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Not to be confused with Expansion rule.

Visualization of \(a \{\{1\}\} b\) using Knuth's up arrow notation.

Expansion refers to the binary function \(a \{\{1\}\} b = a \{a \{\cdots \{a\} \cdots\}a\}a\), where there are b a's from the center out.[1] It is \(\{a,b,1,2\}\) in BEAF and a{X+1}b in X-Sequence Hyper-Exponential Notation. The notation a{c}b means {a,b,c}, which is a "c + 2"-ated to b, using the bracket operator.

The function eventually dominates any hyper-operator, such as tetration, pentation, or even centation.

Graham's number is defined using a very close variant of expansion. It is \(3 \{\{1\}\} 65\) with the central 3 replaced with a 4.

\(a\{\{1\}\}b\) is exactly equal to [a,a,a,b-1] in the Graham Array Notation.

By Bird's Proof, \(a \{\{1\}\} b > a \rightarrow a \rightarrow (b-1) \rightarrow 2\) using chained arrow notation.

Examples

  • \(2\ \{\{1\}\}\ 2\) = 4
  • \(2\ \{\{1\}\}\ 3 = 2\{2\{2\}2\}2 = 2\{4\}2 = 2\{3\}2 = 2\{2\}2 = 2\{1\}2 = 4\)
    • In fact, if the base is equal to 2 and prime is ≥ 2, then the result will always be 4.
  • \(3\ \{\{1\}\}\ 2 = \{3,2,1,2\} = 3 \{3\} 3 = 3\uparrow\uparrow\uparrow 3\) (tritri)
  • \(a\ \{\{1\}\}\ 2 = \{a,2,1,2\} = a \{a\} a = a\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{a}a\)
  • \(3\ \{\{1\}\}\ 3 = \{3,3,1,2\} = 3 \{3 \{3\} 3\} 3 = 3\{\text{tritri}\}3 = 3\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\text{tritri}}3\)
  • \(4\ \{\{1\}\}\ 3 = \{4,3,1,2\} = 4 \{4 \{4\} 4 \}4 = 4 \{\)\(\text{tritet} \)\(\} 4 = 4\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\text{tritet}}4\)
  • \(a\ \{\{1\}\}\ 3 = \{a,3,1,2\} = a \{a \{a\} a\} a = a\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{a\{a\}a}a\)
  • \(3\ \{\{1\}\}\ 4 = \{3,4,1,2\} = 3 \{3 \{3 \{3\} 3\} 3 \}3 = 3 \{3 \{\text{tritri}\} 3\} 3 = 3\{\ \{3,3,1,2\}\ \}3 = 3\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\{3,3,1,2\} }3\) = \(3 \uparrow^{3 \uparrow^{3 \uparrow^{3}3}3}3\)
  • \(a\ \{\{1\}\}\ 4 = \{a,4,1,2\} = a \{a \{a \{a\} a\} a \}a = a\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{a\{a\{a\}a\}a}a\)
  • \(10 \{\{1\}\} 100 = \{10,100,1,2\} = \{10,10,\{10,99,1,2 \}\} = 10 \underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\{10,99,1,2 \} }10\) (corporal)
  • \(a \{\{1\}\} b = \{a,b,1,2\} = \{a,a,\{a,b-1,1,2 \}\} = a \underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\{a,b-1,1,2\} }a\)

Pseudocode

Below is an example of pseudocode for expansion.

function expansion(a, b):
    result := a
    repeat b - 1 times:
        result := hyper(a, a, result + 2)
    return result

function hyper(a, b, n):
    if n = 1:
        return a + b
    result := a
    repeat b - 1 times:
        result := hyper(a, result, n - 1)
    return result

Approximations in other notations

Notation Approximation
Hyper-E notation \(\textrm Ea\#\#a\#b\)
Chained arrow notation \(a \rightarrow a \rightarrow b \rightarrow 2\)
X-Sequence Hyper-Exponential Notation \(a \{X+1\} b\) (exact)
Fast-growing hierarchy \(f_{\omega+1}(b)\)
Hardy hierarchy \(H_{\omega^{\omega+1}}(b)\)
Slow-growing hierarchy \(g_{\Gamma_0}(b)\)

Sources

See also

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