Every ‘point’ (event/ /fix/vertex) has three directional components relative to another point:

(1) it moves towards the other point,

(2) it maintains an initial distance from another point, or

(3) it moves away from the other point.

Fig.1:Three special case components of ("spatial") transformations

(1)Towards other point (3)away from other point

ß ----------------------------------------- 0 ---------------------------------------------à

Two points can be considered to define the endpoints of a ‘line’. Given the three special cases for each point described above, a line has 3 x 3 = 9 special cases of directional component combinations. These are:

Figure. 2. Line:
 
 
 
 

point 1 point 2

1) (=)___a line__ (=) This special case is the initial line, neither point changes its initial motion

2)<------ (=) (=)------> This special case models an expanding line

3) (=)------> <------(=) This special case models a contracting line

4)<------ (=) <------ (=) This special case models a sliding line

5) (=)------> (=)------> This special case models a sliding line

6) (=) <------ (=) This special case models a contracting line

7) (=) (=)------> This special case models an expanding line

8)<------ (=) (=) This special case models an expanding line

9) (=)------> (=) This special case models a contracting line
 
 
 
 
 
 
 
 

Imagine two hands playing a piano. In terms of their movement up and down the keyboard, they will always be found to be doing one of these nine special case combinations..

Where V = the number of considered vertices (event/ fixes), there are:

V ( V – 1 )

3

special case combinations of velocity invariant components .

A "point" has three special case combinations of (velocity invariant) components characterizing all transformations, the "line" has (3^v(v-1) =) nine special cases, the triangle has ( 3^3(3-1) =) 3^6 = 729 special cases , and the tetrahedron, with 4 vertices, displays (3^4(4-1) = 3^12=) 531, 441 special case component combinations characterizing all possible (velocity invariant) transformations.

When an event moves either towards or away from another event, it does so at a velocity. There are three special cases of velocity : an event’s velocity can be (1) invariant, or it can (2)accelerate or (3)decelerate .

This results in these seven special cases of transformation of an event relative to another event:
 
 

ß -------- (1)

diminishing velocity toward other point

ß ------------------- (2)

invariant velocity toward other point

ß ----------------------------------------- (3)

accelerating toward other point
 
 

Imagine two hands playing a piano. Each hand will be in one of the seven states listed above.
 
 

There are

v (v-1)

7

special case component combinations for the number of vertices comprising the considered system where an acceleration component is considered.

The seven special cases listed above can be varied in terms of their frequency (F) and amplitude (A).

Image it’s Jerry Lee Lewis pounding on the keyboard with his fists (not with his fingers). Each of his hands can vary in the frequency (F) with which he pounds the keyboard, and each hand can vary the volume (A) by varying the strength at which the keyboard is pounded.

Where frequency is a component , as in the example above, there are 3 special case variations: the frequency will be (1)invariant, or (2)increasing or (3)decreasing.

Where amplitude is a component there are an additional 3 special case variations : the amplitude will be (1)invariant, or (2)increasing or (3)decreasing.

So:

To ascertain the number of special cases of transformation where velocity, amplitude and frequency are components, one combines:

1(initial event position) + (2 (directions) x 3 (velocity special cases) = 7

x 3 (Amplitude special cases)

=21

x 3 (Frequency special cases)

= 63

There are 63 special cases of transformation of a point relative to any other point where velocity, amplitude and frequency are components.
 
 

V (V – 1)

63

special case component combinations characterize all transformations where V is the number of vertices in the considered system.

Where two events move relative to each other, and where velocity(V), amplitude(A) and frequency(F) are variable components, there are 63 x 63 special cases of transformation.

So in the example of Jerry Lee Lewis playing the piano, his two hands can perform:

63 x 63 (=7^2 x 3^4) = 3969

unique special case combinations of transformational behaviors up and down the keyboard, at varying rates, tempos and volumes.

The triangle

The tetrahedron:

4 (4 – 1)

63

where 4 events transform relative to each other there are 7 ^12(13841287201) x 3 ^24 (282429536481 ) = 3,909,188,328,478,827,879,681 special case component combinations to characterize all transformations.
 
 

Invariance:

All transformations are noteworthy for what elements remain invariant throughout a transformation, AND what elements are , or can be , variable.

For example, if a structure maintains its initial shape during a transformation it is symmetry invariant, if it maintains its initial size it is described as scale invariant, and if it maintains its initial location or orientation it is described as translation or orientation invariant.

In the example of two vertices, the "line" CAN remain scale-invariant in only 3 special cases (Fig. 2: #1,4. 5)and WILL expand in three other special cases (Fig. 2: #2, 7, 8 ) it CAN remain translation-invariant in only 3 special cases (Fig. 2: #1,2,3 ) but WILL translate in 6 special cases (Fig. 2:#4,5,6,7,8,9) and MAY translate in an additional two special cases (Fig. 2:#2,3).

In all special cases two points remain symmetry-invariant (it’s always a line) and orientation-invariant (since there’s no other points, at this point, to move relative to). So the total of all transformations will be both variant and invariant. But no special case, in terms of its components , remains unaccounted for. I speculate that this fact is very relevant to quantum considerations.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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