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− | '''Multiexpansion '''refers to the function \(a \{\{2\}\} b = \{a,b,2,2\} = \underbrace{a \{\{1\}\} a \{\{1\}\} \ldots \{\{1\}\} a \{\{1\}\} a}_{\text{b a's}}\), using [[BEAF]].<ref>[http://www.polytope.net/hedrondude/array.htm]</ref> Or ''a'' [[Expansion|expanded]] to ''a'' for ''b'' times. They are solved from right to left. |
+ | '''Multiexpansion '''refers to the function \(a \{\{2\}\} b = \{a,b,2,2\} = \underbrace{a \{\{1\}\} a \{\{1\}\} \ldots \{\{1\}\} a \{\{1\}\} a}_{\text{b a's}}\), using [[BEAF]].<ref>[http://www.polytope.net/hedrondude/array.htm Array Notation by Jonathan Bowers]</ref> Or ''a'' [[Expansion|expanded]] to ''a'' for ''b'' times. They are solved from right to left. |
In the [[fast-growing hierarchy]], \(f_{\omega+2}(n)\) corresponds to multiexpandal growth rate. |
In the [[fast-growing hierarchy]], \(f_{\omega+2}(n)\) corresponds to multiexpandal growth rate. |
Revision as of 15:08, 16 September 2013
Multiexpansion refers to the function \(a \{\{2\}\} b = \{a,b,2,2\} = \underbrace{a \{\{1\}\} a \{\{1\}\} \ldots \{\{1\}\} a \{\{1\}\} a}_{\text{b a's}}\), using BEAF.[1] Or a expanded to a for b times. They are solved from right to left.
In the fast-growing hierarchy, \(f_{\omega+2}(n)\) corresponds to multiexpandal growth rate.
Examples
- {a,3,2,2} = a{{1}}a{{1}}a. This is equal to a expanded to (a expanded to a). Let A = a{a{a...a{a{a}a}a...a}a}a (a a's), then {a,3,2,2} = a{a...a{a}a...a}a (A a's)
- {3,2,2,2} = 3{{2}}2 = 3{{1}}3 = {3,3,1,2}
- {4,2,2,2} = {4,4,1,2}
- {3,3,2,2} = 3{{2}}3 = 3{{1}}3{{1}}3 = {3,{3,3,1,2},1,2}
- {4,3,2,2} = {4,{4,4,1,2},1,2}
Pseudocode
Below is an example of pseudocode for multiexpansion.
function multiexpansion(a, b): result := a repeat b - 1 times: result := expansion(a, result) return result 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
Sources
See also
Main article: hyper operator
Hyper-operators: successor · addition · multiplication · exponentiation · tetration · pentation · hexation · heptation · octation · enneation · decation (un/doe/tre) · vigintation (doe) · trigintation · centation (tri) · docentation · myriation · circulationBowers' extensions: expansion · multiexpansion · powerexpansion · expandotetration · explosion (multi/power/tetra) · detonation · pentonation
Saibian's extensions: hexonation · heptonation · octonation · ennonation · deconation
Tiaokhiao's extensions: megotion (multi/power/tetra) · megoexpansion (multi/power/tetra) · megoexplosion · megodetonation · gigotion (expand/explod/deto) · terotion · more...