User blog:GamesFan2000/Hyper Tower Notation, Pt. 1 (Inspired by Hyper-E notation)

'''Hyper Tower Notation is an exponentiation-based notation. The expressions of this notation will decompose into power-towers.'''


 * 1) ...n#1=...n
 * 2) ...n#m+1=...n^(...n#m)
 * 3) ...n#k+1m+1=...n#kn#k...n n’s...n#kn^(...n#k+1m); #1=#
 * 4) ...a#mn#k+11=...a#mn#kn#k...n n’s...n#kn^(...a#mn); n#k+11=n#kn#k...n n’s...n#kn

Let’s look at some examples:

4#3=4^(4#2)=4^4^4

5#3#3=5#3^(5#3#2)=5#3^(5#3^(5#3))=5#3^(5#(3^5^5^5))=5#3^5^5^...3^5^5^5 5’s...5^5

4#4#4#3=4#4#4^(4#4#4#2)=4#4#4^(4#4#4^(4#4#4))

'''n#m is basically n^^m, and each new sign and term added roughly moves the expression up a hyper-operator level. n#n#n…#n#n roughly corresponds to recursion level omega(w) in the FGH.'''

4#23=4#4#4#4^(4#22)=4#4#4#4^(4#4#4#4^(4#4#4#4))

4#24#4=4#24^(4#24#3)=4#24^(4#24^(4#24^(4#24)))

4#24#2=4#24#4#4#4^(4#24#4#4#4^(4#24#4#4#4^(4#24)))

4#33=4#24#24#24^(4#24#24#24^(4#24#24#24))

'''Because of the fact that every # added will increase the recursion level, I’d say that it’s safe to assume that n#3n roughly corresponds to recursion level w^2. As such, the limit of n#nn is w^w.'''