2
You tell a second person: Tree 3
And now. Number 3. The Larch.
The...
Larch.
Very nice
Beautiful.
And now......
THe LArcH
Which is bigger?
G(TREE(3)) or TREE(Graham’s Number)?
TREE(Graham’s Number) >> TREE(4) >> G(TREE(3))
well, the tree function does grow faster than grahams number does with increasing iterations.
G(TREE(3) is much much much much smaller than TREE(Graham's Number)
Congrats! You've got a video!
They made a video answering it!
You were being awfully cheeky there lol. Your explanation of TREE(2) and then the graphic of TREE(3) showing a node with 5 coming off THEN 4 coming off THEN 3 three coming off as a way of getting around the common ancestry. I saw that, thought about it for a second, then my head almost exploded. That is crazy!
The definition of "that escalated quickly"
*Spends 6 minutes playing a math game* “So yeah, this number tree3 is so big”
TREE(3) is around between this two big numbers represented in BEAF. {10,100(1)2} & 10
This must explain why I sometimes call 3 tree.
One, two, tree, four...
Double Dare Fan are you Irish by any chance
TREE(TREE) Aha!
tree it's 3 (три) in russian, lol
stop looking at my profile pic TREE(TREE(TREE))
its one, two, TREE(3), four
Graham's Number is alot easier to understand than TREE(3) but TREE(3) is much cooler because it is WAY WAY WAY WAY bigger than Graham's Number and Graham's Number is already unimaginably huge!
Tree(3) is my favorite of all these giant numbers. It a proof of an old Chinese idiom: 1 generates 2; 2 generates 3; and 3 generates everything!
The size of a tree(3) number of Planck volumes is unimaginably larger than if the entire observable universe were Graham's number times wider
6:36 Could somebody explain why the 4th one isn't contained within the 6th? Both have 3 blacks and a red as a chain.
If you're talking about the 6th's left side, it could be also that we only trace it vertically, not on a V shape
Most people have no clue how massive this is, and we cant tell them.
Friend: What's your favorite number?
Me: Oh it's just Tree, nothing much.
I do not know the last digit of TREE(3), the first digit of TREE(3), or how many digits are in TREE(3).
But I do know that 2 * arctan(TREE(3)) = 3.141592653589793 rounded to 15 decimal places.
I don’t understand, it was said that TREE(3) could only have 3 colors and 3 “seeds” every iteration, but in the examples on screen at 6:33 show way more seeds
Him: Tree(3) is so big! U can't imagine anything bigger!
Me: Ok, so what about Tree(3)+1 ?
PhantomGaming
Tree(tree(tree ........ (tree 3))
Tree 3 times
TREE(3) is {3, 6, 3 [1 [2 \ 3 ¬ 1, 2] 2] 2}
I'm about to cry, I can't find a simple explanation for notations stronger than than Ackermann one
But what if we play that game with Grahams cubes?
With 1 color the upper bound is 1.
With 2 colkrs it is already g(12) (mich smaller than G(64) but still huge).
And with 3 colors?
Tony Padilla is on fire here.
This weirdly mirrors chemistry with simple carbon compounds [ Carbon, hydrogen and oxygen ]
When you look at the trees you notice they are actually only using two colors, because they need to use one in the first step and then can never use that again. And the second one where they use one color two times is another huge reduction of possibilities...chemistry (I dont know much about molecules) I think actually likes to reuse earlier structures?
6:48 Very gladdening to hear a mathematician describe a number's bigness as "really really really really really really really really ...... "
At 6:38 isn't the 3rd contained in 5,6,7,8,9,10 and 12?
I know tree(3) is already so ridicolously huge that cannot be processed but I wonder... do we have an idea on "how quickly this function grows"?
I mean, what is the growth rate from tree(3) to tree(4)? Is the difference somehow proportional to the distance we have from tree(2) to tree(3)? Is it growing much faster? Does someone has an idea and does this really matters since tree(3) is already out of every scale?
TREE(4) is even bigger than putting TREE(3) in the repeated G sequence namely GGG…..G(TREE(3)) with TREE(3) number of G’s
This shows how insane the function grows! 🤯
in the fast growing hierarchy it is between the SVO and LVO ordinals
"I can't express how really big it is. It's off the scale big"
That's what he said.
"We're going to try to build a forest, one tree at a time."
[Australia bushfires have entered the chat.]
Tree(1): I'm weak...
Tree(2): I'm just 2 more than the weak one...
Tree(3): Graham's number? Oh, You mean my younger brother?
Graham's Number? You mean that ant in my yard?
Graham's number? Do you mean that tiny cell in my body?
Tree(7): Graham’s number? oh you mean that atom through the microscope?
I relatively recently discovered this channel.
It has a great spirit!. TREE(3)...
Thank you!
Thanks
I still love that you can try picturing the numbers on a visual plane. still so impossible like grahams number. awesome!
Even Tree(Graham's Number) is closer to 0 than it is to infinity. Goes to show how big infinity really is. 😂
Infinity is not a number. It is a concept of something that has no end
@@R3cceTHANK YOU!
imagine the number "TREE(3)" representing the number of nodes allowed. So, TREE(TREE[3])
You need to stop.
What have you done
this Nummer would be SO big we couldnt even *Imagine* it i think
TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
@@AlbertKekstein we can't imagine this one either
Great video, I nearly understood what you're talking about.
TREE(TREE(3))
Boi
TREE(3) can be beaten using beaf notation. It is most likely somewhere at the legion array notation. Although beaf is ill defined at this point
the important rule of the tree function: the nth tree can't have more than n seeds.
What about Tree(Graham's Number)
Have fun
Boring. Build a *novel* simple series where s(3) >>> TREE(3).
See the extra footage..
@@merek6986 ya I have. My comment was before the newer vid
3:08 nearest COMMON ancestor! I get it!
TREE(2) actually has 4 possiblities where we first make a tree of 2 green nodes then one green node then two red nodes and then one red node
The nth tree can only be n nodes (maximum). Albeit your explanation does make sense, the first tree being 2 nodes doesn't work because of the first statement.
Oh thx forgot that part😅
TREE(TREE(G(skews number)))
Is my thought up number
~f^2(ψ((Ω^ω)*ω),f(ω+1,f(2,10)))
TREE(TREE(G(TREE(G64))))
Salad
This was a let down as it did not, neither did the extra footage attempt to calculate the value of tree(3). Maybe a proof that g64
There has been no attempt to calculate it. It is too big for that.
In a sense, given a simple and consise calculation of a number, the probability that it will be Tree(3) is tiny because Tree(3) is so big; hence it is difficult to give it any defining features apart from this definition alone.
To prove that it'll take tooooooooooooooooooooooo many A4 papers, like in the order of 10^100 A4 papers. Can't fit into our Universe
what a smart trick playing with 3 seeds is to use the 1st type of seed only to start the game and never use it again after that. so, basically, the game goes on only with 2 types of seed, giving us more of a tree(2) than (3), and it still heads somewhere to the infinity... now imagine what crazy horror starts when we ACTUALLY have 3 different types playing the Tree(4)
what happens when the forest has a TREE-some
sorry i suk at these
the door is over there
I agree