User blog:Antares.I.G.Harrison/Dimensional Graham Array Notation

Eh-hem. Here we are. dimensional arrays.

Before we discuss on that matter, I'll going to change the base rule in the Notation.

[a,a,a,a,a] used to be [a,a,a,[a,a,a,a-1,a],a-1]. Now it's [a,a,a,a-1,[a,a,a,a,a-1]]. That way the array would degenerate towards d in [a,b,c,d], creating a larger array.

Let's get into it.

First we need to put things into the second row.

[a,b,c] is the first row and [x,y,z] is the second row.

Heck, why not bring a third row [l,m,n]?

Hmmm. We'll need to degenerate those ararys into a single entry. So

The first row is a↑b (c Arrows). Let's call this A

The second row is x↑y (z Arrows) Let's call this B

The third row is l↑m (n Arrows) Let's call this C

Now we need to define a function. [a^b] = [a,a,a,.....,a,a,a] with b a's. [a^^b] = [a^a^a....a^a^a] with b a's

We'll get back to the single entries.

Defination : The above array is [A^[B^C]]

In Planar arrays, we'll use the ^ mark.

Bam! 2-Dimensional Array notation.

To continue beyond this point, we'll need to create a dimensional Mark, like {b,p(0,1)2} in BEAF.

 is the notation we'll use. c is the number used to build the array. A is the lenght and B is the dimension. But have we not defined higher dimensions yet?

For third dimensions, we'll going to use the ^^ mark. Again, let's have an example. <3,3/3>. It's a 3x3 cube of 3's. Now, we need to calculate the arrays


 * 3,3,3|


 * 3,3,3|


 * 3,3,3|

first, than move on to the next dimension. These are [Tritri^[Tritri^Tritri]]. Eh-hem. A pretty big number. Let's call that L.

Because the width of the cube is 3, The final product is [L^^[L^^L]]. A pretty larger number.

What about 4 dimensions? <4,4/4>?

You probably know what that number is by now. It's [<4,3/4>^^^<4,3/4>^^^<4,3/4>^^^<4,3/4>].

Now we can re-write this argument :

[] = [^^...^^^^...^^.........^^...^^] with n-1 's

Bam Bam! Dimensional Array notation.