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!
.
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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