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 postmodern 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 dataspaces. This table shows the beginning of a large system of generatable Formulas to use for multiparametrical 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 ComBINATORical 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)
x^{3} + y^{3} + z^{3} = w^{3}
The very 1st try!
.
Some new ideas considering an universal formula building principicle behind
Multiples of Powers of Natural Quantums
m x^{n}
.
The most simple GeoMetrical Sums, such as the Pythagorean Theorem
a^{2} + b^{2} = c^{2}
reduces to
c^{2} = 2x^{2}
when
a=b=x.
This, for instance, looks like the known construction formula for
the periodic table of chemical elements, 2n^{2} 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):
x^{3} + y^{3} + z^{3} = w^{3}
we find
3x^{3}.
Looking for an universal key formula to that type of equations, we find
mx^{n}
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 mx^{n} in general :
mx^{n} 
x^{n }/inf 
x^{n }/n 
... 
x^{n }/ 5 
x^{n }/ 4 
x^{n }/ 3 
x^{n }/ 2 
x^{n} 
2x^{n} 
3x^{n} 
4x^{n} 
5x^{n} 
... 
nx^{n} 
inf x^{n} 


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 

x^{0} / n 
... 
x^{0} / 5 
x^{0}^{ }/ 4 
x^{0}^{ }/ 3 
x^{0} / 2 
1x^{0} 
2x^{0} 
3x^{0} 
4x^{0} 
5x^{0} 
... 
nx^{0} 



+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} 



+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} 



+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} 



+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} 



+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} 



... 


... 




... 




... 




+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} 



+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
mx^{i}.
From that statement we can go to have a look at what we can do with this formula(s).
A formula like
ix^{i}
can be derived/expanded directly into i single operands:
x^{i}_{1}+x^{i}_{2}+x^{i}_{3}+...+x^{i}_{i}
so that we get out the following system of equations:
ix^{i}  resulting formula  description  STF GeoM. KeyScale 

i  
0  0x^{0}  0  ZERO  Point  singulary  atom  Point  Source 
1  1x^{1}  x  NUMBER of x  Line  linear  element  Line  1d Location 
2  2x^{2}  x^{2}_{1}+x^{2}_{2}  Pythagorean Theorem  TriAngle  quadratical  molecule  Plane  2d Location 
3  3x^{3}  x^{3}_{1}+x^{3}_{2}+x^{3}_{3}  ExTension 1  TetraHedron(?)  cubical  compound  Space  3d Location 
4  4x^{4}  x^{4}_{1}+x^{4}_{2}+x^{4}_{3}+x^{4}_{4}  ExTension 2  Pyramid  linear  Motion  Velocity  
5  5x^{5}  x^{5}_{1}+x^{5}_{2}+x^{5}_{3}+x^{5}_{4}+x^{5}_{5}  ExTension 3  spat. HexaGon  quadratical  Oscillation  Frequency  
6  6x^{6}  x^{6}_{1}+x^{6}_{2}+x^{6}_{3}+x^{6}_{4}+x^{6}_{5}+x^{6}_{6}  ExTension 4  cubical  Radiation  Energy  
...  ...  
n  nx^{n}  x^{n}_{1}+x^{n}_{2}+x^{n}_{3}+ .... + x^{n}_{n}  ExTension n 
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