PART 5: THE FEARSOME COMMANDERS
order type w^w to Γ0
Things get really crazy here with huge numbers too big to represent with large number notations of order type w^w like a xappol, a gongulus and a tethrathoth - some preliminary knowledge of googological notations is HIGHLY recommended.
The Two-Row-Array Range
{10,100(1)2} ~ {10,99(1)(1)2}
(order type w^w ~ w^(w2))
Entries: 78
Goobol
{10,10,10......10} with 100 10s or {10,100(1)2}
A goobol is a linear array of 100 tens. It's the next of Bowers' googol extensions, and the smallest of his dimensional array googolisms (though this can still be written out using linear arrays). It's a good starting point for dimensional arrays.
Godgahlah
E100####......####100 with 100 #s or E100#^#100
This number is the end of Hyper-E and the beginning of Cascading-E, and was created before Cascading-E was invented. It can be written E100#^#100 in Cascading-E, and its name has a sense of far-reaching glory, unlike the goobol, a comparable googolism.
Giatrixul
200![200,200,200]
Roughly equal to a linear array of 202 200’s. The next official HAN number (giaquaxul) will take a very long time for us to reach.
Godgahlahgong
E100,000#######........#######100,000 with 100,000 #s or E100,000#^#100,000
This is the gong version of a godgahlah. It has even more of a glorious sense than the godgahlah.
Googahlah
E100##########..............#######100 with 10^100 #s or E100#^#(E100)
This is a special googolism by Sbiis Saibian, like a godgahlah but with a googol #s - that chain of #s would reach all the way through the observable universe and way beyond. This number is comparable to a linear array of a googol 100's.
Output of pete-7.c
~ fw^w(2^^35)
This number is the largest entry submitted by Pete in Bignum Bakeoff. Here, he uses the system in pete-5.c and pete-6.c, but extends it to calculate with arbitrary numbers of arguments. Here, Pete gives the function a tetrational number of arguments, creating a number comparable to a 2^^35-entry array. The number is much larger than a godgahlahgong but falls short of a dupertri.
Pete-7.c fell in third place in Bignum Bakeoff - the second place entry reaches past epsilon zero in the fast-growing hierarchy and the first place entry is so large that we don't know how to represent it in the fast-growing hierarchy!
Dupertri
{3,3,3...3} with tritri 3’s or {3,tritri(1)2} or {3,3,2(1)2}
This number is a very low-level two-row array, a linear array of tritri threes. There will be plenty of 3-based Bowersisms coming your way. Their decimal expensions all end in the same way Graham’s number ends.
Iteralplex/duperdecal
{10,10,10....10} with iteral 10’s or {10,iteral(1)2}
A linear array of an iteral tens. This can also be written {10,3,2(1)2}.
Goobolplex
{10,10,10........10} with goobol 10’s or {10,goobol(1)2}
As glorious as this seems, we will be really taking this a lot further than plain old recursion.
Grand godgahlah
E100##########...................#########100 with godgahlah #s or E100#^#100#2
This is the next step in recursion from a godgahlah. It's comparable to a goobolplex.
Grand godgahlahgong
E100,000#############.......................#################100,000 with godgahlahgong #s or E100,000#^#100,000#2
This number, as stated by Saibian, has a real nice ring to it, bringing a sense of far-reaching glory like a googolgong once had for Sbiis
Truperdecal/iteralduplex
{10,duperdecal(1)2}
Grand grand godgahlah
E100#grand godgahlah100 or E100#^#100#3
This number can also be called "two-ex-grand godgahlah", although that doesn't really shorten the name.
Grand grand godgahlahgong
E100,000#grand godgahlahgong100,000 or E100,000#^#100,000#3
This feels like we’re really putting loads and loads of improvement over the godgahlah ... but it’s still barely scratching the surface of Cascading-E! It was at one point the largest number on Sbiis Saibian's large number list.
Grand grand grand godgahlah / Three-ex-grand-godgahlah
E100#^#100#4
Grand grand grand grand godgahlah / Four-ex-grand godgahlah
E100#^#100#5
Gibbol
{10,100,2(1)2}
Stage 1 is 10. Stage 2 is {10,10,10,10,10,10,10,10,10,10} (iteral). Stage 3 is {10,10,10,10,10.......,10,10,10} with stage 2 10s. Keep going and a gibbol is stage 100. This still isn't much further recursion from a goobol though
Grandgahlah
E100#^#100#100
Also known as 99-ex-grand godgahlah, and comparable to gibbol. It’s the first new number defined with Cascading-E, as the previous godgahlah-based numbers were already created before Cascading-E.
Googol-ex-grand godgahlah
E100#^#100^(googol+1)
Googolplex-ex-grand godgahlah
Grangol-ex-grand godgahlah
Greagol-ex-grand godgahlah
Gigangol-ex-grand godgahlah
Gugold-ex-grand godgahlah
Saladgahlah
[[(E100#^#100#(G(64)+1))^^(10^100,000,000,000,000,000))!2[2[5]]]^[(E100#^#100#(G(64)+1))^^(10^100,000,000,000,000,000))!2[2[5]]]]{9001}[[(E100#^#100#(G(64)+1))^^(10^100,000,000,000,000,000))!2[2[5]]]^[(E100#^#100#(G(64)+1))^^(10^100,000,000,000,000,000))!2[2[5]]]]
This number is psychedelically huge, which can be obtained in the following process: first you take a Graham’s-number-ex-grand godgahlah, and then make a power tower of 1 followed by 100 quadrillion zeroes of that number, and then take its factorial, then take that number’s factorial, then take THAT number’s factorial, continue Moser's number of times! After that take the number to the power of itself! Finally, go and take that number, 9001 up-arrows, and that number again, and evaluate that! That is the saladgahlah!
Nah... just kidding. This number is an example of a salad number, which I coined just for the hell of it and also as a number a naive googologist may come up with. What is a salad number, you may ask? It’s a number defined with a ridiculous number of inelegant steps in an attempt to be majestically large. These numbers are most often created by inexperienced googologists, who think that by combining a lot of different functions they’d get an amazingly huge number. This doesn’t work because to get anywhere, you need to define whole new functions, as old functions will always become obsolete at one point or another. In fact, Graham's-number-ex-grand godgahlah is so large that all the operators applied to it here have no real effect, so this number is only between gugold- and graatagold-ex-grand godgahlah. So much for a crazy hierarchy of steps.
Hell, even if they aren't familiar with the concept of salad numbers, people can generally intuitively recognize numbers like this as sloppy. Moving on...
Graatagold-ex-grand godgahlah
Gugolthra-ex-grand godgahlah
Gugoltesla-ex-grand godgahlah
Throogol-ex-grand godgahlah
Tetroogol-ex-grand godgahlah
Dektoogol-ex-grand godgahlah
Godgahlah-ex-grand godgahlah
As impressive as this sounds, it’s still a naive extension, since it's only one generation of recursion.
Latri
{3,3,3(1)2} = {3,3(1)3}
Stage 1 = 3
Stage 2 = {3,3,3} = Tritri
Stage 3 = {3,3,3.......3} with tritri 3’s = dupertri
Stage 4 = {3,3,3,3...........3} with stage 3 3’s
...
Latri is Stage Stage 3.
Latri is notable for being the smallest non-degenerate non-linear array that can be expressed with Bowers' array of operator - in this case, it's 2+1 & 3, since a+b & x is {x,x,x.....x,x,x(1)x,x,x,x......x.x,x} with a x's before (1) and b x's before (1).
Grand godgahlah-ex-grand godgahlah
Grand grandgahlah / grandgahlah-minus-one-ex-grand godgahlah
E100#^#100#2
Grandgahlah-ex-grand godgahlah
Godgahlah-ex-grand godgahlah-ex-grand godgahlah
Despite this sounding like the potential next level, this is not where we’ll be going from here.
Gabbol
{10,100,3(1)2}
If you go back to the gibbol stages, this number is stage stage stage ... ... stage 1, with 100 stages.
Greagahlah
E100#^#100#100#100
A bit better than grandgahlah and comparable to gabbol, but still recycling old operators. If we want to go anywhere we’ll need to define new operators. This is 99-ex-grand godgahlah-ex-grand godgahlah-ex-grand ... ... godgahlah with 100 godgahlahs.
Geebol
{10,100,4(1)2}
Gigangahlah
E100#^#100#100#100#100
Gorgegahlah
E100#^#100#100#100#100#100
Gulgahlah
E100#^#100#100#100#100#100#100
Gaspgahlah
E100#^#100#100#100#100#100#100#100
Ginorgahlah
E100#^#100#100#100#100#100#100#100#100
Boobol
{10,10,100(1)2}
Gugoldgahlah
E100#^#100##100
Comparable to a boobol.
Bibbol
{10,10,100,2(1)2}
Gugolthragahlah
E100#^#100##100##100
Troobol
{10,10,10,100(1)2}
Throogahlah
E100#^#100###100
Quadroobol
{10,10,10,10,100(1)2}
Yottoogahlah
E100#^#100########100
Gootrol
{10,100(1)3}
Quite a bit better than the previous attempts, this number is to goobol as goobol is to googol.
Gotrigahlah
E100#^#100#^#100
Slightly better, but here we hit a problem. Instead of recycling old operators we are still just reusing our fresh operators. We’ll need to define new operators along the way if we want to get anywhere. This number is comparable to gootrol. Its order type in the fast-growing hierarchy is twice that of a godgahlah.
Gitrol
{10,100,2(1)3}
Here are some more numbers:
Gatrol
Geetrol
Gietrol
Gotrol
Gaitrol
Bootrol
Trootrol
Quadrootrol
Grangotrigahlah
E100#^#100#^#100#100
Of course, after that we can have a greagotrigahlah, a gugoldgotrigahlah, a gugolthragotrigahlah, a throogotrigahlah, a tetroogotrigahlah, and so on.
Gooquadrol
{10,100(1)4}
We can coin a series of -quadrol numbers, as well as -quintol, -sextol, -septol, etc.
Gotergahlah
E100#^#100#^#100#^#100
Comparable to gooquadrol. You can add a root of a Hyper-E number right before the name to create ten billion different numbers.
Gopeggahlah
E100#^#100#^#100#^#100#^#100 = E100#^#*#5
In general Ea@*#b (where a is any delimiter such as #^#) is Ea@a@a@a....@a with b a's. #*# can actually simplify to ##, since multiple #s in a row are interpreted as a product of #s. Here are some more numbers:
Gohexgahlah
Gohepgahlah
Go-ahtgahlah
Go-enngahlah
Godekahlah
Emperal
{10,10(1)10}
This googolism solves to {10,10,10,10,10,10,10,10,10,10(1)9}. To solve this you would need to decompose this into { ... ... ... (1)8}, and then that to { ... ... ... ... ... ... ... (1)7} until we get a huge linear array. Yikes! This is comparable to godekahlah.
Gossol
{10,10(1)100}
Evaluates to {10,10,10,10,10,10,10,10,10,10(1)99}.
Godgoldgahlah
E100#^#*#100
Or E100#^#100#^#100#^#100..........#^#100 with 100 100’s. This number’s name makes sense because it is analogous to a gugold (E100#100#100.......#100 with 100 100’s) compared to the googol (E100).
Gotrigoldgahlah
E100#^#*#100#^#*#100
This number is comparable to a gissol and also can be written as E100#^#*##3.
Gissol
{10,10(1)100,2}
Gotergoldgahlah
E100#^#*##4
Comparable to gassol.
Gassol
{10,10(1)100,3}
Geesol
{10,10(1)100,4}
Gussol
{10,10(1)100,5}
Hyperal
{10,10(1)10,10}
This number can also be written as:
/10,10\
\10,10/.
It's a 2x2 array of tens, which can also be called a 2+2, 2^2, 2^^2, etc. array of 10s. It evaluates to {10,10,10,10,10,10,10,10,10,10(1)9,10}.
Fish number 3
~ fα(63), where α = (ww+1)*63+1
This is the next of the Fish numbers, coined in 2002. It's a big jump from the previous, transcending Bowers' linear arrays and beginning dimensional arrays. It uses a function called s(n) mapping that reaches w^w in the fast growing hierarchy (the power of linear arrays), and then extends upon it to get a function that reaches low level dimensional arrays. In terms of Bowers' notation, it's about {63,63(1)2,63}.
Fish number 4 applies that idea to the busy beaver function, creating an uncomputable number. However, the next Fish number in numerical order, Fish number 5, uses a different system that reaches epsilon-zero in the fast-growing hierarchy.
Gongulus
Mossol
{10,10(1)10,100}
Evaluates to {10,10,10,10,10,10,10,10,10,10(1)9,100}.
Godthroogahlah
E100#^#*##100
Solves to E100#^#*#100#^#*#100#^#*#100.........#^#*#100 with 100 100’s. This is comparable to mossol.
Missol
{10,10(1)10,100,2}
Bossol
{10,10(1)10,10,100}
Godtetroogahlah
E100#^#*###100
Comparable to bossol.
Trossol
{10,10(1)10,10,10,100}
Godpentoogahlah
E100#^#*####100
Goddektoogahlah
E100#^#*#^#9
Diteral
{10,10(1)(1)2} = {10,10,10,10,10,10,10,10,10,10(1)10,10,10,10,10,10,10,10,10,10}
This evaluates to:
/10,10,10,10,10,10,10,10,10,10\
\10,10,10,10,10,10,10,10,10,10/.
Right around the limit of 2-row arrays. Now on to higher-level planar arrays!
The Planar-Array Range
{10,100(1)(1)2} - {a,b(2)2}
(order type w^(w2) ~ w^w^2)
Entries: 25
Dubol
{10,100(1)(1)2}
Or
/10,10,10,10.........10\
\10,10,10,10.........10/ with 100 10s in each row. Yikes! This number is the start of 3-row arrays. By the way, (2) means move to the next plane, and {a,b(2)c} = {a,b(1)(1)(1)(1)........(1)c-1} with b (1)s, or a b*b array of a’s with c-1 behind the first a.
Also, note how for the low and high bound of the number range I’m using variables. This is because strict limits don’t feel right and I want the limit to just be a number expressible in the way shown. It also makes sense to use order types in the fast-growing hierarchy, because it's a very common way to approximate googologically large numbers.
Deutero-godgahlah
E100#^#*#^#100
This number is comparable to dubol, and not too far from E100#^##100. Now we’re really starting to get somewhere with our godgahlah extensions.
Dutrol
{10,100(1)(1)3}
Deutero-gotrigahlah
E100#^#*#^#100#^#*#^#100
Comparable to dutrol.
Admiral
{10,10(1)(1)10}
This is comparable to deutero-godekahlah, another military-themed Bowerian googolism.
Dossol
{10,10(1)(1)100}
Deutero-godgoldgahlah
E100#^#*#^#*#100
If we want to go anywhere we’ll need new delimiters (#....# expressions) altogether, like we’re doing. This number is comparable to dossol.
Cyperal
{65,googolplex(1)1000,googol(1)7quadragintiquingentillion,42}
This is just another one of the deleted googolisms on Googology Wiki, deleted because no external sources were given. It's a strange number alright.
Dutritri
{3,3(2)2} = {3,3,3(1)3,3,3(1)3,3,3}
Or
/3,3,3\
|3,3,3|
\3,3,3/ in array form. This is a square of nine threes.
Dutridecal
{10,3(2)2}
Or
/10,10,10\
|10,10,10|
\10,10,10/, This is replacing every entry in the enormous dutritri with a 10, but really not too much of an improvement. We’ll need to go better, and faster.
Triubol
{10,100(1)(1)(1)2}
A 3x100 array of 10s, and a number I coined.
Trito-godgahlah
E100#^#*#^#*#^#100 = E100#^##3
At last we’re in the realm of #^##. Whew, that was a long way. But we’re still a long way to #^#^# and an even longer way to #^^#.
A deutero-godgahlah is about a 2-row array, trito- is 3 rows.....you can guess what a teterto-godgahlah would be.
Tetrubol
{10,100(1)(1)(1)(1)2}
Comparable to teterto-godgahlah.
Teterto-godgahlah
E100#^#*#^#*#^#*#^#100 = E100#^##4
Comparable to a 100-length 4-row array of 10s.
Pepto-godgahlah
E100#^##5
Exto-godgahlah
E100#^##6
Epto-godgahlah
E100#^##7
Ogdo-godgahlah
E100#^##8
Ento-godgahlah
E100#^##9
Xappol
{10,10(2)2} = 10^2 & 10
The xappol is the result of solving 10x10 array of 10's in Bowers' notation. Here is that array written out:
/10,10,10,10,10,10,10,10,10,10\
|10,10,10,10,10,10,10,10,10,10|
|10,10,10,10,10,10,10,10,10,10|
|10,10,10,10,10,10,10,10,10,10|
/ 10,10,10,10,10,10,10,10,10,10 \
\ 10,10,10,10,10,10,10,10,10,10 /
|10,10,10,10,10,10,10,10,10,10|
|10,10,10,10,10,10,10,10,10,10|
|10,10,10,10,10,10,10,10,10,10|
\10,10,10,10,10,10,10,10,10,10/
YIKES! This number is a good example of a big planar array. It would require an inconceivably large number of levels of decomposition just to get rid of the last row, and even more unfathomably long to completely solve!
The name xappol has been honored in the name XappolBot of the bot (programmed by Vel!) in Googology Wiki's IRC chat.
Dekato-godgahlah
E100#^##10
Isosto-godgahlah
E100#^##20
Goxxol
{10,100(2)2}
Or a 100x100 array of 10s. Saibian coined this number as a number comparable to the next number.
Gridgahlah/hecato-godgahlah
E100#^##100
This is the first real good extension to a godgahlah. It’s much further than any naive extension of Hyper-E could dream of going, and yet it just looks like a slight extension. Bigger and better numbers are coming our way, be prepared. This number is also in the same general domain of numbers as Bowers’ xappol, even though this is significantly bigger.
Xappolplex
{10,xappol(2)2}
The planar array for this number would be too big to fit in the observable universe, or even for x observable universes where x is the number of Planck volumes in the universe - these numbers transcend any such analogy anyway. This number is the next step in recursion from a xappol.
The Multidimensional-Array Range
{a,b(2)2} - {a,b(c)2}
(order type w^w^2 ~ w^w^w)
Entries: 43
Grand xappol
{10,10(2)3}
Remember the array used to define a xappol? Put a 2 behind the first 10 and you get this. When you solve it to a planar array it would be horrendously huge, and then you'll need to go through insane decompositions just to even turn it into a linear array! However, with numbers this large this is a pretty modest improvement, as it's only doubling the order type!
Gridtrigahlah
E100#^##100#^##100
Notice how we’re zooming a lot quicker than previously. This number is comparable to grand xappol as gridgahlah is to xappol.
Gridtergahlah
E100#^##*#4
E100#^##*#100
Saibian does not define this and goes straight to bigger numbers, This number could probably be called a gridgoldgahlah.
E100#^##*##100
Could probably be called gridthroogahlah.
E100#^##*#^#100
This might be called a gridgodgahlah, I don't know. Saibian doesn't give a name for this number.
Deutero-gridgahlah
E100#^##*#^##100
Just as a deutero-godgahlah is 2 rows, a deutero-gridgahlah is 2 planes.
Dimentri
{3,3(3)2} = 3^3&3
Or {3,3,3(1)3,3,3(1)3,3,3(2)3,3,3(1)3,3,3(1)3,3,3(2)3,3,3(1)3,3,3(1)3,3,3} in full. This is a 3x3x3 cube of 27 3s and it's an unspeakable sized number in its own right. But it pales in comparison to the next numbers.
Trito-gridgahlah
E100#^##*#^##*#^##100
About a 100x100x3 cube of 10s. Now let’s skip to a Bowersism:
Colossol
{10,10(3)2}=10^3&10
This colossal number is a 10x10x10 CUBE of 10’s! It's a horrendous sized number of order type w^w^3, and decomposing it leads to insane levels of recursion. It was formerly called colossal.
Dekato-gridgahlah
E100#^###10
The closest Saibianism to a colossol.
Coloxxol
{10,100(3)2}=100^3&10
Instead of just a 10x10x10 cube of 10’s, this is a 100x100x100 cube of a million 10s! Saibian coined this to have a number comparable to kubikahlah.
Kubikahlah
E100#^###100
About a 100x100x100 CUBE of 10s! We’re really getting to awesome stuff with these epicly huge numbers. This number is reasonably close to a colossol.
Deutero-kubikahlah
E100#^###*#^###100
About a 100x100x100x2 tesseract of 10s! This number is heading straight into 4 dimensional arrays. How about....
Terossol
{10,10(4)2}=10^4&10
A 10x10x10x10 tesseract of 10s! Tesseracts can be drawn by hand quite easily, but still quite mind-boggling to imagine. They aren’t too bad compared to higer dimensions. Anyway, solving this number would requitre solving 10 cubes, each bigger than the previous! Each cube would requite lots and lots of squares to solve, and lots more linear arrays just to decompose the squares! You’d end up with a vast linear array, and it’ll take a grueling number of steps to solve that into the final number!
Dekato-kubikahlah
E100#^####10
Comparable to terossol.
Teroxxol
{10,100(4)2}
Instead of just 10 cubes, solving this would take 100 enormous cubes! Just imagining this boggles your mind greatly. But it pales in comparison to the insanity of tetrational arrays (seen in a bit).
Quarticahlah
E100#^####100
We’re almost to the realm of #^#^# numbers. This mind-boggling number is about a 100^4 tesseract of 10s! Solving the Cascading-E expressions isn’t much better than the arrays; you’ll need to correctly sort out the hyper-operators and probably mess up a bunch of times!
Petossol
{10,10(5)2}
A 10^5 penteract of 10s. It requires solving multiple tesseracts, which in turn requires solving a mind-boggling number of cubes, and an even more mind-boggling number of squares, and rows, and we just can’t grasp how anyone would solve this! I suppose he still could, but at this point the computation is becoming abstract in itself.
Quinticahlah
E100#^#####100
Or E100#^#^#5. At last we reached the realm of #^#^#! Not too long to E100#^#^#100 anymore. You can guess what comes next...
Ectossol
{10,10(6)2}
A 10^6 hexeract of 10s! While we can draw tesseracts and penteracts quite eaily and still make them out, hexeracts and beyond will start to look like a mess even when mapped out by a computer program. The levels of decomposition needed to solve this are getting more and more horrendously huge.
Sexticahlah
E100#^#^#6
Zettossol
{10,10(7)2}
Septicahlah
E100#^#^#7
Yottossol
{10,10(8)2}
Octicahlah
E100#^#^#8
Ogdo-octicahlah
E100#^#^#98
This is Sbiis Saibian's 3000th googolism, defined using Cascading-E notation. It's approximately {10,100(8)(8)(8)(8)(8)(8)(8)(8)2} (8 (8)s) using Bowers' dimensional array notation.
Xennossol
{10,10(9)2}
Xenna- is one of the unofficial SI prefixes used for 1000^9, or an octillion. Bowers seems to like using this one.
Nonicahlah
E100#^#^#9
Dimendecal
{10,10(10)2} = 10^10 & 10
A 10-dimensional array of 10s! Solving this would require continuously going through hypercubes, back and forth as the array slowly decreases in complexity.
Decichlah
E100#^#^#10
Viginticahlah
E100#^#^#20
Nonaginticahlah
E100#^#^#90
Saibian defines more -cahlah numbers, but I’m skipping most of them because I don’t want to be repetitive. This is the last Saibianism smaller than the gongulus.
{10,10(100)2} = 10^100 & 10
A gongulus is an unspeakably huge number coined by Jonathan Bowers. It's a 100-dimensional array of tens, which has exactly a googol entries when the array is expanded out. Bowers seems to bring that number up a lot more than other numbers of his (see negative gongulus, part 1).
At this point, just to shake you up a little let's take a moment to discuss how INCREDIBLY HUGE a gongulus is. Tthis number makes Graham look adorable, to say the least. Graham’s number is so small that once you are introduced to it, you can explain it to anyone and they can understand how it would be computed! A gongulus, however, is far far beyond weeny little numbers whose computation you can understand. It’s much much much much much ... ... ... MUCH larger than Graham's number. A 100-dimensional cube may not seem too overwhelming, but once we get into it your mind will be blown.
To start off, a gongulus will look a bit like this:
x x x x x x x x x where each x is a 99-dimensional cube
And each 99-dimensional cube will consist of 10 98-dimensional cubes, and each of those will consist of 97-dimensional cubes, and each of THOSE will consist of 96-dimensional cubes, etc. The starting array for a gongulus will have a googol 10’s!
Still not impressed? Then let’s go inside the centeract (100-dimensional cube) and start seeing how it would be solved! First, you’ll need to solve the last 99-dimensional cube to turn the hypercube into an x*x*x*x.......*x*9 array of tens. But to solve that, you’ll need to solve a bunch of 98-dimensional cubes, and for each of those you’ll need to solve a bunch of 97-dimensional hypercubes, and for each one a 96-cube ... ... somewhere in the middle there’s 50, 49, 48, 47, etc. cubes ... ... until you get to the lines that decompose the squares, which in turn decompose the cubes, then the tesseracts ... each dimemsion would be more horrendous to solve than the previous! And that’s only the first 99-dimensional cube! Once you’ve got that 99-cube out of the way, you’ll need to solve the next incomprehensibly large 99-cube, but that would require solving lots and lots of 98-cubes, which requires a huge amount of 97-cubes, and a huge amount of 96-cubes, 95-cubes ... ... 29-cubes, 28-cubes ... ... tesseracts, cubes, squares, lines! Then the third 99-cube will need to be solved, and the fourth 99-cube next, each bigger than the previous, continuing with the fifth, sixth, seventh, eighth, ninth, and then the last 99-dimensional cube! To solve that cube, you’ll need a seemingly endless amount of 98-cubes, followed by 97, 96-, etc. cubes, but you’ve done that before a bunch of times, just this time it’s even bigger! Once you’re down to a 98-cube solve that to a 97-cube (which still requires 96-, 95-, 94-cubes ... ... ... tesseracts, cubes, squares, lines)! Once that 97-cube is over and you have a 96-cube, you’ll need to continue and continue through 94 more dimensions until you get an insanely big planar array, and you’ll need to solve that until you get a very, very long linear array! You’ll ned to solve that into gradually smaller linear arrays, and then into an up-arrow notation where you’ll need to repetitively evaluate the expresssions until you finally get a huge power of 10! Evaluate that, and that is a gongulus! And that isn’t even the full picture! It seems to never come to an end when trying to solve, as Bowers says.
That long explanation above is much longer and more overwhelming than any explanation of Graham’s number! And a full explanation of Graham’s number is wimpy compared to this mind-boggling overview of the computation of a gongulus! The gongulus is much much too mighty for us mortals to comprehend. Yet it’s only the beginning of a series of much worse numbers! Eventually we won’t be able to even imagine anything of what the arrays would be like! In other words, the gongulus, as mighty as it is, is easily trumped by FAR more unfathomable numbers!
All in all, this number is a great example of a number that really crushes Graham's number - Sbiis Saibian describes the huge difference in a blog post, and suggests using a gongulus to replace the fame held by Graham's number, since Graham's number, by googological standards, is quite tiny and easy to analyze. Though I personally would transfer Graham's honor to something like the even bigger TREE(3), which like Graham's number has serious use in mathematics.
Second gongulus / Gonguxxus
{10,100(0,1)2} = {10,100(100)2} = 100^100 & 10
Bowers’ gongulus extensions (such as {10,100(0,3)2} or {10,100(0,0,0,0,1)2}) don’t really match the gongulus, since there’s no way to express a gongulus as {a,b(x,y)c}. To fix this problem, Aarex coins the second gongulus, which matches better with the extensions.
Sbiis Saibian calls this number gonguxxus continuing the goxxol, coloxxol, teroxxol, etc, idea.
Godgathor
E100#^#^#100
We’re all in all done with the godgahlah, gridgahlah, kubikahlah, and -icahlah series (collectively the godgahlah gang) and we’ve moved on to a whole new gang. Behold the godgathor gang. This Saibianism is comparable to the gongulus but closer to the second gongulus.
Even though #^#^# looks a lot friendlier than an x-dimensional array, guess what? It’s just as bad. Each #^#99 (#^####....#### with 99 #s) is equivalent to a 99-cube, each #^#98 is equal to a 98-cube, etc. Decomposing this will be quite akin to the horrendous decomposition of a gongulus!
Goober bunch
E100#^#101100
This irregular number was defined by Saibian to break the monotony of his numbers. It’s defined as E100#^#100*#^#100*#^#100......#^#100100 with 100 #^#100s, simplifying to the expression above.
Gongulusplex
{10,10(gongulus)2}
Remember the 100 dimensions we went through in the long gongulus paragraph? Now imagine having to go through a gongulus dimensions! We’ll pass through every whole number in this list up to a gongulus when working with the number of dimensions. Since you can’t even imagine how big a gongulus would be, it’s even more mind-boggling to imagine solving a gongulus-dimensional cube!
As impressive as this number is, it’s still a naive extension because anyone introduced to the gongulus could come up with that. We’ll need to move much faster than ANY recursion to get anywhere!
Grand godgathor
E100#^#^#100#2
Or E100#^#godgathor100. It looks absolutely amazing, but it’s the same situation as a gongulusplex.
Gibbering goober bunch
E100#^#goober bunch+1(goober bunch)
Defined as E100#^#goober bunch*#^#goober bunch#^#goober bunch......#^#goober bunch100 with a goober bunch #^#goober bunchs. It looks ambitious, but once again it isn’t a good improvement.
Gongulusduplex
{10,10(gongulusplex)2}
Just imagine......you’ll need to go through a gongulusplex dimensions to solve the number. Doing that and growing through such a gigantic number of dimensions brings a sense of vast darkness. Kinda creepy if your think about it.
Gongulustriplex
{10,10(gongulusduplex)2}
Gongulusquadriplex
{10,10(gongulustriplex)2}
Superdimensional Array Numbers
{a,b(c)2}~{a,b((1)1)c}
(order type w^w^w ~ w^w^w^w)
Entries: 50
Gotrigathor
E100#^#^#100#^#^#100
Notice how we’re skipping stuff like E100#^#^#100##100 and heading straight to the good stuff.
Also, superdimensional arrays are arrays which have dimensions of dimensions. It’s the first level of tetrational arrays. Following are trimensions (dimensions of superdimensions) and quadramensions, which are the higher tetrational arrays.
Deutero-godgathor
E100#^#^#*#^#^#100
Or E100#^(#^#*#)2.
Trito-godgathor
E100#^#^#*#^#^#*#^#^#100
Or E100#^(#^#*#)3.
Hecato-godgathor / Godgathorfact
E100#^(#^#*#)100
Note that E100#^#^#*#^#^#*#^#^#*#^#^#......*#^#^#100 with 100 #^#^#s is NOT E100#^#^##100. It simplifies to only E100#^(#^#*#)100 .
Godgridgathor / Godgathordeuterfect
E100#^(#^#*##)100
Decomposes to E100#^(#^#*#)*#^(#^#*#)*#^(#^#*#)*......#^(#^#*#)100, with 100 #^(#^#*#)s. Just imagine how many multiplication strings you’ll need to go through to decompose that.
Dulatri
{3,3(0,2)2}
This Bowersism serves to help us better understand superdimensional arrays. To picture this array, imagine a 3x3x3 cube, but with each of the 27 slots filled with a 3x3x3 cube of 3’s, making it a number that scratches the surface of tetrational arrays. It’s also equal to an X^2X array of 3’s, with 729 threes.
Godcubicgathor / Godgathortritofact
E100#^(#^#*###)100
Godquarticgathor / Godgathortetrifact
E100#^(#^#*####)100
Gingulus
{10,100(0,2)2}
A 100^(2*100) array of 10’s. This is the first group of superdimensions. To solve visualize this array, imagine a 100^100 dimensional array, but each slot is filled with a gongulus array. The array has 10^400 entries in total.
Godgathordeus
E100#^(#^#*#^#)100
Comparable to the gingulus. We’re almost to E100#^#^##100! Also, we can define the hecato-godgathordeus, and godgrid-, -cubic-, and -quarticgathordeus.
Gangulus
{10,100(0,3)2}
A 100^(3*100) array of 10’s. The array can be visualized as a gongulus array of ginguluses.
Godgathortruce
E100#^(#^#*#^#*#^#)100
Simplifies to E100#^(#^##)3, or just E100#^#^##3. Comparable to gangulus.
Geengulus
{10,100(0,4)2}
Godgathorquad
E100#^#^##4
Gowngulus
{10,100(0,5)2}
Godgathorquid
E100#^#^##5
Gungulus
{10,100(0,6)2}
Godgathorsid
E100#^#^##6
Godgathorseptuce
E100#^#^##7
Godgathoroctuce
E100#^#^##8
Godgathornonice
E100#^#^##9
Godgathordecice
E100#^#^##10
Bongulus
{10,100(0,0,1)2}
Or {10,100(0,100)2}. A 100^100^2 array of 10’s, but still a low level superdimensional array. If stage 1 is the second gongulus, stage 2 is the gingulus, stage 3 is the gangulus, stage 4 is the geengulus, etc, then this is stage 100.
Gralgathor
E100#^#^##100
Here’s a really epic number, comparable to bongulus. This number simplifies to E100#^(#^#*#^#*#^#*#^#*#^#......#^#)100 with 100 #^#s! To simplify it, you would start off with E100#^(#^#*#^#*#^#........*#100)100, which first decomposes to E100#^(#^#*#^#*#^#........*#99)*#^(#^#*#^#*#^#........*#99)*#^(#^#*#^#*#^#........*#99).......*#^(#^#*#^#*#^#........*#99)100 with 100 #^(#^#*#^#*#^#........*#99)s! The last (but ONLY the last) #^(#^#*#^#*#^#........*#99) decomposes to #^(#^#*#^#*#^#........*#98*#^(#^#*#^#*#^#........*#98)*#^(#^#*#^#*#^#........*#98)....*#^(#^#*#^#*#^#........*#98), then the last one of those decomposes to #^(#^#*#^#*#^#........*#97*#^(#^#*#^#*#^#........*#97)*#^(#^#*#^#*#^#........*#97)....*#^(#^#*#^#*#^#........*#97), then we'll have to repeat the process 97 more times to end with a #^(#^#*#^#*#^#*#^#.....*#^#) with 99 #^#s instead of 100. Right then, we can decompose that very last #^# to #100 ... just so we can repeat that whole process all over again. Then we have to repeat it 100 times until it ends with a ridiculously long product of delimiters, but then we'll need to decompose that, and repeat again, and decompose that, repeat again ... you can scream now ...
Absolutely mind-boggling!!!! And the godgathor (and gongulus) seemed mighty just moments ago. They are now in the dust with the wimpy little grand godgahlahgong! I told you numbers that make the gongulus look wimpy are headed our way - but you ain't seen nothing yet!
Graltrigathor
E100#^#^##100#^#^##100
Deutero-gralgathor
E100#^#^##*#^#^##100
Or E100#^(#^##*#)2.
Hecato-gralgathor
E100#^(#^##*#)100
Hecato-x is no longer very effective. We’re going much faster now.
Gralgridgathor
E100#^(#^##*##)100
Decomposes to E100@100@100@100.......@100 where each @ is #^(#^##*#).
Gralcubicgathor
E100#^(#^##*###)100
Gralgodgathgathor
E100#^(#^##*#^#)100
Gralgodgathordeusgathor
E100#^(#^##*#^#*#^#)100
Bingulus
{10,100(0,0,2)2}
An X^(2*X^2) array of 10s. To imagine this array, you'll need a bongulus array of bongulus arrays - keep in mind that a bongulus array is a 100x nested array of gongulus arrays.
Gralgathordeus
E100#^(#^##*#^##)100
Or E100#^#^###2. Comparable to bingulus.
Trimentri
{3,3(0,0,0,1)2}
Remember megafuga-three, which was equivalent to about 7.6 trillion? Well, that number is used to define this number, specifically as a 3^3^3 or 3^^3 array of 3’s. Trimentri can be expressed in various other ways. Here they are:
{3,3(0,0,3)2}
{3,3((1)1)2}
A 3-superdimsional array of 3’s
A 1-trimensional array of 3’s
3^3^3&3
3^^3&3
3^^^2&3
A 3 tetrated to 3 array of 3’s
A 3 pentated to 2 array of 3’s
Bangulus
{10,100(0,0,3)2}
An X^(3*X^2) array of 10s.
Gralgathortruce
E100#^#^###3
Beengulus
{10,100(0,0,4)2}
We saw this number’s reciprocal a loooooooooong time ago in part 1.
Gralgathorquad
E100#^#^###4
Trongulus
{10,100(0,0,0,1)2}
In the lower superdimensional arrays but still super insane. This number is about an X^X^3 array of 10’s!
Thraelgathor
E100#^#^###100
The numbers just keep getting crazier. To compute this, you’ll need to follow the same general thing we did with the gralgathor, then do it again and this time even longer, and it's difficult to bring to words. Keep in mind that we can coin plenty of numbers based on this like an isosto-thraelgathor and a thraelgathorseptuce.
Quadrongulus
{10,100(0,0,0,1)2}
A four-superdimensional array of tens.
Terinngathor
E100#^#^####100
Comparable to the quadrongulus.
Pentaelgathor
E100#^#^#####100
Hexaelgathor
E100#^#^#^#6
Heptaelgathor
E100#^#^#^#7
Octaelgathor
E100#^#^#^#8
Ennaelgathor
E100#^#^#^#9
Dekaelgathor
E100#^#^#^#10
Goplexulus
{10,100((1)1)2}
Also imaginable as:
{10,100(0,0,0,0.......0,1)2} with 100 zeros
100^100^100&10
A 100-superdimensional array of tens
A 100 tetrated to 3 array of tens
This number is right around the end of superdimensional arrays and the beginning of higher tetrational arrays. It's formed with some pretty insane array nesting.
Godtothol
E100#^#^#^#100
This number simplifies to E100#^#^#100100 (hectaelgathor perhaps), and once solving it further the expression expands rapidly in length. We saw how insane the gralgathor was, so this has to be much much more crazy. By the way, this is comparable to goplexulus.
The Ordinal-Tetration Range
{a,b(x,y.....n)c} ~ X^^X & a
(order type w^w^w^w ~ ε0)
Entries: 22
Graltothol
E100#^#^#^##100
The gralgathor was crazy, so is there a reason for this not to be super ultra mega crazy? I think not. To solve this, you’ll need to first go through the whole gralgathor fun zillions of times just to get anywhere. This is about an X^X^X^2, or 2-trimensional, array of tens!!
Hyper-gralgodgathorseptucegathor
E100#^#^(#^##*#^##)7
This is Sbiis Saibian's 4000th googolism, defined using Cascading-E notation. It's a member of the graltothol squadron in the godtothol regiment. Using Bowers' tetrational array notation this is about {10,100((2)(1)(1)(1)(1)(1)(1)(1)1)2}, and using the fast-growing hierarchy it's approximately fw^w^w^(w^2*w7)(100). Read Sbiis Saibian's list of Cascading-E googolisms for why it's named like that.
Thraeltothol
E100#^#^#^###100
Terinntothol
E100#^#^#^####100
Goduplexulus
{10,10((100)1)2} = 100^^4 & 10
We’re heading further and further into the tetrational arrays - this is a 100-trimensional array of tens.
Godtertol
E100#^#^#^#^#100
We’re almost done with the Cascading-E numbers and are getting ready to head into Extended Cascading-E. This is comparable to goduplexulus. I could include all the numbers in the godtertathol gang and the rest of the -tathol gangs, but I don’t want to be repetitive.
Gotriplexulus
100^^5 & 10 = {10,100(((1)1)1)2}
A 100-quadramensional array of tens.
Godtopol
E100#^#^#^#^#^#100
With these insane numbers, it may be tricky to remember that we're blasting through them at light speed, a lot faster than we could - actually we're going WAY FASTER than light speed through these monsters!
Godhathor
E100#^#^#^#^#^#^#100
Godheptol
E100#^#^#^#^#^#^#^#100
Godoctol
E100#^#^#^#^#^#^#^#^#100
Godentol
E100#^#^#^#^#^#^#^#^#^#100
Godekatothol
E100#^#^#^#^#^#^#^#^#^#^#100
Quintuple-hyper-terinntopoldeus
E100#^#^#^#^#^#^(#^#^#^#^####*#^#^#^#^####)100
This is Sbiis Saibian's 5000th googolism, a decent tetrational-array level number. It is a member of the godektathol regiment and it's approximately fw^w^w^w^w^w^(w^(w^w^w^4*2))(100) using the fast-growing hierarchy.
Tethrathoth
E100#^#^#^#^#^#^# ... ... ... ^#^#100 with 100 #s = E100#^^#100
The tethrathoth is one of Sbiis Saibian's milestone googolisms. It's comparable to and a little smaller than Bowers’ goppatoth. It’s approximately equal to an X^^99 array of 10s. Its name comes from tetration + Thoth, the Egyptian god of mathematics, since it's formed by tetrating #s, and it's on the order of epsilon-zero in the fast-growing hierarchy.
Goppatoth
10^^100 & 10
A goppatoth is a large tetrational array googolism by Jonathan Bowers. It's a ten-tetrated-to-100 array of tens, which would be 10 99-mensional array of 10s, and a 1 100-mensional array of 10s. There are a giggol 10s in the starting array, and the number is a little bigger than a tethrathoth.
Second Goppatoth
X^^X & 100 = 100^^100 & 100
Just as the gongulus does not match with its extensions, the goppatoth will not match with my extensions. To make it match, I’ve defined the second goppatoth to match with those extensions, just as Aarex defined the second gongulus. That’s the only reason this number exists.
Giaquaxul
200![200,200,200,200]
This number is pronounced gia-quazzle, and it shows how 4-entry hyperfactorial arrays grow surprisingly quickly. In general, [a,b,c,2] hyperfactorial arrays are on the realm of dimensional arrays, [a,b,c,3] arrays are superdimensional, [a,b,c,4] are trimensional, [a,b,c,5] are quadramensional, etc. The next major Hollomism (hugexul) is much much much larger and is a very long way from here.
fε0(1000)
This is just a placeholder entry to discuss order-type epsilon-zero in the fast-growing hierarchy. It's an interesting tipping point for googologically large numbers.
Why is that? Because after numbers of order type epsilon-zero, they quickly become harder and harder to work with - even epsilon-one has two notably different fundamental sequences, making comparing numbers with the fast-growing hierarchy a lot harder to work with. There seems to be a general natural way to get up to epsilon-zero in the FGH, but soon afterwards there are several different paths you can take, many of which will need ways to work around the annoying +1's that may crop up (e.g. w^(ε0+1)). After order-type epsilon-zero, BEAF also becomes harder to work with - because of ambiguity, it is up to the reader to interpret the numbers. Up to a certain point the numbers nowadays are pretty clear, but later levels (discussed in parts 6-7) are still pretty unclear.
Epsilon-zero is the growth rate achieved by Cascading-E notation and Bowers' tetrational arrays, as well as m(n) mapping used in Fish number 5 - it's a common growth rate for functions.
I should point out that this is one of the largest levels of number size mentioned on Wikipedia: in its article on the slow-growing hierarchy (a relative of the fast-growing hierarchy) it says:
"The slow-growing hierarchy grows much more slowly than the fast-growing hierarchy. Even gε0 is only equivalent to f3 and gα only attains the growth of fε0 (the first function that Peano arithmetic cannot prove total in the hierarchy) when α is the Bachmann–Howard ordinal."
However, Wikipedia discusses still larger numbers later, and the largest computable number to appear on Wikipedia may be the famous TREE(3).
Tethrathothigong
E100,000#^#^#^#^#.......^#100,000 with 100,000 #s = E100,000#^^#100,000
This is the gong version of a tethrathoth, for old time's sake as Sbiis Saibian said. It's technically insanely larger than a tethrathoth with 100,000 cascade levels instead of just 100, but it's a modest improvement compared to what's next.
Grand tethrathoth
E100#^#^#^#^#^#^#...........^#100 with tethrathoth #s = E100#^^#100#2
The next step in recursion for a tethrathoth. This might seem utterly insane, but it's still a naive extension because anyone who encountered the tethrathoth can come up with such a number. We need to go MUCH faster to really get anywhere.
Goppatothplex
10^^goppatoth & 10
A goppatoth is a ten tetrated to a goppatoth array of tens, the next step in recursion after a goppatoth.
NOTE: If you are a seasoned googologist reading this list, you may object to me including Bowersisms past a goppatothplex beyond this list. I know that Bowers' notation past tetrational arrays is not fully defined (and that there isn't even full consensus), and I don't object to that statement. However, I still feel a need to include the numbers.
Therefore I am going to put them in based on the value using Bowers' and Saibian's theory of the climbing method (see Bowers' hypernomials page and Saibian's Extended Cascading-E page) for numbers up to {X,X(1)2} arrays. After that point I will rely on the approximate value (in terms of the fast-growing hierarchy) that seems most accepted in the googology community, based mainly on Hyp cos's BEAF "analysis" (for more on that see entries in part 6 right before TREE(3)).
Grand grand tethrathoth
E100#^^#grand tethrathoth = E100#^^#100#3
The Epsilon Range
X^^X&a - X^^(X+1)&a
(order type ε0 ~ εw)
Entries: 61
Fish number 5
~fε0+1(63)
This number was defined by Kyodaisuu of Googology Wiki in 2003. It uses a system called m(n) mapping that reaches epsilon-zero in the fast-growing hierarchy, the same power of Bowers' tetrational arrays. The number falls just under a grantethrathoth, which is about fε0+1(100).
Grangol-carta-tethrathoth / grantethrathoth / 99-ex-grand tethrathoth
E100#^^#100#100
This number is also equal to 99-ex-grand tethrathoth.
Greagol-carta-tethrathoth/greatethrathoth
E100#^^#100#100#100
Gigangol-carta-tethrathoth/gigantethrathoth
E100#^^#100#100#100#100
Gugolda-carta-tethrathoth
E100#^^#100##100
Gugolthra-carta-tethrathoth
E100#^^#100##100##100
Output of marxen.c
~ fε0+w2(1,000,000)
This is Heiner Marxen's entry in Bignum Bakeoff, the second place entry in Bignum Bakeoff. Unlike other submissions, this one uses a fast-growing function called Goodstein sequences that reaches epsilon-zero in the fast-growing hierarchy, then iterates it to produce a number that makes it a little further than order type epsilon-zero. It can be approximated in Extended Cascading-E as E1,000,000#^^#1,000,000##1,000,000##1,000,000, placing it just above a gugolthra-carta-tethrathoth. It's difficult to use only 512 characters to create a recursive function in C like the other people did that reaches epsilon-zero, which is why using a fast-growing function like Goodstein sequences is clever.
The output of marxen.c is far far smaller than the winning entry. The winning entry is the output of loader.c, Ralph Loader's entry, which is often called Loader's number. We don't even know how to approximate Loader's number in the fast-growing hierarchy because it's so huge. Loader's number can be found in part 7 of this list (COMING SOON).
Throogola-carta-tethrathoth
E100#^^#100###100
Teroogola-carta-tethrathoth
E100#^^#100####100
Fish number 6
Godgahlah-carta-tethrathoth
E100#^^#100#^#100
Gridgahlah-carta-tethrathoth
E100#^^#100#^##100
Kubikahlah-carta-tethrathoth
E100#^^#100#^###100
Quarticahlah-carta-tethrathoth
E100#^^#100#^####100
Godgathor-carta-tethrathoth
E100#^^#100#^#^#100
Godtothol-carta-tethrathoth
E100#^^#100#^#^#^#100
Tethrathoth-carta-tethrathoth/tethratrithoth
E100#^^#100#^^#100 = E100#^^#*#3
Tethraterthoth
E100#^^#100#^^#100#^^#100 = E100#^^#*#4
Tethrathoth-by-hyperion
E100#^^#*#100
Tethrathoth-by-deutrohyperion
E100#^^#*##100
Evaluates to E100#^^#*#100#^^#*#100.....#^^#*#100 with 100 100’s.
Tethrathoth-by-tritohyperion
E100#^^#*###100
Tethrathoth-by-godgahlah
E100#^^#*#^#100
Tethrathoth-by-godgathor
E100#^^#*#^#^#100
Deutero-tethrathoth
E100#^^#*#^^#100 = E100(#^^#)^#2
Trito-tethrathoth
E100#^^#*#^^#*#^^#100 = E100(#^^#)^#3
Hecato-tethrathoth/tethrafact
E100(#^^#)^#100
This number is called tethrafact because the tethrathoth delimiter is the only factor of the whole expression. It’s still not really that great of an improvement, but kind of the base for what will come.
Grideutertethrathoth
E100(#^^#)^##100
Or E100(#^^#)^#*(#^^#)^#*(#^^#)^#.....*(#^^#)^# with 100 (#^^#)^#s. This number can also be called hecato-tethrafact, but Saibian does not give this name.
Kubicutethrathoth
E100(#^^#)^###100 = E100(#^^#)^#^#3
Tethragodgathor / Centicutethrathoth
E100(#^^#)^#^#100
Tethragodtothol
E100(#^^#)^#^#^#100 = E100(#^^#)^(#^^#)3
Tethraduliath/tethra-tethrathoth/tethrathoth-dopplux/tethrathoth-dubletetrate
E100(#^^#)^(#^^#)100 = E100(#^^#)^^#2
This is a milestone point in Extended Cascading-E - it's just starting on tetrating the tethrathoth delimiter. Pretty soon the tetration will be iterated. It's interesting to point out that Saibian explicity gives four different names for this number.
Tethradulifact
E100(#^^#)^(#^^#*#)100
Evaluates to E100(#^^#)^(#^^#)*(#^^#)^(#^^#)*(#^^#)^(#^^#).....*(#^^#)^(#^^#) with 100 (#^^#)^(#^^#)s.
Grideutertethraduliath
E100(#^^#)^(#^^#*##)100
Tethraduli-godgathor
E100(#^^#)^(#^^#*#^#)100
Tethrathruliath
E100(#^^#)^(#^^#*#^^#)100
Tethraterliath
E100(#^^#)^(#^^#*#^^#*#^^#)100 = E100(#^^#)^(#^^#)^#3
Monster-Giant
E100(#^^#)^(#^^#)^#100
The somewhat irregular Monster-Giant is a pretty awesome number. It decomposes to E100(#^^#)^(#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*
#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*
#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*
#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*
#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*
#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*
#^^#*#^^#*#^^#*#^^#*#^^#*#^^#)100.
This is pretty insane now (comparable to a (X^^X)X array), but only starting a whole new realm of insanity.
Grand Monster-Giant
E100(#^^#)^(#^^#)^#100#2
Or E100(#^^#)^(#^^#*#^^#*#^^#*#^^#*#^^#*......................*#^^#)100 with Monster-Giant+1 #^^#s! As impressive as this sounds, it's still a naive extensions. We can further extend upon the Monster-Giant with numbers like:
Deutero-Monster-Giant
E100(#^^#)^(#^^#)^#*(#^^#)^(#^^#)^#100
Trito-Monster-Giant
E100(#^^#)^((#^^#)^#*#)3
Hecato-Monster-Giant
E100(#^^#)^((#^^#)^#*#)100
Tethra-Monster-Giant
E100(#^^#)^((#^^#)^#*#^^#)100
Tethraduli-Monster-Giant
E100(#^^#)^((#^^#)^#*#^^#*#^^#)100 = E100(#^^#)^((#^^#)^#*(#^^#)^#)2
Monster-Monster-Giant = Two-ex-Monster-Giant
E100(#^^#)^((#^^#)^#*(#^^#)^#)100 = E100(#^^#)^(#^^#)^##2
Three-ex-Monster-Giant
E100(#^^#)^((#^^#)^#*(#^^#)^#*(#^^#)^#)100 = E100(#^^#)^(#^^#)^##3
Hundred-ex-Monster-Giant/Monster-Grid
E100(#^^#)^(#^^#)^##100
Monster-Giant-ex-Monster-Giant
E100(#^^#)^(#^^#)^##(Monster-Giant)
Grand Monster-Grid
E100(#^^#)^(#^^#)^##100#2
Monster-Cube
E100(#^^#)^(#^^#)^###100
Monster-Tesseract
E100(#^^#)^(#^^#)^####100
Monster-Hecateract
E100(#^^#)^(#^^#)^#^#100
Tethrathoth-trebletetrate
E100(#^^#)^(#^^#)^(#^^#)100 = E100(#^^#)^^#3
Super Monster-Giant
E100(#^^#)^(#^^#)^(#^^#)^#100
Evaluates to:
E100(#^^#)^(#^^#)^(#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*
#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*
#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*
#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*
#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*
#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*
#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#)100
Super Monster-Grid
E100(#^^#)^(#^^#)^(#^^#)^##100
Tethrathoth-quadrupletetrate
E100(#^^#)^^#4
Tethrathoth-quintupletetrate
E100(#^^#)^^#5
Tethrathoth-decupletetrate
E100(#^^#)^^#10
Terrible tethrathoth
E100(#^^#)^^#100
Evaluates to E100(#^^#)^(#^^#)^(#^^#)^(#^^#)^(#^^#).......^(#^^#) with 100 (#^^#)s.
Dubletetrated terrible tethrathoth
E100((#^^#)^^#)^((#^^#)^^#)100 = E100(((#^^#)^^#)^^#)2
Trebletetrated terrible tethrathoth
E100(((#^^#)^^#)^^#)3
Double-terrible tethrathoth
E100(((#^^#)^^#)^^#)100
Triple-terrible tethrathoth
E100((((#^^#)^^#)^^#)^^#)100
The Hyper-Epsilon Range
X^^(X+1) & a - X^^X^2 & a
(order type εw ~ ζ0)
Entries: 68
Tethriterator / Tethrathoth ba’al / 99-ex-terrible tethrathoth
E100(....((#^^#)^^#)........^^#)100 with 100 #^^#s = E100#^^#>#100
This is Sbiis Saibian's smallest googolism to use the > operator, a special operator called the "caret top" which iterates the function applied to the # before it. Terrible tethrathoth is E100#^^#>#2, double-terrible tethrathoth is E100#^^#>#3, triple-terrible tethrathoth is E100#^^#>#4, and so on with the > operator.
This number called tethriterator because it iterates the smallest tetrational delimiter, or #^^#.
Grand tethriterator / Great and Terrible Tethrathoth / tethrathoth-ba'al-minus-one-ex-terrible tethrathoth
E100(... ...((#^^#)^^#)^^#)^^#) ... ... ^^#)100 with tethrathoth ba’al #^^#s = E100#^^#>#100#2
As insane as this looks, you should know by now that such recursion, unfortunately, is not really progress anymore.
Quadrule-tetrated-tethrathoth-ex-terrible tethrathoth
E100#^^#>#(1+E100(#^^#)^(#^^#)^(#^^#)^(#^^#)100)
This is Sbiis Saibian's 6000th googolism, defined using Extended Cascading-E. It's about fεw(fε0^ε0^ε0^ε0(100)) using the fast-growing hierarchy, or about X^^(X+1) & ((X^^X)^^4 & 100) using the climbing method interpretation of the array-of operator in BEAF. It's an example of a number you can form by repeatedly applying the terrible operator, but it's much smaller than even...
Grangol-carta-tethriterator
E100#^^#>#100#100
Or 99-ex-grand tethriterator. Although this is mind-crushingly larger than something like Great-and-Terrible-Tethrathoth-ex-terrible tethrathoth, it's not really progress. Let's jump to:
Tethritertri
E100#^^#>#100#^^#>#100 = E100#^^#>#*#3
Tethriterhecate/tethritera-by-hyperion
E100#^^#>#*#100
Tethritera-by-deutero-hyperion
E100#^^#>#*##100
Tethritera-by-godgahlah
E100#^^#>#*#^#100
Tethritera-by-tethrathoth
E100#^^#>#*#^^#100
Deutero-tethriterator
E100#^^#>#*#^^#>#100
Trito-tethriterator
E100#^^#>#*#^^#>#*#^^#>#100 = E100(#^^#>#)^#3
Hecato-tethriterator/tethriterfact
E100(#^^#>#)^#100
Grideutertethriterator/hecato-tethriterfact
E100(#^^#>#)^##100
Tethriter-godgathor
E100(#^^#>#)^#^#100
Tethriter-tethrathoth
E100(#^^#>#)^#^^#100
Tethriter-terri-tethrathoth
E100(#^^#>#)^(#^^#)^^#100
Tethriter-dubletetrate
E100(#^^#>#)^(#^^#>#)100 = E100(#^^#>#)^^#2
Terrible tethriterator
E100(#^^#>#)^^#100
Double-terrible tethriterator
E100((#^^#>#)^^#)^^#100
Tethriditerator / 100-ex-terrible tethriterator
E100#^^#>(#+#)100
Terrible tethriditerator
E100(#^^#>(#+#))^^#100
Tethritriterator / 100-ex-terrible tethritriterator
E100#^^#>(#+#+#)100 = E100#^^#>##3
Tethriquaditerator
E100#^^#>##4
Tethrigriditerator
E100#^^#>##100
Tethricubiculator
E100#^^#>###100
Tethriquarticulator
E100#^^#>####100
Tethrispatialator
E100#^^#>#^#100
Tethrispatial-squarediterator
E100#^^#>(#^#*#^#)100
Tethrispatial-cubiculator
E100#^^#>#^##3
Tethrideuterspatialator
E100#^^#>#^##100
Tethritritospatialator
E100#^^#>#^###100
Tethritetertospatialator
E100#^^#>#^#^#4
Tethri-superspatialator
E100#^^#>#^#^#100
Tethri-quadratetratediterator
E100#^^#>#^#^#^#100
Dustaculated-tethrathoth
E100#^^#>#^^#100
Can also be written as E100#^^##2.
Gippatoth
X^^2X & 100
This is a googolism I coined as a fairly low-level pentational array number. I'm putting this and related numbers under the climbing method interpretation - although the non-climbing interpretation is more commonly used among the googology community, the climbing method intentionally matches nicely with Extended Cascading-E, and is largely advocated by Bowers. This is comparable to a dustaculated-tethrathoth.
Tethriter-turreted-tethrathoth
E100#^^#>#^^#>#100
Territethriter-turreted-tethrathoth
E100#^^#>(#^^#>#)^^#100
Not to be confused with the much smaller terrible tethriter-turreted-tethrathoth.
Tethriditer-turreted-tethrathoth
E100#^^#>#^^#>(#+#)100
Tethritriter-turreted-tethrathoth
E100#^^#>#^^#>##3
Tethrigriditer-turreted-tethrathoth
E100#^^#>#^^#>##100
Tristaculated-tethrathoth
E100#^^#>#^^#>#^^#100
Gappatoth
X^^3X & 100
Comparable to a tristaculated-tethrathoth.
Tetrastaculated-tethrathoth
E100#^^#>#^^#>#^^#>#^^#100
Geepatoth
X^^4X & 100
Pentastaculated-tethrathoth
E100#^^##5
Dekastaculated-tethrathoth
E100#^^##10
Tethracross / Tethrasquare
E100#^^##100
A tethracross is another one of Sbiis Saibian's milestone googolisms. It's on the order of zeta-zero in the fast-growing hierarchy. It evaluates to:
E100#^^#>#^^#>#^^# ... ... ... >#^^#>#^^#100 with 100 copies of #^^#
It is comparable to Bowers' kungulus using the more common interpretation of pentational arrays, although using the climbing method (which I'm using on this list) it's much smaller than a kungulus.
Boppatoth
X^^X^2 & 100
Another extrapolation from the tethrathoth, based on Bowers' number, the bongulus. It's comparable to a tethrasquare.
Giaquixul
200![200,200,200,200,200]
A number I coined, which is in the realm of X^^X^2 arrays. The next HAN number after this will be the hugexul.
Berlin Wall
E100#^^##100,000,000
An irregular number defined by Saibian. The expression it expands to, or E100#^^#>#^^#>#^^#>#^^#.....#^^#>#^^#>#^^#100 with 100,000,000 #^^#>s wouldn’t be much longer than the real Berlin Wall, but that also depends on the text size.
Grand tethracross
E100#^^##100#2
Grand Berlin Wall
E100#^^##100,000,000#2
~ fζ0+1(63)
Fish number 6 is the largest computable Fish number, defined by Kyodaisuu of Googology Wiki in 2007. It extends upon Fish number 5's system with a two-argument function that reaches zeta-0 in the fast growing hierarchy (the power of X^^X^2 arrays), so it falls just under a grangol-carta-tethrasquare, equal to about fζ0+1(100)
The Zeta-Eta Range
X^^^X^2 & a - X^^X^X & a
(order type ζ0 ~ φ(w,0))
Entries: 121
Grangol-carta-tethrasquare
E100#^^##100#100
Godgahlah-carta-tethrasquare
E100#^^##100#^#100
Tethrathoth-carta-tethrasquare
E100#^^##100#^^#100
Tethriterator-carta-tethrasquare
E100#^^##100#^^#>#100
Tethrasquare-by-deuteron
E100#^^##100#^^##100
Tethrasquare-by-triton
E100#^^##100#^^##100#^^##100 = E100#^^##*#4
Tethrasquare-by-hyperion
E100#^^##*#100
Tethrasquare-by-deuterohyperion
E100#^^##*##100
Tethrasquare-by-godgahlah
E100#^^##*#^#100
Tethrasquare-by-tethrathoth
E100#^^##*#^^#100
Tethrasquare-by-tethriterator
E100#^^##*#^^#>#100
Deutero-tethrasquare
E100#^^##*#^^##100
Trito-tehtrasquare
E100#^^##*#^^##*#^^##100 = E100(#^^##)^#3
Hecato-tethrasquare/tethrasquarorfact
E100(#^^##)^#100
Grideutertethrasquare
E100(#^^##)^##100
Centicutethrasquare
E100(#^^##)^#^#100
E100(#^^##)^#^#^#100
E100(#^^##)^(#^^#)100
E100(#^^##)^(#^^#>#)100
Dutetrated-tethrasquare
E100(#^^##)^(#^^##)100
Tritetrated-tethrasquare
E100(#^^##)^(#^^##)^(#^^##)100 = E100(#^^##)^^#3
Terrible tethrasquare
E100(#^^##)^^#100
Tethriterated-tethrasquare
E100(#^^##)^^#>#100
Tethrastaculated-tethritertethrasquare
E100(#^^##)^^#>#^^#100
Tethriterstaculated-tethritertethrasquare
E100(#^^##)^^#>#^^#>#100
Dustaculated-tethraturreted-tethritertethrasquare
E100(#^^##)^^#>#^^#>#^^#100
Tristaculated-tethraturreted-tethritertethrasquare
E100(#^^##)^^#>#^^#>#^^#>#^^#100 = E100(#^^##)^^#>#^^##3
Tetrastaculated-tethraturreted-tethritertethrasquare
E100(#^^##)^^#>#^^##4
Tethrasquare-turreted-tethritertethrasquare
E100(#^^##)^^#>#^^##100
Territethrasquare-turreted-tethritetrtethrasquare / Dustaculated-tethritertethrasquare
E100(#^^##)^^#>(#^^##)^^#100
Sorry that the names sound absurdly long, but it kind of needs to be that way if we want to have a useful system to name any such number. After all, naming systems are a big part of googology, and almost by necessity the names can get pretty long.
Tristaculated-tethritertethrasquare
E100(#^^##)^^#>(#^^##)^^#>(#^^##)^^#100 = E100(#^^##)^^##3
Tetrastaculated-tethritertethrasquare
E100(#^^##)^^##4
Dekastaculated-tethritertethrasquare
E100(#^^##)^^##10
Secundo-tethrated tethrasquare
E100(#^^##)^^##100
Dustaculated-secundo-tethrated tethrasquare
E100(#^^##)^^##>(#^^##)^^##100
Tristaculated-secundo-tethrated tethrasquare
E100(#^^##)^^##>(#^^##)^^##>(#^^##)^^##100 = E100((#^^##)^^##)^^##3
Thrice-tethrasecunda
E100((#^^##)^^##)^^##100 = E100#^^##>#3
Quatrice-tethrasecunda
E100#^^##>#4
Tethrasquarediterator
E100#^^##>#100
Tethrasquared-diterator
E100#^^##>(#+#)100 = E100#^^##>##2
Tethrasquared-triterator
E100#^^##>##3
Tethrasquared-griditerator
E100#^^##>##100
Tethrasquared-cubiculator
E100#^^##>###100
Tethrasquared-spatialator
E100#^^##>#^#100
Tethrasquared-superspatialator
E100#^^##>#^#^#100
Tethrasquared-quadratetratediterator
E100#^^##>#^^#4
Tethraturreted-tethrasquare
E100#^^##>#^^#100
Dustacu-tethraturreted-tethrasquare
E100#^^##>#^^#>#^^#100
Tristacu-tethraturreted-tethrasquare
E100#^^##>#^^#>#^^#>#^^#100 = E100#^^##>#^^##3
Dustaculated-tethrasquare
E100#^^##>#^^##100
Tristaculated-tethrasquare
E100#^^##>#^^##>#^^##100 = E100#^^###3
Tetrastaculated-tethrasquare
E100#^^###4
Dekastaculated-tethrasquare
E100#^^###10
Tethracubor / tethratertia
E100#^^###100
A tethracubor is of the order of eta-zero in the fast-growing hierarchy. It's the next of Sbiis Saibian's milestone googolisms, and its name begins the idea of naming numbers after multidimensional figures.
Troppatoth
X^^X^3 & 100
Comparable to tethracubor.
Grand tethracubor
E100#^^###100#2 = E100#^^###tethracubor
Deutero-tethracubor
E100#^^###*#^^###100
Hecato-tethracubor/tethracuborfact
E100(#^^###)^#100
Grideutertethracubor
E100(#^^###)^##100
Tethracubor-godgathored
E100(#^^###)^#^#100
Tethracubor-godtotholed
E100(#^^###)^#^#^#100
Tethracubor-isptethrathoth
E100(#^^###)^#^^#100
Tethracubor-isptethriterator
E100(#^^###)^#^^#>#100
Tethracubor-isptethrasquaror
E100(#^^###)^#^^##100
Dutetrated-tethracubor
E100(#^^###)^(#^^###)100
Tritetrated-tethracubor
E100(#^^###)^(#^^###)^(#^^###)100 = E100(#^^###)^^#3
Terrible tethracubor
E100(#^^###)^^#100
Terrible terrible tethracubor
E100((#^^###)^^#)^^#100 = E100(#^^##)^^#>#2
Tethritertethracubor
E100(#^^###)^^#>#100
Dustaculated-territethracubor
E100(#^^###)^^#>(#^^###)^^#100
Tristaculated-tethracubor
E100(#^^###)^^#>(#^^###)^^#>(#^^###)^^#100 = E100(#^^###)^^##3
Terrisquared-tethracubor
E100(#^^###)^^##100
Territerated terrisquared-tethracubor
E100((#^^###)^^##>#100
Double-terrisquared-tethracubor
E100((#^^###)^^##)^^##100
Triple-terrisquared-tethracubor
E100(((#^^###)^^##)^^##)^^##100 = E100(#^^###)^^##>#3
Terrisquarediter-tethracubor
E100(#^^###)^^##>#100
Dustaculated-terrisquared-tethracubor
E100(#^^###)^^##>(#^^###)^^##100
Tristaculated-terrisquared-tethracubor
E100(#^^###)^^##>(#^^###)^^##>(#^^###)^^## = E100(#^^###)^^###3
Tethraducubor
E100(#^^###)^^###100
Tethratricubor
E100((#^^###)^^###)^^###100
Tethratetracubor
E100(((#^^###)^^###)^^###)^^###100 = E100#^^###>#4
Tethracubiter
E100#^^###>#100
Tethracubor-diterator
E100#^^###>(#+#)100 = E100#^^###>##2
Tethracubo-gridulator
E100#^^###>##100
Dustaculated-tethracubor
E100#^^###>#^^###100
Tristaculated-tethracubor
E100#^^####3
Tethrateron
E100#^^####100
Now we reach a bit of trouble in naming the numbers. What should we call this? An obvious choice seems to be tethratesseract, but that’s too long. 4-dimensional figures are often called polychorons, so another choice would be tethrapolychoron, but that’s also too long. The name "polychoron" litterally means "many rooms" and polychorons are composed of multiple 3-dimensional figures, so a "choron" can be considered to be a 3-D figure. Then we can take the word “polyteron”, which is a 5-dimensional figure. We can extract the root “teron” from there and use the convenient name “tethrateron”. This is also convenient because the name “teron” directly implies four.
Quadroppatoth
X^^X^4 & 100
Comparable to tethrateron.
Tethrateron-by-deuteron
E100#^^####100#^^######100
Tethrateron-by-dekatonon
E100#^^####*#11
This is Sbiis Saibian's 7000th googolism, defined using Extended Cascading-E. It's approximately fφ(4,0)*11(100) using the fast-growing hierarchy, and X^^X^4*11 & 100 using the climbing interpretation of BEAF.
Deutero-tethrateron
E100#^^####*#^^####100
Tethrateronifact
E100(#^^####)^#100
Dutetrated-tethrateron
E100(#^^####)^(#^^####)100
Terrible tethrateron
E100(#^^####)^^#100
Territerated-tethrateron
E100(#^^####)^^#>#100
Dustaculated-territethrateron
E100(#^^####)^^#>(#^^####)^^#100 = E100(#^^####)^^##2
Terrisquared-territethrateron
E100(#^^####)^^##100
Terricubed-tethrateron
E100(#^^####)^^###100
Tethraduteron / territesserated-tethrateron
E100(#^^####)^^####100
Tethratriteron
E100((#^^####)^^####)^^####100
Tethriterteron
E100#^^####>#100
Dustaculated-tethrateron
E100#^^####>#^^####100
Tristaculated-tethrateron
E100#^^#####3
Tethrapeton
E100#^^#^#5 = E100#^^#5100
Similar in name to tethrateron, this is comparable to an X{6}100 array of tens. We’ll breeze through this number’s group.
Deutero-tethrapeton
E100#^^#5*#^^#5100
Tethrapetonifact
E100(#^^#5)^#100
Terrible tethrapeton
E100(#^^#5)^^#100
Terrisquared tethrapeton
E100(#^^#5)^^##100
Tethradupeton
E100(#^^#5)^^#5100
Tethriterpeton
E100#^^#5>#100
Dustaculated-tethrapeton
E100#^^#5>#^^#5100
Tethrahexon
E100#^^#^#6
Tethrahepton
E100#^^#^#7
Tethra-ogdon
E100#^^#^#8
Tethrennon
E100#^^#^#9
Tethradekon
E100#^^#^#10
Tethratope / tethrahecton
E100#^^#^#100
This is the beginning of the last regiment among Sbiis Saibian's hyperion-tetration numbers. Since a figure of any dimensions is called a polytope, we can just call this the tethratope. This is also known as a tethrahecton from Greek "hecto-" meaning 100. It's comparable to a X^^X^X array.
Grand tethratope
E100#^^#^#100#2
The Binary-Phi Range
X^^X^X ~ X^^^X arrays
(order type φ(w,0) ~ Γ0)
Entries: 72
Grangol-carta-tethratope
E100#^^#^#100#100
Gugolda-carta-tethratope
E100#^^#^#100##100
Godgahlah-carta-tethratope
E100#^^#^#100#^#100
Tethrathoth-carta-tethratope
E100#^^#^#100#^^#100
Tethratope-by-deuteron
E100#^^#^#100#^^#^#100
Tethratope-by-triton
E100#^^#^#100#^^#^#100#^^#^#100
Tethratope-by-teterton
E100#^^#^#*#5
Tethratope-by-hyperion
E100#^^#^#*#100
Tethratope-by-deuterohyperion
E100#^^#^#*##100
Tethtratope-by-godgahlah
E100#^^#^#*#^#100
Tethratope-by-godgathor
E100#^^#^#*#^#^#100
Tethratope-by-tethrathoth
E100#^^#^#*#^^#100
Tethratope-by-tethriterator
E100#^^#^#*#^^#>#100
Tethratope-by-tethrasquaror
E100#^^#^#*#^^##100
Tethratope-by-tethracubor
E100#^^#^#*#^^###100
Tethratope-by-tethrateron
E100#^^#^#*#^^####100
Deutero-tethratope
E100#^^#^#*#^^#^#100
Trito-tethratope
E100#^^#^#*#^^#^#*#^^#^#100
Tethratopofact
E100(#^^#^#)^#100
Terrible tethratope
E100(#^^#^#)^^#100
Terrisquared tethratope
E100(#^^#^#)^^##100
Tethradeutertope
E100(#^^#^#)^^#^#100
Tethratritotope
E100((#^^#^#)^^#^#)^^#^#100 = E100#^^(#^#)>#3
Tethritertope
E100#^^(#^#)>#100
Tethriditertope
E100#^^(#^#)>(#+#)100
Tethritritertope
E100#^^(#^#)>##3
Tethrigriditertope
E100#^^(#^#)>##100
Tethricubiculitertope
E100#^^(#^#)>###100
Tethraspatialitertope
E100#^^(#^#)>#^#100
Tethrathoth-turreted tethratope
E100#^^(#^#)>#^^#100
Dustaculated-tethratope
E100#^^(#^#)>#^^(#^#)100
Tristaculated-tethratope
E100#^^(#^#)>#^^(#^#)>#^^(#^#)100 = E100#^^(#^#*#)3
Tethratopothoth
E100#^^(#^#*#)100
Tethratoposquaror
E100#^^(#^#*##)100
Tethratopocubor
E100#^^(#^#*###)100
Tethratopodeus
E100#^^(#^#*#^#)100
Tethratopotruce
E100#^^(#^#*#^#*#^#)100 = E100#^^#^##3
Tethratopoquad
E100#^^#^##4
Tethralattitope
E100#^^#^##100
Tethralattitopodeus
E100#^^(#^##*#^##)100
Triakulus
3^^^3 & 3
This is the next official Bowersism after the goppatoth. This mind-crushingly large number is an X^^^3 pentational array with tritri threes, and is the first pentational array Bowers defines. It can also be written as 3&3&3. This is the first three-based number we’ve had in quite a long time. This googologism is approximately E3#^^#^###3, placing it between tethralattitopodeus and tethralattitopotruce.
This number is also notable as the first non-degenerate legion array. The first part of legion arrays is {a,b/2} = a&a&a&a....&a with b a's - {n,2/2} arrays degenerate into linear arrays, and {1,3/2} and {2,3/2} both degenerate into 1 and 4, respectively. Therefore, triakulus is the first non-degenerate legion array. Legion arrays are further explored in part 7.
Tethracubitope
E100#^^#^###100
Tethraquarticutope
E100#^^#^####100
Tethrato-godgathor
E100#^^#^#^#100
Tethrato-gralgathor
E100#^^#^#^##100
Tethrato-godtothol
E100#^^#^#^#^#100
Tethrato-tethrathoth / tethrarxitri
E100#^^#^^#100
This is also expressible as E100#^^^#3.
Blooshker bundle
E100#^^(#^^(100))^^#[100]100
This is one of Sbiis Saibian's irregular googolisms. Since #^^(100) decomposes to #^#^#^#^#^#^#....^# with 100 #s, so this is E100#^^(#^#^#^#........^#)^^#100 with 102 #s. Despite looks, this number isn’t much of an improvement over tethrarxitri. In fact, it’s less than E100#^^(#^#^#^#^#^#.......#^#^##)100 with 102 #s, let alone E100#^^#^^#101.
Grand tethrarxitri
E100#^^#^^#100#2
Blistering blooshker bundle
E100#^^(#^^(blooshker bundle))^^#[blooshker bundle]100
Or E100#^^(#^#^#^#......^#)^^#(blooshker bundle) with blooshker bundle+2 #s. Despite how amazing this sounds, it’s still less than even grand grand tethrarxitri, or E100#^^#^^#100#3.
Hectastaculated-tethrato-tethrathoth
E100#^^(#^^#*#)100
Tethrato-tethrathothosquaror
E100#^^(#^^#*##)100
Tethrato-tethrathothocubor
E100#^^(#^^#*###)100
Tethrato-deutero-tethrathoth
E100#^^(#^^#*#^^#)100
Tethrato-tethrafact
E100#^^(#^^#)^#100
Tethrato-territethrathoth
E100#^^(#^^#)^^#100
Tethrato-tethriterator
E100#^^#^^#>#100
Not to be confused with E100(#^^#^^#)>#100. which is much smaller.
Tethrato-tethrasquare
E100#^^#^^##100
Tethrato-tethratope
E100#^^#^^#^#100
Brother-Giant
E100#^^#^#^^#^#100
Remember the Monster-Giant, which is constructed as E100(#^^#)^(#^^#)^#100? Well, this is its brother, which is constructed the same as Monster-Giant, but with the parentheses removed, which means that it’s computed as E100#^^(#^(#^^(#^#)))100. This makes the number incomprehensibly larger. Brother-Giant is not significantly bigger than tethrato-tethratope, because #^#^^#^# is not significantly more than #^^#^#...at least from a googological point of view. It’s all too mind-boggling to really speak of.
Tethrato-tethralattitope
E100#^^#^^#^##100
Tethrato-tethracubitope
E100#^^#^^#^###100
Tethrarxitet
E100#^^#^^#^^#100
Super-Brother-Giant
E100#^^#^#^^#^#^^#^#100
A further extension of the Brother-Giant.
Tethrarxipent
E100#^^^#5
Tethrarxihex
E100#^^^#6
Tethrarxihept
E100#^^^#7
Tethrarxi-ogd
E100#^^^#8
Tethrarxi-enn
E100#^^^#9
Tethrarxideck
E100#^^^#10
Tethrarxicose
E100#^^^#20
Tethrarxigole
E100#^^^#50
So now that that’s that ... how about some super incomprehensible higher-level arrays? Welcome to the realm of the Mighty Royals! That class is most of the whole gap between a humongulus and a golapulus, and THEN some.