# BEAF

BEAF, or Bowers Exploding Array function, is an extremely fast-growing function.

## Defination

The base, or b, is the first entry in the array.

The prime, or p, is the second entry in the array.

The pilot is the first non-1 entry after p.

The copilot is the entry before the pilot. It is sometimes p.

A previous entry is entries before the pilot, but is on the same row as the pilot; a previous row is rows before the pilot, but is on the same plane as the pilot; and so on.

1. Base rule: If any entry that exist after p is 1, then the value of the array (called $$v(A)$$) is $$b^p$$.

2. Prime rule: If p is 1, then $$v(A)=1$$.

3. Catastrophic rule: If Rules 1-2 don't apply, then pilot decreases by 1, copilot becomes the original array but with p decreased by 1, and every previous structure $$X^n$$ is filled with a $$\underbrace{b \times b \cdots \times b}_n$$ hypercube.

## Examples

\begin{eqnarray*} \{3,3,3,3\} &=& \{3,\{3,2,3,3\},2,3\} \\ &=& \{3,\{3,\{3,1,3,3\},2,3\},2,3\} \\ &=& \{3,\{3,3,2,3\},2,3\} \\ &=& \{3,\{3,\{3,2,2,3\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,\{3,1,2,3\},1,3\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,1,3\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,2,1,3\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,3,\{3,1,1,3\},2\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,3,3,2\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,\{3,\{3,3,1,2\},1,2\},2,2\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,\{3,\{3,3,\{3,2,1,2\}\},1,2\},2,2\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,\{3,\{3,3,\{3,3,\{3,1,1,2\}\}\},1,2\},2,2\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,\{3,\{3,3,\{3,3,3\}\},1,2\},2,2\},2\},1,3\},2,3\} \\ &=& \{3,\{3,\{3,3,\{3,\{3,3\uparrow^{3\uparrow\uparrow\uparrow3}3,1,2\},2,2\},2\},1,3\},2,3\} \end{eqnarray*}

\begin{eqnarray*} \{3,3(1)3,3\} &=& \{3,3,3(1)2,3\} \\ &=& \{3,\{3,2,3(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,3,2(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,2,2(1)2,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,3(1)2,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,3,3(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,2,3(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,3,2(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,2,2(1)1,3\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3(1)1,3\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,2(1)1,3\},2\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,3(1)3,2\},2\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,3,3(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,2,3(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,3,2(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,2,2(1)2\}(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,3(1)2\}(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,3,3\}(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,3\uparrow\uparrow\uparrow3(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{\underbrace{3,\cdots,3}_{3\uparrow\uparrow\uparrow3}\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\} \end{eqnarray*}

## Hyperdimentional arrays

$$\{3,3(0,1)2\}=???$$

What does (0,1) even mean? It is a $$X^X$$ structure, or $$X^p$$. So we can replace (0,1) with (p), or (3). The process is something like diagonalization, but it isn't.

\begin{eqnarray*} \{3,3(0,1)2\} &=& \{3,3(3)2\} \\ &=& \{3,3,3(1)3,3,3(1)3,3,3(2)3,3,3(1)3,3,3(1)3,3,3\} \end{eqnarray*}

You get the idea how big this number is.