This array notation was created by Jonathan Bowers, and is known as the Bowers's operators. It was created to help represent very large numbers, and was first published to the web in 2002.
Let , the hyperoperation (see Square bracket notation, this is the same as , they are just different notations of hyperoperation). That is
The function means , i.e. is equal to for every ∈ .
The first operator is and it is defined:
Bowers calls the function "m expanded to n".
Thus, we have
The number inside the brackets can change. If it's two
Bowers calls the function "m multiexpanded to n".
Operators beyond can also be made, the rule of it is the same as hyperoperation:
Bowers continues with names for higher operations:
is "m powerexpanded to n"
is "m expandotetrated to n"
The next level of operators is , it to behaves like is to .
and beyond will work similarly.
Bowers continues to provide names for the functions:
is "m exploded to n"
is "m multiexploded to n"
is "m powerexploded to n"
is "m explodotetrated to n"
and beyond will follow similar recursion. Bowers continues with:
is "m detonated to n"
is "m pentonated to n"
For every fixed positive integer , there is an operator with sets of brackets. The domain of is , and the codomain of the operator is .
Another function means , where is the number of sets of brackets. It satisfies that for all integers , , , and . The domain of is , and the codomain of the operator is .
Pentetrical operators and beyond
Bowers generalizes this towards 5+ entries with the following ruleset:
- if there are 0 entries (like Conway chained arrow notation, the value of the empty chain is 1)
- if there are only 1 or 2 entries
- if the last entry is 1
- if the second entry is 1
- if the 3rd entry is 1
- if none of the above rules apply
Bowers does not provide operator names for and beyond, but he describes a notation for up to 8 entries:
"a,b, and c are shown in numeric form, d is represented by brackets (as seen above), e is shown by [ ] like brackets, but rotated 90 degrees, where the brackets are above and below (uses e-1 bracket sets), f is shown by drawing f-1 Saturn like rings around it, g is shown by drawing g-1 X-wing brackets around it, while h is shown by sandwiching all this in between h-1 3-D versions of [ ] brackets (above and below) which look like square plates with short side walls facing inwards."
Numbers like TREE(3) are unattainable with Bowers's operators, but Graham's number lies between and .