Katie's

[@LeoStaley]

Judging from your shirt, you might be happy to learn that untitled goose game is getting a co-op DLC in September.

[@m1lkweed]

Leo Staley Honk

[@TheGreatPurpleFerret]

Two-in-a-circle-gon might be my new favorite shape.

[@garrettducat5769]

If each side was a plank length then how much space would it take to actually draw that polygon?

[@PheonyxPhyre73]

Garrett Ducat funnily enough even with this condition the megagon would be so much drastically bigger than the observable universe it's not even definable. If memory serves right the scale from Planck length to the observable universe is about 10^(60 0r 70) while 2 in a circle is defined as 256 in 256 triangles. even just 256 in one triangle is 256^256 so it's already bigger than the scale of the observable universe! Imagine how big it gets over 256 triangles!!!

[@EebstertheGreat]

@@garrettducat5769 Mega is defined as a 2 in a circle, or a 2 in two squares, or a 256 in a square, or a 256 in 256 triangles. Unpacking this definition, Mega is defined by the following sequence: m₀=256, m₁=256^256, m₂=(256^256)^(256^256), ..., mₙ=mₙ₋₁^mₙ₋₁, ..., where Mega = m₂₅₆. This is vastly larger than a power tower 256^(256^(256^(...(256^256))...) with height 256. Even thinking about this very small number for comparison, 256^256 is 617 digits long, 256^(256^256) is over 7×10⁶¹⁹ digits long (storing such a number would require stashing entire universes inside every elementary particle in the observable universe several layers deep), 256^(256^(256^256)) is vastly large still, and so on. You have to get to a height of 256 before you even reach my "small" number used for comparison to Mega. The number in question here is Moser's Number, a 2 in a megagon (or a 2 in a "two-in-a-circle-gon"). It's important to realize that the method by which the notation grows here is FAR faster than anything discussed up to this point. Adding even a single side to the polygon surrounding the 2 is like jumping up from 256 to Mega. Every time you add a side, you are entering a whole other universe of numbers to which the previous one seems so negligible as to be incomprehensibly puny. Even that hardly gives it credit. And we have to do this over and over again, so many times, we can't even describe the number of times we repeat this process except to just call the number "Mega." Needless to say, it's pretty big. Interestingly, it is not all that difficult to show that Moser's Number is much smaller than some other famously large integers like Graham's Number.

[@carbrickscity]

Yet it's proven that even Moser < G2 of Graham's number.

[@PapaFlammy69]

Great job Katie! =D

[@AyushYadav-mr5jj]

I am happy to inform that I successfully calaculated 1 in TREE(3)-gon .

[@gabbo396]

Ahahahahah nice job!

[@randomviewer896]

This is very similar to Conway's Arrow Notation! That's an interesting notation for it.

[@andreybashkin9030]

Alongside with Knuth's up-arrow notation, Ackerman function and some others.

[@hatacoyama1246]

@@andreybashkin9030 yessss

[@petros_adamopoulos]

A quite cumbersome arrow notation. Its merit would be to precede the latter by 1976-1938 years.

[@andreybashkin9030]

@@petros_adamopoulos Foundational ideas matter. They open field for later improvements.

[@carbrickscity]

It's growth rate is actually similar to up arrow notation. N in N-gon grows similar to N(N up arrows)N

[@astrixistheman]

WoWza. I like the idea of a book filled of random maths.

[@JacksonBockus]

Cool number. Cooler notation. Coolest shirt.

[@sk8rdman]

The pentagon thing was my first instinct. When you jumped to a circle I thought, well that's surely infinite, since a circle is basically a regular infinigon.

[@ScottCroom]

Circle-gon is now my favorite shape.

[@leefisher6366]

Only because squircles haven't been mentioned. (Squares with 'corners' that are 90 degree arcs of a quartered circle instead).

[@LeRodriguesTattoo]

Thank you very much to share this with us! It helps me a lot! =)

[@WilliamLeeSims]

I remember finding this notation in high school. I thought it was cool until I took up a few dozen pages just trying to expand something simple!

[@willk7184]

Cool video, thanks for sharing. It's amazing how fast the numbers can grow.

[@srwapo]

Nice honk shirt. 👍

[@IWouldLikeToRemainAnonymous]

WHAAAAT??? why don't you have more views and subs? this was explained in a very understandable way and very professional! Keep up the good work!

[@joshhickman77]

Seems like this notation is a bit similar to diagonalized Ackermann's function?

[@KatieSteckles]

It's similar enough that Ackerman's function is listed under "See also" on the Wikipedia page for Steinhaus-Moser notation :)

[@Lotrfan1991]

Just found your channel and am so stoked to watch your videos. Awesome hair by the way.

[@xyz.ijk.]

Megiston was in the Guinness Book of World Records in the 1960s as the largest written number. I don't know when it changed. For many years I was unable to find it again. Glad to see its reemergence. Thank you Katie!

[@glowingfish]

I remembered this video from a year ago and decided to watch it to cure my insomnia bevause those immense numbers are somewhat sedating. And I did make a video but it came in late and was at the end of the playlist.

[@Fanny-Fanny]

Superb video. You have a new sub. Love your enthusiasm for maths. Thanks!

[@lorenzobianchi1896]

I don't know what I like more, the tattoo, the hair color, the t-shirt or the maths discussed in the video! But why choose? Keep up the great work, Katie!

[@cosinev1265]

I found out about Steinhaus-Moser notation on Wikipedia about 7 years ago when I got really interested in math and I was frustrated that I couldn't find it on youtube. It's finally here after so long :)

[@rogerkearns8094]

I've subscribed to a megiston of mathematical uploaders since MegaFavNumbers started.

[@gladhobo]

This reminded me of a letter that I wrote to the editor of the Toronto Globe and Mail (published on 26 January 1985) wherein, after restating the definition of a "moser", I let H-1 be a moser inside a moser-gon; H-2 be H-1 inside an H-1-gon; and, in general, H-n be H-(n-1) inside an H-(n-1)-gon. I concluded by defining a "hoser" as the number represented by H-moser.

[@sthubbar]

Thank you for introducing me to this fun notation.

[@lunafoxfire]

Hey I didn't know you had a channel, but I found you in the playlist! Also I'm definitely gonna have to get a copy of that book.

[@Ganerrr]

ignore the following i was having a stroke using language you can describe crazy stuff but it kinda breaks down, so you gotta use concepts you can also iterate those megatons right, and then define a process of creating more iterations, and define a process of creating iteration of iterations, and then one that encompasses the encompassing of that. and then that that that that, itterate

[@RolandWolf]

Hmm. Sounds like a fun project doing some coding trying to crunch this... Though my usual approach of slapping together some JavaScript is doomed to fail. Also, love the hair!

[@columbus8myhw]

I'm thinking you'll need more digits than can be stored in your computer.

[@NightKev]

@@columbus8myhw s/your computer/the universe You'll get there quite quickly, in fact.

[@anonymoususer9837]

Try 256 double-arrow 256 first (Knuth up-arrow notation). That'd just be a tower of powers of 256, that goes 256 high, which is smaller than mega because once it takes 256^256, that becomes the base as well as the power, whereas 256 double-arrow 256 keeps the base at 256 each step of the way. Still, I suspect mega is smaller than 3 quadruple-arrow 3, also known as G1 where G64 is Graham's Number (G[n+1]=3 and 3, separated by G[n] arrows).

[@razieldolomite698]

I'm so glad I found your comment after doing just this in python. I was surprised to find repl.it could actually handle circle(2). but when I changed it to pentagon(2) so I could give hexagon(2) a go, repl.it killed it instantly like I was hoping all along xD Code is below id anyone wants to run it in their python IDE: import sys def triangle(x): return float(x)**float(x) def square(x): return triangle(float(x))**triangle(float(x)) def pentagon(x): return square(float(x))**square(float(x)) def hexagon(x): return pentagon(float(x))**pentagon(float(x)) print("WARNING: These operations grow so fast that using the word 'rapidly' would be an understatement of truly galactic proportions. Input your digits responsibly, your processor will thank you.") num = input("gib number: ") op = input("type 1 for triangle, 2 for square, 3 for pentagon, 4 for hexagon: ") if float(op) == 1: print(triangle(float(num))) elif float(op) == 2: print(square(float(num))) elif float(op) == 3: print(pentagon(float(num))) elif float(op) == 4: print(hexagon(float(num))) else: sys.exit() Edit: Changed the code to also work with non-integers

[@RolandWolf]

It seems obvious that no computer can give me the exact number these notation represent. But maybe I could approximate how they relate to other huge numbers. I guess I just want to see it fail for myself in a desperate try to internalize this.

[@danielstephenson7558]

Reminds me of how Numberphile explained how to get to Graham's Number. It just explodes so fast!

[@ASSamiYT]

Now I have heard two very scientific and explicitly accurate mathematical expressions on this channel, that I will be using. "Timesing" and "something-something-triangles". 👌

[@gerrykavanagh]

Thanks! Just got the book. Looking forward to reading it & boring my kids with mathematical wonders

[@paulzagieboylo7315]

Hi Katie! I was getting flashbacks to an old Mystery Hunt puzzle (web.mit.edu/puzzle/www/2016/puzzle/identify_sort_index_solve/) throughout this whole video. My favorite line from the answer key to that one: "We're veering dangerously away from mathematics and into philosophy, but if you believe in a Platonic ideal of "truth", many of these describable numbers do "exist" even though we can never know or verify them."

[@NoriMori1992]

Cool! It's fun to learn about non-standard notations, even if I can't quite wrap my head around them and will probably never use them! This makes me want to rewatch the Numberphile videos about Graham's Number! Edit: And then I did just that. All five of them.

[@kcmichaelm]

This was really cool, thank you!!

[@Petertronic]

Good to see a new video, very interesting!

[@redplayer4821]

I did a bit of a puzzle for myself and interestingly enough, I found a (somewhat simple ?) way to describe 2 in a pentagon / circle as a way to understand just how massive it is if you take the operation of a succession of triangles in a more normalised mathematical notation you will find that the pattern is actually quite easy to follow you start at 0 triangles, with your value (here 2) and then for each triangle you add, you take the previous answer, and you mirror it around an exponent sign 2 : 2 2 in a triangle : 2 ^ 2 2 in 2 triangles : (2 ^ 2) ^ (2 ^ 2) 2 in 3 triangles : ((2 ^ 2) ^ (2 ^ 2)) ^ ((2 ^ 2) ^ (2 ^ 2)) . . . and following that logic, with 2 in a pentagon being equal to 256 in 256 triangles, or 2 in 259 triangles you end up with 259 mirrorings of 2 so to write that sequence, you would need to write 2^259 times the number 2 which is a bit under 10^78 926,336,713,898,529,563,388,567,880,069,503,262,826,159,877,325,124,512,315,660,672,063,305,037,119,488 to be exact and I think THIS number is my #megafavnumbers

[@Lexivor]

You're not doing the math correctly, 2 inside 259 triangles is much bigger than 2^259. 2 in 1 triangle is 2^2, 2 in 2 triangles is 2^8, 2 in 3 triangles is 2^2048, 2 in 4 triangles is 2^(2^2057) etc. You were right about 2 in a pentagon being 259 mirrorings of 2 but that's just a gigantic power tower with a height of roughly 259. So it's approximately 2^^259, not 2^259.

[@redplayer4821]

@@Lexivor I never said 2 in 259 triangles = 2^259 I said 2 in 259 triangles would require that you write the number "2" 2^259 times if you follow the mirroring pattern that simplifies the writing of 2 in 259 triangles but then to find 2 in 259 triangles you have to do the math of that weirldy shaped power tower of 2^259 twos

[@Lexivor]

@@redplayer4821 OK, I see now, it wasn't clear to me from what you wrote, but looking again I see what you meant.

[@draztiqmeshaz6226]

Good to see your face again! OK listening...

[@senseidei]

I was going down the megfavnumbers playlist, and for some unknown reason, remembered that it's SGDQ week :)

[@pooyataleb2514]

Damn that's a cool shirt!

[@TheOfficialPolo]

You just convinced me to order myself a Steinhaus's book :)

[@cyrilio]

Love this

[@eamonnsiocain6454]

Yes, I too have found dot dot dot to be very mathematical. Why not? LOL! Fascinating video. I love to shop at used bookstores, looking for such gems.

[@davidalearmonth]

I need to know approximately how large is 2 in a circle? Larger than a googol?

[@MichaelWarman]

I don't think there's any real way to express 2circle meaningfully without an industrial level of computer power, but I believe a gogol would be negligible by comparison. I think a gogolplex would also be negligible by comparison, though I find a gogolplex hard to conceptualise.

[@KatieSteckles]

Significantly more. According to Wikipedia, it's bigger than 10^(10^(10^…^10))) where there are 257 10s in the power tower. (But smaller than a stack of 258 10s).

[@lak395]

Much, much bigger than googol, if my understanding is correct.

[@davidalearmonth]

Thanks! I did search it afterwards, but initially was only finding answers in notations I was unfamiliar with.

[@TheoEvian]

@@lak395 2 circle is ridiculously big it is according to wiki somewhere between 10 arrow arrow 257 and 10 arrow arrow 258, so it has even googolplex beaten quite easilly. But even the 2 megagon makes it only as big as if I am not mistaken 4th grahams number and the Graham's number proper is 64th number in that sequence so it isn't THAT big in comparison :D

[@GenericAnimeBoy]

Liking the video just for that shirt. The maths is a bonus. Edit: didn't Knuth do something similar/related with up-arrow notation?

[@LuizBHMG]

I like the Steinhaus-Moser notation more because _a_ in a *circle* is equal to _a_ in infinite *circles.*

[@AureliusR]

The 2 in two triangles, for some reason, my immediate interpretation was 2^2^2

[@somniad]

Oh my god your shirt is so good

[@nrellis666]

How do 'a in a triangle' and 'a in a square' numbers relate to arrow notation?

[@shruggzdastr8-facedclown]

To hold the grammatitians (sp?) at-bay, put the a's in quotation marks

[@karlwaugh30]

It'd be nice to see how these relate to bit Up Arrow and/or Conway Chains? Also is there a way to "level up" this notation by saying, for example, a in a tetrahedron means a in an a-gon, a in a square means a in a tetrahedrons etc.

[@m1lkweed]

I'm fond of bracket notation, a[n]b because the names for the higher hyperoperations (tetration, pentation, etc.) line up with n whereas they're 2 off with up-arrow counting

[@Qermaq]

I'm in love with the shape of megagon.

[@geekjokes8458]

first of all, your shirt is amazing

[@geekjokes8458]

second, THIS ONE IS JUST TOO BIG aaaaaAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

[@danielroder830]

It's basically like the fast-growing hierarchy. You could go on and make a special shape and say something like : the number you get from the inner function defines how many edges the shape has. And then just put some number like "2 in pentagon" in and your special shape deconstructs this to a shape with "2 in a pentagon" edges with the number "2 in a pentagon" written inside....but thats not enough is it? We now define another shape that defines how ofthe this special shape is nested in itself, so we got "2 in a pentagon" rings of this special shape around the "2 in a pentagon". Each one has to be deconstructed from the middle. Just the first ring in the middle is this enourmous number, but it just tells you how many edges the next shape has.

[@shruggzdastr8-facedclown]

Q&A @Katie: In terms of largeness, how does 2 in a megagon compare to Graham's Number or Tree-3?

[@ceselb]

I'd love to see that.

[@thegreatgario3478]

three years later i can say that graham's number is unfathomably larger than moser's number, and TREE(3) is unfathomably larger than G(64) (graham's number) they're... big

[@Dogo.R]

This seems related to arrow notation... or the same? Am I crazy?

[@whatelseison8970]

I was thinking the same thing. It seems to share some similarity to tetration and pentation as well; especially in the sense that I don't really understand them at all.

[@calvincrady]

a-triangle = a^a = a↑↑2, so they're definitely related, but they're not the same. 2↑↑↑↑...2 will always equal 4 no matter how many arrows you put in, but 2-in-a-square = 256

[@petros_adamopoulos]

@@calvincrady Let's say they're still if comparable power but arrows are easier to draw in print.

[@lucascisneros8147]

Calvin Crady 2 double arrow 2 wouldnt be 4, nor would the next iterations. For example, 2doublearrow2 is 2arrow2 2 times, which is 2arrow2arrow2 or 2^4.

[@Pacvalham]

@@lucascisneros8147 For any number of arrows, write the second number of copies of the first number and put groups of one fewer arrow between them. 2 ↑↑ 2 = 2 ↑ 2 = 2 ^ 2 = 4

[@SafetyBoater]

Any estimate on the size of Megiston?

[@LeftPinkie]

Has anyone calculated some of these notations to integer numbers?

[@Miaumiau3333]

my fav megafavnumber is 91 because its one of the largest prime numbers probably

[@livedandletdie]

This would be a hilarious project to do with non-integers in the span of 0 to 1. They would be infinitely small in the end, but still. Could be interesting. I mean 1 in any ngon is just 1 except for any n greater than or equal to aleph0.

[@microtubules]

So how what size polynomial shape does 2 have to be in before the result is bigger than Graham's number?

[@andrewbradley9052]

Given the 'threeness' of Graham's number, maybe a more fun question would be what polygon would you have to chuck a three into to get a number of the order of g64. I don't know the answer to either question though.

[@the2ndblunder]

Is a semi-circle a**2 and a circle just a in Steinhaus-Moser notation

[@fibbooo1123]

I have mathematical snapshots! Its an awesome book!

[@godwin972]

can't believe all the cirno fans were using fancy math notation this entire time

[@Tsskyx]

Funny coincidence, I've been exploring googology recently. Seeing all these videos about large numbers really made my day. And, I must say, googology is a fascinating subject, albeit slightly pointless from a mathematical standpoint.

[@nickparkyn3561]

What even is googology? It sounds cool

[@Tsskyx]

@@nickparkyn3561 Well, in a nutshell, obsessing over super large numbers and excessively powerful notations.

[@nickparkyn3561]

Tsskyx I already do that! It sounds perfect for me, thx for introducing me to this awesome concept

[@coloripple]

This is really interesting to me! I've wondered at Graham's number for a while, and this seems similar in how stupidly fast it grows, compared to arrow notation. And both work with power towers. To me it seems like Grahams number is still bigger, as Moser's number seems comparable to g2, because that "only" does the insane incomprihensible step once. (If that makes sence) However, obviously these numbers are way too big to understand. I'd love to know how big it really is compared to the G-sequence

[@coloripple]

Ok now I had to subscribe in hope of possibly a video about this...? I do think I'm not the only one that would be really interested by this comparison

[@gudmundurjonsson4357]

a in an a-gon also sounds fun

[@davewilson13]

How are large primes written? With +1 on a large number?

[@KatieSteckles]

If you can find one large enough that this notation can be used, then maybe - but there aren't many known even nearly as big as 2 in a circle! I don't think this notation is very widely used, and you can only use it to write specific numbers.

[@nerdyjoe314]

Is there a comparison between this notation and Conway Up-arrow notation?

[@convindix9638]

A rough approximation is that "a in a square" approximates tetration and "a in a circle" approximates pentation, and this is more accurate for larger a

[@fulla1]

That's exactly, what I was thinking!

[@Lexivor]

Knuth had the up-arrow notation, Conway's chained-arrow notation had the arrows pointing right.

[@PhilBoswell]

Here's a couple of links that folks might find interesting and/or useful: en.wikipedia.org/wiki/Steinhaus%E2%80%93Moser_notation googology.wikia.org/wiki/Megiston The latter is to a page on the "Googology Wiki" which is a tarpit if you're into that kind of thing ❣

[@Pacvalham]

This reminds me of Knuth's Up-Arrow Notation.

[@Pacvalham]

How does this big number compare to Graham's number and TREE(3)?

[@sinisternightcore3489]

How can we extend this to calculate non-integers in a circle?

[@karito1358]

what would N inside a circle inside a triangle be?

[@KatieSteckles]

It'd be n in a circle to the power of n in a circle, I think?

[@omerd602]

Two things: 1. If 3 in a triangle is 3^3, and it is also 3 in 3 two-gons, what would the two-gon operation be? Would such an operation exist? If it does, what would a one-gon mean? 2. We should turn this into a proper notation that doesn't involve drawing a Megagon to calculate Moser's number: How about ᵇa to mean "a in a b-gon"?

[@KatieSteckles]

Sadly, I don't think being in an n-gon is defined for n

[@omerd602]

Yeah... that's kinda what I thought. Also, I wanted to add on to the notation, but I couldn't figure something out when I posted the original comment -- but now I've got something: Define a new system where "a in a triangle" means "a in an a-gon in the original system", and apply similar rules to this system: "a in a square" means "a in a triangles", etc. Call this System 2. Then make a similar system where "a in a triangle" means "a in an a-gon in System 2", call it System 3, and keep going. Now, ₛa (that's a tiny S) means "a in an a-gon in System S". We can keep going with this - define a new "system of systems", call the systems "System [n] in System 2", and keep everything the same except "a in a triangle in System 1 in System 2" means "a in an a-gon in system A in System 1". Now create a System 3 where "a in a triangle in System 1 in System 3" means "a in an a-gon in system A in System 2", etc. Keep going. I'm not even going to try to make notation now. We can have systems of systems of systems... (All of this for a ≥ 3 of course.) Finally, redefine "a in a circle" ("a in a squares" was assigned to the pentagon, if we recall) to mean "a in an a-gon in System A of System A of System A of ... where there are A of these "nested systems". THAT should get you some massive numbers really quick. (The reader may try to explain the "Mega-Megiston" given by ➉.) (If you didn't tl:dr this I congratulate you.)

[@straaths]

I'd throw a limit on 2-in-a-circle and make it infinitely big.

[@jimmyh2137]

But that's not even close to infinitely big! What about 3-in-a-circle? And what about (2-in-a-circle)-in-a-circle?

[@straaths]

I meant, triangle, square, pentagon, ... N-gon where N goes to infinity. So: 2-in-a-N-gon ~ 3-in-a-N-gon ~ ... ~ infinity In other words those are the same thing in a limit which goes to infinity. Lim(2-in-a-N-gon) where n goes to infinity = 2-in-a-circle

[@straaths]

Whatever, it's late and my life has no sense anyway...

[@timh.6872]

Here's a question. If we take "circle" to mean "aperiogon" (infinitely many sides with inner angle of 180°, blah-de-blah long lines, etc.), wouldn't that be asking for a fixed point with the second notation discussed? Because it's essentially the function family f_3(a) = a^a, f_{n+1}(a) = f_{n}^a(a). That is, when we have an n+1 gon around a, we repeat n-gons a times around a. But with an aperiogon, the number of sides remains the same when removing a side, as it has aleph-null sides. f_∞(a) = f_∞^a(a). Actually, that's completely undefined because any idempotent function satisfies that equation, nevermind!

[@hangugeohaksaeng]

How does this compare to Knuth arrow notation and Grams number? Or Tree? Really liked the video. Thanks for sharing a cool find. I think I'll get the book. :) Cheers!

[@chrissekely]

I wanted to ask this as well. I hope someone provides a real answer.

[@dlevi67]

See en.wikipedia.org/wiki/Steinhaus%E2%80%93Moser_notation#Moser's_number

[@nomekop777]

I wonder how large these get compared to knuth's notation, tree(n), and Graham's numbers

[@carbrickscity]

Mega < Megiston < G1 < Moser < G2 < Graham's number < TREE(3)

[@greyed]

Kind of feels like how Graham did his double arrow notation.

[@Fanny-Fanny]

What does 0.5 in a Megagon work out as?

[@petros_adamopoulos]

Something very close to 1 from below, no?

[@Fanny-Fanny]

@@petros_adamopoulos you know what, I think you might be right! Thanks!

[@andreybashkin9030]

For anyone interested in large numbers, start here and branch out: en.m.wikipedia.org/wiki/Large_numbers

[@andrewbradley9052]

So I can get this straight in my head what is 3 in a square? I think it works out as (27^27)^(27^27) but have I got that wrong?

[@KatieSteckles]

No, that's right - it's 3 in 3 triangles, or 27 in 2 triangles, or 27^27 in a triangle, or what you said :)

[@andrewbradley9052]

@@KatieSteckles Thank you. It's always a pleasant warm feeling when I get stuff right.

[@andrewbradley9052]

And congrats on marvellous Only Connect win. Just watched it :)

[@timh.6872]

I'm curious about how this might generalize downwards. If we take "a in a triangle" as f_3(a), then the obvious recurrence for higher sided shapes is f_{n+1}(a) = f_n^a(a), that is, "a in an n+1-gon is a in a n-gons". So what's f_2(a)? f_1(a)? It's not addition or multiplication, you end up taking power by multiple of two or multiplying by powers of two, not getting a^a for f_3. Is there even a sane function that maps all the positive integers such that applying it a times to a gives a^a?

[@wolfelkan8183]

In Knuth Arrow Notation, I think the Megiston ⑩ would be written as 10 ↑↑↑ 10.

[@WtbgoldBlogspot]

Neat

[@alexeecs]

What about a in an a-gon?

[@NikopolAU]

is Moser's number bigger than Graham's number? nvm, found the answer: en.wikipedia.org/wiki/Steinhaus%E2%80%93Moser_notation#Moser's_number

[@rentzepopoulos]

Mega means big in Greek. Megiston means biggest. Nice names by the way :)

[@certainlynotthebestpianist5638]

The most important question: how big is Moser's number compared to TREE(3)?

[@alan2here]

Tiny I think that you can't describe Tree 3 with other large number formats.

[@alan2here]

It's the Mth hyper-operation with N and N?

[@westerp]

Isn't 10 in a circle is the same as 10↑↑↑10 in Knuth's up-arrow notation?

[@alessandrolongo6738]

Is the Mosei’s number bigger than the Graham’s number or even bigger than Tree(3)? And going even further... which the ascending order of the following numbers: g(Tree(64)); Tree(g(64)); g(2 in a 64-sided polygon); 2 in a g(64)-sided polygon; 2 in a Tree(3)-sided polygon; Tree(2 in a 3-sided polygon). I need a video

[@dlevi67]

Much smaller than G, never mind TREE(3)

[@forna4090]

Nexttt

[@forna4090]

But wait no i want to see them :( time to abuse my calculator

[@AlbertoSaracco]

Nice video! A question:you say "2 is the smallest sensible number to ask how big circle-2 is". You are right, but... is there a way to extend the triangle, square and circle operation to real positive numbers, just as exponantiation (which is repeated multiplication) can be extended to positive real basis and real exponent (making possible to give a meaning to e^pi, e.g.)?

[@KatieSteckles]

Well, given a^a is defined for any real a already, putting a non-integer real number in a triangle works, but it's not possible to go further with this notation (what does it mean for something to be in 1.2 triangles?) - that only works with whole numbers.

[@AlbertoSaracco]

@@KatieSteckles you are obviously right for triangles. But maybe there is a logical way to extend also the definition of square and circle... After all... what does it mean to multiply e for itself pi times? ;-)

[@AlbertoSaracco]

Of course, one must decide what properties of the square and circle operations one wants to conserve... and maybe study the properties of those operations first... Nevermind... I was just curious!

[@KatieSteckles]

@@AlbertoSaracco There might be, I guess!

[@darthrainbows]

These numbers are incomprehensibly large for our puny brains, but how big are they really? Is 2-in-a-pentagon bigger than Graham's number? The idea of Moser's notation is similar to arrow notation used to construct Graham's number, but I have no idea which one ends up bigger. How does 2-in-a-n-gon grow compared to TREE(n)?

[@dlevi67]

Much more slowly. Moser's number (2 in a Megagon, and bear in mind that Mega = 256 ↑↑ 256, so it's quite a few more sides than a pentagon!) is much much smaller than Graham's number. TREE is another thing altogether...

[@Viruzzz]

This reminds me of arrow notation ( en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation ) I'm not sure if it's exactly the same but the idea is very very similar if not exactly the same.

[@AgentM124]

so by Moser's notation, any positive integer 2 or larger in a circle is basically infinite, because a circle is an infinite-agon

[@Science-sx8ho]

Is megiston bigger than tree(10)?

[@carbrickscity]

Mega < Megiston < G1 < Moser < G2 < Graham's number < TREE(3)

[@nathanderhake839]

Isn’t this a lot like arrow notation?

[@mydroid2791]

2 in a circle, is that an infinite sided polygon (re: Moser notation). Wonder how big that infinity is? Probably equals -PI/12.

[@ig2d]

is megagon > TREE(3)?

[@carbrickscity]

Mega < Megiston < G1 < Moser (2 in a Mega-gon) < G2 < Graham's number < TREE(3)

[@augustinfrancotte3163]

my fav mega number is omega

[@mendheimjw]

The square seems to be the same as tetration