User blog:Tetramur/Possible explanations to tetrational arrays on BEAF

Hello to all! I want to introduce my possible explanation of tetrational arrays of BEAF. I'll provide some examples, of course. If you have something to say to me, I'm always glad to hear y'all. Let's start!

Part 1 - Small arrays
First of all, I want to look at relatively small array, which is array for Bowers' goplexulus - {10,100 ((1)1) 2}. What is it? It is a stage 100, if stage 1 is second gongulus (100-dimensional hypercube with sidelength 100 filled in with 10's from just one 10), stage 2 is bongulus, stage 3 is trongulus etc.

What is {10,100 ((1)(1)1) 2}? It is giplexulus - and it is array that grows from the array of goplexulus with the same operations as getting the entire goplexulus from one 10 alone. We'll iterate this operation.

So, we have:

Stage 1 - {10,100 ((1)1) 2}

Stage 2 - {10,100 ((1)(1)1) 2}

Stage 3 - {10,100 ((1)(1)(1)1) 2}

...

Stage 100 - {10,100 ((2)1) 2} - boplexulus.

Whoops, this is stage 100 - and we are on a plane separator!

But we iterate further - from one 10 into boplexulus, from boplexulus into biplexulus etc.

Stage 1 - {10,100 ((2)1) 2}

Stage 2 - {10,100 ((2)(2)1) 2} - biplexulus

Stage 3 - {10,100 ((2)(2)(2)1) 2} - baplexulus

...

Stage 100 - {10,100 ((3)1) 2} - troplexulus.

So, we ran into only two groups. Let's see what happens with 99th group...

Stage 1 - {10,100 ((99)1) 2}

Stage 2 - {10,100 ((99)(99)1) 2}

Stage 3 - {10,100 ((99)(99)(99)1) 2}

...

Stage 100 - {10,100 ((100)1) 2} = {10,100 ((0,1)1) 2}

Uh-oh, THIS is enormously huge goduplexulus!

Part 2. From goduplexulus to higher arrays...
This is complicated because when we multipicate, for example, 100th degree with itself, it'll produce only 200th degree. But we are not scared with difficulties, right?

When we raise 100^100^100 into 100^100th degree, we only get 100^100^(100*2). And we must iterate until multiplier in the third level raises to 100. That is only one group. The exponent in fourth level now has value 2. Second group raises this multiplier to 3, ..., 99-th group raises it to 100. Now we have four levels, each is equal to 100 - and we want to expand this to five levels. How will we do this?

1. Raise 100^100^100^100 to 100^100^100th degree. Multipler 2 now is "sitting" in third level.

2. Continue raising until multipler reach 100. Now we have 100^100^100^101.

3. Raise to 100^100^101th degree (all exponent minus first 100, this is important)...

4. Continue... have 100^100^100^102.

5. Raise... continue... 100th iteration of this process will shift 2 into fourth level (100^100^100^(100*2))

6. Continue... we have 100th iteration of THIS (points 1-5) and multiplier will increase to 3.

7. 99-th iteration of this will make fifth level to appear, but it has value only 2. In BEAF, the whole array is {10,100 ((0,100)1) 2} = {10,100 ((0,0,1)1) 2}

8. Repete steps 1-7 98 times - and we finally have array for gotriplexulus - {10,100 (((1)1)1) 2}.

We can define the sixth, seventh level of exponentiation... by the same way. The limit of this is epsilon-zero in FGH - and with tetrational arrays we can't define anything higher, we need pentational arrays. ​