alright guys, here’s another old article from my large number site. it’s about andre joyce’s googo- and googolple- system, and it was in fact the very first article on my site. for a while it lurked around as an unfinished 3-part page, but i have now finished a new article on that system. for history’s sake i will archive the old unfinished googo- and googolple- article on this blog.

**PART 1**

Googologist Andre Joyce has made some odd observations about the googol and googolplex, and devised a googo- and googolple- number system. This page will showcase the two.

__Googo-__

**googoi**(goo-goy) = 2^1 = 2, which is trivial even in everyday life.

**googoij**(goo-goyj), or 4^2 = 16. In day to day life it’s a sizable number (for example, having 16 kids is outrageous), but easy for us to comprehend, and trivial on the larger scale.

**googoiji**(goo-goy-jee). Googoiji evaluates to 6^3 = 216, which is sizable in the everyday world and generally impressive, but still palpable to the mind.

**googoiv**” (goo-goyv). It evaluates to 8^4 = 4,096, around the limits of what our minds can visualize.

**googov**(goo-gawv). It’s equal to 10^5, or exactly 100,000 – called lakh in the Indian number system.

**googovi**(goo-gaw-vye) = 12^6 = 2,985,984, or about three million. It’s approximately the population of Missisippi.

**googovij**(goo-gawv-ij) = 14^7 = 105,413,504, a bit less than the population of Mexico.

**googoviji**(goo-guh-vee-jee) = 16^8 = 4,294,967,296, around the population of Asia

**googoix**(goo-goyx) = 18^9 = 189,359,290,368. Comparable to the number of stars in the Milky Way Galaxy.

**googox**(goo-gawx) = 10,240,000,000,000, or 10 trillion. Comparable to the number of human cells in your body, and slightly more than the googologically important megafuga-three (7,625,597,484,987).

**googoxi**(goo-gawx-ee) = 584,318,301,411,328, or 584 trillion. This many miles is around 10 light years, or the distance to the star Sirius.

**googoxij**(goo-gawx-ij) = 36,520,347,436,056,576, or 36 quadrillion. This many miles is around 6,000 light years, or the distance to the Perseus Spiral Arm in the Milky Way Galaxy.

**googoxiji**(goo-gawx-ee-jee) = 2,481,152,873,203,736,576, or 2.5 quintillion. This many miles is around 425,000 light years, or four times the diameter of the Milky Way.

**googoxiv**(goo-gawx-iv) = 182,059,119,829,942,534,144, or 182 quintillion. This many miles is around 31 million light years, or the distance to the Sombrero Galaxy.

**googoxvy**“, therefore, is pronounced “googox-vee”. It’s approximately 1.435*10^22, or 14 sextillion. At this scale it’s harder to come up with good and precise examples for the magnitude of these numbers.

**googoxvi**(goo-gawx-vye) ~ 1.209*10^24, or 1.2 septillion.

**googoxvij**(goo-gawx-vij) ~ 1.084*10^26, or 108 septillion.

**googoxviji**(goo-gawx-vee-jee) ~ 1.031*10^28. This number is bigger than the diameter of the observable universe in meters!

**googoxix**(goo-gawx-ix) ~ 1.037*10^30.

**googoxex**(Joyce allows us to use “ex” instead of “x” to make it pronounceable) ~ 1.100*10^32.

**googoxexi**,

**googoxexij**,

**googoxexiji**,

**googoxexiv**,

**googoxexvy**,

**googoxexvi**,

**googoxexvij**,

**googoxexviji**, and

**googoxexix**, but by now we can go at a faster rate and take jumps of tens.

**googoxexex**~ 2.210*10^53. That’s roughly the amount of atoms in Jupiter.

**googoxel**~ 1.329*10^76.

**Googoxeliji**is the first googo- number more than the number of atoms in the observable universe, but still less that the number of small volumes in large spaces (like Planck volumes in the universe), and much less than many combinatorical numbers such as the odds of a monkey typing out Hamlet on its first try.

**googol**= 10^100. It’s often regarded as impractically large, but that isn’t true – it’s still less than, say, the number of Planck volumes in the universe. It’s also not too much less than (for instance) the number of ways to arrange a 6x6x6 Rubik’s cube (that’s around 10^116), so if you happen to get your hands on one, you have a real example of a number bigger than a googol RIGHT ON YOUR FINGERTIPS!

**googolex**, a number Joyce writes on his website. It’s around 10^124.

**googolexex**~ 10^150. That’s almost as much as 4^^3 = 4^4^4 ~ 10^153.

**googolexexex**~ 10^176.

**googoxcy**, since “cy” (as in Nancy) is offered as an alternative to c. In pronunciation, it could be confused with googoxi, so I’ll make it pronounced “goo-gawks-kee” instead. This number is around 10^203, or slightly more than a gargoogol (googol^2 = 10^200). In reality it’s a thousand times larger, but you get the point. It’s more than the number of Planck volumes in the observable universe!

**googoc**(goo-gawk) ~ 10^230. This one seems to be brought up particularly often, probably due to its simple name. It’s equal to bingol (2^100) times gargoogol (10^200).

**googoci**(goo-gawk-ee) ~ 10^232

**googocex**(goo-goh-sex) ~ 10^258

**googocel**(goo-goh-sel) ~ 10^372

**googocc**(goo-gawch) ~ 10^520

**googocci**(goo-gawch-ee). Its name is reminiscent of Italian, and it’s about 10^523.

**googoccc**. That’s pronounced goo-goshk. It’s about 10^833.

**googocdy**(goo-gawk-dee) ~ 10^1161. That’s more than Joyce’s great googol (10^1000)! Joyce offers “dy” (as in lady) as an alternative to “d”.

**googod**. That’s exactly 10^1500, and it has an awesome name, doesn’t it?

**googodcy**(goo-god-see) ~ 10^1847

**googodecelexvi**(goo-goh-dek-uh-lex-vye) ~ 10^2081.

**googodccy**(goo-god-chee) ~ 10^2202

**googodcccy**(goo-god-shkee) ~ 10^2563

**googocem**(goo-goh-sem) ~ 10^2930

**googom**~ 10^3,301

**googomcccexexexvij**(goo-gawm-shkex-ex-ex-vij) ~ 10^4582.

**googomem**~ 10^7204

**googomemem**~ 10^11,334

__Googolple-__

**googolplei**(goo-guhl-play, not to be confused with Google Play) = 1^1^2 = 1^1 = 1. Heh, pathetically trivial.

**googolpleij**(goo-guhl-playj) = 2^2^4 = 2^16 = 65,536. A good jump from before, but not quite … scary, to say the least. It’s between googoiv and googov, closer to googov.

**googolpleiji**[goo-guhl-play-jee]) is much larger. It’s equal to 3^3^9 = 3^19,683 =

**googolpleiv**(goo-guhl-playv) = 4^4^16 = 4^4,294,967,296.

**googolplev**, which is equal to 5^5^25, which is around 5^(2.98*10^17). In terms of 10 it’s around 10^(2.08*10^17), or around E17#2 (that’s 10^10^17) in Hyper-E Notation. In Hyper-E, Ea#b = 10^10^10…..^10^b, with a 10s. The number is around 200 quadrillion digits long. It’s possible to compute both the leading and ending digits easily, but let’s do that later. Storing this number is possible on a supercomputer with around an exabyte (exabyte is 1 million terabytes) of space or around a million terabyte files, but it’s hard and time-consuming to do so.

**Googolplevi**is next in line – it’s equal to 6^6^36 ~ 10^(8.01*10^27) ~ E28#2 – that’s 8 octillion digits long! It can’t be stored on all the data in the world (somewhere in the zettabytes, a zettabyte is 10^21 (sextillion) bytes or a billion terabytes), but it probably will be possible in the future … but it’s safe to say that it’s quite a while before we are able write this monster in full. It would take up over a brontabyte of space to store in full.

**googolplevij**, equal to 7^7^49 ~ 10^(2.7*10^41) ~ E41#2. Writing this number is near impossible, but it’s hard to say whether it’s “possible” or “impossible”. It will require technology over a QUINTILLION (that’s a million million million, or a billion billion) TIMES more powerful than what we have today (keep in mind that a quintillion is HUGE), and let’s just say that.

**googolpleviji**= 8^8^64 ~ 10^10^58 = E58#2. The number of digits is a billion times the number of atoms on Earth!!! It’s impossible to write out, as we’ll need a billion Earths at LEAST, but definitely more because most atoms on planets are in the inaccessible core. The number of atoms on Earth can be considered a limit of “writable”, but it’s still kind of arbitrary.

**googolpleix**(goo-guhl-playx) = 9^9^81 ~ 10^10^78 = E78#2 – the number of digits is approximately the number of atoms in the observable universe, and a hard limit of “writable” – even numbers at this scale are guaranteed to be unwritable.

**googolplex**= 10^10^100. It’s a well-established statement that it’s impossible to write a googolplex, but unlike the notion that “there isn’t a googol of anything”, this one is true. But it can still represent real-world things – for instance Sbiis Saibian’s promaxima is around 10^10^343, and it’s an estimate of the number of possible histories of the universe, start to finish, assuming Planck units aren’t meaningful.

**Googolplexi**is next in line. It’s equal to 11^11^121 ~ E126#2.

**googolplexvy**(goo-guhl-plex-vee) = 15^15^225 ~ E265#2 – the number of digits this has is more than the number of PLANCK VOLUMES in the observable universe!

*googolgong*(10^100,000) digits!!!

__Conclusion__

**PART 2**

Remember the googo- system? Well, Joyce decided to continue it with a crazy scheme. Here it is:

A | a |

0 | o (pronunciation depends on context) |

1 | oo (as in food) |

2 | ee (as in feet) |

3 | or (as in born) |

4 | ie (i as in fine) |

5 | i (as in kid) |

6 | e (as in leg) |

7 | ei (a as in game) |

(the pronunciations were by me)

__1. Examples with “o” as vowels__

**gogoi**(goh-goy). First we determine A, B, C, and D:

**gogoij**(goh-goyj):

**gogoiji**:

**2x**

**2**(that’s two times the square of x)

x | number name | value |

4 | gogoiv | 32 |

5 | gogov | 50 |

6 | gogovi | 72 |

7 | gogovij | 98 |

8 | gogoviji | 128 |

9 | gogoix | 162 |

10 | gogox | 200 |

11 | gogoxi | 242 |

… | ||

15 | gogoxvy | 450 |

20 | gogoxex | 800 |

30 | gogoxexex | 1,800 |

40 | gogoxel | 3,200 |

50 | gogol* | 5,000 |

51 | gogoli | 5,202 |

52 | gogolij | 5,409 |

… | ||

60 | gogolex | 7,200 |

70 | gogolexex | 9,800 |

80 | gogolexexex | 12,800 |

90 | gogoxcy (goh-gox-kee) | 16,200 |

100 | gogoc | 20,000 |

Hmmm. These first 100 aren’t that impressive, even by everyday standards. The gogo- function only has modest quadratic growth – it’s literally a quadratic function. But let’s continue.

*Not to be confused with googol, or Jonathan Bowers’ goggol (that’s 10{6}100). Though that could be a continuation of the giggol, gaggol, geegol, gigol, goggol, gagol series (giggol, gaggol, geegol, etc are 10{x}100 where x is 2, 3, 4, etc, so giggol is 10{2}100 = 10^^100, gaggol is 10^^^100, geegol is 10^^^^100, etc). None of these numbers are to be confused with any of my googo- numbers (like googoxex vs gogoxex).

101 | gogoci | 20,402 |

102 | gogocij | 20,808 |

110 | gogocex | 24,200 |

120 | gogocexex | 28,800 |

150 | gogocel | 51,200 |

200 | gogocc (goh-goch) | 80,000 |

300 | gogoccc (goh-goshk) | 180,000 |

400 | gogocdy | 320,000 |

500 | gogod | 500,000 |

600 | gogodec | 720,000 |

700 | gogodecc (goh-goh-dech) | 1,008,200 |

800 | gogodeccc (goh-goh-deshk) | 1,280,000 |

900 | gogocem | 1,620,000 |

1,000 | gogom | 2,000,000 |

1,100 | gogomec | 2,420,000 |

1,500 | gogomed | 4,500,000 |

2,000 | gogomem | 8,000,000 |

3,000 | gogomemem | 18,000,000 |

**goggoi**(gog-oy). First we find A, B, C, and D:

**3x^2**(that’s three times the square of x),where x is written in Roman numerals (same variations allowed as previous).

x | number name | value |

3 | goggoiji | 27 |

4 | goggoiv | 48 |

5 | goggov | 75 |

10 | goggox | 300 |

20 | goggoxex | 1,200 |

50 | goggol* | 7,500 |

100 | goggoc | 30,000 |

200 | goggocc | 120,000 |

500 | goggod | 750,000 |

1,000 | goggom | 3,000,000 |

2,000 | goggomem | 12,000,000 |

3,000 | goggomemem | 27,000,000 |

*Goggol is also the name of a Bowersism equal to 10^^^^^^100 (that’s 6 ^s), which is FAR FAR FAR UNFATHOMABLY larger. I shall call this “goggol” the Joycian Goggol, as it’s created with a Joycian system. But this number is so small that it’s full decimal expansion (7,500) is easy to remember, so why bother naming it?

__2. Examples with “oo” as vowels__

**googoi**

x | number name | value |

2 | googoij | 16 |

3 | googoiji | 216 |

4 | googoiv | 4,096 |

5 | googov | 100,000 |

6 | googovi | 2,985,984 |

7 | googovij | 105,413,504 |

Wow! The seventh member of the sequence already transcends ANY number we’ve covered so far. Let’s continue:

8 | googoviji | 4.3E9 |

9 | googoix | 1.9E11 |

10 | googox | 1.0E13 |

11 | googoxi | 5.8E14 |

12 | googoxij | 3.7E16 |

13 | googoxiji | 2.5E18 |

14 | googoxiv | 1.8E20 |

15 | googoxvy | 1.4E22 |

… | ||

20 | googoxex | 1.1E32 |

30 | googoxexex | 2.2E53 |

40 | googoxel | 1.3E76 |

50 | googol | 1E100 (exactly equal) |

51 | googoli | 3E102 |

60 | googolex (1) |
E124 |

70 | googolexex | E150 |

80 | googolexexex | E176 |

90 | googoxcy | E203 |

100 | googoc |
E230 |

101 | googoci |
E232 |

110 | googocex | E258 |

150 | googocel | E372 |

200 | googocc | E520 |

201 | googocci |
E523 |

299 | googoccexecix (googoccic) (2) |
E830 |

300 | googoccc | E833 |

400 | googocdy | E1161 |

500 | googod | E1500 (exactly equal) |

600 | googodec | E1847 |

700 | googodecc | E2202 |

800 | googodeccc | E2563 |

900 | googocem | E2930 |

990 | googocemexec (googoxem) (2) |
E3264 |

1000 | googom | E3301 |

1003 | googomiji (googolmox) (3) |
E3312 |

2000 | googomem (googomump) (4) |
E7204 |

3000 | googomemem | E11,334 (5) |

(2) Joyce offers alternative number naming (example: ic here means 99, because c is 100 and i is 1, 100 – 1 = 99) to make numbers easier to pronounce (I don’t offer this, for consistency with Roman numerals’ sake).

**l**mox.

x | number name | value |

1 | googgoi (goo-guh-goy)* | (3*1)^1 = 3 |

2 | googgoij (goo-guh-goyj) | (3*2)^2 = 36 |

3 | googgoiji (goo-guh-goyj-ee) | (3*3)^3 = 729 |

* Sbiis Saibian wrote in his article on Joyce’s googology (which serves mostly to show how broken it is) that he doesn’t know how to distinguish googol, googgol, and googgool in speech, and doubts Joyce cares. Here is my answer – just pronounce each repeated g individually as “guh”.

The first three members of the googo- sequence (3, 36, 729) are already noticeably larger than their googo- counterparts (2, 16, 216). Let’s continue:

4 | googgoiv | 20,736 – that’s 5x larger than googoiv. |

5 | googgov | 759,375 – 7.5x larger than googov |

6 | googgovi | 34,012,224 – 11x larger than googovi |

7 | googgovij | 1.8E9 – 18x larger than googovij |

8 | googgoviji | 1.1E11 – 44x larger than googoviji |

9 | googgoix | 7.6E12 = 7,625,597,484,987 – that’s exactly equal to the important megafugathree, which is 3^3^3 = 3^27. Megafugathree shows up a lot when working with powers of 3. |

10 | googgox | 5.9E14 – that’s almost 600x larger than googox. |

As you can see, these numbers grow notably faster than googo-. Let’s continue:

11 | googgoxi | 5.1E16 |

12 | googgoxij | 4.7E18 |

13 | googgoxiji | 4.8E20 |

14 | googgoxiv | 5.3E22 |

15 | googgoxvy | 6.3E24 |

… | ||

20 | googgoxex | 3.7E35 |

30 | googgoxexex | 4.2E58 |

40 | googgoxel | 1.5E83 |

47 | googgoxelvij | 3E101 – this is the first larger than a googol. |

50 | googgol (goo-guh-guhl) |
6E108 – defined by Joyce himself. |

60 | googgolex | 2E135 |

70 | googgolexex | 4E162 |

80 | googgolexexex | 3E190 |

90 | googgoxec | 7E218 |

100 | googgoc | 5E247 |

….. | ||

200 | googgocc | 4E555 |

300 | googgoccc | 2E866 |

500 | googgod | E1588 |

1000 | googgom | E3477 |

2000 | googgomem | E7556 |

3000 | googgomemem | E11,863 |

x | number name | value |

1 | googggoi (goo-guh-guh-goy) | 4 |

2 | googggoij | 64 |

3 | googggoiji | 1,728 |

4 | googggoiv | 65,536 |

5 | googggov | 32,000,000 |

6 | googggovi | 1.9E8 |

7 | googggovij | 1.3E10 |

8 | googggoviji | 1.1E12 |

9 | googggoix | 1.0E14 |

10 | googggox | 1.0E16 |

15 | googggoxvy | 4.7E26 |

20 | googggoxex | 1.2E38 |

25 | googggoxexv | E50 (exactly equal) |

30 | googggoxexex | 2.4E62 |

45 | googggoxelvy | 3E101 (first term more than a googol) |

50 | googggol | 1E115 |

75 | googggolexexv | 6E185 |

100 | googggoc | 2E260 |

200 | googggocc | 4E580 |

250 | googggoccel | E750 (exactly equal) |

500 | googggod | E1651 |

1000 | googggom | E3602 |

3000 | googggomemem | E12,238 |

number name | value |

googgggoi | (5*1)^1 = 5 |

googgggoij | (5*2)^2 = 100 |

googgggox | (5*10)^10 ~ 9.8E16 |

googgggoxex | E40 (exactly equal) |

googgggol | 8E119 |

googgggoc | 8E269 |

googgggocc | E600 (exactly equal) |

googgggom | E3699 |

googgggomem | E8000 (exactly equal) |

googgggomemem | E12,528 |

Like you probably expected, this one grows ever so slightly faster than the previous. Now let’s try a few with five or more g’s:

googggggol | (6*50)^50 ~ 7E123 |

googggggoc | 7E277 |

googggggom | E2778 |

googgggggom | E3845 |

googggggggom | E3903 |

Since it’s hard to keep track of all the g’s now, we can safely say that numbers with “oo” then “o” as vowels are at a breaking point. These numbers reach their breaking point between a great great googol (10^10,000) and a googolgong (10^100,000), and have exponential growth rates overall. They can be upper-bounded by a googolgong.

**2^x^2**(note that this is 2^(x^2), NOT (2^x)^2). Let’s try some examples:

x | number name | value |

2 | googooij | 2^2^2 = 2^4 = 16. Interestingly, this is exactly equal to googoij. |

3 | googooiji | 2^3^2 = 2^9 = 512 |

4 | googooiv | 2^4^2 = 2^16 = 65,536 |

5 | googoov | 33,554,432 |

6 | googoovi | 6.9E10 |

7 | googoovij | 5.6E15 |

8 | googooviji | 1.9E19 |

9 | googooix | 2.4E24 |

10 | googoox* | 1.3E30 |

11 | googooxi | 2.7E36 |

12 | googooxij | 2.2E43 |

13 | googooxiji | 7.5E50 |

14 | googooxiv | 1.0E59 |

15 | googooxvy | 5.4E67 |

16 | googooxvi | 1.2E77 |

17 | googooxvij | 9.9E86 |

18 | googooxviji | 3.4E97 |

19 | googooxix | 5E108 |

*This is why I don’t like using “ox” in place of “iii” – the third member of the googo- series (googo-3) would be called googoox with Joyce’s system, but this is already an interpretation of googoox. Joyce’s googology is ambiguity galore!

20 | googooxex | 3E120 |

30 | googooxexex | 9E270 |

40 | googooxel | 4E481 |

50 | googool | 4E752 – that’s more than googol to the seventh! |

60 | googoolex | E1084 – passed a great googol! |

70 | googoolexex | E1475 |

80 | googoolexexex | E1927 |

90 | googooxcy | E2438 |

100 | googooc | E3010 |

….. | ||

150 | googoocel | E6773 |

200 | googoocc | E12,041 |

300 | googooccc | E27,093 – larger than anything we’ve seen so far! |

400 | googocdy | E48,615 |

500 | googood | E75,258 – good number, isn’t it? |

….. | ||

1000 | googoom | E301,030 – bigger than a googolgong! |

2000 | googoomem | E1,204,199 – even passes a millionplex (also called four-ex-great googol) |

3000 | googoomemem | E2,709,270 |

x | number name | value |

1 | googgooi (goo-guh-goo-ee) | 3 |

2 | googgooij | 81 |

3 | googgooiji | 19,683 – sizable difference so far (googooiji, for comparison, is 512) |

4 | googgooiv | 43,046,721 |

5 | googgoov | 8.5E11 |

6 | googgoovi | 1.5E17 |

7 | googgoovij | 2.4E23 |

8 | googgooviji | 3.4E30 |

9 | googgooix | 4.4E38 |

10 | googgoox | 5.2E47 |

….. | ||

15 | googgooxvy | 2E107 – first one bigger than googol |

20 | googgooxex | 7E190 |

30 | googgooxexex | 3E429 |

40 | googgooxel | 3E763 |

50 | googgool |
E1193 – defined by Joyce himself |

60 | googgoolex | E1718 |

100 | googgooc | E4771 |

200 | googgoocc | E19,085 |

300 | googgooccc | E42,941 |

500 | googgood | E119,280 |

1000 | googgoom | E477,121 |

2000 | googgoomem | E1,908,485 |

3000 | googgoomemem | E4,294,091 |

x | number name | value |

1 | googggooi | (4^1)^1 = 4 |

2 | googggooij | (4^2)^2 = 256 |

3 | googggooiji | 262,144 |

4 | googggooiv | 4.3E9 – interestingly, exactly equal to googoviji from part 1 and earlier here |

5 | googggoov | 1.1E15 |

6 | googggoovi | 4.7E21 |

….. | ||

10 | googggoox | 1.6E60 |

13 | googggooxiji | E102 – first one larger than googol |

15 | googggooxvy | E135 |

20 | googggooxex | E241 |

50 | googggool | E1505 |

100 | googggooc | E6021 |

500 | googggood | E150,515 |

1000 | googggoom | E602,060 |

3000 | googggoomemem | E5,418,540 |

… and then with more g’s …

googgggool | E1748 |

googgggooc | E6990 |

googgggoom | E698,970 |

googggggoom | E778,151 |

googgggggoom | E845,098 |

googgggggoomemem | E7,605,882 |

x | number name | value |

1 | gogooi (goh-goo-ee) | 2 – note how the frst member of a sequence with one g in a row is 2, with two g’s in a row it’s 3, with three g’s it’s 4, etc. |

2 | gogooij | 8 |

3 | gogooiji | 24 |

4 | gogooiv | 64 – so far not very impressive. |

5 | gogoov | 160 |

6 | gogoovi | 384 |

7 | gogoovij | 896 |

8 | gogooviji | 2048 – also the name of an infamous game |

9 | gogooix | 4608 |

10 | gogoox | 10,240 |

….. | ||

15 | gogooxvy | 491,520 |

20 | gogooxex | 20,971,520 |

30 | gogooxexex | 3.2E10 |

40 | gogooxel | 4.4E13 |

50 | gogool | 5.6E16 |

75 | gogoolexexv | 2.8E24 |

100 | gogooc | 1.3E32 |

150 | gogoocel | 2.1E47 |

200 | gogoocc | 3.2E62 |

324 | gogoocccexexiv (goh-goosh-kex-ex-iv) | 1.1E100 – first to pass a googol |

500 | gogood | E153 |

1000 | gogoom | E304 |

2000 | gogoomem | E605 |

3000 | gogoomemem | E906 |

One thing to point out: gogoo-x is EXACTLY equal to f2(x) in the fast growing hierarchy. (if you don’t know, f0(x) = x+1, f1(x) = f0(f0(f0…..(f0(x))….)) nested x times = 2x, f2(x) = f1(f1(f1…..(f1(x))….)) = x*2^x, f3(x) = f2(f2(f2…..(f2(x))….)) nested x times, etc. It has been extended to far far bigger proportions involving bigger infinite ordinals, might discuss that in other pages.)

This system’s numbers start out rather unimpressive, but they do go faster: they eventually reach the astronomical range, and then something beyond that! It’s certainly far far faster than numbers with only “o” as vowels, but far weaker than googoo-, googgoo-, and the like and even slower than googo-. This one doesn’t make it to a great googol. Nevertheless, let’s try numbers with more g’s here.

x | number name | value |

1 | goggooi | 3 |

2 | goggooij | 18 |

3 | goggooiji | 81 |

4 | goggooiv | 324 |

5 | goggoov | 1215 |

6 | goggoovi | 4374 – note that gogoovi is only 384 |

7 | goggoovij | 15,309 |

8 | goggooviji | 52,488 – not to be confused with 524,288, which is the 19th power of 2 |

9 | goggooix | 177,147 |

10 | goggoox | 590,490 |

….. | ||

15 | goggooxvy | 215,233,605 |

20 | goggooxex | 7.0E10 |

30 | goggooxexex | 6.1E15 |

40 | goggooxel | 4.9E20 |

50 | goggool (gog-ool) | 3.6E25 – compare to gogool, which is around 5.6E16 |

75 | goggoolexexv | 4.6E37 |

100 | goggooc | 5.2E49 |

200 | goggoocc | 5.3E97 |

205 | goggooccvy (go-gooch-vee) | 1.3E100 – first to pass a googol. This sequence passes a googol 36% earlier than the previous. |

300 | goggooccc | E146 |

500 | goggood | E241 |

1000 | goggoom | E480 |

2000 | goggoomem | E958 |

3000 | goggoomemem | E1435 |

gogggoomemem | E1810 |

goggggoomemem | E2100 |

gogggggoomemem | E2338 |

goggggggoomemem | E2538 |

None of them, not even the last, reach a millillion – it takes nine g’s in a row to make it to that number, and by then the g’s get hard to manage – let’s just say the go-x-oo-y family can be bounded by a millillion.

__2.5. Hierarchy of numbers so far__

**levels**so far: those with o as the highest vowel and those with oo as the highest vowel. The higher “ee”, “or”, etc. levels will come later.

**family**(the o-o family, like gogol and goggol), and the second has three (from lowest to highest, o-oo (like gogool), oo-o (like googol), and oo-oo (like googool)).

**series**, depending on how much the g is repeated. For example, the oo-o family has the googo-, googgo-, googggo-, etc. series.

**numbers**. For example, the googo- series starts with googoi, googoij, googoiji, etc. and also includes higher members like googox, googol, and googoc.

**PART 3**

__3. Numbers with “ee” as vowels__

**HUGE!**

etc.

*tetrational growth rate*. What is tetrational growth? The growth rate of a function like this – the size of the power tower increases by one in each step, usually not until after a few steps. Here’s another few functions with tetrational growth:

x | number name | value |

20 | gogeexex | E19,278#17 |

30 | gogeexexex | E19,278#27 |

40 | gogeexel | E19,278#37 |

50 | gogeel | E19,278#47 |

….. | ||

100 | gogeec | E19,278#97 |

200 | gogeecc | E19,278#197 |

500 | gogeed | E19,278#497 |

1000 | gogeem | E19,278#997 |

2000 | gogeemem | E19,278#1997 |

3000 | gogeememem | E19,278#2997 |

__EASILY__dwarf anything from parts one and two, even the largest members of the googolple- series!

*terabytes*of space.

this is the end of the googo- and googolple- article series. as i said it’s mostly archived for history’s sake – the new content is currently under construction and will be much better.