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Jonathan Bowers

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Jonathan Bowers, mathematician (November 27, 1969–)

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[edit] 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.

[edit] 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 Failed to parse (Cannot write to or create math temp directory): {\ ^{b}a = \atop {\ }} {{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}} \atop b} , and its higher order analogues: pentation, sexation, heptation. some examples of which are

  • Failed to parse (Cannot write to or create math temp directory): \{a,b,1\} = a \{1\} b = a+b\;
  • Failed to parse (Cannot write to or create math temp directory): \{a,b,2\} = a \{2\} b = a*b\;
  • Failed to parse (Cannot write to or create math temp directory): \{a,b,3\} = a \{3\} b = a^b\;
  • Failed to parse (Cannot write to or create math temp directory): \{a,b,4\} = a \{4\} b = \ ^{b}a
a tetrated to b.
  • Failed to parse (Cannot write to or create math temp directory): \{a,b,5\} = a \{5\} b = \ ^{\ ^{\ ^{\ ^a\cdot}\cdot}a}a
- a pentated to b - a tetrated to itself b times.
  • Failed to parse (Cannot write to or create math temp directory): \{a,b,c,2\} = a \{\{c\}\}b\;
  • Failed to parse (Cannot write to or create math temp directory): \{a,2,1,2\} = a \{\{1\}\}2 = a \{a\}a\;
  • Failed to parse (Cannot write to or create math temp directory): \{a,3,1,2\} = a \{\{1\}\}3 = a \{a \{a\}a\}a\;
  • Failed to parse (Cannot write to or create math temp directory): \{a,b,1,2\} = a \{\{1\}\}b = a \{a\ldots\{a\}\ldots a\}a\;
- a expanded to b.
  • Failed to parse (Cannot write to or create math temp directory): \{a,b,2,2\} = a \{\{2\}\}b\;
- a expanded to itself b times.
Group Name Value Array notation
Googol group Googol Failed to parse (Cannot write to or create math temp directory): 10^{100}=10^{10^2} {10,100,3} = 10 {3} 100
Googolplex Failed to parse (Cannot write to or create math temp directory): 10^{10^{100}}=10^{10^{10^2}} {10,{10,100,3},3} = 10 {3} 10 {3} 100
Googolplexian Failed to parse (Cannot write to or create math temp directory): 10^{10^{10^{100}}} {10,{10,{10,100,3},3},3} = 10 {3} 10 {3} 10 {3} 100
Giggol group Failed to parse (Cannot write to or create math temp directory): \ ^{10}10 {10,10,4} = 10 {4} 10
Giggol Failed to parse (Cannot write to or create math temp directory): \ ^{100}10 {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.

[edit] 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 Failed to parse (Cannot write to or create math temp directory): {10}^6
Billion Failed to parse (Cannot write to or create math temp directory): {10}^9
Trillion Failed to parse (Cannot write to or create math temp directory): {10}^{12}
Quadrillion Failed to parse (Cannot write to or create math temp directory): {10}^{15}
Quintillion Failed to parse (Cannot write to or create math temp directory): {10}^{18}
Sextillion Failed to parse (Cannot write to or create math temp directory): {10}^{21}
Septillion Failed to parse (Cannot write to or create math temp directory): {10}^{24}
Octillion Failed to parse (Cannot write to or create math temp directory): {10}^{27}
Nonillion Failed to parse (Cannot write to or create math temp directory): {10}^{30}
Decillion Failed to parse (Cannot write to or create math temp directory): {10}^{33}
Undecillion Failed to parse (Cannot write to or create math temp directory): {10}^{33}
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 Failed to parse (Cannot write to or create math temp directory): {10}^{6000}
Myrillion Failed to parse (Cannot write to or create math temp directory): {10}^{6000}
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)
Googolplexian 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 Failed to parse (Cannot write to or create math temp directory): {10}^{{3 * {{10}^{3 * {{10}^{30,000}}}}} + 3}
Lakhillion Failed to parse (Cannot write to or create math temp directory): {10}^{{3 * {{10}^{3 * {{10}^{300,000}}}}} + 3}
Crorillion Failed to parse (Cannot write to or create math temp directory): {10}^{{3 * {{10}^{3 * {{10}^{30,000,000}}}}} + 3}
Awkillion Failed to parse (Cannot write to or create math temp directory): {10}^{{3 * {{10}^{3 * {{10}^{300,000,000}}}}} + 3}
Bentrizillion Failed to parse (Cannot write to or create math temp directory): {10}^{6 * {{10}^{6 * {{10}^{6 * {{10}^{6 * {{10}^9}}}}}}}}

[edit] 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
Googolplexian 10^10^10^100
Googoltriplex 10^10^10^10^100
Googolquadriplex 10^10^10^10^10^100
Googolquinplex 10^10^10^10^10^10^100
Googolcentplex Googolplexian and (98 more plexes)
Googolmilleplex Googolplexian 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 {10,100,4} = 10 {4} 100: 10 tetrated to 100
Giggolplex {10,Giggol,4} = 10 {4} Giggol: 10 tetrated to Giggol
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}

[edit] 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:

Failed to parse (Cannot write to or create math temp directory): \left\langle\begin{matrix}a&b\\2&\end{matrix}\right\rangle=\{a,a,\ldots,a\}
with a repeated b times.
Failed to parse (Cannot write to or create math temp directory): \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
with a repeated b times.
Name Value
Emperal Failed to parse (Cannot write to or create math temp directory): \left\langle\begin{matrix}10&10\\10&\end{matrix}\right\rangle
Emperalplex Failed to parse (Cannot write to or create math temp directory): \left\langle\begin{matrix}10&10\\emperal&\end{matrix}\right\rangle
Hyperal Failed to parse (Cannot write to or create math temp directory): \left\langle\begin{matrix}10&10\\10&10\end{matrix}\right\rangle
Hyperalplex Failed to parse (Cannot write to or create math temp directory): \left\langle\begin{matrix}10&10\\10&Hyperal\end{matrix}\right\rangle
Dutritri Failed to parse (Cannot write to or create math temp directory): \left\langle\begin{matrix}3&3&3\\3&3&3\\3&3&3\end{matrix}\right\rangle
Dutridecal Failed to parse (Cannot write to or create math temp directory): \left\langle\begin{matrix}10&10&10\\10&10&10\\10&10&10\end{matrix}\right\rangle
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 "# #"

[edit] See also

[edit] External links

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