Jonathan Bowers

Jonathan Bowers, mathematician (November 27, 1969–)

Polychora

One of the participants in the Uniform Polychora Project, an attempt to name higher-dimensional polychora. He independently began his search for polychora in 1990. Circa 1993, he invented his short names for the uniform polychora, which have come to be known as the Bowers style acronyms (or pet name). In 1997, he contacted others who are also interested in the subject, such as Magnus Wenninger, Vincent Matsko, and George Olshevsky.

Very large numbers

Bowers has proposed a series of names (including giggol, gaggol, geegol, goggol, tridecal, tetratri, dutritri, xappol, dimendecal, gongulus, trimentri, goppatoth, golapulus, golapulusplex, golapulusplux, big boowa and guapamonga) for extremely Large numbers, which he terms infinity scrapers (a pun on skyscraper), many of which are so unspeakably and unprecedentedly large as to require special newly-devised extended mathematical notations in order to be expressed.infinity scrapers

He is also the inventor of the Array Notation as a means to represent very large numbers. This notion is very similar to the Conway chained arrow notation and relies on the tetration operator <math>{\ ^{b}a = \atop {\ }} {{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}} \atop b}</math>, and its higher order analogues: pentation, sexation, heptation. some examples of which are

• <math>\{a,b,1\} = a \{1\} b = a+b\;</math>
• <math>\{a,b,2\} = a \{2\} b = a*b\;</math>
• <math>\{a,b,3\} = a \{3\} b = a^b\;</math>
• <math>\{a,b,4\} = a \{4\} b = \ ^{b}a</math> a tetrated to b.
• <math>\{a,b,5\} = a \{5\} b = \ ^{\ ^{\ ^{\ ^a\cdot}\cdot}a}a</math> - a pentated to b - a tetrated to itself b times.
• <math>\{a,b,c,2\} = a \{\{c\}\}b\;</math>
• <math>\{a,2,1,2\} = a \{\{1\}\}2 = a \{a\}a\;</math>
• <math>\{a,3,1,2\} = a \{\{1\}\}3 = a \{a \{a\}a\}a\;</math>
• <math>\{a,b,1,2\} = a \{\{1\}\}b = a \{a\ldots\{a\}\ldots a\}a\;</math> - a expanded to b.
• <math>\{a,b,2,2\} = a \{\{2\}\}b\;</math> - a expanded to itself b times.

Group	Name	Value	Array notation
Googol group	Googol	<math>10^{100}=10^{10^2}</math>	{10,100,2} = 10 {2} 100
	Googolplex	<math>10^{10^{100}}=10^{10^{10^2}}</math>	{10,{10,100,2},2} = 10 {2} 10 {2} 100
	Googolduplex	<math>10^{10^{10^{100}}}</math>	{10,{10,{10,100,2},2},2} = 10 {2} 10 {2} 10 {2} 100
Giggol group		<math>\ ^{10}10</math>	{10,10,4} = 10 {4} 10
	Giggol	<math>\ ^{100}10</math>	{10,100,4} = 10 {4} 10: 10 to the power of itself 100 times.
	Giggolplex		10 {4} 10 {4} 10: 10 tetrated to a giggol
	Giggolduplex		10 {4} 10 {4} 10 {4} 100.
Gaggol group	Gaggol		{10,100,5} = 10 {5} 100: 10 pentated to 100.
	gaggolplex		10 {5} gaggol = 10 {5} 10 {5} 100: 10 tetrated to itself a gaggol times.
	geegol		{10,100,6}=10 {6} 100
	geegolplex		{10,geegol, 6}
	gygol		{10,100,7}
	gygolplex		{10,gygol,7}
	goggol		{10,100,8}
	goggolplex		{10,goggol,8}
	gagol		{10,100,9}
	gagolplex		{10,gagol,9}
Infinity Scrapers	Tridecal		{10,10,10} = 10 {10} 10 = 10 decated to 10
	boogol		{10,10,100} = 10 {100} 10

Bowers notation extends beyond even these numbers and includes the Corporal {10,100,1,2}, Grand Tridecal {10,10,10,2}, Tetratri = {3,3,3,3}, General = {10,10,10,10}, Pentadecal = {10,10,10,10,10}, Iteral = {10,10,10,10,10,10,10,10,10,10}. Still larger numbers require an extended array notation and include the Emperal Group, Hyperal Group, Dutritri and Dutridecal. The largest number mentioned is the Guapamongaplex which is still considerably less than infinity, but is much larger than even Graham's number.

Illion group

Jonathan Bowers has also created a number of names for large numbers that extend the -illion family of much smaller numbers than those mentioned above. Here is a list of the names of large -illion numbers on the short scale.

Name	Short scale value
Million	<math>{10}^6</math>
Billion	<math>{10}^9</math>
Trillion	<math>{10}^{12}</math>
Quadrillion	<math>{10}^{15}</math>
Quintillion	<math>{10}^{18}</math>
Sextillion	<math>{10}^{21}</math>
Septillion	<math>{10}^{24}</math>
Octillion	<math>{10}^{27}</math>
Nonillion	<math>{10}^{30}</math>
Decillion	<math>{10}^{33}</math>
Undecillion	<math>{10}^{33}</math>
Duodecillion	1039
Tredecillion	1042
Quattuordecillion	1045
Quindecillion	1048
Sexdecillion	1051
Septendecillion	1054
Octodecillion	1057
Novemdecillion	1060
Vigintillion	1063
Trigintillion	1093
Googol (Ten Duotrigintillion)	10100
Quadragintillion	10123
Quinquagintillion	10153
Sexagintillion	10183
Septuagintillion	10213
Octogintillion	10243
Nonagintillion	10273
Centillion	10300
Platillion	<math>{10}^{6000}</math>
Micrillion	10^ 3000003
Nanillion	10^ 3billion3
Picillion	10^ 3trillion3
Femtillion	10^ 3quadrillion3
Attillion	10^ 3 quintillion3
Zeptillion	10^ 3sextillion3
Yoctillion	10^ 3septillion3
Xonillion	10^ 3octillion3
Vecillion	10^ 3nonillion3
Mecillion	10^ 3decillion3
Duecillion	10^ 3undecillion3
Trecillion	10^ 3doedecillion3
Tetrecillion	10^ 3tridecillion3
Pentecillion	10^ 3quattuordecillion3
Hexecillion	10^ 3quindecillion3
Heptecillion	10^ 3sexdecillion3
Octecillion	10^ 3septendecillion3
Ennecillion	10^ 3octodecillion3
Icosillion	10^ 3novemdecillion3
Triacontillion	10^ (3x10^90+3)
Googolplex	10^10^100
Tetracontillion	10^ (3x10^120+3)
Pentacontillion	10^ (3x10^150+3)
Hexacontillion	10^ (3x10^180+3)
Heptacontillion	10^ (3x10^210+3)
Octacontillion	10^ (3x10^240+3)
Ennacontillion	10^ (3x10^270+3)
Hectillion	10^ (3x10^300+3)
Killillion	10^ (3x10^3000+3)
Megillion	10^ (3x10^3million +3)
Gigillion	10^ (3x10^3billion +3)
Terillion	10^ (3x10^3trillion +3)
Petillion	10^ (3x10^3quadrillion +3)
Exillion	10^ (3x10^3quintillion +3)
Zettillion	10^ (3x10^3sextillion +3)
Yottillion	10^ (3x10^3septillion +3)
Xennillion	10^ (3x10^3octillion +3)
Vekillion	10^ (3x10^3nonillion +3)
Duekillion	10^ (3x10^3undecillion +3)
Trekillion	10^ (3x10^3doedecillion +3)
Tetrekillion	10^ (3x10^3tridecillion +3)
Pentekillion	10^ (3x10^3quattuordecillion +3)
Hexekillion	10^ (3x10^3quindecillion +3)
Heptekillion	10^ (3x10^3sexdecillion +3)
Octekillion	10^ (3x10^3septendecillion +3)
Ennekillion	10^ (3x10^3octodecillion +3)
Twentillion	10^ (3x10^(3x10^60) +3)
Triatwentillion	10^ (3x10^(3x10^69) +3)
Thirtillion	10^ (3x10^(3x10^90)+3)
Googolduplex	10^10^10^100
Fortillion	10^ (3x10^(3x10^120)+3)
Fiftillion	10^ (3x10^(3x10^150)+3)
Sixtillion	10^ (3x10^(3x10^180)+3)
Seventillion	10^ (3x10^(3x10^210)+3)
Eightillion	10^ (3x10^(3x10^240)+3)
Nintillion	10^ (3x10^(3x10^270)+3)
Hundrillion	10^ (3x10^(3x10^300)+3)
Thousillion	10^ (3x10^(3x10^3000)+3)
Manillion	<math>{10}^{{3 * {{10}^{3 * {{10}^{30,000}}}}} + 3}</math>
Lakhillion	<math>{10}^{{3 * {{10}^{3 * {{10}^{300,000}}}}} + 3}</math>
Crorillion	<math>{10}^{{3 * {{10}^{3 * {{10}^{30,000,000}}}}} + 3}</math>
Awkillion	<math>{10}^{{3 * {{10}^{3 * {{10}^{300,000,000}}}}} + 3}</math>
Bentrizillion	<math>{10}^{6 * {{10}^{6 * {{10}^{6 * {{10}^{6 * {{10}^9}}}}}}}}</math>

Googol, giggol and gaggol groups

Jonathan Bowers defines the googol, giggol, and gaggol groups as being lower than the infinity scrapers. Here's a list of the names of numbers in these groups:

Name	Value
Googol	10^100
Googolplex	10^10^100
Googolduplex	10^10^10^100
Googoltriplex	10^10^10^10^100
Googolquadriplex	10^10^10^10^10^100
Googolcentplex	Googolduplex and (98 more plexes)
Googolmilleplex	Googolduplex and (998 more plexes)
Googolmegaplex	Googolplex and (999,999 more plexes)
Googolgigaplex	Googolplex and (999,999,999 more plexes)
Googolteraplex	Googolplex and (999,999,999,999 more plexes)
Googolpetaplex	Googolplex and (999,999,999,999,999 more plexes)
Fzgoogolplex	Googolplex to the power of a googolplex
Giggol	{10,100,5} = 10 {5} 100: 10 pentated to 100
Giggolplex	{10,100,5} = 10 {5} 100: 10 pentated to 100
Mega	...
Gaggol	{10,100,5} = 10 {5} 100: 10 pentated to 100
Gaggolplex	10 {5} gaggol = 10 {5} 10 {5} 100: 10 tetrated to itself a gaggol times
Megaston	...
Tripent	{5,5,5} = 5 {5} 5 = 5 {4} 5 {4} 5 {4} 5 {4} 5
Trisept	{7,7,7} = 7 {7} 7 (7 heptated to 7)
Geegol	{10,100,6}=10 {6} 100
Geegolplex	{10,geegol, 6}
Gygol	{10,100,7}
Gygolplex	{10,gygol,7}
Goggol	{10,100,8}
Goggolplex	{10,goggol,8}
Gagol	{10,100,9}
Gagolplex	{10,gagol,9}

Infinity scrapers

Numbers higher than those in the Gaggol group are referred to by Jonathan Bowers as the infinity scrapers. Here's a list of the names of those numbers that are infinity scrapers:

Name	Value
Tridecal	{10,10,10} = 10 {10} 10 = 10 decated to 10
Boogol	{10,10,100} = 10 {100} 10
Boogolplex	{10,10,boogol}
Moser's number	...
Graham's number	...
Corporal	{10,100,1,2}
Corporalplex	{10,corporal,1,2}
Grand Tridecal	{10,10,10,2}
Tetratri	{3,3,3,3}
General	{10,10,10,10}
Generalplex	{10,10,10,general}
Pentatri	{3,3,3,3,3}
Pentadecal	{10,10,10,10,10}
Pentadecalplex	{10,10,10,10,pentadecal}
Hexatri	{3,3,3,3,3,3}
Hexadecal	{10,10,10,10,10,10}
Hexadecalplex	{10,10,10,10,10,hexadecal}
Iteral	{10,10,10,10,10,10,10,10,10,10}
Ultatri	{3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Iteralplex	{10,10,10,10,10,10,...........,10,10,10}

The following numbers requre an extended array notation to define. These are defined recursively, using rules such as:

<math>\left\langle\begin{matrix}a&b\\2&\end{matrix}\right\rangle=\{a,a,\ldots,a\}</math> with a repeated b times.

<math>\left\langle\begin{matrix}a&b\\k&\end{matrix}\right\rangle=\left\langle\begin{matrix}a&a&\ldots&a\\k-1\end{matrix}\right\rangle</math> with a repeated b times.

Name	Value
Emperal	<math>\left\langle\begin{matrix}10&10\\10&\end{matrix}\right\rangle</math>
Emperalplex	<math>\left\langle\begin{matrix}10&10\\emperal&\end{matrix}\right\rangle</math>
Hyperal	<math>\left\langle\begin{matrix}10&10\\10&10\end{matrix}\right\rangle</math>
Hyperalplex	<math>\left\langle\begin{matrix}10&10\\10&Hyperal\end{matrix}\right\rangle</math>
Dutritri	<math>\left\langle\begin{matrix}3&3&3\\3&3&3\\3&3&3\end{matrix}\right\rangle</math>
Dutridecal	<math>\left\langle\begin{matrix}10&10&10\\10&10&10\\10&10&10\end{matrix}\right\rangle</math>
Xappol	10 by 10 array of 10's
Xappolplex	xappol by xappol array of 10's
Dimentri	3 x 3 x 3 array of 3's
Colossal	10 x 10 x 10 array of 10's
Colossalplex	colossal x colossal x colossal array of 10's
Dimendecal	10x10x10x10x10x10x10x10x10x10 array of 10's
Gongulus	100 dimensional array of 10's (10^100 array that is)
Gongulusplex	gongulus dimensional array of 10's (10^gongulus array)
Dulatri	(3^3)^2 array of 3's
Trimentri	3^(3^3) array of 3's
Goppatoth	10 tetrated to 100 array of 10's
Goppatothplex	{10,goppatoth,4} array of 10's
Tridecatrix	{10,10,10} array of 10's
Gongulus	a "10^100 array of 10's" array of 10's.
Golapulus	a * "10^100 array of tens" array of tens* array of tens.
Big Boowa	X3, {X3,dutritriX, 2} X
Great Big Boowa	X3,3,3X
Wompogulus	10^10 "100th level" exploded array of 10's
Wompogulusplex	10^10 "wompogulusth level" exploded array of 10's!!
Guapamonga	10^100 array of B's within "# #"
Guapamongaplex	10^100 array of B's within guapamonga-level "# #"

See also

• Uniform polychoron
• Knuth's up-arrow notation

External links

• Jonathan's home page