Pointless Gigantic List of Numbers - Part 4 (order type w to w^w)
Ultatri
Output of pete-5.c
Fish number 2
Fish number 1
Output of chan.c
Output of chan-2.c
Output of ioannis.c
fw+1(100)
Graham's number times a googolplex
Graham’s number / Graham-Gardner Number
PART 4: THE BIGGER INTERMEDIATES
order type w ~ w^w
Here you'll find some really big numbers like graatagold, supertet, and iteral, expressible with functions of order types between w and w^w in the fast-growing hierarchy. Prepare for a frenzied hurry towards infi - oh wait this is just the beginning of what googology has to offer!
The Expansion Range
10{10^100}10 ~ 10{{1}}10{10}10
(order type w+1)
Entries: 61
Gammagoogolplex
g(2000,g(2,2,50,100),50,100) ~ 100{10^5000-1}10,000
This number is one of Joyce's more recent googolisms. In summer 2014 he created a lot of crazy numbers (see also the gamoogol family), and this is one of them. Most of Joyce's new numbers are not on this list, as I find a lot of them to be silly or salad numbers. However, I decided to add this one to this list because of its cool name.
The double m in the name means 2000 (in Roman numerals), googolplex comes from a false definition of the googolplex incorporated in the definition, and the rest of the names's origin is unclear.
Deltagoogolplex
g(20000,g(2,2,50,100),50,100) ~ 100{10^5000-1}100,000
If there's one thing Andre Joyce loves to do, it's extrapolating from the names of numbers. Here, he noticed the Greek letter "gamma" in gammagoogolplex, and extrapolated to get "deltagoogolplex", since delta is the Greek letter after gamma.
Note: Normally I would put a comma in 20000 so that it would be 20,000, but that comma can be confused with a comma separating arguments in the g function, so I had to omit the comma.
Omegagoogolplex
g(2*10^24,g(2,2,50,100),50,100) ~ 100{10^5000-1}10^25
Andre Joyce decided to extrapolate further from deltagoogolplex to epsilongoogolplex, zetagoogolplex, etc all the way to omegagoogolplex. Really, he only lists gammagoogolplex, then deltagoogolplex, then skips all the way to omegagoogolplex. He actually messed up the definition, but I'd rather be logical than verbatim here.
Gongplexol
10{10^10^100-2}100
A number I coined for fun in analogy to the gongol - this is equal to 10 googolplexated to the 100th.
Moser
2[2[5]]
Also called Moser’s number, this gigantic number is 2 in a mega-gon (polygon with mega sides) in Steinhaus-Moser notation. Unlike the mega and megiston, Leo Moser coined this number, taking advantage of his idea of polygons with any number of sides instead of just triangles, squares, and circles, and plugging in a huge number of sides to form this giant.
The Moser just transcends the mega and megiston in a whole new way - it actually holds an interesting place among the googolisms. That is because the number of up-arrows usable to approximate this number is a high tetrational number, and therefore in a way it can be imagined as laying at the edge of the land of familiar up-arrows and gazing into a land of epic iteration and recursion, where Graham's number is just the beginning!
The Moser's last digits are ...1056, and it's equal to about 2^^^ ... (mega-2 ^s) ... ^^^3 in terms of Knuth's arrows.
Joyce's tetratri
g(2,1,1,4,3,3) = g(g(4,3,3),3,3)
Not to be confused with my Joycian Tetratri (in part 3). Joyce attempted to define Bowers' larger numbers as well. Joyce's definition is equal to 3^^^ ... ^^^3 with 3^^^3 - 1 ^s. It looks pretty insane, but it's NOT EVEN CLOSE to the real tetratri! The real tetratri makes this look tiny.
Tritriplex
3{tritri}3 = 3{3{3}3}3 = 3{{1}}3
This number shows up a lot when working with 3’s in 4-entry arrays - it’s much more than the Moser and much less than G(2). On this scale, it's virtually indistinguishable from the previous number.
G(2)
3{3^^^^3}3
Or 3^^^^^^^ ... ^^^^^^^3 with G(1) ^s. Yikes. This number is the second term in the road to Graham. Aarex calls this graham grahal.
10{10{10}10}10 = {10,3,1,2}
This is ten expanded to the third, which can be written using Bowers' hyper-operator notation as 10{{1}}3 or in array notation as {10,3,1,2}, since a expanded to b is a{{1}}b = {a,b,1,2} = a{a{a ... {a{a}a} ... a}a}a with b a's from the inside out. Therefore this number simplifies to 10{{1}}3 = 10{10{10}10}10 = 10^^^...(tridecal ^s)...^^^10.
Great googondol
E100##10#2
This number decomposes to E100#100#100#100 ... #100 with a googondol 100s. It's comparable to 10{10{10}100}100 in terms of up-arrows. It was coined by Saibian as a hint on what will come next in his numbers.
g(2)
2{2{12}3}
The second step in calculating Little Graham.
Magthree
A(A(A(3,3),A(3,3)),(A(A(3,3),A(3,3))) = A(A(61,61),A(61,61))
After the gag- prefix was defined, someone else suggested to continue with a mag- prefix, equal to gag-gag-gag-gag.....-gag-n n times. Therefore, magthree is equal to gag-gag-gag-3 = gag-gag-61 ~ 10{10{59}10}10. The value falls between g(2) and a boogolplex.
Boogolplex
10{10{100}10}10
Or 10^^^^ ... ^^^^10 with boogol ^s. The next level of recursion from a boogol.
Gugolda-suplex
E100##100#2
With numbers above a gugold, we need to define a new array of suffixes. -suplex is the first one. x-suplex, where x=Ea##b#c, is Ea##b#(c+1).
This number is also equal to E100#100#100....#100 with a gugold 100s.
Kilohyperfaxul
(200![1])[1]
After that there’s a mega-, giga-, etc. hyperfaxul.
Joyce's pentatri
g(3,1,1,4,3,3)
Not to be confused with my Joycian Pentatri (earlier in this list). This is equal to 3^^^^^^..........^^^^^^3, with 3^^^.....(3^^^3 - 1 ^s).....^^^3 - 1 ^s, so it iterates the number of up-arrows twice. The real pentatri is UNFATHOMABLY larger than this or even the tetratri - it can't remotely be compared to this number, as they aren't even in the same realm of numbers!
Tritriduplex
3{3{3{3}3}3}3 = 3{{1}}4
A lower bound to my Joycian Hexatri, equal to 3{tritriplex}3. Also an upper bound to Joyce's pentatri.
Joycian Hexatri
g(3,3,3,3,3,3)
Another g-function number I coined. This is approximately 3{3{E14#5#2}3}3, placing it between 3{3{3{3}3}3}3 and 3{3{3{4}3}3}3 (G(3)). It’s around E3##(E14#5#2)#2 in Extended Hyper-E.
Joycian Heptatri
g(3,3,3,3,3,3,3)
Yet another g-function number. Because of the g-function’s erratic behavior, this is almost no bigger than the Joycian Hexatri.
Joycian Octatri
g(3,3,3,3,3,3,3,3)
Ditto.
G(3)
3{3{3{4}3}3}3
This number is 3^^^....^^^3 with G(2) ^s. It also serves as an upper bound to the Joycian Hexatri.
Joycian grand tridecal
g(3,1,1,11,10,10)
The grand tridecal is the next of Joyce's epic fails at trying to define Bowerian numbers - this is approximately equal to 10{10{10{10}10}10}10 = 10{{1}}4 - the real grand tridecal is far far bigger, at 10{{10}}10. Hell, it isn't even close to as big as the pentatri or even the tetratri!
Gugolda-dusuplex
E100##100#3
This number is also E100#100#100....#100 with a gugolda-suplex 100s.
Little Graham / Graham-Rothschild Number
g(7) = 2{2{2{2{2{2{2{12}3}3}3}3}3}3}3
This number is the original version of Graham's number. Sbiis Saibian calls it "Little Graham", and Robert Munafo calls it the Graham-Rothschild number (he calls the number now known as Graham's number the Graham-Gardner number). It was defined by Ronald Graham and Bruce Rothschild in a paper about Ramsey theory, as an example demonstrating how hard it is to find upper bounds in certain areas of that field, even when the problem is that simple. It was once the best upper bound to the problem where it came up, but now the best bound 2^^2^^2^^9. It is defined similarly to Graham's number, like so:
g(1) = 2^^^^^^^^^^^^3 (12 ^s)
g(2) = 2^^^...(g(1) ^s)...^^3
g(3) = 2^^^...(g(2) ^s)...^^3
etc.
g(7) is Little Graham
Little Graham is FAR less well known than Graham's number, and many people who know of Graham's number don't know that Little Graham was the real upper-bound and Graham's number is an alternate easier-to-explain version created to popularize the number. Even Little Graham is no longer the best upper-bound, and many people still think the best upper-bound is Graham's number.
G(8)
The first member of Graham's sequence bigger than Little Graham.
Joycian dimentri
g(27,1,1,4,3,3)
Agghhhh ... what was Joyce THINKING here? This definition is approximately equal to 3{{1}}28, upper-bounded by G(27) in Graham's function. Here Joyce is so wrong it hurts. He probably thought dimentri is equal to {3,3,3,3.....3,3,3,3} with 27 3's, but the threes are arranged in a 3x3x3 CUBE! Either way he's not even REMOTELY close.
3{{1}}65 = {3,65,1,2}
A BEAF lower-bound to Graham. This number is 3{3{3.....{3{3}3}......3}3}3 with 65 3’s from the center to the right end. It’s like Graham but starting with tritri instead of G(1).
g(64,1,1,5,3,3)
A very (on this scale) exacting lower bound for Graham's number in Joyce's g function. It can be defined similarly to Graham's number like so:
GB(1) = 3^^^^3
GB(2) = 3^^^...^^^3 with GB(1)-1 ^s
GB(3) = 3^^^.....^^^3 with GB(2)-2 ^s
....
The approximation is GB(64). If it weren't for the offset from up-arrows, the g function could exactly express Graham.
G(64)
Graham's number is a famous number that is typically thought of as an upper-bound to the solution of a problem in Ramsey theory, a field of mathematics dealing with the minimal conditions in which certain conditions must necessarily arise. The problem asks, what is the minimum number of dimensions N of a hypercube such that all 2-colorings of all vertex pairs of the hypercube would necessarily have a coplanar complete graph of 4 vertices? That problem sounds complicated, but it's actually quite simple. Read the article on Graham's number for an explanation of the problem.
Here is the definition of Graham's number:
G(n)=3{G(n-1)}3
G(1)=3^^^^3
Graham’s number = G(64)
Or more visually:
G(1)=3^^^^3
G(2)=3^^^...^^^3 with G(1) ^s
G(3)=3^^^.....^^^3 with G(2) ^s
...
Graham’s number is G(64), or G for short. This number is famous because it was at one time the largest number used in any serious mathematics, which basically means the largest number to most people. The number was made famous in an article by Martin Gardner describing the number, and it even found its way to Guinness World Records, replacing Skewes' numbers (both have entries earlier in part 3). Because Gardner made this version of Graham's number famous, Robert Munafo calls this number the Graham-Gardner Number.
Graham's number did indeed appear in a mathematical proof in Ramsey theory by mathematician Ronald Graham in 1977, as an upper bound in a problem where the lower bound was six. However, the story is different from what most people think. The original Graham's number was Little Graham, the number that appeared in the 1971 proof. When Martin Gardner noticed the problem where Graham's number arose, he was intrigued by its size, and decided to devise an easier-to-explain but larger number that Graham proved in an later unpublished paper to share to the world. Therefore the number we know as Graham's number is more attributable to Gardner than Graham!
Nonetheless, Graham's number made it to Guinness World Records as the largest number used in a mathematical proof. Both bounds have since been improved, the high one from Little Graham to the relatively humble 2^^2^^2^^9, and the low one to 11 and then to 13. Many people still seem to think that Graham's number is still the best upper-bound, even though it never was.
Now, there are larger numbers used in mathematics which have dethroned Graham, such as TREE(3), SCG(13), and other terms in far-reaching sequences. Nevertheless, Graham’s number remains famous, and is still mistakenly thought by many as "the largest useful number". This is because the "largest number used in a mathematical proof" title was taken off of Guinness World Records in 1980, not long after Graham's number was created. That's a shame, because now more people can't hear of bigger better numbers like TREE(3), which is actually better than Graham's number since it isn't an upper bound, but an actual answer to a fairly simple problem that ends up surprisingly huge (and mind-crushingly larger than Graham's number mind you) - but in all fairness, TREE(3) is much harder to explain than Graham's number. In fact, the hype about Graham's number is not entirely a bad thing - Graham's number can be seen as an eye-opener, a relatively simple number that grasps newcomers' attention, introducing people to the wonderful world of googology.
An interesting fact about Graham's number is that even though it's WAY too huge to compute, we can use modular exponentiation to find its last digits without too much trouble - this is because Graham's number is a gigantic power tower of threes. The last 500 digits of Graham's number are:
02425950695064738395657479136519351798334535362521 43003540126026771622672160419810652263169355188780 38814483140652526168785095552646051071172000997092 91249544378887496062882911725063001303622934916080 25459461494578871427832350829242102091825896753560 43086993801689249889268099510169055919951195027887 17830837018340236474548882222161573228010132974509 27344594504343300901096928025352751833289884461508 94042482650181938515625357963996189939679054966380 03222348723967018485186439059104575627262464195387
These digits don't just end Graham's number, but they also end G(63), G(62), G(61), ... G(3), G(2), and even G(1) and 3^^^3. In fact, those end any power tower of over 500 3’s, such as G(65), G(66), G(67), ... G(G(64)), and bigger 3-based googolisms like tetratri, ultatri, latri, dimentri, triakulus, or the outrageous big boowa.
G(64)*10^10^100
Sbiis Saibian, at the beginning of his number list, gives Graham's root of googolminex and one (the Graham's numberth root of the sum of one and the reciprocal of a googolplex) as an example of a very small large number. That creates a notable apparent paradox - googologists have it implanted in their mind that, for example, graham's number times a googolplex is indistinguishable from Graham's number in googological terms. But on the other hand, raising Graham's root of googolminex and one to the power of Graham's number gives googolminex and one, which is still very close to one, but raising googolminex and one to a googolplex (i.e. raising Graham's root of googolminex and one to the power of (Graham's number times a googolplex) gives a much larger number, almost equal to the mathematical constant e!
So you may now think that Graham's number and Graham's number times a googolplex are once again in two different realms of numbers! However, googologists can really forget about that kind of stuff, as that doesn't change that GRAHAM'S NUMBER TIMES A GOOGOLPLEX IS INDISTINGUISHABLE FROM GRAHAM'S NUMBER! It's all about the counter-intuitivity of numbers this big ... but hold on now, we're just getting started with some FAR BIGGER NUMBERS!
If you want to make Graham's number times a googolplex NOT seem much bigger than Graham's number, start by imagining all the digits of Graham's number, and then adding a googol zeros at the end. With the huge number of digits in Graham's number (which is approximately equal to Graham's number), since a googol is so much smaller than the number of digits in Graham's number, adding a googol more digits is barely a blip on the map!
Graham's number factorial
G(64)!
This number is a common retort to Graham's number based on a common misguided idea - it's that no matter how big the number is, a powerful (by layman standards) function like the factorial will always be a huge improvement on the number. Well guess what - that's wrong, and at this point Graham's number factorial is far less than even G(65)! Don't believe me?! I'll show you how that can be, using Sbiis Saibian's Knuth Arrow Theorem, which states that for a≥2, b≥1, c≥1, x≥2, (a{x}b){x}c < a{x}(b+c).
Consider that n! = n*(n-1)* ... *3*2*1 < n^n = n*n* ... *n*n*n. Then:
G64!
< G64^G64
= (3{G63}3)^(3{G63}3)
<< (3{G63}3){G63}(3{G63}3)
< 3{G63}(3+3{G63}3)
<< 3{G63}(3{G63}3{G63}3)
= 3{G63}4
<<< 3{G63}3{G63}3 = 3{G64}3 = G65
Graham's number to the power of Graham's number
G(64)^G(64)
Ditto. Besides, why should you build upon Graham's number when it isn't hard to build FAR FAR FAR LARGER NUMBERS from the ground up with things like the Ackermann function? Iterating the Ackermann function can easily crush Graham's number, for instance.
E(3)3##4#64
A Hyper-E upper bound to Graham's number. This can be visually represented like so:
E(3)3#3#3#3........#3
w/ E(3)3#3#3#3........#3 3's
w/ E(3)3#3#3#3........#3 3's
w/ E(3)3#3#3#3........#3 3's
.........
w/ E(3)3#3#3#3........#3 3's
w/ E(3)3#3#3#3 3's
with 64 layers total. It's a relatively exacting bound - it's less than even the next upper bound.
Graham-Conway Number
{4,65,1,2} = 4{{1}}65
In Conway and Guy's Book of Numbers, Conway described Graham's number same as how it's usually described, but being based upon fours instead of threes. This alternate version of Graham's number is known as the Graham-Conway number. This alternate version of Graham's number is usually believed to be an error, but in fact it isn't: Conway has said that "Ronald Graham had originally used fours", according to Sbiis Saibian personally communicating with Conway (source). This however opens a whole new array of questions: did he refer to in the 1971 paper with Rothschild or the 1977 paper with the new Graham's number, and when/how was it changed to the threes version? Therefore this is really a mysterious version of Graham's number - see my article on Graham's number for more on that.
This number, unlike the other two versions of Graham's number, can be compactly expressed in Bowers' array notation as {4,65,1,2}. None of them can be expressed compactly in terms of Conway's chain arrows though.
3{{1}}66 = {3,66,1,2}
A BEAF upper-bound to Graham. This is like Graham’s number, but you start with 3{tritri}3 instead of 3^^^3. It’s equal to 3{3{3.....{3{3}3}......3}3}3 with 66 3’s from the center to the either end.
xkcd number
A(G64,G64)
An xkcd comic titled "What xkcd Means" said in one panel "It means calling the Ackermann function with Graham's number as the arguments just to horrify mathematicians". We saw how much of a hassle it was to directly work with the Ackermann function in the entry for 61, so using Graham's number as the arguments would be fucking scary, right?
It turns out that this number, representable in up-arrow notation as 2{G-2}(G+3) - 3 (G is Graham's number), turns out to fall just under G(65) - only two levels of recursion less (i.e. pretty much the same value in googological terms). It's a borderline case of a salad number.
G(65)
A common retort to Graham's number. People who come up with numbers like this think, Graham's number is the largest number ... or it was, until I cleverly came up with G(65) instead of G(64).
G(64) {G(64)} G(64)
This is an example of a retort to Graham's number that is at least measurably different from Graham's number. It seems to be an amazing improvement, and yet it's still smaller than G(66). To see why consider this:
G(64) {G(64)} G(64)
= (3{G63}3){G64}G64
< (3{G64}3){G64}G64
< 3{G64}(3+G64)
< 3{G64}G65
= 3{G64}3{G64}3 = 3{G65}3 = G(66)
This seems utterly paradoxical, but it's still a relatively tame example of the strange behavior with numbers this big!
Corporal
10{{1}}100 = {10,100,1,2}
This is the smallest Bowersism bigger than Graham's number. It's equal to ten expanded to the hundredth. It’s like a little corporal but with 100 layers, and it can be imagined as:
Stage 1 = 10
Stage 2 = 10^1010
Stage 3 = 10^Stage 210
...
Corporal is stage 100
It's a number very much like Graham's number but notably larger, falling between G(99) and G(100).
I have also developed my own extensions to this number, which appear on this list.
G(100)
Another obvious attempt at trouncing Graham. This passes the corporal but is still smaller than Saibian’s graatagold.
This entry is used to discuss the fast-growing hierarchy's order type omega plus one. It's the same order type Graham's function has, and it's a major turning point in large numbers.
Why is that? Because from numbers of the scale of w+1 in the fast growing hierarchy onwards, the fast-growing hierarchy becomes pretty much the universal approximation notation for numbers among googologists - numbers of this size onwards are usually approximated with the fast-growing hierarchy.
It's notable because before order type w+1, up-arrow notation and Hyper-E notation are often used to approximate large numbers, but after that point, one special notation becomes nigh universal, mainly because of its simplicity and growth speed.
Side note: In Bowers' notation, this number is about 99{{1}}101.
Graatagold
E100##100#100
Let stage 1 be E100#100#100....#100 with 100 100’s.
Stage 2 is E100#100#100....#100 with Stage 1 100’s.
Stage 3 is E100#100#100....#100 with Stage 2 100’s.
...
This number, short for gratuitous golden googol, is stage 100. It’s comparable to the corporal. It’s the smallest Saibianism bigger than Graham, but it’s still not too much bigger...relatively speaking.
G(101)
The smallest number in Graham’s sequence bigger than a graatagold.
~ fw+1(115), or between G(115) and G(116) in terms of Graham's function
This is an entry in Bignum Bakeoff submitted by a man named Ioannis. It's a big jump from the previous entry - unlike the tetrational numbers, this number leaves tetration and even up-arrow notation in the dust, and it's bigger than Graham's number, or even G(115) in the sequence! It's actually the smallest entry in Bignum Bakeoff to be bigger than Graham's number. In terms of Bowers' arrays, this number is about 9{{1}}116.
Ioannis achieves this magnitude by defining a 3-argument function similar to the Ackermann function noted a(x,y,z), and then creates a function called d(n), equal to a(n,n,n). Then Ioannis applies d(n) to 9 115 times making a number, d(d(d(d(d(d(d(.........d(9).......))))) with 115 d's. Even though this number is already bigger than Graham's, the next Bignum Bakeoff number is far far larger than Graham's number and utterly leaves it in the dust!
200![2]
This is a hyperfactorial array notation number in the same spirit of Hollom's googolisms which falls between hyperfaxul and giaxul. It evaluates to 200![1]![1]![1] ... ![1] with 200 repetitions of ![1]. In general, x![y]~x{{y-1}}x.
Gamamamamoogood
g(1000,1000,1000,1000,1000,1000)
This number is the largest member of the gamoogol family. It's approximately G(1000) in Graham's function - it's still less than Joyce's googolplux, -pluc, and -plum.
Joycian dulatri
g(19683,1,1,4,3,3)
This is an ESPECIALLY stupid Joycian attempt at expressing a Bowersism. First off, the array does NOT contain 19,683 threes, it contains merely 729 threes! Joyce probably thought it was 19683 because he read the (3^3)^2 array as 3^3^2 (he is said to be a mathematician, but he really doesn't know his exponential laws). Even though he thought the number of entries was larger than what it actually is, this is still UNFATHOMABLY smaller than the real dulatri.
The real dulatri is formed with a much more complicated array. It's past even iterating the number of dimensions - in Bowers' arrays, infinite-dimensional space forms a block. Then you can have multiple blocks, and the dulatri is a cube of THESE blocks! And this definition is merely basic recursion! It's upper-bounded by G(19,683).
Graatagoldagong
E100,000##100,000#100,000
Yay more gong numbers.
Forcal
G(1,000,000)
A forcal is a googolism coined by Aarex, equal to G of one million in Graham's function. He actually has a whole family of forcal-based numbers coined with a notation he calls Aarex's Graham Generator, which extends on Graham's function and goes up to the level of w^w in the fast-growing hierarchy.
Joycian trimentri
g(3^27,1,1,4,3,3)
This is not even close to the trimentri, needless to say. Although he got the number of entries right, this can be upper-bounded by G(3^27) - he can't go past a single recursive step from Graham.
How is the trimentri array formed? It's harder to explain. First, call those infinite-dimensional blocks X1. After a cube of those blocks, we can have a tesseract, 5-D cube, 6-D cube, etc. of these blocks. Then an infinite-sized infinite dimensional block of those blocks is X2. After X2, we can have an infinite-dimensional block of THOSE, and call that X3, then an infinite dimensional block of X3s is an X4 - continue with X5, X6, X7, and continue infinitely - call that block Y. Then you can have a row of Ys, grid of Ys, cube, tesseract, 5-D cube, 6-D cube of Ys.....infinite dimensional block, X2 block, X3 block of Ys, and a Y block of Ys - call that Y block of Ys Y2. Then we can have a Y block of Y2s, and call that Y3 - continue with Y4, Y5, Y6, and continue infinitely. That block is Z, the block required for a trimentri. It's OK if this is hard to understand.
G(googol)
Yet another extension to Graham - also a fairly accurate lower bound for the maggoogol and bed.
Bed
trooga(googol)
Wait, whoa now. What the heck is trooga?!?! This number used to have an article on the Googology Wiki, but its article was deleted due to no proper sources. Cloudy176 of the wiki, however, has a list he calls The Best of Deleted Googologisms. Most of them are either salad numbers, numbers with ridiculous names, or ridiculous definitions, or any combination of those. Some of them don’t even have a meaningful definition, like Sam's number (which is only described as so big that it can't be described and far far larger than Rayo, and therefore it isn't even a number)! Many such definitions are very inconsistent as well. The bed is one of the few with a logical definition and name, but still....bed is a weird name for the number. Let’s hope you don’t sleep for this many hours!
But what IS trooga? Trooga(x) is booga(booga(booga(booga..........(booga(x))....)))), nested x times - a few booga- numbers were seen earlier. The function achieves growth rates of w+1 in the fast-growing hierarchy, the same growth rate achieved by Graham's function.
Trooga(x) is slightly less than {x,x+1,1,2}, so this number is about {googol,googol+1,1,2}, between G(googol) and G(googol+1).
Maggoogol
gag-gag-gag-.....gag-googol (googol gags)
Another number nameable with the mag- prefix. The value is (on a googological scale) barely bigger than a bed, and falls between the cracks of G(googol) and G(googol+1).
G(googol+1)
An upper bound to the maggoogol and bed.
G(googolplex)
A lower bound to Joyce’s googolplux, and another common retort to Graham's number.
Joyce's googolplux
g(g(3,2,g(2,100,10)),1,1,2,100,10)
Here’s a weird Joycian googolism - god knows what he was thinking when he coined it. It evaluates to:
g(g(3,2,10^100),1,1,2,100,10)
g((10^100)^^2,1,1,2,100,10)
g(10^10^102,1,1,2,100,10)
Now we need to define star notation. It’s an offest version of up-arrow notation where a*b = a*b, a**b = a^b, a***b = a^^b, etc. Now this number can be visually represented as:
10**100 (googol)
10****.....*****100
10*******.......*******100
10*******.......*******100
10*******.......*******100
: : : :
: : : :
10*******.......*******100 with 10^10^102 layers, where each layer has as many stars as the number of the previous layer.
Weird, yet pretty cool.
G(10^10^102)
An upper bound to Joyce’s googolplux.
Yudkowsky's Number
G(3^^^^3)
This number was mentioned by Eliezer Yudkowsky in an article of his titled "Staring at the Singularity". It's a pretty cool number that he gave as an example of a number that you can easily make which is mind-bogglingly larger than Graham's number. It can be expressed as G(G(1)), which is interestingly among the HUGE gulf of numbers between things like G(googolplex) and things like G(Graham's number). As Sbiis Saibian said, this begs the question: why stop here?
Yudkowsky has had quite some involvement with googology, and he's been on the googology wiki a few times as well - in googology he is perhaps best known for a post on his Tumblr blog: Why isn't googology a recognized field of math? It's an interesting discussion on the obscurity of googology, but some people, such as the founder of Googology Wiki have quite different views on the subject. Nonetheless it's interesting that someone as famous as Yudkowsky (yes he even has his own Wikipedia article) got involved in the obscure world of googology beyond the popular stuff like Graham's number and Conway chain arrows.
G(gagol) = G(10{7}100)
A rough lower bound to Joyce’s googolpluc.
Joyce’s googolpluc
g(g(g(2,2,3),2,g(2,100,10)),1,1,2,100,10)
A Joycian googolism weirder than the googolplux, apparently using some sort of extrapolation with Roman numerals analogous to the googo- prefix from "googolplux". This number evaluates to:
g(g(3^2,3,10^100),1,1,2,100,10)
g(g(9,3,googol),1,1,2,100,10)
g(googol{8}2,1,1,2,100,10)
g(googol{7}googol,1,1,2,100,10)
10**100
10****.....*****100
10*******.......*******100
10*******.......*******100
10*******.......*******100
: : : :
: : : :
: : : :
: : : :
: : : :
10*******.......*******100 with googol{7}googol layers.
Pretty crazy, but nothing too impressive in googology - in fact, it’s still less than a single step of recursion from Graham's number.
G(gagolplex) = G(10{7}10{7}100)
A rough upper bound to Joyce’s googolpluc.
The Superexpansion Range
10{{1}}10{10}10 ~ 10{{10^100}}10
(order type w+1 ~ w2)
Entries: 41
Joycian decaltrix
g(g(11,10,10),1,1,11,10,10)
This is Joyce's last attempt to define a Bowersism - here he didn't even get the name right. The number is called the tridecatrix. Despite passing the googolplux and googolpluc, this is nowhere close - it's upperbounded by G(10{10}10+1), or Graham's number with a tridecal plus one layers.
How much is the tridecatrix? Well, we can have Z blocks of Z blocks of Z blocks of Z........blocks of Z infinitely, call that A4, A4 blocks of A4 blocks of A4 blocks of.........A4 infinitely, call that A5, then A6, A7, A8, A9.......then go infinitely, and call that B, then continue in that manner for B and call that C, continue infinitely.............ah fuck it, it's nowhere close. It's formed as a dodecational array, far far beyond what we've described. It's beyond nesting levels of nesting levels of nesting, or iterating hyper-levels or whatever mumbo jumbo.
G(10{10^12}10)
A rough lower bound to Joyce’s googolplum.
Joyce’s googolplum
g(g(g(3,3,3),2,g(2,100,10)),1,1,2,100,10)
This is Andre Joyce's largest googolism, besides the baggoogol family if you interpret them as extrapolations from the gag- and mag- prefixes rather than the ambiguous definitions of them Joyce gave with the g-function. A googolplum solves to:
g(g(3^^3,2,10^100),1,1,2,100,10)
g(googol{7,625,597,484,985}googol,1,1,2,100,10)
Visually:
10**100
10****.....*****100
10*******.......*******100
10*******.......*******100
10*******.......*******100
: : : :
: : : :
: : : :
: : : :
: : : :
: : : :
: : : :
10*******.......*******100 with googol{3^^3-2}googol layers.
This is an even crazier number, but Joyce doesn't go any further with his googolisms (besides one interpretation of the baggoogol family).
{3,3,2,2} = 3{{2}}3 = 3{{1}}3{{1}}3
This number serves as an upper-bound to all of Joyce's googolisms (except for the baggoogol family). It's equal to three expanded to the tritriplexth.
Joycian Enneatri
g(3,3,3,3,3,3,3,3,3)
Yet another of my g-function numbers. This is between 3{{1}}3{{1}}4 and 3{{1}}3{{1}}5, and between G(G(2)) and G(G(3)).
G(G(26))
In Graham’s function, a lower-bound to Conway’s tetratri.
Conway’s tetratri
3->3->3->3
This number is a very large number mentioned in John Conway's Book of Numbers as an example of a number larger than Graham's number. The book mainly discusses recreational mathematics, and its the same book where he introduces his -illion system. In his book, Conway uses a few pages to describe some very large numbers, where he briefly discusses Knuth's up-arrows and Graham's number, and a notation of his own called chained arrow notation.
Chain arrows are a googological notation that, like Bowers' arrays, extends on up-arrows, but isn't quite as powerful and has some quirks of its own. I discuss it in detail here.
Here’s how this giant number would be solved:
3->3->3->3
3->3->(3->3->2->3)->2
3->3->(3->3->(3->3->1->3)->2)->2
3->3->(3->3->(3->3)->2)->2
3->3->(3->3->(3^3)->2)->2
3->3->(3->3->27->2)->2
Note that 3->3->x->2 can be expressed in Bowers' operator notation as 3{3{3{......3{27}3....}3}3}3 with x-1 3's.
Therefore, 3->3->27->2 can be imagined as:
3^^^...^^^3
w/ 3^^^...^^^3 ^s
w/ 3^^^...^^^3 ^s
.......
w/ 3^^^...^^^3 ^s
w/ 27 ^s
where there are 26 lines
and just that number exceeds G(26)!
But the full number, 3->3->3->3 = 3->3->(3->3->27->2)->2 is represented as:
3^^^...^^^3
w/ 3^^^...^^^3 ^s
w/ 3^^^...^^^3 ^s
.......
.......
.......
.......
w/ 3^^^...^^^3 ^s
w/ 27 ^s
where there are (3->3->27->2)-1 lines
and just that is very difficult to comprehend!! Conway's tetratri is clearly a number that leaves Graham in the dust....but we're still just getting started with the power of Bowers' four-entry arrays!
G(G(27))
In Graham’s function, an upper-bound to Conway’s tetratri.
G(G(64))
Another common retort to Graham’s number. It's a bit better than the previous retorts, but it's still naive as it's only one recursive step away from Graham's number. If we want to really extend Graham’s number well we’ll need to develop an epic notation to extend this number far far beyond Graham.
Kudi-Chan's Number
~ G(G(64)+64)
This is a number Sbiis Saibian found somewhere on the Internet as a humorous retort to Graham's number. All I can gather about the number is that it jokingly measures "Kudi-Chan"'s genitalia in light-years, so I call it Kudi-Chan's number. It is defined like so (where G is Graham's number):
G^^^... ... ... ^^^G
G^^^... ... ... ^^^G
G^^^... ... ... ^^^G
: : : :
: : : :
G^^^... ...^^^G
G^^^...^^^G
G^^^^G
(with G layers)
With the popular visualization of Graham's number as:
3^^^... ... ... ^^^3
3^^^... ... ... ^^^3
3^^^... ... ... ^^^3
: : : :
: : : :
3^^^... ...^^^3
3^^^...^^^3
3^^^^3
(with 64 layers)
this number seems almost like the sequel to Graham's number. However it's really little more than yet another salad number. G^^^^G is less than G(65), G^G^^^^GG is less than G(66), etc, so the whole number is actually upper-bounded by G(G(64)+64), which is itself a bit of an amusing salad number.
Corporalplex
10{{1}}corporal
This is like a corporal, but with a corporal iterations of up-arrows instead of just 100 (in other words, the visualization has a corporal layers rather than 100). It's ten expanded to the corporalth, and the next recursive step from a corporal.
Graatagolda-sudex
E100##100#100#2
The next step in recursion from a graatagold. In the stages used to visualize a graatagold, this is the graatagoldth stage. It's comparable to a corporalplex.
Graatagolda-dusudex
E100##100#100#3
Mulporal
10{{2}}100 = {10,100,2,2}
Thie first of my entensions to the corporal. This number’s name comes from multiexpansion + corporal, as it's ten multiexpanded to the hundredth. It's comparable to Saibian's greegold.
Greegold
E100##100#100#100
This number is also a graatagolda-99-sudex. It’s name is short for greedy golden googol.
Hyper hyper hyperfaxul / three-ex-hyper-faxul
200![3]
Continuing the hyperfaxul.
Baggoogol
mag-mag-mag-.....mag-googol (googol mags)
Andre Joyce decided to further extrapolate from mag- to give bag-x = mag-mag-mag-mag....-mag-x (x mags), trag-x = bag-bag-bag-bag....-bag-x (x bags), etc, continuing with Latin roots to form quadrag-, quintag-, etc. I should note that Andre Joyce actually gives a fucked up definition of baggoogol and its relatives, and defines them as something nowhere close to what was probably meant.
In the fast-growing hierarchy, baggoogol is about fw+2(10^100), and in Bowers' notation it's about googol{{2}}googol.
Hypergraham
G(G(G(G(G......(G(64))...)))) (Graham's number of G's)
Hypergraham is a googolism coined by SpongeTechX of Googology Wiki, which extends on Graham's function. It's approximately equal to 65{{2}}Graham's number in Bowers' operator notation.
Greegolda-suthrex
E100##100#100#100#2
The next step in recursion from a greegold.
Conway's tetratet
4->4->4->4
In John Conway's book of numbers, Conway also introduces 4->4->4->4 as an example of a very large number expressible in chain arrows (and one bigger than Graham's number). This number, by Chris Bird's proof that compares Conway chain arrows against Bowers' arrays, falls between 4{{3}}4 and 4{{3}}5. It's the largest number explicitly mentioned in Conway's book.
Conway's tetratet is the fourth Conway number, a member of a sequence coined by Conway in his book. Conway numbers are members of the sequence:
1
2->2 = 4
3->3->3 = 3^^^3 = tritri
4->4->4->4 ~ 4{{3}}5
5->5->5->5->5 ~ 5{{{4}}}6
etc.
Powporal
10{{3}}100
I like the sound of the operation used to define this number, called powerexpansion. This number is 10 powerexpanded to 100. But we’re still far from reaching a biggol!
Grinningold
E100##100#100#100#100
This gargantuan number is short for grinning golden googol. Saibian says it is grinning because of how infinitesimal it makes you seem.
Four-ex-hyperfaxul
200![4]
We can easily continue the "ex-hyper" series.
Traggoogol
bag-bag-bag-.....bag-googol (googol bags)
A Joycian googolism. In the fast-growing hierarchy, this number is about fw+3(10^100), and in Bowers' notation it's about googol{{3}}googol.
Grinningolda-sutetrex
E100##100#100#100#100#2
You can guess what suffixes there would be for the next numbers in the gugold group.
Terporal
10{{4}}100
Here it’s important to make names sound good and pronounceable, so I’m using terporal, which sounds much nicer than tetrporal or tetporal.
Golaagold
E100##100#100#100#100#100
Short for golem golden googol. Here “golem” refers this number being so big it has no shape or form.
Quadraggoogol
trag-trag-trag-.....trag-googol (googol trags)
A Joycian googolism. In the fast-growing hierarchy, this number is about fw+4(10^100), and in Bowers' notation it's about googol{{4}}googol.
Pepporal
10{{5}}100
Whatever you do, don’t sprinkle this much pepper on your pasta or whatever you make. You will probably sneeze up a tornado or something stupid like that.
Gruelohgold
E100##100#100#100#100#100#100
This number is short for “grueling golden googol”, and Saibian calls it that because it would take so long to count to. Of course, which -illion system would you use? Regardless, the time wouldn’t vary at all at this scale; in fact, no matter which units you use, the time would be almost exactly a gruelohgold.
Hexporal
10{{6}}100
After that we have a hepporal, an ocporal, an enporal, and a dekporal.
Gaspgold
E100##100#100#100#100#100#100#100
Short for *gasp* golden googol. We’re wrapping up the gugold series.
Ginorgold
E100##100#100#100#100#100#100#100
Short for ginormous golden googol. Also E100##100##8.
Gargantuuld
E100##100#100#100#100#100#100#100#100
Grand tridecal
{10,10,10,2} = 10{{10}}10
This Bowerian googolism is equal to ten expandodecated to the tenth. It first simplifies to 10{{9}}10{{9}}10{{9}}10{{9}}10{{9}}10{{9}}10{{9}}10{{9}}10{{9}}10, which you’ll need to simplify further and further, analogous to a tridecal - the two numbers behave quite similarly.
Googondold
E100##100#100#100#100#100#100#100#100
Comparable to a grand tridecal.
Joycian hectatri
g(3,3,3,3,3,3,3......,3,3,3) with 100 3s
This is my last g-function number. It’s around 3{{32}}3, placing it snugly between the grand tridecal and biggol, and just under the output of chan-2.c.
~ fw+47(10^39) or about 10{{47}}10^39
This is the output of chan-2.c, Tak-Shing Chan's second submission to Bignum Bakeoff. Chan-2.c, unfortunately, produces a smaller value than chan.c, Chan's original submission. Chan defines this number with a 2-argument function similar to the Ackermann function that achieves growth rates of w in the fast-growing hierarchy, and then applies 47 levels of iteration to get growth rates of w2 in the fast-growing hierarchy. The value is far far larger than Graham's number, and is a decent sized 4-entry array number.
Gugolthra
E100##100##100
This number ends in -thra because of the three 100s in its definition. It's comparable to biggol. It also starts the third row in our googol extension table; the first two were googol through googondol and gugold throughgoogondold.
Biggol
{10,10,100,2} = 10{{100}}10
This is 10 expandocentated to the 10th, a Boweria. After that, we have a baggol, beegol, etc, which we’ll see when we get there. Those numbers are quite a long way!
Giaxul
200![200] = 200![1,2]
This number’s name means “giant faxul”. It’s an offical Hollomism equal to about 200{{199}}200. It ends the faxul series and begins its own series. It's roughly comparable to the biggol.
The Higher-Tetrentrical Range
10{{10^100}}10 - {10,10,10,10^100}
(order type w2+1 ~ w^2)
Entries: 45
Corplodal
10{{{1}}}100 = {10,100,1,3}
This number’s name (which I made up) comes from “explosion”+”corporal”. It's equal to 10{{10{{10......{{10}}.......}}10}}10}}10 with 100 10s from the center out.
Graatagolthra
E100##100##100#100
This is comparable to corplodal.
200![2,2]
A HAN construction to make the notation more clear. Here's how to calculate it:
x![1,2]=x![x]
x![n,2]=(.....(x![n-1,2])![n-1,2]).......![n-1,2])
Then:
Stage 1 = 200![1,2] = 200![200] = giaxul
Stage 2 = giaxul![1,2] = giaxul![giaxul]
Stage 3 = (Stage 2)![1,2] = (Stage 2)![Stage 2]
........
This number is stage 200. It’s about 200{{{1}}}200 with Bowers' operator notation.
Mulplodal
10{{{2}}}100
From multiexplosion+corplodal. To calculate this number:
Stage 1 = 10
Stage 2 = 10{{10{{10{{10......{{10}}......10}}10}}10}}10 with 10 10s from the center to one end
Stage 3 = 10{{10{{10{{10......{{10}}......10}}10}}10}}10 with Stage 2 10s from the center to one end
......
Mulplodal is stage 100.
Greegolthra
E100##100##100#100#100
Comparable to a mulplodal.
200![3,2]
Or (((200![2,2])![2,2])![2,2])....![2,2]).
Bowers' Tetratri
{3,3,3,3}=3{{{3}}}3
This is the number Bowers calls tetratri. It's equal to 3 powerexploded to the 3rd. Powerexplosion has a real nice ring to it, giving a sense of far-reaching glory of the insanely large numbers. Here’s how this number is calculated:
Stage 1 = 3
Stage 2 = 3{{3{{3}}3}}3 (3 3’s from the center out) = 3{{x}}3, where x (that's 3{{3}}3) is:
3^^^...^^^3 with 3^^^...^^^3 with 3^^^...^^^3 ... ... with 3^^^3 with 3 ^s
with 3^^^...^^^3 with 3^^^...^^^3 with 3^^^...^^^3 ... ... with 3^^^3 with 3 ^s
with 3^^^...^^^3 with 3^^^...^^^3 with 3^^^...^^^3 ... ... with 3^^^3 with 3 ^s
: : : : :
: : : : :
with 3^^^...^^^3 with 3^^^3 with 3 ^s
with 3^^^...^^^3 where there are 3^^^...^^^3 with 3^^^...^^^3 with 3^^^...^^^3 ... ... with 3^^^3 with 3 ^s, where there are 3{tritri}3 steps steps. And that’s just the number within the double curly brackets!
As you can see stage 2 is alreardy SCARY!!! But it pales in comparison to stage 3:
Stage 3 = 3{{3{{3{{3.......{{3}}........3}}3}}3}}3 (Stage 2 3s from the center out)
THAT’S INSANE!!!! But it’s still much less than Stage 4:
Stage 4 = 3{{3{{3{{3.......{{3}}........3}}3}}3}}3 (Stage 3 3s from the center out)
... and so on ... continue to Stage 100, stage googol, stage googolplex, stage tritri, stage tritriplex, stage 3{{2}}3, stage 3{{3}}3, stage 3{{3{{3}}3}}3 ... ... ...
Are you ready? Tetratri is Stage Stage 3 - Mother of God it's HUGE!!! Graham's number seems quite modest now, doesn't it? But this is still rather small amongst the googolisms, as even Conway chain arrows are enough to approximate this number ...
Among Saibian's googolisms the falls between greegolthra and grinningolthra.
Powplodal
10{{{3}}}100
10 powerexploded to the 100th.
Grinningolthra
E100##100#100#100#100
5->5->5->5->5
This is the fifth Conway number (see 4->4->4->4 for more), a Conway chain of five fives. It is not explicitly named by Conway but it is still a valid construction. The value is comparable to 5{{{4}}}6 in Bowers' operator notation.
Terplodal
10{{{4}}}100
We can continue this series of numbers with pepplodal, hexplodal, then hepplodal, ocplodal, ennplodal, and dekplodal.
Golaagolthra
E100##100##100#100#100#100#100
Upper bound of output of chan-3.c
~ fw2+4(199,999) or 10{{{4}}}200,000
Chan-3.c is Chan's third submission to Bignum Bakeoff - here, Chan starts off with a 3-argument function similar to previously, but this time he extends with two functions that use complicated recursion to iterate each other, achieving growth rates in the ballpark of w2+4 in the fast-growing hierarchy.
Gruelohgolthra
E100##100##100#100#100#100#100#100
Gaspgolthra
E100##100##100#100#100#100#100#100#100
Ginorgolthra
E100##100##100#100#100#100#100#100#100#100
Gargantuulthra
E100##100##100#100#100#100#100#100#100#100#100
Googondolthra
E100##100##100#100#100#100#100#100#100#100#100#100
The end of the third row of extended Hyper-E numbers. Now on to:
Gugoltesla
E100##100##100##100
Not to be confused with an electric car company. It is the start of the fourth row of Saibianisms, and is also comparable to baggol.
Baggol
{10,10,100,3} = 10{{{100}}}10
Or 10 explodocentated to the 10th. Another 4-entry array Bowersism.
200![1,3]
Or 200![200,2]. In hyperfactorial arrays (like in Bowers' arrays), the later entries clearly matter more than the earlier ones, and that effect is only more profound with larger arrays.
Cordetal
10{{{{1}}}}100
Or 10 detonated to the 100th. Once again, I named this number. Here are some more numbers we can name:
Muldetal
Powdetal
Terdetal
Pendetal
Hexdetal
Hepdetal
Ocdetal
Enndetal
Dekdetal
Graatagoltesla
E100##100##100##100#100
Here are some more numbers that we can name:
Greegoltesla
Grinningoltesla
Golaagoltesla
Gruelohgoltesla
Gaspgoltesla
Ginorgoltesla
Gargantuultesla
Googondoltesla
Supertet
{4,4,4,4}=4{{{{4}}}}4
These numbers are getting more and more outrageous by the minute. A supertet can be computed like so:
Stage 1 = 4
Stage 2 = 4{{{4{{{4{{{4}}}4}}}4}}}4 - note that the number inside the triple braces, 4{{{4}}}4, is MONSTROUSLY LARGER than 3{{3}}3 - hell, it's FUCKING UNFATHOMABLY BIGGER than even a tetratri! And that's merely the second step!
Stage 3 = 4{{{4{{{4{{{4.......{{{4}}}........4}}}4}}}4}}}4 (Stage 2 4s from the center out) - hold on, don't scream yet
Stage 4 = 4{{{4{{{4{{{4.......{{{4}}}........4}}}4}}}4}}}4 (Stage 3 4s from the center out)
etc.
Now continue with stage 5, 6, 7, 8, 9, 10.....100....1000.....stage googol, googolplex, giggol, tritri, tridecal, boogol, biggol, tetratri..........stage stage 2, stage stage 3, stage stage 4...........then go to stage x, where x is stage y, where y is stage 4 - that's the horrifyingly huge stage stage stage 4!!
But that ISN'T EVEN CLOSE to supertet. Imagine it like so:
Super-Stage 1 = 4
Super-Stage 2 = Stage Stage Stage 4 - that's the number described previously
Super-Stage 3 = Stage Stage Stage Stage...........Stage Stage 4 with (Super-Stage 2) - 1 stages
Super-Stage 4 = Stage Stage Stage Stage...........Stage Stage 4 with (Super-Stage 3) - 1 stages - THAT is a supertet......feel free to scream now!!!
This number will easily be humbled by EVEN MORE GIGANTIC numbers!
Gugolpeta
E100##100##100##100##100
Starting the fifth row of extended Hyper-E numbers.
Beegol
10{{{{100}}}}10
Not to be confused with a dog, as Bowers says on his website. This is ten detonocentated to the tenth.
200![1,4]
In general, a![b,c] ~ {a,a,b-1,c+1}. 200![1,4] is also equal to 200![200,3], so this is about 200{{{{199}}}}200.
Corpental
10{{{{{1}}}}}100
Here’s some more numbers:
Mulpental
Powpental
Terpental
Peppental
Hexpental
Heppental
Ocpental
Ennpental
Dekpental
Gugolhexa
E100##100##100##100##100##100 or E100###6
Bigol
10{{{{{100}}}}}10 = {10,10,100,5}
10 pentonohectated to the tenth.
200![1,5]
Corhexal
10{61}6100
Related numbers include:
Mulhexal
Powhexal
Terhexal
Penhexal
Hex-hexal
Hephexal
Ochexal
Ennhexal
Dekhexal
Corheptal
Mulheptal
Powheptal
Terheptal
Penheptal
Coroctal
Corennal
Cordekal
You may notice that 10{x5}x100’s names can start pen- or pep-. Here’s the general rule: if the middle syllable starts with a p, you say pep-. Otherwhise, you say pen-.
Gugolhepta
E100##100##100##100##100##100##100
Boggol
{10,10,100,6}
Jonathan Bowers says about this number, “boggles the mind, doesn’t it?”
Gugolocta
E100##100##100##100##100##100##100##100
Bagol
{10,10,100,7}
Bowers says about this number, “whatever you do, don’t eat this many bagels”. Just imagine...
Gugolenna
E100###9
General/tetradecal
{10,10,10,10}
In Bowers' operator notation, this is equal to 10{{{{{{{{{{10}}}}}}}}}}10. Solving this would take 10 levels of decomposing the curly bracket groups before we hit the number. The old man is getting a little tired from solving all those numbers, but they’re still quite easy. Just imagine how huge it is!
Gugoldeka
E100###10
This was originally a number found in a hidden article of Saibian's (see grand throogol), but now it's a public Saibianism. It expands to E100##100##100##100##100##100##100##100##100##100, and is used as an upper bound to a general.
Gargantuuldeka
E100##100##100##100##100##100##100##100##100##100#100#100#100#100#100#100#100#100#100
The largest possible Extended Hyper-E number using ## and # (about {10,100,9,10} in BEAF), but there’s still a lot more.
Output of pete-4.c
~ fw33(100,000) or about {10,10,100000,33}
This is the output of Pete's fourth submission in Bignum Bakeoff - here, he uses a 3-argument function based on exponentiation that achieves growth rates of w^2, the same level of power of Conway chained arrow notation and Bowers' four-entry arrays. It's much larger than his third, eighth, and ninth submissions, but much less than his far larger fifth, sixth, and seventh submissions.
~ fw50(100,000) or about {10,10,100000,50}
This number is the output of Chan's first submission in Bignum Bakeoff. Here, he defines a 3-argument recursive function and uses further recursion that's more powerful than in his next attempts. Like pete-4.c, he achieves growth rates of w^2.
Throogol
E100###100
This crazy number is short for “third googol”. It starts the second table of Hyper-E numbers, beginning with throogol and ending with thrinorgolocta. The second table would go behind out first table, starting the third dimension. When we get to E100####100 we’ll need to go through higher dimensions. This number is comparable to and slightly smaller than a troogol.
Troogol
{10,10,10,100}
Or 10{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{10}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}10 in full. That’s 100 pairs of { }! Solving this number would require a hundred levels of decomposition! These numbers are getting more and more outrageous, and by now we've neared the breaking point of Conway chained arrow notation - this is comparable to a chain of 100 tens.
Giabixul
200![200,200]
This can also be written as 200![1,1,2], and it's about {200,200,199,201}. The next official Hollomism, giatrixul, is much much larger.
Hollom's hyperfactorial arrays are actually a lot more complicated than they seem, and something that really would take a whole article to even discuss. The arrays are more powerful than Bowers' idea because a different, more complex, approach is taken, reaching growth rates of φ(w,0) in the FGH instead of just w^w.
The Pententrical Range
{10,10,10,10^100} - {10,10,10,10,10^100}
(order type w^2+1 ~ w^3)
Entries: 39
Grand throogol
E100###100#2 = E100###throogol
This number is equal to E100##100##100##100......##100 with throogol 100’s. It is the next step in recursion after a throogol.
Troogolplex
{10,10,10,troogol}
Or 10{{{{{.....{{{{10}}}}.....}}}}}10 with a troogol pairs of { }. This number needs a troogol levels of decomposition! But it's still comprehensible recursion, so in a sense we're still yet to reach the real whoppers.
~ fw^2+1(63)
Fish number 1 is the smallest googolism of Kyodaisuu of Googology Wiki. It was defined by him on 2ch.net in 2002, using an extension of the Ackermann function, which is discussed in part 1 and earlier in part 3. Kyodaisuu said that what he knew about googology at the time was only the Ackermann function and Graham's number, and he decided to combine elements from both to get this number. The extension to the Ackermann function Kyodaisuu used order type w^2, the same power of chained arrow notation. by iterating it, we reach order type w^2+1. Therefore this number falls just below a thrangol, which is about fw^2+1(100).
Thrangol
E100###100#100
From throogol + grangol. You can see we’re recycling old number names now.
BOX_M~
Approximately fw^2+1(G(G(64)))
This number was coined by Marco Ripa in January 2012, which was naively thought by him to be the largest named number. It is defined as follows:
n = GL (xL is defined later, and G is Graham's number)
n$ = (n!)^^(n!)
nY = ((n$)^^(n$))^(((n-1)$)^^((n-1)$)).....^((2$)^^(2$))^((1$)^(1$)) (note: supposed to be the yen sign, but I used Y as an ASCII substitute)
nL = ((nY)^^(nY))^(((n-1)Y)^^((n-1)Y)).....^((2Y)^^(2Y))^((1Y)^(1Y)) (note: supposed to be the pound sign, but I used L as an ASCII substitute)
A1 = nL, Ak = (A(k-1))^^(A(k-1))
Mk(a) = a{{1}}(k+1)
k1= MnL(AnL)!
kn = n{k(n-1)}n
R~ = kkkkk.....kGL, where G is Graham's number (the tilde is on top of the R, but doesn't work in ASCII)
M~1 = (GL{R~}GL) -> (GL{R~}GL) -> (GL{R~}GL) ..... -> (GL{R~}GL), with GL{R~}GL horizontal arrows in chained arrow notation
M~k = (M~(k-1)) -> (M~(k-1)) -> (M~(k-1)) ..... -> (M~(k-1)) with M~(k-1) horizontal arrows (the tilde is on top of the M, but that doesn't work in ASCII)
BOX_M~ = M~(M~1+1)
This whole number, as it turns out, is a very sloppy salad number, as seasoned googologists are sure to notice. It throws in so many gratuitous factorials and mishmashes a variety of popular notations, rather than making an original notation beyond a simple set of sequences that each build upon the previous with a totally random step. What's weird about this number is it was created by the founder of an Italian high-IQ society, who once wrote a book about the hyper-operators, but it's exactly the kind of number googologists laugh at. I'd expect a guy like this to do better. I think this serves to show how obscure of a field googology is, with its ways as natural as can be to those who know it but completely misunderstood by the majority of people.
Threagol
E100###100#100#100
From throogol + greagol. You can see where this is going...
Thrigangol
E100###100#100#100#100
Throrgegol
E100###100#100#100#100#100
Thrulgol
E100###100#100#100#100#100#100
Thraspgol
E100###100#100#100#100#100#100#100
Thrinorgol
E100###100#100#100#100#100#100#100#100
Thrargantuul
E100###100#100#100#100#100#100#100#100#100
Throogondol
E100###100#100#100#100#100#100#100#100#100#100
Thrugold
E100###100##100
Thraatagold
E100###100##100#100
3 #s, then 2, then 1, This number has a neat construction.
Thrugolthra
E100###100##100##100
Thrugoltesla
E100###100##100##100##100
Thrinorgolocta
E100###100##100##100##100##100##100##100##100##8
The largest Hyper-E number before E100###100###100 - now there are 200 possible number names.
How about...
Throotrigol
E100###100###100
This is the next extension to a throogol once we exhaust the ones directly based on the googol, gugold, and gugolthra series. It’s comparable to triggol.
Triggol
{10,10,10,100,2}
Yep. That much between a troogol and a triggol.
Thrantrigol
E100###100###100#100
Threatrigol
E100###100###100#100#100
Thrutrigold
E100###100###100##100
Pentatri
{3,3,3,3,3}
A pentatri is a horrendously huge number equal to a linear array of five threes. It's mind-crushingly larger than a tetratri, and mind-crushing to imagine the computation - it's too complex to really understand! In terms of Aarex's extension to Bowers' hyper-operator names, a pentatri is equal to three powergigoexploded to the third.
Thrutrigolthra
E100###100###100##100##100
Throotergol
E100###100###100###100
To name every potential number here, just infix -ter- in any number from a throogol to a throogondoldeka.
Traggol
{10,10,10,100,3}
Throopetol
E100###100###100###100###100
This name drops the "g" from throogol because throopetgol is awkward to pronounce.
Treegol
{10,10,10,100,4}
Whatever you do, don’t even think about planting this many trees.
You can guess what a trigol, troggol, and tragol are and what they’re comparable to.
Superpent
{5,5,5,5,5}
This is the next super-x number. It’s also equal to {5,{5,{5,{5,5,4,5,5},4,5,5},4,5,5},4,5,5}, which will get complicated quickly when we try to solve it. More visual representations would help us, but there’d be too many to directly list. In terms of Aarex's hyper-operators, this number is five pentapetopentonated to the fifth.
Throohexol
E100###100###100###100###100###100
Throoheptol
E100###100###100###100###100###100###100
Throogogdol
E100###100###100###100###100###100###100###100
Throogentol
E100###100###100###100###100###100###100###100###100
Pentadecal
{10,10,10,10,10}
Throodekol
E100###100###100###100###100###100###100###100###100###100
The highest possible throogol number is throogondekoldeka. Now on to...
~ fw^3(63)
Later in 2002, Kyodaisuu discovered more googology, such as Chris Bird's array notation. To compete better with that, he extended Fish number 1's system to reach order type w^3 in the fast-growing hierarchy, and came up with this number. It falls just below a teroogol, which is about fw^3(99).
Tetroogol
E100####100
This number is also known as the fourth googol and is comparable to quadroogol, but the prefix tera- is used for convenience. It begins the second cube of numbers and starts to go four-dimensional in the hyper-grid of Hyper-E numbers.
Quadroogol
{10,10,10,10,100}
First we have a googol, then a boogol, then a troogol, then this. This number is horrendously big and is starting to get a bit harder for the old man. But it’s still doable, but pretty ... damn ... tiring.
The Higher-Linear-Array Range
{10,10,10,10,10^100} - {10,99(1)2}
(order type w^3+1 ~ w^w)
Entries: 40
Quadroogolplex
{10,10,10,10,quadroogol}
The next step in recursion for a quadroogol.
Tetrangol
E100####100#100
Time to skim through the tetroogol series.
Tetreagol
E100####100#100#100
Tetrugold
E100####100##100
Tetrugolthra
E100####100##100##100
Tetraatagolthra
E100####100##100##100#100
This is Sbiis Saibian's 2000th googolism in his multi-part list of Extensible-E numbers. It's approximately {10,100,1,3,1,2} using Bowers' array notation.
Tetrithroogol
E100####100###100
This is a neat trick we’ll be using to name higher Hyper-E numbers.
Tetrithrangol
E100####100###100#100
Tetrithrootrigol
E100####100###100###100
Now we can have the first two cubes named. The first cube will have the googol series and the gugol/gugolthra series in its first plane, the throogol series in its second, the throotrigol series in its third, etc. The second cube will have tetroogol to tetrinorgolocta in its first plane, tetrithroogol to tetrithrinorgolocta in its second, the tetrthrootrigol series in its third, etc.
Quadriggol
{10,10,10,10,100,2}
Quadraggol, quadreegol, etc. should be obvious.
Tetrootrigol
E100####100####100
Once again, recycling an old trick for the throogol series. This number starts the third cube and is comparable to quadriggol.
Tetrootrithroogol
E100####100####100###100
Tetrootergol
E100####100####100####100
The largest possible tetroogol series number would be tetroogogogdithroogongodgoldeka. The number names are getting crazy.
Tetroopetol
E100####100####100####100####100
Superhex
{6,6,6,6,6,6}
A linear array of six sixes.
Tetroodekithroogondekugoldeka
E100####100####100####100####100####100####100####100####100####100###100###100##
#100###100###100###100###100###100###100##100##100##100##100##100##100##100##100##
100#100#100#100#100#100#100#100#100#100
The largest possible teroogol series number.
Pentoogol
E100#####100
This is where the megacube of numbers hits the 5th dimension. A few sample pentoogol numbers are coming up. By the way, this is comparable to quintoogol.
Quintoogol
{10,10,10,10,10,100}
The outrageousness is almost unbearable now. It’s starting to become a challenge for the old man to solve these numbers.
Pentangol
E100#####100#100
Pentuelohgoltesla
E100#####100##100##100##100#100#100#100#100#100
Pentithroogol
E100#####100###100
Pentitetrithroopetangol
E100#####100####100###100###100###100###100#100#100#100
From a googol to the pentoogol group, 100,000 Hyper-E numbers can be named.
Pentolaahexold
E100#####100#####100#####100#####100#####100##100#100#100#100#100
These numbers are but a few of these many possibilities.
Supersept
{7,7,7,7,7,7,7}
A linear array of seven sevens.
Hexoogol
E100######100
This number is comparable to sextoogol and is where the “Hyper-E-Grid” becomes 6-dimensional. It's not to be confused with sextoogol.
Sextoogol
{10,10,10,10,10,10,100}
Superoct
{8,8,8,8,8,8,8,8}
A linear array of eight eights.
Heptoogol
E100#######100
Also called the seventh googol. Not to be confused with septoogol.
Septoogol
{10,10,10,10,10,10,10,100}
Superenn
{9,9,9,9,9,9,9,9,9}
A linear array of nine nines.
Ogdoogol
E100########100
Once was the last simple Hyper-E number, so to speak. But recently Sbiis Saibian coined some extra Hyper-E googolisms after this.
Octoogol
{10,10,10,10,10,10,10,10,100}
The last number of Bowers’ prime googol series. The next extensions will move into planar arrays.
Ogdiheptihexipentitetrithroogol
E100########100#######100######100#####100####100###100
This number is directly mentioned on Saibian’s website as an example of how many different numbers can be named with his system. How about:
Ogdatrihexapentateritetrapetithrinortrigolthra
E100########100########100######100#####100#####100#####100####100####100####100
####100###100###100##100##100#100#100#100#100#100#100#100
Ditto.
Entoogol
E100#########100
The "ninth googol".
Iteral
{10,10,10,10,10,10,10,10,10,10}
This Bowerian googolism is ten tens in a linear array, a number near the breaking point of linear arrays.
Dektoogol
E100##########100
This number is comparable to an iteral and is the last milestone Hyper-E number.
Dektadekientadekiogdadekiheptadekihexadekipentadekitetradekithrooogondekoldeka
E100##########100##########100##########100##########100##########100########
##100##########100##########100##########100#########100#########100#########
100#########100#########100#########100#########100#########100#########100###
#####100########100########100########100########100########100########100####
####100########100#######100#######100#######100#######100#######100#######100
#######100#######100#######100######100######100######100######100######100###
###100######100######100######100#####100#####100#####100#####100#####100#####1
00#####100#####100#####100####100####100####100####100####100####100####100####1
00####100###100###100###100###100###100###100###100###100###100##100##100##100##10
0##100##100##100##100##100#100#100#100#100#100#100#100#100#100
This is the largest possible Hyper-E number. It’s as crazy as it looks. In total, 10,000,000,000 (formerly 16,777,216) such numbers can be named, as mentioned in part 2. Now on to bigger and better numbers
~ fw^11+16(1000) or about {1000,1000,1,1,1,1,1,1,1,1,1,1,1,17}
This is Pete's fifth submission to Bignum Bakeoff, far larger than his previous two submissions. Here, he uses the a function that can have any number of arguments that achieves growth rate of w^w, the same growth rates of Bowers' linear arrays! The value is comparable to a 14-entry array, making it much more than an iteral but much less than an ultatri.
Output of pete-6.c
~ fw^23(10^3011)
This is Pete's sixth submission to Bignum Bakeoff. It uses the same system as pete-5.c but with a different value of the function. In terms of Bowers' arrays, this number is comparable to a 25-entry array of 10^3011s.
{3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} = {3,27(1)2}
Another official Bowersism, and the last one before we hit the 2-row array numbers. Just imagine trying to solve this - it's a linear array of 27 threes. In Bowers' old array notation, this number showed up again and again when working with dimensional arrays of threes, because it was equal to {3,3,3........3} with {3,3,3} threes.
We've exhausted linear arrays now - welcome to the next level of array notation, dimensional arrays - now we'll encounter dimensional arrays and similarly powerful notations in part 5.