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Googology Wiki

Graham Array Notation[1] is a googological notation devised by Antares Harrison. It was designed as a notation able to express Graham's number, though the later parts of the notation are meant for other googolisms.

This notation[2] has been defined until Operator Dimensional arrays.

At the core of this notation is Arrow notation and BEAF.


"Arrays" are lists of numbers. An example is \([3,3,4,6,5]\)

Arrays, like BEAF, BAN or HAN, can be multidimensional or multirowed.


First, we must set some set-definitions.

1. @ means an array.

2. # means an operator.

3. M is the "Main".

4. C is the "Controller".

Now, we'll set some base rules.

  • 1. The default bracket is \([ ]\). An array must be inside \([ ]\).
  • 2. If 1 is the second entry the 1 and anything to the right of it are cropped off. \([a,1,@]\) = \([a]\)
  • 3. Any ones that are not the first or second entry may be deleted.
  • 4. An empty bracket \([ ]\) may be deleted. (IF there is one)
  • 5. The same rules for ones applies for zeroes as well. (Yes, we treat zeroes as a valid entry before destroying them)

Here's an important rule.

  • Let there be an array \([3!2,1,3!1,2]\). Here, we must calculate the @ mark before calculating the comma marks.

BEAF defines a pilot as the first non-1 entry after the prime. Here, all ones are cropped off at the beginning. So we will just define a "main" as the second-to-last entry in the array. The "Controller" is the last entry in the array.

  • An example : \([3,3,3,3,4,[ ],1!4]\) = \([3,3,3,3,4!4]\) = \([[3,3,3,3,4]!4]\)

Simple Linear Arrays

We'll start with 4 or less entry arrays

  • \([a]\) = a
  • \([a,b]\) = a↑b
  • \([a,b,c]\) = a{c}b or a↑cb (Using Knuth's Up-Arrows and BEAF)

BEAF starts degenerating arrays at 4 entries. However, the notation needs to keep the namesake, so we'll start degeneration at 5 entries.

\([a,b,c,d]\) is somewhat simple, you start with a↑...↑b (c arrows) and the next step is a↑...↑b with "a↑...↑ b (c arrows)", and keep this continuing until d layers.

So, we'll introduce some comparisons.

Advanced Linear Arrays


Remember the @(Array)? Recognize the Main(m) and the Controller(c)? Well we'll start off by introducing this rule.

\([@,m,c]\) = \([@,m-1,[@,m,c-1]]\)

The c is transformed into the same array where the c is decreased by 1. The main array's m decreases by 1.

Here's an example.

\([3,3,3,3,3]\) = \([3,3,3,2,[3,3,3,3,2]]\)

Future arrays

Although it has been stated that the notation is a work in progress, it has been said that the creator is currently working at extending the notation to Dimensional arrays, Tetrational Arrays, Pentational Arrays, Operator, Linear Dimensions, Nested Dimensions, Brackets and Beyond Nested Brackets stages.

Connection with other G(n) related functions

\(G(64)\) in Graham's Function is exactly \([3,3,4,64]\).

So[3] \(G(n)\) in Graham's Function is \([3,3,4,n]\). Graham's Number is \([3,3,4,64]\) because this is 3↑↑↑↑3 repeated for 64 layers, or \(G(64)\).

In the same way the Little Graham function would be \([2,3,7,n]\) in \(F(n)\).

In the same way the Graham-Conway number function would be \([4,4,4,n]\) in \(GC(n)\).


External links