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Topic: BIG NUMBERS #1
Replies: 19   Last Post: Feb 11, 2003 9:59 AM

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anonymous

Posts: 2,580
Registered: 12/4/04
Re: BIG NUMBERS #1
Posted: Feb 5, 2003 4:12 PM
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On 04 Feb 2003, C .Stevens wrote:
>Dave:I read with great interest your post on BIG NUMBERS.I made a (curious) construction based on a similiar construction I saw concerning Graham's #.My little construction is based on Conway's chained arrows notation and the colossi 3->3->2->2->2
>

>>>>>>>>>>>>>>>>>>>>>>>>>=3->3->2->2->2!
>^
>^
>^ 1)27 { {1)27 {1)27 } }
>^ 2)3^..1)..^3 { {2)3^..1)..^3 {2)3^..1)..^3 } }
>^ . {.. { . { . } }
>^ . { . { . { . }27 }
>^ . { . { . { . } levels}
><<<........... { ..< .....levels < .....levels } }
>\________________________________________________/ }
> | }
> . }
> . }
> . }

>>>>>>>>>>>>>>>>>>>>>>>>>. }
>^ }
>^ }
>^ 1)27 { {1)27 {1)27 } }
>^ 2)3^..1)..^3 { {2)3^..1)..^3 {2)3^..1)..^3 } }
>^ 3)3^..2)..^3 {.. {3)3^..2)..^3 {3)3^..2)..^3 } >3^27
>^ . { . { . { . } } levels
>^ . { . { . { . >27 }
>^ . { . { . { . } levels}
><<<<.....BAYS { ..< ......levels < .....levels } }
> { { { } }
> { { { } }
>\________________________________________________/ }
> | }

>>>>>>>>>>>>>>>>>>>>>>>>>| }
>^ }
>^ }
>^ 1)27 { {1)27 {1)27 } }
>^ 2)3^..1)..^3 { {2)3^..1)..^3 {2)3^..1)..^3 } }
>^ 3)3^..2)..^3 {.. {3)3^..2)..^3 {3)3^..2)..^3 } }
>^ . { . { . { . >27 }
>^ . { . { . { . } levels}
>^ . { . { . { . } }
><<<......Bays { ..< ......levels < ......levels} }
> { { { } }
>\________________________________________________/ }
> | }
> 27 bays }
>
>Notes:Reads from right to left, bottom to top with the final amount of arrows top left. Why start with 27? I calculated this using Robert Munafo's c3(N):chain: function at http://home.earthlink.net/~mrob/pub/math/ln-notes1.html


>
>The top of each "bay" should really read 1)27
> 2)3^..^3 with 1) arrows,in other words 27 arrows and so forth on down the bay vertically.
>
>Thanks to Munafo's function above,I was able to make a rough estimate of 3->3->2->4 and worked from there. One thing about 3->3->2->2->2 is,three steps into the expansion,it changes into a 4-number notation similiar to 3->3->2->4 and I worked from there.
>Having said all this,the construction above may very well be wrong.I would appreciate anyone posting if they find an error (or errors!).One thing for sure,3->3->2->2->2 is one "BIG NUMBER"!
>
>
>
>
>
>
>

Whoops.screwed the pooch again.Another try at 3->3->2->2->2:

1)27 { {1)27 {1)27 } }
2)3^..1)..^3 { {2)3^..1)..^3 {2)3^..1)..^3 } }
3)3^..2)..^3 { {3)3^..2)..^3 {3)3^..2)..^3 } }
. ... . . . . . .
. . . . . . . > 27
. . . . . . . . levels.
3->3->2->2->2 < ....< .....levels < ....levels } }
{ { { } }
\_________________________________________________/ }
| .
. .
. .
. .
. .
>>>>>>>>>>>>>>>>>>>>>>>>>>bays .
^ }
^ }
^ 1)27 { {1)27 {1)27 } }
^ 2)3^..1)..^3 { {2)3^..1)..^3 {2)3^..1)..^3 } }
^ 3)3^..2)..^3 { {3)3^..2)..^3 {3)3^..2)..^3 } }
^ . ... . . . . . .
^ . . . . . . . >27 >(3^27)-4
^ . . . . . . . .levels . levels
<<<<....bays < ....< .....levels < ....levels } }
\__________________________________________________/ }
| }
>>>>>>>>>>>>>>>>>>>>>>>>>>>bays }
^ }
^ }
^ 1)27 { {1)27 {1)27 } }
^ 2)3^..1)..^3 { {2)3^..1)..^3 {2)3^..1)..^3 } }
^ 3)3^..2)..^3 { {3)3^..2)..^3 {3)3^..2)..^3 >27 }
^ . ... . . . . . levels.
^ . . . . . . . . .
^ . . . . . . . . .
<<<<......bays < ....< .....levels < .....levels . .
\__________________________________________________/ }
| }
27 bays }

< or > alone are meant to be part of a long vertical }. <<<<< or >>>> are just long arrows. "......." are meant to mean etc.,etc..
This diagram is just an expansion horizontally and vertically of
a diagram of Graham's # thusly:

1) 3^^^^3 }
2) 3^..^3 where the arrows are }
represented by the }
# in 1) }
3) 3^..^3 where the arrows are }
represented by the }
# in 2) > 64 layers
4) ............................}
. .
. .
. .
. .
Graham's number! }
Although Graham's number is vast, it is still dwarfed by a 5 number
chained arrow construction!


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