Yod Zain

- -- --- ---- ----- Commentary ON ----- ---- --- -- -

- -- --- ---- ----- ------ ----- ---- --- -- -

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

xinf / inf

....

xinf / 5

xinf / 4

xinf / 3

xinf / 2

1xinf

2xinf

3xinf

4xinf

5xinf

....

infxinf

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:

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+ .... + xnn ExTension n

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