T-cube = W1

On Breaking-down any Triangle into
a pair of Mirror-Imaged Equilateral Triangles

Section XXIII - Solving for [ T ] and [ U ] based on [ W1 ]

In this Section , we shall solve for the solution values of [ x ] based on this equatiom :

Let us recall that we did set up [ W1 ] in this format :
Let us now also set up the cube-roots of [ W1 ] as being [ T1 ] , [ T2 ] and [ T3 ] respectively .

We then have :
- where :
  - [ R-sub-T-123 ] is the real-number given by :
    - with :
    and
  - the angles [ THETA-sub-T1 ] / [ THETA-sub-T2 ] / [ THETA-sub-T3 ] are given by :
Thus , we have now established this relation :
And we take special note at this point that :
- the complex values [ T1 ] / [ T2 ] / [ T3 ] are in the formation of an equilateral triangle on the Complex Plane ,
  - with the { T1-T2-T3 rotation-order } being a { counter-clockwise rotation-order } .

Let us recall again { Condition ONE } and { Condition TWO } from Section XXI that brought us here :
- Condition ONE :
- Condition TWO :
And consequently , we have :
We then set up :
- [ U1 ] corresponding to [ T1 ] :
- [ U2 ] corresponding to [ T2 ] :
- [ U3 ] corresponding to [ T3 ] :
Let us recall that we did previously set up [ M ] in this format :
We then have :
Let us now bring in the { Special Type of Reciprical Relation } established in the last Section , Section XXII :
We can then re-write the above 3 equations in this manner :
Let us now bring in the { Algebraic Identity on Complex Multiplication } established also in the last Section , Section XXII :
We then have :
And we take special note at this point that :
- the complex values [ U1 ] / [ U2 ] / [ U3 ] are in the formation of an equilateral triangle on the Complex Plane ,
  - with the { U1-U2-U3 rotation-order } being a { clockwise rotation-order } .