A Tribute to Charles William Johnson's: Alternative ExTENsion of the Pythagorean Theorem

Yod Zain

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dedicated to
Charles William Johnson

Visiting this great site on some of the most important relationships between ancient ArtWork and PostModern scientific view the very first time, I knew to have found one of the major pieces of the essential contents of a post-modern type of Holonome Integrated Science, I call HighEndResearch, as claimed by the whole DeSign of this Joiner'Site. Making Contact very fast, we both, Charles & me, found to look at the things under the Sun a very similar way and to speak a very similar language. Thank you Charles, I hope we will find some more of what the Earth, our PlaNETary Home, needs for to pass the next Millenium. Now, let me take this great occasion to use this page here to show & to discuss the flow of ideas ON the final JOINment of the Things ToGetHER OnLine, as it comes each and every day out of my fingers.
4th of JANuary 2001

02.01.2001 at 3:33 a.m.
There is a whole series of Monomial, BiNomial, Trinomial and... "EXP"-Nomial Formulas, and their reciproce versions, seemingly building a type of interconnection key between the TENSOR Algebra (mostly used by EINSTEIN) and the explicite written Algebra for GeoMetrical advance resp. building of "geometrical SUMs" of different parametricalized data-spaces. This table shows the beginning of a large system of generatable Formulas to use for multi-parametrical models.

Point Line Pythagorean
(TriAngle, Circle) Pythagorean
(TetraHEdron?, Sphere)

EXP

-INF

....

-n

-1

0

1

2

3

4

+n

+INF

Res

0

....

DIV <-

1/u

1

u

-> MUL

Dim

1

u^-inf

....

u^-n (:)

u^-1

u^0

u^1

u^1+v^1

u^1+v^1+w^1

u^1+v^1+w^1+x^1

2

u^2+v^1

u^2+v^1+w^1

u^2+v^1+w^1+x^1

3

u^1+v^2

u^3+v^1+w^1

u^3+v^1+w^1+x^1

4

u^2+v^2

u^1+v^2+w^1

u^4+v^1+w^1+x^1

5

u^2+v^2+w^1

....

6

u^3+v^2+w^1

etc. ... till 4^4 (=EXP^2)

7

u^1+v^3+w^1

8

u^2+v^3+w^1

9

u^3+v^3+w^1

10

u^1+v^1+w^2

11

u^2+v^1+w^2

12

u^3+v^1+w^2

13

u^1+v^2+w^2

14

u^2+v^2+w^2

15

u^3+v^2+w^2

16

u^1+v^3+w^2

17

u^2+v^3+w^2

18

u^3+v^3+w^2

19

u^1+v^1+w^3

20

u^2+v^1+w^3

21

u^3+v^1+w^3

22

u^1+v^2+w^3

23

u^2+v^2+w^3

24

u^3+v^2+w^3

25

u^1+v^3+w^3

26

u^2+v^3+w^3

27

u^3+v^3+w^3

....

(=EXP^2)

Understanding the Formulas as to have free running variable values and combinatorical running Exponents, we get out this Combina-Torical set of geometrical SUMS.

HAPPY NEW MILLENIUM !
January 5th 2001, 23:44
A Tribute to
Charles William Johnson's
Alternative Extension of the Pythagorean Theorem
(Cfr., Earth/matriX: Essay No.58)

x³ + y³ + z³ = w³

The very 1st try!

.
January 6th 2001
Some new ideas considering an universal formula building principicle behind

Multiples of Powers of Natural Quantums

m xⁿ

.
The most simple GeoMetrical Sums, such as the Pythagorean Theorem

a² + b² = c²

reduces to

c² = 2x²

when

a=b=x.

This, for instance, looks like the known construction formula for the periodic table of chemical elements, 2n² in its standard form.
At least, all polynomial equations of any order and complexity are reducing this way when all their elements are equal to each other. Considering the Alternative Extension of the Pythagorean Theorem (Cfr., Earth/matriX: Essay No.58):

x³ + y³ + z³ = w³

we find

3x³.

Looking for an universal key formula to that type of equations, we find

mxⁿ

wherein

m = n.

This could stand for an equal mathematical ExPression of Multiples of Powers of Natural Numbers or Quants.

Now we can formulate a new table, showing a MaTRiX of the derivations of mxⁿ in general :

mxⁿ

xⁿ/inf

xⁿ/n

...

xⁿ/ 5

xⁿ/ 4

xⁿ/ 3

xⁿ/ 2

xⁿ

2xⁿ

3xⁿ

4xⁿ

5xⁿ

...

nxⁿ

inf xⁿ

m

1/inf

1/n

...

1/5

1/4

1/3

1/2

1

2

3

4

5

...

n

inf

n

-inf

x^-inf / inf

....

x^-inf / 5

x^-inf/ 4

x^-inf/ 3

x^-inf / 2

1x^-inf

2x^-inf

3x^-inf

4x^-inf

5x^-inf

....

inf x^-inf

-n

x^-n / n

x^-n/ 5

x^-n/ 4

x^-n/ 3

x^-n / 2

1x^-n

2x^-n

3x^-n

4x^-n

5x^-n

nx^-n

...

...

...

...

-5

x^-5 / n

...

x^-5/ 5

x^-5/ 4

x^-5/ 3

x^-5 / 2

1x^-5

2x^-5

3x^-5

4x^-5

5x^-5

...

nx^-5

-4

x^-4 / n

...

x^-4 / 5

x^-4/ 4

x^-4/ 3

x^-4 / 2

1x^-4

2x^-4

3x^-4

4x^-4

5x^-4

...

nx^-4

-3

x^-3 / n

...

x^-3 / 5

x^-3/ 4

x^-3/ 3

x^-3 / 2

1x^-3

2x^-3

3x^-3

4x^-3

5x^-3

...

nx^-3

-2

x^-2 / n

...

x^-2 / 5

x^-2/ 4

x^-2/ 3

x^-2 / 2

1x^-2

2x^-2

3x^-2

4x^-2

5x^-2

...

nx^-2

-1

x^-1 / n

...

x^-1 / 5

x^-1/ 4

x^-1/ 3

x^-1 / 2

1x^-1

2x^-1

3x^-1

4x^-1

5x^-1

...

nx^-1

0

x⁰ / n

...

x⁰ / 5

x⁰/ 4

x⁰/ 3

x⁰ / 2

1x⁰

2x⁰

3x⁰

4x⁰

5x⁰

...

nx⁰

+1

x¹ / n

...

x¹ / 5

x¹/ 4

x¹/ 3

x¹ / 2

1x¹

2x¹

3x¹

4x¹

5x¹

...

nx¹

+2

x² / n

...

x² / 5

x²/ 4

x²/ 3

x² / 2

1x²

2x²

3x²

4x²

5x²

...

nx²

+3

x³ / n

...

x³ / 5

x³/ 4

x³/ 3

x³ / 2

1x³

2x³

3x³

4x³

5x³

...

nx³

+4

x⁴ / n

...

x⁴ / 5

x⁴/ 4

x⁴/ 3

x⁴ / 2

1x⁴

2x⁴

3x⁴

4x⁴

5x⁴

...

nx⁴

+5

x⁵ / n

...

x⁵ / 5

x⁵/ 4

x⁵/ 3

x⁵ / 2

1x⁵

2x⁵

3x⁵

4x⁵

5x⁵

...

nx⁵

...

...

...

...

+n

xⁿ / n

xⁿ / 5

xⁿ/ 4

xⁿ/ 3

xⁿ / 2

1xⁿ

2xⁿ

3xⁿ

4xⁿ

5xⁿ

nxⁿ

+inf

x^inf / inf

....

x^inf / 5

x^inf/ 4

x^inf/ 3

x^inf / 2

1x^inf

2x^inf

3x^inf

4x^inf

5x^inf

....

infx^inf

Inerestingly, this table, respectively the placement of equations wherein m=n, are looking a bit like what José Arguelles called the Mayan's "weaving loom".

In order to avoid a double use of the same variables, as it happens for n within this table, the free running variable for the exponents we should call i, so that our formula reads

mxⁱ.

From that statement we can go to have a look at what we can do with this formula(s).

A formula like

ixⁱ

can be derived/expanded directly into i single operands:

xⁱ₁+xⁱ₂+xⁱ₃+...+xⁱ_i

so that we get out the following system of equations:

ixⁱ resulting formula description STF GeoM.
KeyScale

i

0 0x⁰ 0 ZERO Point singulary atom Point Source

1 1x¹ x NUMBER of x Line linear element Line 1d Location

2 2x² x²₁+x²₂ Pythagorean Theorem TriAngle quadratical molecule Plane 2d Location

3 3x³ x³₁+x³₂+x³₃ ExTension 1 TetraHedron(?) cubical compound Space 3d Location

4 4x⁴ x⁴₁+x⁴₂+x⁴₃+x⁴₄ ExTension 2 Pyramid linear Motion Velocity

5 5x⁵ x⁵₁+x⁵₂+x⁵₃+x⁵₄+x⁵₅ ExTension 3 spat. HexaGon quadratical Oscillation Frequency

6 6x⁶ x⁶₁+x⁶₂+x⁶₃+x⁶₄+x⁶₅+x⁶₆ ExTension 4 cubical Radiation Energy

... ...

n nxⁿ xⁿ₁+xⁿ₂+xⁿ₃+ .... + xⁿ_n ExTension n

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A Tribute to
Charles William Johnson's
Alternative Extension of the Pythagorean Theorem
(Cfr., Earth/matriX: Essay No.58)

x³ + y³ + z³ = w³

The very 1st try!

Some new ideas considering an universal formula building principicle behind

Multiples of Powers of Natural Quantums

m xⁿ

The most simple GeoMetrical Sums, such as the Pythagorean Theorem

a² + b² = c²

reduces to

c² = 2x²

when

a=b=x.

x³ + y³ + z³ = w³

we find

3x³.

Looking for an universal key formula to that type of equations, we find

mxⁿ

wherein

m = n.

This could stand for an equal mathematical ExPression of Multiples of Powers of Natural Numbers or Quants.

Now we can formulate a new table, showing a MaTRiX of the derivations of mxⁿ in general :

Inerestingly, this table, respectively the placement of equations wherein m=n, are looking a bit like what José Arguelles called the Mayan's "weaving loom".

In order to avoid a double use of the same variables, as it happens for n within this table, the free running variable for the exponents we should call i, so that our formula reads

mxⁱ.

From that statement we can go to have a look at what we can do with this formula(s).

A formula like

ixⁱ

can be derived/expanded directly into i single operands:

xⁱ₁+xⁱ₂+xⁱ₃+...+xⁱ_i

so that we get out the following system of equations:

A Tribute to Charles William Johnson's Alternative Extension of the Pythagorean Theorem (Cfr., Earth/matriX: Essay No.58)

x3 + y3 + z3 = w3

The very 1st try!

Some new ideas considering an universal formula building principicle behind

Multiples of Powers of Natural Quantums

m xn

The most simple GeoMetrical Sums, such as the Pythagorean Theorem

a2 + b2 = c2

reduces to

c2 = 2x2

when

a=b=x.

x3 + y3 + z3 = w3

we find

3x3.

Looking for an universal key formula to that type of equations, we find

mxn

wherein

m = n.

This could stand for an equal mathematical ExPression of Multiples of Powers of Natural Numbers or Quants.

Now we can formulate a new table, showing a MaTRiX of the derivations of mxn in general :

Inerestingly, this table, respectively the placement of equations wherein m=n, are looking a bit like what José Arguelles called the Mayan's "weaving loom".

In order to avoid a double use of the same variables, as it happens for n within this table, the free running variable for the exponents we should call i, so that our formula reads

mxi.

From that statement we can go to have a look at what we can do with this formula(s).

A formula like

ixi

can be derived/expanded directly into i single operands:

xi1+xi2+xi3+...+xii

so that we get out the following system of equations:

A Tribute to
Charles William Johnson's
Alternative Extension of the Pythagorean Theorem
(Cfr., Earth/matriX: Essay No.58)

x³ + y³ + z³ = w³

m xⁿ

a² + b² = c²

c² = 2x²

x³ + y³ + z³ = w³

3x³.

mxⁿ

Now we can formulate a new table, showing a MaTRiX of the derivations of mxⁿ in general :

mxⁱ.

ixⁱ

xⁱ₁+xⁱ₂+xⁱ₃+...+xⁱ_i