Triakulus

Triakulus is equal to \(3\&3\&3\) or \(\{ 3,3 /2 \}\) or \(\{L,1\}_{3,3}\) using the array of operator and BEAF. Since the array of operator solves from left to right, \(3\&3\&3 = \{ 3,3,3 \} \& 3 \) = \(3 \uparrow\uparrow\uparrow 3 \& 3\), a pentational array. The structure of an array which defines triakulus is supposedly X^^^X and the array is intended to have tritri entries. 
It must be remembered that although it has tritri entries, a triakulus is in no way the same thing as a linear array with tritri repetitions of 3. That would be dupertri, which is a much smaller number. This is because you have to group the terms in the arrays, as an array of size X+1 & 10, when the terms are grouped, is equal to {10, 10(1)10}. If the terms are ungrouped, the array will be a linear array of size 10+1, which will be {10, 10+1(1)2}, which is just a linear array with size 11. One of the major challenges behind defining this level of BEAF would be successfully conceive a representation of triakulus that captures all its tritri entries while setting it apart from a dupertri, as a dupertri is an ungrouped 3^^^3 & 3, while triakulus is X^^^X & 3, where the X's are grouped. 
If it's well-defined, this would be the smallest number named by Jonathan Bowers which lies in the pentational-array range.Sbiis Saibian, Ultimate Finite Numbers List part 3 This is the first and the smallest Bowerism which is not well-defined. See Introduction to BEAF#Pentational arrays for more details of that level.
==Approximations==
Since it is not defined by now, the following approximation is just an expected value under climbing method (strong bound):
Notation 	Approximation
Bird's array notation 	\(\{3,3 [1 [1 [1 \backslash 2] 2 ¬ 2] 2] 2\}\)
Extended Cascading-E notation	E[3]3#^^#^^#3
= E[3]3#^^^#3
Fast-growing hierarchy 	\(f_{\varphi(\varepsilon_0,0)}(3)\)
Hardy hierarchy 	\(H_{\varphi(\varepsilon_0,0)}(3)\)
==Sources==
==See also==
ja:トリアクルス
Category:Numbers
Category:Numbers by Jonathan Bowers
Category:Higher array notation
Category:Greek-based name
Category:BEAF
Category:Powers of 3
Category:Binary phi level
Category:Unformalised