Jonathan Bowers

From Free net encyclopedia

Jonathan Bowers, amateur mathematician (November 27, 1969–) has worked on Polychora (higher dimensional analogues of polyhedra), and representing very large numbers.

Contents

[edit]

Polychora

Bowers is one of the participants in the Uniform Polychora Project, an attempt to name higher-dimensional polychora, higher dimensional analogues of uniform polyhedra. He independently began his search for polychora in 1990. Circa 1993, he invented his short names for the uniform polyhedra and 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]

Illion group

Bowers has also invented names for very large numbers that extend the -illion family in names of large numbers.

Name Short scale value
Vigintillion <math>10^{63}</math>, not coined by Bowers.
Trigintillion <math>10^{93}</math>, not coined by Bowers.
Googol <math>10^{100}</math>, not coined by Bowers.
Quadragintillion <math>10^{123}</math>
Quinquagintillion <math>10^{153}</math>
Sexagintillion <math>10^{183}</math>
Septuagintillion <math>10^{213}</math>
Octogintillion <math>10^{243}</math>
Nonagintillion <math>10^{273}</math>
Centillion <math>10^{303}</math>, not coined by Bowers.
Ducentillion <math>10^{603}</math>
Trecentillion <math>10^{903}</math>
Quadringentillion <math>10^{1203}</math>
Quingentillion <math>10^{1503}</math>
Sescentillion <math>10^{1803}</math>
Septingentillion <math>10^{2103}</math>
Octingentillion <math>10^{2403}</math>
Nongentillion <math>10^{2703}</math>
Millillion <math>10^{3003}=10^{3*10^3+3}</math>
Micrillion <math>10^{3000003}=10^{3*10^6+3}</math>
Nanillion <math>10^{3\ \mathrm{billion}+3}=10^{3*10^9+3}</math>
Picillion <math>10^{3\ \mathrm{trillion}+3}=10^{3*10^{12}+3}</math>
Femtillion <math>10^{3\ \mathrm{quadrillion}+3}=10^{3*10^{15}+3}</math>
Attillion <math>10^{3\ \mathrm{quintillion}+3}=10^{3*10^{18}+3}</math>
Zeptillion <math>10^{3\ \mathrm{sextillion}+3}=10^{3*10^{21}+3}</math>
Yoctillion <math>10^{3\ \mathrm{septillion}+3}=10^{3*10^{24}+3}</math>
Xonillion <math>10^{3\ \mathrm{octillion}+3}=10^{3*10^{27}+3}</math>
Vecillion <math>10^{3\ \mathrm{nonillion}+3}=10^{3*10^{30}+3}</math>
Mecillion <math>10^{3\ \mathrm{decillion}+3}=10^{3*10^{33}+3}</math>
Duecillion <math>10^{3\ \mathrm{undecillion}+3}=10^{3*10^{36}+3}</math>
Trecillion <math>10^{3\ \mathrm{duodecillion}+3}=10^{3*10^{39}+3}</math>
Tetrecillion <math>10^{3\ \mathrm{tredecillion}+3}=10^{3*10^{42}+3}</math>
Pentecillion <math>10^{3\ \mathrm{quattuordecillion}+3}=10^{3*10^{45}+3}</math>
Hexecillion <math>10^{3\ \mathrm{quindecillion}+3}=10^{3*10^{48}+3}</math>
Heptecillion <math>10^{3\ \mathrm{sexdecillion}+3}=10^{3*10^{51}+3}</math>
Octecillion <math>10^{3\ \mathrm{septendecillion}+3}=10^{3*10^{54}+3}</math>
Ennecillion <math>10^{3\ \mathrm{octodecillion}+3}=10^{3*10^{57}+3}</math>
Icosillion <math>10^{3\ \mathrm{novemdecillion}+3}=10^{3*10^{60}+3}</math>
Triacontillion <math>10^{3*10^{90}+3}</math>
Googolplex <math>10^{10^{100}}</math>, not coined by Bowers.
Tetracontillion <math>10^{3*10^{120}+3}</math>
Pentacontillion <math>10^{3*10^{150}+3}</math>
Hexacontillion <math>10^{3*10^{180}+3}</math>
Heptacontillion <math>10^{3*10^{210}+3}</math>
Octacontillion <math>10^{3*10^{240}+3}</math>
Ennacontillion <math>10^{3*10^{270}+3}</math>
Hectillion <math>10^{3*10^{300}+3}</math>
Killillion <math>10^{3*10^{3000}+3}</math>
Megillion <math>10^{3*10^{3\ \mathrm{million} }+3}=10^{3*10^{3*10^6}+3}</math>
Gigillion <math>10^{3*10^{3\ \mathrm{billion}}+3}=10^{3*10^{3*10^9}+3}</math>
Terillion <math>10^{3*10^{3\ \mathrm{trillion} }+3}=10^{3*10^{3*10^{12}}+3}</math>
Petillion <math>10^{3*10^{3\ \mathrm{quadrillion} }+3}=10^{3*10^{3*10^{15}}+3}</math>
Exillion <math>10^{3*10^{3\ \mathrm{quintillion} }+3}=10^{3*10^{3*10^{18}}+3}</math>
Zettillion <math>10^{3*10^{3\ \mathrm{sextillion} }+3}=10^{3*10^{3*10^{21}}+3}</math>
Yottillion <math>10^{3*10^{3\ \mathrm{septillion} }+3}=10^{3*10^{3*10^{24}}+3}</math>
Xennillion <math>10^{3*10^{3\ \mathrm{octillion} }+3}=10^{3*10^{3*10^{27}}+3}</math>
Vekillion <math>10^{3*10^{3\ \mathrm{nonillion} }+3}=10^{3*10^{3*10^{30}}+3}</math>
Mekillion <math>10^{3*10^{3\ \mathrm{decillion} }+3}=10^{3*10^{3*10^{33}}+3}</math>
Duekillion <math>10^{3*10^{3\ \mathrm{undecillion} }+3}=10^{3*10^{3*10^{36}}+3}</math>
Trekillion <math>10^{3*10^{3\ \mathrm{duodecillion} }+3}=10^{3*10^{3*10^{39}}+3}</math>
Tetrekillion <math>10^{3*10^{3\ \mathrm{tredecillion} }+3}=10^{3*10^{3*10^{42}}+3}</math>
Pentekillion <math>10^{3*10^{3\ \mathrm{quattuordecillion} }+3}=10^{3*10^{3*10^{45}}+3}</math>
Hexekillion <math>10^{3*10^{3\ \mathrm{quindecillion} }+3}=10^{3*10^{3*10^{48}}+3}</math>
Heptekillion <math>10^{3*10^{3\ \mathrm{sexdecillion} }+3}=10^{3*10^{3*10^{51}}+3}</math>
Octekillion <math>10^{3*10^{3\ \mathrm{septendecillion} }+3}=10^{3*10^{3*10^{54}}+3}</math>
Ennekillion <math>10^{3*10^{3\ \mathrm{octodecillion} }+3}=10^{3*10^{3*10^{57}}+3}</math>
Twentillion <math>10^{3*10^{3*10^{60}} +3}</math>
Triatwentillion <math>10^{3*10^{3*10^{69}} +3}</math>
Thirtillion <math>10^{3*10^{3*10^{90}}+3}</math>
Googolduplex or Googolplexian <math>10^{10^{10^{100}}}</math>, not coined by Bowers.
Fortillion <math>10^{3*10^{3*10^{120}}+3}</math>
Fiftillion <math>10^{3*10^{3*10^{150}}+3}</math>
Sixtillion <math>10^{3*10^{3*10^{180}}+3}</math>
Seventillion <math>10^{3*10^{3*10^{210}}+3}</math>
Eightillion <math>10^{3*10^{3*10^{240}}+3}</math>
Nintillion <math>10^{3*10^{3*10^{270}}+3}</math>
Hundrillion <math>10^{3*10^{3*10^{300}}+3}</math>
Thousillion <math>10^{3*10^{3*10^{3000}}+3}</math>
[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 large that they can only be expressed using a special set of extended mathematical notations which he has devised.

These notations are very similar to the hyper operators and Conway chained arrow notation, and rely 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 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.
[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 some numbers in these groups:

Name Value
Googol 10^100: "roughly" 10 tetrated to 2 = 10^10
Googolplex 10^10^100: roughly 10 tetrated to 3 = 10^(10^10)
Googolduplex or Googolplexian 10^10^10^100: roughly 10 tetrated to 4
Googoltriplex 10^10^10^10^100: roughly 10 tetrated to 5
Googolquadriplex 10^10^10^10^10^100: roughly 10 tetrated to 6
Giggol {10,100,4} = 10 {4} 100: 10 tetrated to 100
Mega roughly 10 tetrated to 258
Giggolplex 10 {4} giggol: 10 tetrated to giggol
Tripent {5,5,5} = 5 {5} 5 = 5 {4} 5 {4} 5 {4} 5 {4} 5: 5 pentated to 5
Megaston roughly 10 pentated to 11
Gaggol {10,100,5} = 10 {5} 100: 10 pentated to 100
Gaggolplex 10 {5} gaggol = 10 {5} 10 {5} 100: 10 pentated to gaggol
Geegol {10,100,6}=10 {6} 100
Geegolplex {10,geegol, 6}
Trisept {7,7,7} = 7 {7} 7 (7 heptated to 7)
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. These require four or more terms in the array notation to represent. An older notation represents four term arrays using multiple pairs of braces about the third term, thus extending the operator notation. Some of the rules for constructing these numbers include:

  • <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.

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
Moser's number ...
Boogolplex {10,10,boogol}
Graham's number roughly {3, 64, 1, 2}
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} (iteral 10's)

The following numbers require 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.

The last few numbers from the previous table are repeated to establish the notation.

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
Golapulus a "10^100 array of 10's" array of 10's.
Golapulusplex 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

Retrieved from "http://www.netipedia.com/index.php/Jonathan_Bowers"