2.2. Knuth's Up-Arrows and the Hyper-Operators


Introduction

We've finished with the "basic numbers" so to speak. We've gone through various illion systems, the magnitude of various numbers, and also the googol family. We've also familiarized ourselves with the concept of recursion. So where to go now? The best start is probably Knuth's up-arrow notation, which is a very well-known notation in googology. Many googological notations are based on it like Bowers' and Bird's arrays, Conway's chain arrows, Hollom's hyperfactorials, Joyce's g function, many of Aarex's notations, and also in the definiton of Graham's number (which we'll discuss in detail a few articles later). It also provides perhaps the most intuitive way to extend upon the ordinary tools for making large numbers. Therefore I think up-arrows are a good starting point for discussing large number notations.

The Hyper-Operators

Up-arrow notation was invented by mathematician and computer scientist Donald Knuth in 1976, as a way to generalize expressing the hyper-operators. What are the hyper-operators? They extend upon the familiar sequence of addition, multiplication, and exponentiation—we looked at each of those in an earlier page.

To understand what that means, first notice that multiplication is repeated addition:

a*b = a+a+a ... +a+a
                     b

and exponentiation is repeated multiplication:

a^b = a*a*a ... *a*a
                     b

But what is a^a^a ... ^a^a?
                            b

That function (when solved right to left!) has a name: tetration. It's sometimes been denoted ba, especially before Knuth's arrows. To better understand that operator let's try a few examples. Note that here the expressions are solved right to left:

13 = 3
23 = 3^3 = 27
33 = 3^3^3 = 3^27 = 7,625,597,484,987, a number that shows up prevalently when working with threes in up-arrows.
43 = 3^3^3^3 = 3^3^27 = 3^7,625,597,484,987 — this number has about 3.68 trillion digits.
53 = 3^3^3^3^3 = 3^3^3^27 = 3^3^7,625,597,484,987 = 3^(3.68-trillion-digit number) — this number has about 43 digits!

I'm assuming you get the pattern. But how would you denote repeated tetration (called pentation)? Or repeated pentation? Or repeated repeated repeated repeated pentation?!

Introducing the Up-Arrows

Though previous functions have been made to express tetration, pentation, and beyond, Knuth's up-arrows give a very convenient way to do so.

They can be defined with only three rules (note that x and y must be whole numbers):

Rule 1: x↑y = x^y

Rule 2: xay = xa-1(xa(y-1)) if y isn't 0

Rule 3: xa0 = 1

(a means ↑...↑ with a ↑s)

From here on out, ↑ will be instead denoted with the common ASCII substitution ^, which also nicely matches with a^b for exponentiation. If you don't get all this here's a way to imagine it:

x tetrated to y = x^^y = x^x^x ... ^x^x^x with y x's
x pentated to y = x^^^y = x^^x^^x ... ^^x^^x^^x with y x's
x hexated to y = x^^^^y = x^^^x^^^x ... ^^^x^^^x^^^x with y x's
etc.

Now that we know how up-arrows are defined, let's look at the hyper-operators and numbers they produce.

Tetration

Properties and Notation of Tetration

Tetration, as we saw earlier, is the hyper-operator that comes after exponentiation. Unlike exponentiation tetration is not present in mainstream mathematics, although it still has had its behavior quite extensively studied. The name "tetration" was coined by Reuben Goodstein from tetra- (Greek prefix for four) plus iteration, meaning that it's the fourth operation in the sequence addition, multiplication, exponentiation, etc. Tetration is iterated exponentiation, so a tetrated to b would be aaa...a with b a's. Expressions of the form aaa...a are known as power towers, so a tetrated to b is a power tower of b a's.

Tetration has even more ways to denote than exponentiation. As we saw earlier, of the more common ways is ba, in analogy to ab for exponentiation. That notation is most often used when there's no need for higher hyper-operators. The most common way, however, is as a↑b, or as a^^b when typing since ^ is easier to type than ↑.

Robert Munafo uses ab to denote tetration. Why is that? He recognizes that there are actually two ways to make a hyper-operator after exponentiation. The stronger way is as a^(a^(a^(a^(...^a)...)))) with b a's, and the weaker way is as ((...(a^a)^a)^a...)^a with b a's. The weaker way (Munafo calls it hyper4is denoted ab (we'll see how it compares against the hyper-operators in the next page) and is not as mathematically interesting as the stronger way (Munafo calls it hyper4), which is is denoted ab. In this article we'll focus on the stronger hyper-operators, and in the next we'll focus on the weaker hyper-operators.

Other ways to denote tetration include:

- a***b, on analogy with a*b for multiplication and a**b for exponentiation in some programming languages. The Big Psi web-book on large numbers uses this notation.
- a{2}b in Bowers' operator notation (we'll examine it in section 3)
- Harvey Friedman has used x[y]
- a->b->2 in Conway chain arrows, and {a,b,2} in Bowers' arrays

Tetration lacks many of the symmetric properties of exponentiation. Like exponentiation it is neither commutative nor associative, but it also doesn't have many laws for simplifying expressions like exponentiation does. One law it does have (which holds for all hyper-operators) is that xx^^n = x^^(n+1), and a law that can be used to upper-bound tetration expressions (shown by Sbiis Saibian as part of his Knuth Arrow Theorem) is that (x^^a)^^b will always be comparable to and less than x^^(a+b). This actually generalizes to any hyperoperator more powerful than exponentiation, and that law (called the Knuth Arrow Theorem) is useful for upper-bounding and estimating expressions with Knuth's arrows.

Googology Wiki, on its article on tetration, mentions an interesting identity for tetration:

(x^^a)(x^^b) = (x^^(b+1))(x^^(a-1))

That identity is quite easy to see, since (x^^a)(x^^b) = (x(x^^(a-1)))(x^^b) = xx^^(a-1)*x^^b = xx^^b*x^^(a-1) = (xx^^b)x^^(a-1) = (x^^(b+1))(x^^(a-1)).

Now that we've familiarized ourselves with the properties of tetration, let's take a look at the numbers tetration produces.

Examples of Tetration

We'll start with cases with the base of two, since cases with the base of 1 always degenerate to 1. For example:

1^^3
= 1^(1^^2) (rule 2)
= 1^(1^(1^^1)) (rule 2)
= 1^(1^(1^(1^^0))) (rule 2)
= 1^(1^(1^1)) (rule 3)
= 1^(1^1) (rule 1)
= 1^1 (rule 1)
= 1 (rule 1)

All such expressions with tetration and higher operators will simply degenerate to power towers of 1, which always evaluate to 1. So let's go on to examples with the base of 2.

First we'll try 2^^1:

2^^1
= 2^(2^^0) (by rule 2)
= 2^1 (by rule 3)
= 2 (by rule 1)

Next let's try 2^^2:

2^^2
= 2^(2^^1)
= 2^2 as we saw earlier
= 4. The case of 4 is very prevalent with twos in the hyper-operators, as you will see.

2^^3:

2^^3
= 2^(2^^2)
= 2^4
= 16 — quite modest so far ...

2^^4:

2^^4
= 2^(2^^3)
= 2^16
= 65,536. This number crops up pretty often when working with twos in up-arrow notation.

2^^5:

2^^5
= 2^(2^^4)
= 2^65,536
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88451545977289516457213189779790749194101314836854463961690460703010759681893374121757598816512700
07612627891695104063158576375347874200702220510708912576123616580268068158584998526314658780866168
00733264676830206391697203064894405628195406190685242003053463156621891327309069687353181641094514
28803660599522024824888671155442910472192913424834643870536850864874909917881267056566538719104972
18200423714927401644609434598453925367061322106165330856620211889682340057526754861014769936887382
09584552211571923479686888160853631615862880150395949418529489227074410828207169303387818084936204
01825522227101098565344481720747075601924591559943107294957819787859057894005254012286751714251118
43564371840535630241812254732660933027103979680910649392727226830354104676325913552796838377050198
55234621222858410557119921731717969804339317707750755627056047831779844447637560254637033369247114
22081551997369137197516324130274871219986340454824852457011855334267526471597831073124566342980522
14554941562527240289153333543493412178620370072603152798707718724912344944771479095207347613854254
85311552773301030342476835865496093722324007154518129732692081058424090557725645803681462234493189
70813889714329983134761779967971245378231070373915147387869211918756670031932128189680332269659445
92862106074388274169194651622676325406650708810710303941788605648937698167341590259251946118236429
45652669372203155504700213598846292758012527715422016629954863130324912311029627923723899766416803
49714122652793190763632613681414551637665655983978848938173308266877990196288693229659737995193162
11872154552873941702436698855938887933167445333631195415184040882838151934212341228200309503133410
50704760159987985472529190665222479319715440331794836837373220821885773341623856441380700541913530
24594391350255453188645479625226025176292837433046510236105758351455073944333961021622967546141578
11271970017386114942795014112532806212547758105129720884652631580948066336876701473107335407177108
76615935856814098212967730759197382973441445256688770855324570888958320993823432102718224114763732
79135756861542125284965790333509315277692550584564401055219264450531207375628774499816364633283581
61403301758139673594273276904489203618803867549557518068900585329272014939235005258451467069826285
48257883267398735220457228239290207144822219885587102896991935873074277815159757620764023951243860
20203259659625021257834995771008562638611823381331850901468657706401067627861758377277289589274603
94039303372718738505369129571267150668966884938808851429436099620129667590792250822753138128498515
26902931700263136328942095797577959327635531162066753488651317323872438748063513314512644889967589
82881292548007642518658649024111112730135719718138160258317850693224400799865663537154408845486639
31817083957357807990597308390948818040609359591909074739609044101505163217496814121007657191774837
67355751000733616922386537429079457803200042337452807566153042929014495780629634138383551783599764
70885134900485697369796523869584599459559209070905895689145114141268450546211794502661175016692826
02509507707782119504326173832235624376017767993627960993689751913949650333585071554184364568526166
74243688920371037495328425927131610537834980740739158633817967658425258036737206469351248652238481
34166380806150570482905989069645193644001859712042572300731641000991698752426037736217776343062161
67448849308109299010095179745415642512048220867145868492551324442667771278637282113315362243010918
24391243380214046242223349153559516890816288487989988273630445372432174280215755777967021666317047
96972817248339284101564227450727177926939992974030807277039501358154514249404902653610582540937311
46531049433824843797186069372144446008267980024712294894057618538922034256083026970528766213773735
94394224114707074072902725461307358541745691419446487624357682397065703184168467540733466346293673
98362000404140071405427763248013274220268539369886978760700959004868465062677136307097982100655728
51013066010107806337433447730734786538817426812307437660666433127753564665786037151929227684404582
73283243808212841218776132042460464900801054731426749260826922155637405486241717031027919996942645
62095561981645454766204502241144940474934983220680719135276798674781345820385957041346617793722853
49400316315995440936840895725334387029867178297703733328068017646395020900239419314991150091052768
21119510999063166150311585582835582607179410052528583611369961303442790173811787412061288182062023
26384986151565645123004779296756361834576810504334176954306753804111392855379252924134733948105053
20257087281863072911589113359420147618726642915640363719276023062838406504254417423354645499870553
18726887926424102147363698625463747159744354943443899730051742525110877357886390946812096673428152
58591992485764048805507132981429935991146323991911395992675257635900744657281019180584180734222773
47213977232182317717169164001088261125490933611867805757223910181861685491085008852722743742120865
24852372456248697662245384819298671129452945515497030585919307198497105414181636968976131126744027
00964866754593456705993699546450055892162804797636568613331656390739570327203438917541526750091501
11988568727088481955316769316812728921430313768180164454773675183534978579242764633541624336011259
60252109501612264110346083465648235597934274056868849224458745493776752120324703803035491157544831
29527589193989368087632768543876955769488142284431199859570072752139317683783177033913042306095899
91373146845690104220951619670705064202567338734461156552761759927271518776600102389447605397895169
45708802728736225121076224091810066700883474737605156285533943565843756271241244457651663064085939
50794755092046393224520253546363444479175566172596218719927918657549085785295001284022903506151493
73101070094461510116137124237614267225417320559592027821293257259471464172249773213163818453265552
79604270541871496236585252458648933254145062642337885651464670604298564781968461593663288954299780
72254226479040061601975197500746054515006029180663827149701611098795133663377137843441619405312144
52918551801365755586676150193730296919320761200092550650815832755084993407687972523699870235679310
26804136745718956641431852679054717169962990363015545645090044802789055701968328313630718997699153
16667920895876857229060091547291963638167359667395997571032601557192023734858052112811745861006515
25988838431145118948805521291457756991465775300413847171245779650481758563950728953375397558220877
77506072339445587895905719156736, which has 19,729 digits

Wait now. Where did that even come from?!

It came from the pattern that occurs with tetration and twos: we took 2 to the 1st power = 2, then 2 to the 2nd power = 4, then 2 to the 4th power = 16, then 2 to the 16th power = 65,536, and then it's only the continuation of the sequence that 2 to the 65,536th power is this huge number.

So it shouldn't come as much of a surprise that 2^^6, or 2 to the power of 2003529930...............5719156736, is a number with about the previous number of digits, so big that you couldn't come close to being able to write it out even if you could fit as many digits as there are Planck volumes in the observable universe into every Planck volume in the observable universe! Or should it? This kind of surprise is common among new googologists. Welcome to the counterintuitive land of googology.

What about cases involving 3? As we saw earlier:

3^^1 = 3 — trivial case here.

3^^2 = 3^3 = 27 — the number 27 shows up often when plugging 3's into googological functions as we will see

3^^3 = 3^27 = 7,625,597,484,987, a number that shows up prevalently when working with threes in up-arrows and other googological functions.

3^^4 = 3^7,625,597,484,987 — this number has about 3.68 trillion digits. Even though it can still be computed, I can't store the trillions of digits on my site — besides, scrolling through its digits, even on a separate page, is a pain in the ass.

3^^5 = 3^(3.68-trillion-digit number) — this number has about 3^^4 digits!

You get the pattern from here.

Now for cases involving 4:

4^^1 = 4 — we can see that n^^1 is always equal to n. To prove this consider this:

n^^1
= n^(n^^0) [rule 2 of up-arrow notation]
= n^1 [rule 3]
= n

That said, we can skip cases of the form n^^1 now. So we continue:

4^^2 = 256 — nothing too scary yet

4^^3 = 4^256 = 13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,030,073,546,9
76,801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,649,006,084,096 — that has 155 digits but pales in comparison to:

4^^4 = 4^(155-digit number) ~ 10^(8.02*10^153)

That's a pretty cool number that can be named megafuga-four or booga-four. It's already bigger than a googolplex. We'll learn more about names like megafuga-four and booga-four in not too long.

We can continue with 4^^5 (4 to the power of 4^^4), 4^^6 (4 to the power of 4^^5), etc. 

Now let's look at base 5:

5^^2 = 3125 — nothing too big

5^^3 = 5^3125 = 19110125979454775203564045597039645991980810489900943371395127892465205302426158030120593865197398
50265586440155794462235359212788673806972288410146915986602087961896757195701839281660338047611225
97553362610100148265112341314776825241149309444717696528275628519673751439535754247909321920664188
30117871691225524210700507090646743828708514499502565861944615431835113798491336917799281274338404
31549236855526783596374102105331546031353725325748636909159778690328266459182983815230286936572873
69142264813129174376213632573032164528297948686257624536221801767322494056764281936007872071383707
23553054463561539464011853484937927195145945055082327492216058489129109451899599486861995431476669
38013037176163592594479746164220050885079469804487133205133160739134230540198872570038329801246050
19701346739717590902738949392381731578699684589979478106804282243609378394633526542281570430283244
23855150823164909672857121717081232327904818172683275101127467823174109858886837085220007117334922
53913322300756147180429007527677793352306200618286012455254243061006894805446584704820650982664319
36096038873625851074707434063628697657670269925864995355797631817390255089133122329474393034395616
13283340728316634982581452268620043077990846881038041873683248009038735962129196336025831207816736
73742533322879296907205490595621406888825991244581842379597863476484315673760923625090371511798941
42426227022006628648686786871018298087280256069310194928083082504419842479679205890881711232719230
14555829167467951974305480264046468540027339938607985944659615017525869658114475685100415686877309
03712482535343839285397598749458497050038225012489284001826590056251286187629938044407340142347062
05578530532503491818958970719930566218851296318750174353596028220103821161604854512103931331225633
22607664362366882968502088394961428304847391139916696226499485636852347128732947966808845094058939
51104650944137909502276545653133018670633521323028460519434381399810561400652595300731790772711065
78349417464268472095613464732774858423827489966875505250439421823219135722305406671537337424854364
56637820457016545932181540535483936142506644985854033074664685418901481343477146503150379541757786
22811776585876941680908203125

Pretty nice number. It has exactly 2185 digits. We can have 5^^4, 5^^5, etc. Now for base 6:

6^^2 = 46,656

6^^3 =  a 36,306 digit number (2659119772.........7863878656). This time I'll be nice and not make you have to scroll through the tens of thousands of digits, but if you want to see them click here.

we can have 6^^4, 6^^5, 6^^6, etc., but now for base 7:

7^^2 = 823,543

7^^3 = a number with 695,975 digits (all the digits)

and base 8:

8^^2 = 16,777,216

8^^3 = a number with over 15 million digits (all the digits)

and base 9:

9^^2 = 387,420,489

9^^3 = a number with 369 million digits (all the digits) — this number is mentioned in the novel Ulysses by James Joyce. For more about that look here.

and finally base 10:

10^^2 = 10,000,000,000 or ten billion. Sbiis Saibian gives this number a name, dialogue.

10^^3 = a number Sbiis Saibian calls a trialogue, equal to 1 followed by ten billion zeros. The number appears in a School House Rock song My Hero, Zero, and I have a page about the number here (as a subpage of part 2 of my number list) which discusses the sequence 1^1^1 to 10^10^10 and the number's appearance in School House Rock.

10^^4 = 1 followed by "1 followed by 10 billion zeros" zeros, a number Sbiis Saibian calls a tetralogue.

10^^5 = 1 followed by "1 followed by "1 followed by 10 billion zeros" zeros" zeros, a number Sbiis Saibian calls a pentalogue. Once again I'm assuming you get the pattern. Just continue with the Greek prefixes for numbers (same ones used to name polygons), and remember to use ennea- for nine and not nona-. :)

And now for two impressive power towers of tens:

10^^10 = 10^10^10^10^10^10^10^10^10^10 (a power tower of 10 tens) is a number Jonathan Bowers calls decker, one of my own favorite power tower numbers. Sbiis Saibian gives this number an alternate name, dekalogue.

10^^100 (a power tower of 100 tens) is a number Jonathan Bowers calls giggol similarly to how 10^100 = googol. Similarly he defines giggolplex to be 10^^10^^100 (remember to solve right to left), a power tower of a giggol tens.

I think you get the idea. As powerful an operator as tetration is, when looking at pentational numbers the tetrational numbers will utterly vanish in the dust!

Pentation

Properties and Notation of Pentation

We've seen and are now familiar with the power of tetration. But what about repeated tetration, known as pentation? Let's first learn about the history of pentation.

Pentation is less known than its little brother tetration, and as we will see it produces far larger values than tetration does. The name was coined by Goodstein from penta- (meaning five) + iteration, indicating that it's the fifth hyper-operator. It does not have a popular notation like ba for tetration, though in 2004 Mark Cutler proposed a for a pentated to b (a bar above the b). Sbiis Saibian instead proposes b<--a for pentation, and he uses that with the hyper-operators when writing them by hand. The most common notation for pentation is by far a^^^b (a pentated to b), though it can also be notated a{3}b in Bowers' operator notation, a->b->3 in Conway chain arrows, and {a,b,3} in Bowers' arrays. Sunir Shah continued ab for exponents and ba for tetration with a(*)b for pentation.

Robert Munafo uses ab for pentation (the hyper5 operator, a(a(a(...aa)...))), and additionally ab for the weaker hyper5 operator (a.k.a. weak pentation), equal to (...(((aa)a)a...)a with b a's. We'll examine this operator, and see how it compares to the normal hyper-operators in the next article. As for the hyper5 operator, a^^^b is a^^a^^a^^a.....^^a with b a's (remember that the expressions are always solved right to left. In terms of power towers this can be visualized as:

aaa...a }  aaa...a } aaa...a ... ... ...  } aaa...a } a with b levels, including a

where the number of a's in each power tower is represented by the number to the right. This means that with each step in pentation we're feeding in the previous step into the height of the power tower. Tim Urban, on his blog post on large numbers titled "From 1,000,000 to Graham's Number", calls this "a power tower feeding frenzy".

Pentation has a few properties similar to those of tetration that can be used to simplify or estimate expressions: a^^(a^^^b) = a^^^(b+1), and (a^^^b)^^^c ~< a^^^(b+c) (as a consequence of the Knuth Arrow Theorem). But other than that, it has even fewer nice properties than tetration has.

As you can see pentation already looks scary, and we haven't examined any of the numbers it produces yet! So now it's time to look at the power of pentation.

Examples of Pentation

What horrifying numbers will pentation bring? Let's jump right in:

2^^^1 = 2^^(2^^0) = 2^^1 = 2 — once again a trivial case. We can skip cases of the form n^^^1 from here on out, as it's clear that n^^^1 will always evaluate to n (in fact this is true for all the hyper-operators).

Now let's try 2^^^2:

2^^^2
= 2^^(2^^^1)
= 2^^2
= 4 — trivial again

2^^^3 now:

2^^^3 = 2^^2^^2 = 2^^4 = 2^2^2^2 = 2^2^4 = 2^16 = 65,536 — we encounter 65,536 again. That's a good jump, but not too bad.

What about 2^^^4?

2^^^4 = 2^^2^^2^^2 = 2^^2^^4 = 2^^65,536

= 2^2^2^2^2^2^2^2^.......................^2^2^2^2^2 with 65,536 2's — YIKES!!! This is so much bigger than ANYTHING previously seen with the only exception being the giggolplex!

Once again, where did that even come from?!? Once again, that jump seemed completely out of nowhere ... until you observe the pattern:

2^^^1 is 2 — meh
2^^^2 is power tower of 2 2's = 4 — meh
2^^^3 is a power tower of 4 2's = 65,536 — hmm
2^^^4 is a power tower of 65,536 2's = 2^2^2^2^2^2^2^2^.......................^2^2^2^2^2 with 65,536 2's — AAAAAAHHHHHHHHHHHHHH!!!!!!!!

Once again, it's only natural now that  2^^^4, a power tower of 65,536 2's, is so utterly unfathomable that it's waaaaaaaay beyond what we can appreciate.

The number 2^^^4, the power tower of 65,536 twos, is a number that never fails to impress me. Even as someone experienced in googology, I find that 2^^^4 is a great example of how horrifying these very large numbers are, and how erratic and jumpy the growth rates appear to be. Once again, welcome to the world of googology.

What about 2^^^5?! Continuing the sequence, this number is a power tower of 2^^^4 twos—in other words, a power tower of "a power tower of 65,536 twos" twos! This time, it's not only impossible to appreciate the number, but it's also impossible to even appreciate the power tower used to represent this number!

Continuing, 2^^^6 is a power tower of 2^^^5 twos, 2^^^7 is a power tower of 2^^^6 twos, etc. I think you get the pattern: pentating a number iterates the height of a power tower, feeding in the previous number into the height of the power tower of the next number.

Now let's try cases involving 3's:

3^^^2 = 3^^(3^^^1) = 3^^3 =  3^3^3 = 3^27 = 7,625,597,484,987 — we encounter this important power of three again.

3^^^3 = 3^^(3^^^2) = 3^^7,625,597,484,987 = a power tower of 7.6 trillion 3's! Jonathan Bowers calls this number tritri.

Now 3^^^3 is a super-awesome number. It's commonly used when giving an example of an unfathomably large number, and it's my own favorite example of a number that's waaaay beyond what we can appreciate. You can imagine it like so:

3^3^3^3^3^3..................^3
             7.6 trillion 3's

So how would you imagine this number? Imagine a power tower of those 7.6 trillion threes stretching all the way from Earth to the sun, with each three about 2 centimeters tall. Then, only the top five threes, only a few inches from the Sun, already make a number insanely bigger than a googolplex! Imagine the insanity with only the top meter of the tower, and that's less than a billionth of the way down the tower—with that in mind, you can hardly imagine how huge solving the entire tower down to the Earth would be!!!

What about 3^^^4? That number, by the sequence, is a power tower of 3^^^3 3's, aka a power tower of "a power tower of 7 trillion 3's" 3's. It can be imagined like so:

3^3^3^3^3^3..................^3
3^3^3^3^3^3..................^3 3's
               7.6 trillion 3's

This time, just like with 2^^^5, the power tower is itself impossible to appreciate now!

Likewise, 3^^^5, a power tower of "a power tower of "a power tower of 7 trillion 3's" 3's" 3's, can be imagined like so:

3^3^3^3^3^3..................^3
3^3^3^3^3^3..................^3 3's
3^3^3^3^3^3..................^3 3's
               7.6 trillion 3's

You get the pattern from here. How about cases with fours?

4^^^2 = 4^^4 = megafugafour ~ 10^(8*10^153).

But 4^^^3 = 4^^4^^4 = a power tower of 10^10^153 fours. It can be imagined like so:

4^4^4^4^4^4..................^4
              10^10^153 4's

That's a power tower, of more than a googolplex fours! By now we've lost all hope for comprehending such gigantic values.

Likewise, 4^^^4 is a power tower of 4^^^3 fours, which can be imagined as:

4^4^4^4^4^4..................^4
4^4^4^4^4^4..................^4 4's
              10^10^153 4's

etc.

Now let's try base 5:

5^^^2 = 5^^5 ~ 10^10^10^2184. That value is bigger than a googolplex! It's actually impossible to even appreciate the gulf between this value and a googolduplex, as you would need to multiply a googolduplex by itself 10^10^2184 times to reach this number, and we have no hopes of comprehending 10^10^2184 as a value! But that's a kind of number we've seen when working with tetration!

5^^^3, on the other hand, is a power tower of 5^^5 5's, imaginable as:

5^5^5^5^5^5..................^5
         10^10^10^2184 5's

That's a power tower of more than a googolduplex fives!

Now let's skip to 5^^^5, named "boogafive" with the booga- prefix (we'll examine it a few articles from now). It is imaginable as:

5^5^5^5^5^5..................^5
5^5^5^5^5^5..................^5 5's
5^5^5^5^5^5..................^5 5's
         10^10^10^2184 5's

Note that this far far larger than boogafour, which is merely about 10^10^153.

I think you get the pattern from here on out. Let's skip to base 10:

10^^^2 = 10^^10, the decker which we saw earlier

10^^^3 = 10^^10^^10 = a power tower of a decker tens. Sbiis Saibian calls this number tria-taxis.

10^^^4 is a power tower of 10^^^3 tens. Sbiis Saibian calls this tetra-taxis; the names for higher numbers of the form 10^^^n continue with the Greek prefixes just like with the -logue numbers.

And now for two final numbers:

10^^^10 is a number Saibian calls deka-taxis. It can be imagined as:


And finally, 10^^^100 is a number Bowers calls gaggol, which can be imagined as:


That's an awesome number alright.

I TOLD you pentation will make tetration look adorable! But things get really crazy with the NEXT next operator, hexation.

Hexation

Properties and Notation of Hexation

Often, in an introduction to hyper-operators people will go no further in describing them than the sixth hyper-operator, known as hexation from hexa- (Greek prefix for six) + iteration. It is the largest hyper-operator Wikipedia mentions at all as of January 2016. Tetration has a long article on Wikipedia, pentation has a short article on Wikipedia, and hexation is mentioned in the articles on Knuth arrows and the hyper-operators. However, it does not mention any higher operators. Hexation is also the most powerful hyper-operator whose Googology Wiki article has examples of the operation.

Hexation is, of course, most often noted a^^^^b with Knuth's up-arrows. It's also denoted a{4}b with Bowers' operators, a->b->4 with Conway chain arrows, and {a,b,4} with Bowers' arrays. Sbiis Saibian proposes a-->b for hexation similar to b<--a for pentation. Sunir Shah uses a(**)b, and in Cutler's bar notation a hexated to b ia. Robert Munafo uses ab for hexation (the hyper6 operator) and ab for the weaker hyper6 operator (equal to (((....(aa)a)a).......)))) with b a's).

a hexated to b is equal to iterating the number of power-towers in pentation, aka a^^^a^^^a^^^a ... ^^^a with b a's. In terms of power-towers it can be visualized like so:

aaa...a}  aaa...a } aaa...a ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...  } aaa...a } a

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aaa...a}  aaa...a } aaa...a ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...  } aaa...a } a

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aaa...a}  aaa...a } aaa...a ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...  } aaa...a } a

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aaa...a}  aaa...a } aaa...a ... ... ... ... ... ...  } aaa...a } a

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a

where there are b layers (including a), and the number of power towers in each layer is the leftmost tower in the previous layer. Tim Urban calls this a "power tower feeding frenzy psycho festival".

Notice the insane progression from each hyper-operator. We have the mundane addition and multiplication, then the more powerful exponentiation, and then the staggering power-tower tetration, then the crazy power-tower-iteration pentation, and now we have the utterly insane power-tower-super-iteration hexation. Now let's look at the numbers hexation produces.

Examples of Hexation

We now know the basics of hexation. Now it's time to find out how large the numbers hexation produces are. Let's go to some examples:

2^^^^1 = 2^^^(2^^^^0) = 2^^^1 = 2. We can skip cases of the form n^^^^1, as they clearly all evaluate to n.

2^^^^2 is, once again, 4

2^^^^3 is 2^^^2^^^2 = 2^^^4 = the power tower of 65,536 twos. A mind-boggling number, but one we've seen before.

2^^^^4, however, is equal to 2^^^2^^^^3 = 2^^^(2^^65,536) = :

2^2^2^2^2..................^2
2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
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2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
              65,536 2's

with 2^^65,536 minus 2 layers

That seems like a HUGE jump yet again, and it is indeed one, but now needless to say, it's all part of the pattern. Nonetheless this is a truly incredible number. Even the number of layers used in this visualization above is an unfathomable number!

How about 2^^^^5? That's:

2^2^2^2^2..................^2
2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
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2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
              65,536 2's

with 2^^^^4 minus 2 layers

Notice that we're feeding the numbers into the number of layers now!

Let's move on to cases with 3's:

3^^^^2 = 3^^^3 = a power tower of 7.6 trillion 3's. Another cool number that we've seen before. The next number, however, is MUCH bigger:

3^^^^3 is equal to 3^^^(power tower of 7.6 trillion 3's). It's the first term in the sequence used to define Graham's number. That's not just a power tower of 3^^^3 threes—it's so much bigger than that. 3^^^^3 can be visualized as:


That's one of my own favorite large numbers. It's a huge number whose computation you can almost be able to imagine by iterating the height of power towers of 3's (starting with 3) 3^^^3 times. When you look at 3^3, 3^^3, 3^^^3, and 3^^^^3 as a family of large numbers, each has its own flavor. Sbiis Saibian describes this progression of numbers like so:

3^3 (palpable), 3^^3 (astonomical), 3^^^3 (unfathomable), 3^^^^3 (more unfathomable still?!?!)

While 3^^^3 is unfathomable, 3^^^^3 is a new kind of unfathomable, which we can call, for lack of a better term, "super unfathomable".

Likewise 3^^^^4 is like 3^^^^3, but with 3^^^^3 layers of power towers instead of just 3^^^3 layers. Unlike 3^^^^3, 3^^^^4 is defnitely past numbers whose computation you can really "see".

Of course, we can continue with cases involving fours. Let's just cut to the chase and describe 4^^^^4, a number Jonathan Bowers calls tritet. Here's a visualization of tritet:


The extra layer of iterated power towers is a big reason why this number is so much harder to understand than 3^^^^3, and also because the number in the righmost third of the picture, 4^^^4, is way harder to understand then 3^^^3. But this number pales in comparison to something like 10^^^^100, a number Bowers calls geegol to continue the googol, giggol, gaggol idea:


Are you dizzy yet?

Heptation, Octation, Enneation ...

After hexation comes heptation, octation, enneation, etc. Each name just uses the Greek prefix for what level of operation it is plus -ation, so heptation is iterated hexation, octation is iterated heptation, and so on.

I think that for heptation we can just give one example. 5^^^^^5 (five heptated to five) is a number Bowers calls tripent, which can be imagined like so:


Ok this is fucking huge.

If you want to imagine octation, just imagine iterating the number of large horizontal layers in the image above. It's OK if that's hard for you to wrap your mind around. Then you can imagine enneation as iterating octation, decation as iterating enneation, etc., etc., etc., etc.

If these were huge, imagine how horrifying a tridecal, 10^^^^^^^^^^10 (that's ten ^s) would be!!! Visualizing a tridecal like the picture above would be difficult, since at that point the numbers exponentiated would require zooming in a lot to see. Or even worse, how about 10^^^^^^^^^^^^^^^^^^^^^^^^^^^10 (that's 27 ^s)?

Or 10^^^^^^... ... (100 ^s) ... ...^^^^^^10, which Bowers calls boogol?

Or 10^^^^^^... ... ... (7,625,597,484,987 ^s) ... ... ... ^^^^^^10?

Or 10^^^^^^... ... ... (3^^^3 ^s) ... ... ... ^^^^^^10?

Or 10^^^^^^... ... ... (5^^^^^5 ^s) ... ... ... ^^^^^^10?!

Or 10^^^^^^... ... ... (10^^^...(100 ^s)...^^^10 ^s) ... ... ... ^^^^^^10 which Bowers calls boogolplex?!?!

Before we get lost imagining iterating up-arrows, let's just say that there are in fact ways to express iterating these up-arrows in a whole new mind-blowing way! The three main ways are Conway's chain arrows, Bird's and Bowers' arrays (which are closely related), and Aarex's extended up-arrows. We'll get to those later, but for now I think these huge up-arrow numbers are satisfying to imagine. Let's wrap up with the sequence of Ackermann numbers, which is based on up-arrow notation.

Review: The Ackermann Numbers

I think a good way to wrap up this article is with the sequence of the Ackermann numbers. The Ackermann numbers are best defined
 with up-arrow notation, and they were described by Conway and Guy in their Book of Numbers. Sources vary on the origin (and definition) of Ackermann numbers, but my best bet is that they were either defined somewhere around 1928—the year the Ackermann function was first defined—or the name was coined by Conway and Guy.

They are defined like so:

the nth Ackermann number = nnn

NOTE: Some sources say that the Ackermann numbers are defined as n n-ated to n, which creates the sequence 1+1, 2*2, 3^3, 4^^4, 5^^^5, etc. However this is errouneous as far as I am aware.

Let's start with visual representations of the first five Ackermann numbers:

First Ackermann number (1^1):


Second Ackermann number (2^^2):


Third Ackermann number (3^^^3):


3^^^3 seems humble now, doesn't it?

Fourth Ackermann number (4^^^^4):


Fifth Ackermann number (5^^^^^5):


... and here's a bonus one ...

Sixth Ackermann number (6^^^^^^6)


(click on the picture above to zoom in)

Notice the insane progression here. Just imagine how big the 7th Ackermann number which Bowers calls trisept is.

Conclusion

Up-arrows are a notation that extends upon the familiar sequence of addition, multiplication, and exponentiation, and quickly becomes scary. I think it makes a good first notation to discuss because it is a building block in many large number notations. Up next, I will go into more detail into the weak hyper-operators, an alternate version of the hyper-operators.

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