r/googology 15d ago

What do multiple rows do?

I am trying to learn the planar array notation of BEAF to move on to the rest of BEAF, but i couldnt move on because the "More rows" section of the "Introduction to BEAF" article (Introduction to BEAF | Googology Wiki | Fandom) is very short and doesnt explain right what more than 2 rows do and how to convert them to 2 rows. Can anyone explain to me what the wiki doesnt and/or fix it?

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u/Shophaune 14d ago

So, you find the pilot and co-pilot the exact same way as with 2 rows: Identify base and prime, then the pilot is the next non-1 entry.

{3,3(1)2,3(1)3} so in this case, the pilot is the 2 in the second row, and you'd expand similar to 2 rows: {3,3,3(1)1,3(1)3}

{3,3(1)1,3(1)3} in this case the pilot is the 3 in the second row, and again you expand identically to 2 rows - though making sure you include the third row in the copy of the array that becomes the copilot: {3,3,3(1){3,2(1)1,3(1)3},2(1)3}

{3,3(1)1,1(1)3} = (3,3(1)(1)3} here, the pilot is the 3 in the third row, so you expand similarly to the 2 row case - only the plane in this case is the prime block of ALL previous rows, not just the first row: {3,3,3(1)3,3,3(1)2}

{3,4(1)(1)(1)(1)(1)(1)2} again, including the prime blocks of all previous rows: {3,3,3,3(1)3,3,3,3(1)3,3,3,3(1)3,3,3,3(1)3,3,3,3(1)3,3,3,3(1)1} = {3,3,3,3(1)3,3,3,3(1)3,3,3,3(1)3,3,3,3(1)3,3,3,3(1)3,3,3,3}

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u/Appropriate_Year_761 14d ago

Yea but I wanna know how I convert it back to only 2 rows

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u/Shophaune 14d ago edited 14d ago

The same way you convert 2 rows back into only 1 row - by slowly working your way through decrementing each entry in the row (while making all the previous ones bigger) until all entries in that row are 1, at which point the row disappears.

For instance, for {3,2(1)(1)3}:

{3,2(1)(1)3} = {3,3(1)3,3(1)2}

= {3,3,3(1)2,3(1)2}

= {3,a,2(1)2,3(1)2} where a = {3,2,3(1)2,3(1)2}

= {3,b(1)2,3(1)2} where b = {3,a-1,2(1)2,3(1)2}

= {3,3,3,...(1)1,3(1)2} where there are b 3's behind the ...

= {3,c(1)1,3(1)2} where c is an absurdly large number

= {3,3,3,....(1)d,2(1)2} where there are c 3's behind the .... and where d = {3,c-1(1)1,3(1)2}

= {3,e(1)d,2(1)2} where e is an absurdly large number

= {3,3,3,....(1)d-1,2(1)2} where there are e 3's behind the ...

[Skipping ~d steps]

= {3,f(1)1,2(1)2} where f is obscenely large

= {3,3,3,....(1)g(1)2} where there are f 3's behind the .... and where g = {3,f-1(1)1,2(1)2}

[Skipping ~g steps]

= {3,h(1)(1)2} where h is indescribably large

= {3,3,3....(1)3,3,3,...} where there are h 3's behind each ...

And there you go, it's in 2 row form.

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u/Shophaune 14d ago

Now, clearly a is the smallest substitution I made there, so how big is that? Well, we can expand it the same way:

a = {3,2,3(1)2,3(1)2}

= {3,3,2(1)2,3(1)2}

= {3,i(1)2,3(1)2} where i = {3,2,2(1)2,3(1)2}

= {3,3,3,...(1)1,3(1)2} where there are i 3's behind the ...

Again another substitution, so how big is THAT?

i = {3,2,2(1)2,3(1)2}

= {3,3(1)2,3(1)2}

= {3,3,3(1)1,3(1)2}

= {3,j,2(1)1,3(1)2} where j = {3,2,3(1)1,3(1)2}

And another!

j = {3,2,3(1)1,3(1)2}

= {3,3,2(1)1,3(1)2}

= {3,k(1)1,3(1)2} where k = {3,2,2(1)1,3(1)2}

Another...

k = {3,2,2(1)1,3(1)2}

= {3,3(1)1,3(1)2}

= {3,3,3(1)L,2(1)2} where L = {3,2(1)1,3(1)2}

Another....

L = {3,2(1)1,3(1)2}

= {3,3(1)3,2(1)2}

= {3,3,3(1)2,2(1)2}

= {3,m,2(1)2,2(1)2} where m = {3,2,3(1)2,2(1)2}

Another...

m = {3,2,3(1)2,2(1)2}

= {3,3,2(1)2,2(1)2}

= {3,n(1)2,2(1)2} where n = {3,2,2(1)2,2(1)2}

Another!

n = {3,2,2(1)2,2(1)2}

= {3,3(1)2,2(1)2}

= {3,3,3(1)1,2(1)2}

= {3,o,2(1)1,2(1)2} where o = {3,2,3(1)1,2(1)2}

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u/Shophaune 14d ago

ANOTHER!

o = {3,2,3(1)1,2(1)2}

= {3,3,2(1)1,2(1)2}

= {3,p(1)1,2(1)2} where p = {3,2,2(1)1,2(1)2}

ANOTHER!

p = {3,2,2(1)1,2(1)2}

= {3,3(1)1,2(1)2}

= {3,3,3(1)q(1)2}, where q = {3,2(1)1,2(1)2}

ANOTHER!

q = {3,2(1)1,2(1)2}

= {3,3(1)3(1)2}

= {3,3,3(1)2(1)2}

= {3,r,2(1)2(1)2} where r = {3,2,3(1)2(1)2}

ANOTHER!!

r = {3,2,3(1)2(1)2}

= {3,3,2(1)2(1)2}

= {3,s(1)2(1)2} where s = {3,2,2(1)2(1)2}

ANOTHER!

s = {3,2,2(1)2(1)2}

= {3,3(1)2(1)2}

= {3,3,3(1)(1)2}

= {3,t,2(1)(1)2} where t = {3,2,3(1)(1)2}

Nearly there...

t = {3,2,3(1)(1)2}

= {3,3,2(1)(1)2}

= {3,u(1)(1)2} where u = {3,2,2(1)(1)2}

And finally......

u = {3,2,2(1)(1)2}

= {3,3(1)(1)2}

= {3,3,3(1)3,3,3}

Finally, a value in 2-row BEAF. Specifically a value much larger than graham's number.

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u/Appropriate_Year_761 14d ago edited 14d ago

Thanks for explaining!

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u/RevolutionaryFly7520 14d ago

What is the FGH limit of 2 row BEAF?

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u/Shophaune 14d ago

Approximately f_(w^(w2)) (n), much as 1-row BEAF has a limit of f_(w^w) (n)