11/18/2011, 12:19 AM

Introduction to NEPT and NOPT structures

Username: MikeSmith

Email: aliwww@mail.com

Hi everybody at tetration forum. Thanks to Bo for all the administration help.

I hope that there are some new ways of looking at familiar things, and some of the material below can be understood and is reasonably clear.

Warning: The material is jam-packed with ideas, but quite difficult to check, so please take your time reading it, and if you’re reading this and new to hyperoperations you should try to work out 3^^3, 3^^^3, 3^^^^3 to get some intuition about the patterns I’ll be discussing below. Also, if you understand Graham’s number construction that can help as well.

Post or email any comments or queries as you wish.

Nept and Nopt structures (Part 1).

A NEPT structure is the representation of a large number as multi-layered Nested Exponentional Power Towers.

A NOPT structure is a generalisation of multi-layered nested power towers. Usually we think of power towers where the operation is exponentiation, but the concept of nested power towers could be used with other operations from the hyperoperation hierarchy as well as any others where height of power tower is well-defined and each power tower in the expression produces a natural number and the operation is a strictly increasing function on the natural numbers.

Other operations are possible for putting into nested power towers for example, such as (+, iterated addition) and (*, iterated multiplication) and (^^, tetration, iterated tetration) and so on. Considering very large numbers, other applications of NOPT structures are possible by using powerful operations such as the gi-sequence from Graham’s number construction where g1=(3)^4(3) (3 hexated to 3) and g64=Graham’s number. Even more powerful operations can be considered such as g-subscript towers where each “subscript power tower” has a well-defined height (of nested g-subscript symbols, ending with g1).

The canonical NOPT structure is the NEPT structure where the operation is exponentiation. The canonical relationship between NEPT and NOPT structures is that an OrderType of a NOPT structure can be well-defined by comparing with the NEPT structure of same order number. In other words, the OrderType of a NOPT structure borrows from the canonical NEPT structure related to some hyperoperation.

So the relationship can be seen like this:

Hyperoperation (n>=4) <-> NEPT (n>=4) <-> NOPT (n>=4)

For example, NOPT structures with OrderType 4 or 5 correspond to the appearance of tetration and pentation numbers when they are written into NEPT from (the form of Nested Exponential Power Towers).

NOPT structures with OrderType 4 or 5 are (the only) Linear NOPT structures.

Hexation NOPT structure has OrderType 6 and has 2-dimensional array structure.

Hepation NOPT structure has OrderType 7 and has (2,1)-dimensional array structure.

Octation NOPT structure has OrderType 8 and has (2,2)-dimensional array structure.

Nonation NOPT structure has OrderType 9 and has (2,2,1)-dimensional array structure. By zigzagging from bottom right corner, in left-up-left-up-left-up-… fashion we can build up NOPT structures with higher dimensions. The computation pathway proceeds from bottom right corner in left-up-left-up-left-up-… fashion until the top left corner.

Linear NOPT structures have Linear Ellipsis Structure.

Non-linear NOPT structures (OrderType>=6) have Multi-layered Ellipsis Structure.

The Ellipsis corresponding to the final top-left corner component of the NOPT structure is where the final computation is achieved. Superficially, it looks like a Linear Nopt structure, but looking at it carefully, notice that the Ellipsis length (of linearly arranged nested power towers) is given by a small number in Tetration and Pentation NOPT structures, but in larger NOPT structures is given by a Multi-layered Ellipsis expression, that is the combined effort of the rest of the interlinked components of the NOPT structure.

The induced Multi-layered Ellipsis Structure from a NOPT structure has a very similar structure to the well-known H-Fractal.

A NOPT-6 structure (hexation) has a 2-dimensional array structure and the induced Multi-layered Ellipsis Structure corresponds with simple H-Fractal at the first level.

A NOPT-7 structure (heptation) has a (2,1)-dimensional array structure and the induced Multi-layered Ellipsis Structure corresponds with the H-Fractal at the second level of resolution.

A NOPT-8 structure (octation) has a (2,2)-dimensional array structure and the induced Multi-layered Ellipsis Structure corresponds with the H-Fractal at the third level of resolution.

A NOPT-9 structure (nonation) has a (2,2,1)-dimensional array structure and the induced Multi-layered Ellipsis Structure corresponds with the H-Fractal at the fourth level of resolution.

You can observe that in terms of written symbolic requirements, going from NOPT-(I) to NOPT-(I+1) structure requires twice as many symbol expressions, due to the symbol folding that proceeds zigzagging L-U-L-U-L-U-L-U from bottom right corner to top right corner.

NOPT structures require a fixed SEEDVALUE, that has the roles of (1) either initiating a Linear NOPT Component OR (2) Controlling Ellipsis Length of a Component.

Username: MikeSmith

Email: aliwww@mail.com

Hi everybody at tetration forum. Thanks to Bo for all the administration help.

I hope that there are some new ways of looking at familiar things, and some of the material below can be understood and is reasonably clear.

Warning: The material is jam-packed with ideas, but quite difficult to check, so please take your time reading it, and if you’re reading this and new to hyperoperations you should try to work out 3^^3, 3^^^3, 3^^^^3 to get some intuition about the patterns I’ll be discussing below. Also, if you understand Graham’s number construction that can help as well.

Post or email any comments or queries as you wish.

Nept and Nopt structures (Part 1).

A NEPT structure is the representation of a large number as multi-layered Nested Exponentional Power Towers.

A NOPT structure is a generalisation of multi-layered nested power towers. Usually we think of power towers where the operation is exponentiation, but the concept of nested power towers could be used with other operations from the hyperoperation hierarchy as well as any others where height of power tower is well-defined and each power tower in the expression produces a natural number and the operation is a strictly increasing function on the natural numbers.

Other operations are possible for putting into nested power towers for example, such as (+, iterated addition) and (*, iterated multiplication) and (^^, tetration, iterated tetration) and so on. Considering very large numbers, other applications of NOPT structures are possible by using powerful operations such as the gi-sequence from Graham’s number construction where g1=(3)^4(3) (3 hexated to 3) and g64=Graham’s number. Even more powerful operations can be considered such as g-subscript towers where each “subscript power tower” has a well-defined height (of nested g-subscript symbols, ending with g1).

The canonical NOPT structure is the NEPT structure where the operation is exponentiation. The canonical relationship between NEPT and NOPT structures is that an OrderType of a NOPT structure can be well-defined by comparing with the NEPT structure of same order number. In other words, the OrderType of a NOPT structure borrows from the canonical NEPT structure related to some hyperoperation.

So the relationship can be seen like this:

Hyperoperation (n>=4) <-> NEPT (n>=4) <-> NOPT (n>=4)

For example, NOPT structures with OrderType 4 or 5 correspond to the appearance of tetration and pentation numbers when they are written into NEPT from (the form of Nested Exponential Power Towers).

NOPT structures with OrderType 4 or 5 are (the only) Linear NOPT structures.

Hexation NOPT structure has OrderType 6 and has 2-dimensional array structure.

Hepation NOPT structure has OrderType 7 and has (2,1)-dimensional array structure.

Octation NOPT structure has OrderType 8 and has (2,2)-dimensional array structure.

Nonation NOPT structure has OrderType 9 and has (2,2,1)-dimensional array structure. By zigzagging from bottom right corner, in left-up-left-up-left-up-… fashion we can build up NOPT structures with higher dimensions. The computation pathway proceeds from bottom right corner in left-up-left-up-left-up-… fashion until the top left corner.

Linear NOPT structures have Linear Ellipsis Structure.

Non-linear NOPT structures (OrderType>=6) have Multi-layered Ellipsis Structure.

The Ellipsis corresponding to the final top-left corner component of the NOPT structure is where the final computation is achieved. Superficially, it looks like a Linear Nopt structure, but looking at it carefully, notice that the Ellipsis length (of linearly arranged nested power towers) is given by a small number in Tetration and Pentation NOPT structures, but in larger NOPT structures is given by a Multi-layered Ellipsis expression, that is the combined effort of the rest of the interlinked components of the NOPT structure.

The induced Multi-layered Ellipsis Structure from a NOPT structure has a very similar structure to the well-known H-Fractal.

A NOPT-6 structure (hexation) has a 2-dimensional array structure and the induced Multi-layered Ellipsis Structure corresponds with simple H-Fractal at the first level.

A NOPT-7 structure (heptation) has a (2,1)-dimensional array structure and the induced Multi-layered Ellipsis Structure corresponds with the H-Fractal at the second level of resolution.

A NOPT-8 structure (octation) has a (2,2)-dimensional array structure and the induced Multi-layered Ellipsis Structure corresponds with the H-Fractal at the third level of resolution.

A NOPT-9 structure (nonation) has a (2,2,1)-dimensional array structure and the induced Multi-layered Ellipsis Structure corresponds with the H-Fractal at the fourth level of resolution.

You can observe that in terms of written symbolic requirements, going from NOPT-(I) to NOPT-(I+1) structure requires twice as many symbol expressions, due to the symbol folding that proceeds zigzagging L-U-L-U-L-U-L-U from bottom right corner to top right corner.

NOPT structures require a fixed SEEDVALUE, that has the roles of (1) either initiating a Linear NOPT Component OR (2) Controlling Ellipsis Length of a Component.