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White-aster notation is a notation for large numbers made by aster in a video on September 17, 2017[1][2].


This is an extension of Steinhaus-Moser Notation, which is one of the notations for large numbers.

It is extended by adding the concept of stars and levels in addition to the usual polygons.

Definition[]

First, we introduce the terminology to explain White-aster notation.

1.To simplify writing down the definitions, A string of integers greater than or equal to 3 or Ast is called a "figure type" as the only terminology used here, and a star is formally called "a Ast-gon".

2.For a figure of type p and integer n, a figure with <n> on the right shoulder of a regular p-gon is called "a p-gon of level n". For example, a regular triangle at level 4 is \(\triangle^{\langle 4 \rangle}\), and a star at level 5 is \(\textrm{☆}^{\langle 5 \rangle}\).

3.Let a be a figure or an integer greater than or equal to 3. A figure with an inside a p-gon of level n is denoted p[a,<n>] and called "a inside a regular p-gon of level n". For example, 3[5,<4>], which is 5 inside a regular 3-gon of level 4, is shown on the left in the figure below, 4[Ast[7,<6>],<5>], a 7 in a star of level 6 in a regular tetragon of level 5, is shown on the right below.

Aster example


Based on this, the calculation of the White-aster notation is shown below.

Rule 1-1. If p = 3 and n=1, \[3[a,\langle 1 \rangle] = a^{a}\]

Rule 1-2. If p = Ast, \[\text{Ast}[a,\langle n \rangle] = a[a,\langle n \rangle]\]

Rule 2-1. If p = 3 and n > 1, \[3[a,\langle n \rangle] = \underbrace{\text{Ast}[\text{Ast}[\cdots[\text{Ast}}_{a}[a,\langle n-1 \rangle],\langle n-1 \rangle],\cdots],\langle n-1 \rangle],\langle n-1 \rangle]\]

Rule 2-2. If Rule 2-1 does not apply, \[p[a,\langle n \rangle] = \underbrace{p-1[p-1[\cdots[p-1}_{a}[a,\langle n \rangle],\langle n \rangle],\cdots],\langle n \rangle],\langle n \rangle]\]

Examples[]

5[2,<1>]

=4[4[2,<1>],<1>]

=4[3[3[2,<1>],<1>],<1>]

=4[3[2^2,<1>],<1>]

=4[3[4,<1>],<1>]

=4[44,<1>]

=4[256,<1>]

This value is equal to Mega.


4[3,<2>]

=3[3[3[3,<2>],<2>],<2>]

=3[3[Ast[Ast[Ast[3,<1>],<1>],<1>],<2>],<2>]

=3[3[Ast[Ast[3[3,<1>],<1>],<1>],<2>],<2>]

=3[3[Ast[Ast[33,<1>],<1>],<2>],<2>]

=3[3[Ast[Ast[27,<1>],<1>],<2>],<2>]

=3[3[Ast[27[27,<1>],<1>],<2>],<2>]

Issues[]

Computation programs[]

See also[]

Original numbers, functions, notations, and notions

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
By aster: White-aster notation · White-aster
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4 original idea
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 formalisation · TR function (I0 function)
By Gaoji: Weak Buchholz's function
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence formalisation · ω-Y sequence formalisation
By Nayuta Ito: N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4 · Googology Wiki can have an article with any gibberish if it's assigned to a number
By Okkuu: Extended Weak Buchholz's function
By p進大好きbot: Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence original idea · YY sequence · Y function · ω-Y sequence original idea


Methodology

By バシク (BashicuHyudora): Bashicu matrix system as a notation template
By じぇいそん (Jason): Shifting definition
By mrna: Side nesting
By Nayuta Ito and ゆきと (Yukito): Difference sequence system


Implementation of existing works into programs

Proofs, translation maps for analysis schema, and other mathematical contributions

By ふぃっしゅ (Fish): Computing last 100000 digits of mega · Approximation method for FGH using Arrow notation · Translation map for primitive sequence system and Cantor normal form
By Kihara: Proof of an estimation of TREE sequence · Proof of the incomparability of Busy Beaver function and FGH associated to Kleene's \(\mathcal{O}\)
By koteitan: Translation map for primitive sequence system and Cantor normal form
By Naruyoko Naruyo: Translation map for Extended Weak Buchholz's function and Extended Buchholz's function
By Nayuta Ito: Comparison of Steinhaus-Moser Notation and Ampersand Notation
By Okkuu: Verification of みずどら's computation program of White-aster notation
By p進大好きbot: Proof of the termination of Hyper primitive sequence system · Proof of the termination of Pair sequence number · Proof of the termination of segements of TR function in the base theory under the assumption of the \(\Sigma_1\)-soundness and the pointwise well-definedness of \(\textrm{TR}(T,n)\) for the case where \(T\) is the formalisation of the base theory


Entertainments

By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Dancing video of a Gijinka of Fukashigi · Dancing video of a Gijinka of 久界 · Storyteller's theotre video reading Large Number Garden Number aloud


See also: Template:Googology in Asia


Sources[]

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