After all those months of absence, I'm back! Well, not in the wiki editing stuff like a Gongulus but back into googology.

I've been into Computer Coding (Javascript), Web Developing, and yes, Studies in other branches of Nature. (Astrophysics and/or Chemistry)

I finally put together my long-abandoned notation. Today I'll just introduce Linear Arrays, as I'm busy.

Eh-hem. Let's start.

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 behind it are cropped off. [a,1,@] = [a]

  This must take place before the array is calculated.

3. Any repeated 1s are merged into 1. The operators connecting the ones are deleted. [1,1,1,1,1] = [1] [@#1,1,1,1,] = [@#1]

  This must take place before the array is calculated.

4. Any ones that are not the first or second entry may be deleted.

  This must take place before the array is calculated.

5. An emptry bracket [] may be deleted.

  This must take place before the array is calculated.


Here's an important rule.

Let there be an array [3#2,1,3#1,2]. Here, we must calculate the comma-connected numbers (2,1,3 and 1,2) before we calculate the # mark.

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.

We'll start with 1/2/3/4 entries.

[a] = a [a,b] = ab [a,b,c] = a^^...^^b where there are c up-arrows. (Using Knuth's Up-Arrows)

BEAF starts degenerating arrays at 4 entries. However, we need 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)". You keep this continuing untill d layers.

So, we'll introduce some comparisions.

[5] = 5 [3,3,3] = Tritri [3,3,4,64] = Graham's Number [2,3,12,7] = Little Graham [10,100] = Googol

Simple (Until now, until now)

We'll get into degenerating.


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

[@,m,c] = [@[@,m-1,c],c-1]

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

Here's an example.

[3,3,3,3,3] = [3,3,3,[3,3,3,2,3],2] = [3,3,3,[3,3,3,[3,3,3,1,3],2],2] = [3,3,3,[3,3,3,[3,3,3,3],2],2]

We'll say [3,3,3,3] = a

= [3,3,3,[3,3,3,a,2],2] = [3,3,3,[3,3,3,[3,3,3,a-1,2],1],2] =[3,3,3,[3,3,3,[3,3,3,a-1,2]],2]

And it continues.

= [3,3,3,[3,3,3,[3,3,3,[3,3,3,a-2,2],1]],2] = [3,3,3,[3,3,3,[3,3,3,[3,3,3,a-2,2]]],2]

It keeps this pattern until "a" becomes 1. And well all know "a" is ridiculously large in itself!

So, if the a become 1, the a will be cropped off and if you crop all the 1 from the Controller 2, you will get

= [3,3,3,[3,3,3,Big Number],2] = [3,3,3,[Bigger Number],2] = [3,3,3,[3,3,3,Bigger Number-1,2],1] = [3,3,3,[3,3,3,Bigger Number-1,2]]

So the final thing we get is this.

"Starting from tritri, create "a 3^^...^^3 with tritri arrows", create a 3^^...^3 with "a 3^^...^^3 with tritri arrows", etc etc etc. Do this [3,3,3,Bigger Number-1,2] Times.


6 arrays are basically much the same, but it (Obviusly) takes more time in degenerating.

Same for 7+ Arrays.

We'll introduce Dimensional arrays (+ Possibly Tetrationals ones too) in the next post.

Let's end this with an example.

[3,3,3,3,4,[],1#4] = [3,3,3,3,4#4] = [[3,3,3,3,4]#4]



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