11/18/2011, 12:51 AM

Nept and Nopt structures (Part 3).

HEFTY NOPT Structures

H-(Ellipsis)-Fractal Type Nested Operational Power Towers.

Transitional numbers between finite and infinite realm.

Keywords: hyperoperations, seed values (starting and controlling), linear nopt structures, gi-sequence from Graham’s number, g-subscript towers.

Summary: NOPT structures give a standard approach to looking at the bizarre world of numbers that straddle the finite and infinite divide. The induced Multi-layered Ellipsis Structure from a NOPT structure has similar component-connection structure to the well-known H-Fractal, given a corresponding degree of resolution.

Moreover, The H-fractal structure is emergent from the NOPT structure and guides the multi-layered nestedly embedded computational pathways. In the representations I give (see the pictures below) the computation starts at the bottom right corner and moves towards the top left corner.

Caveat: This mathematical material is very abstract and not practically computable on any computer, if accurate magnitude is a desired goal, because it’s in the area of hyperoperations, the boundary of practical maths.

When the operation is not specified, NOPT structures are floaty and esoteric in nature, but nevertheless they still say something about the patterns arising from unlimited multilayered nested recursion, with SeedValue being the standard stage level phase transition marker.

A comparison can be made with Hereditary Base n, where the base value is used at other exponent levels apart from the level of standard positional notation, resulting in a treelike number structure. Perhaps, another comparison can be made with CantorNormalForm for infinite ordinals.

In the pictures below:

The NOPT structure we consider below and is shown in the 4 pictures below is the nonation NOPT structure, remembering that OrderType=9 is derived from considering the corresponding NEPT structure arising from the nonation hyperoperation.

All of the “formal power towers” in the NOPT structure are given the symbol “theta” (Note: by “formal” expression I mean an unevaluated symbolic expression that could be evaluated given more information).

When the SeedValue is supplied, in theory, the power tower can be evaluated because the height information about the power tower is given, then each succeeding power tower in the linear Nopt structure can be evaluated with each power tower evaluation supplying the height of the next power tower to be evaluated.

In the first picture the SeedValues have the form “theta ) n”

In the second picture the SeedValues have the form “n”.

Sometimes, we may want to ensure that the next linear component in the NOPT structure has nontrivial ellipsis value and so we use the first style of NOPT structure.

However, I think that for a standard definition of NOPT structure it is better to use the second style of NOPT structure where SeedValue=n.

By some kind of careful inspection argument, one can notice ellipsis values that are the same magnitude within a NOPT structure, for example where the ellipsis is equal to SeedValue (a controlling seed value, not a starting seed value).

(Remember, starting seed values inform the height of the adjacent power tower, theta, in the linear NOPT component. Controlling seed values give the length of an ellipsis within a component of the NOPT structure.)

Also, by careful inspection argument, one can order ellipsis lengths by increasing order of magnitude. Although it is very difficult to quantify “how much” longer one ellipsis is compared to another, the magnitudes increase very quickly as is the nature of fast-growing functions. The idea of visualising the ellipsis lengths in increasing magnitude order has been color-coded in Picture 3.

Finally, Picture 4 shows the induced Multi-layered Ellipsis Structure from a NOPT structure and is color-coded in the same way as in Picture 3. The basic structural pattern of the well-known H-Fractal can be seen somewhat clearly.

HEFTY NOPT Structures

H-(Ellipsis)-Fractal Type Nested Operational Power Towers.

Transitional numbers between finite and infinite realm.

Keywords: hyperoperations, seed values (starting and controlling), linear nopt structures, gi-sequence from Graham’s number, g-subscript towers.

Summary: NOPT structures give a standard approach to looking at the bizarre world of numbers that straddle the finite and infinite divide. The induced Multi-layered Ellipsis Structure from a NOPT structure has similar component-connection structure to the well-known H-Fractal, given a corresponding degree of resolution.

Moreover, The H-fractal structure is emergent from the NOPT structure and guides the multi-layered nestedly embedded computational pathways. In the representations I give (see the pictures below) the computation starts at the bottom right corner and moves towards the top left corner.

Caveat: This mathematical material is very abstract and not practically computable on any computer, if accurate magnitude is a desired goal, because it’s in the area of hyperoperations, the boundary of practical maths.

When the operation is not specified, NOPT structures are floaty and esoteric in nature, but nevertheless they still say something about the patterns arising from unlimited multilayered nested recursion, with SeedValue being the standard stage level phase transition marker.

A comparison can be made with Hereditary Base n, where the base value is used at other exponent levels apart from the level of standard positional notation, resulting in a treelike number structure. Perhaps, another comparison can be made with CantorNormalForm for infinite ordinals.

In the pictures below:

The NOPT structure we consider below and is shown in the 4 pictures below is the nonation NOPT structure, remembering that OrderType=9 is derived from considering the corresponding NEPT structure arising from the nonation hyperoperation.

All of the “formal power towers” in the NOPT structure are given the symbol “theta” (Note: by “formal” expression I mean an unevaluated symbolic expression that could be evaluated given more information).

When the SeedValue is supplied, in theory, the power tower can be evaluated because the height information about the power tower is given, then each succeeding power tower in the linear Nopt structure can be evaluated with each power tower evaluation supplying the height of the next power tower to be evaluated.

In the first picture the SeedValues have the form “theta ) n”

In the second picture the SeedValues have the form “n”.

Sometimes, we may want to ensure that the next linear component in the NOPT structure has nontrivial ellipsis value and so we use the first style of NOPT structure.

However, I think that for a standard definition of NOPT structure it is better to use the second style of NOPT structure where SeedValue=n.

By some kind of careful inspection argument, one can notice ellipsis values that are the same magnitude within a NOPT structure, for example where the ellipsis is equal to SeedValue (a controlling seed value, not a starting seed value).

(Remember, starting seed values inform the height of the adjacent power tower, theta, in the linear NOPT component. Controlling seed values give the length of an ellipsis within a component of the NOPT structure.)

Also, by careful inspection argument, one can order ellipsis lengths by increasing order of magnitude. Although it is very difficult to quantify “how much” longer one ellipsis is compared to another, the magnitudes increase very quickly as is the nature of fast-growing functions. The idea of visualising the ellipsis lengths in increasing magnitude order has been color-coded in Picture 3.

Finally, Picture 4 shows the induced Multi-layered Ellipsis Structure from a NOPT structure and is color-coded in the same way as in Picture 3. The basic structural pattern of the well-known H-Fractal can be seen somewhat clearly.