July 22, 2024
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R2 – Return and Orthogonal Rules

This post describes the first 2 universal rules of rotation in R2, the Return and Orthogonal.

Standard Rotation Definition

Rotation is a fundamental operation in Wave Numbers and is supported by the facts expressed in its axioms. A rotation moves a point represented by Opposite Values to a location represented by other Opposite Values by a circular movement around  a single point axis which is represented by Opposite Values. The point of rotation remains fixed during rotation.

The Return Rule in R2

The first rule of rotation is the Return Rule that states that continuous rotation of a point by the same rotation will return a point to its original location when the total rotation is a multiple of 360o.  Furthermore, all other axes are reachable by continuous rotation of a point by some rotation. 

These rotations routes are called Return Rings. There is no free axis of rotation in R2. This results in no rotation when rotating/multiplying by 1^ and 180o rotation when rotating/multiplying by 1v. Even with this, it is possible to say that the return rule applies after one turn by 1^ which is 0*360o and two turns by 1v at 180o each giving 360o.

Examples of Return Rule with Unitary Rotations

  • Return Rule applies to multiplication by 1^ because the point remains at the same location.
    • 1^*1^ = 1^
    • 1^*1v = 1v
    • 1^*i^ = i^
    • 1^*iv = iv

  • Return Rule also applies to multiplication by 1v .
    • 1v*1^ = 1v, 1v*1v = 1^
    • 1v*1v = 1^; 1v*1^ = 1v
    • 1v*i^ = iv, 1v*iv = i^
    • 1v*iv = i^; 1v*i^ = iv

  • Finally, the Return Rule applies to multiplication by i^ and iv
    • i^*1^ = i^, i^*i^ = 1v, i^*1v = iv, i^*iv=1^          
    • i^*1v = iv, i^*iv = 1^, i^*1^ = i^, i^*i^ = 1v     
    • i^*i^ = 1v, i^*1v = iv, i^*iv=1^, i^*1^ = i^            
    • i^*iv = 1^, i^*1^ = i^, i^*i^ = 1v, i^*1v = iv 
    • iv*1^ = iv, iv*iv = 1v, iv*1v = i^, iv*i^=1^           
    • iv*1v = i^, iv*i^ = 1^, iv*1^ = iv, iv*iv = 1v              
    • iv*i^ = 1^, iv*1^ = iv, iv*iv=1v, iv*1v = i^            
    • iv*iv = 1v, iv*1v = i^, iv*i^ = 1^, iv*1^ = iv

Examples of Return Rule with Non-Unitary Rotations

  • ^/3 ↺5^ 1v = (8.0^ + 5.19615iv)
    • ^/3 ↺5^ (8.0^ + 5.19615iv) = (8.0^ + 5.19615i^)
    • ^/3 ↺5^ (8.0^ + 5.19615i^) = 1v

  • ^/3 ↺5^ 4i^ = (4.0359^ + 6.33013iv)
    • π^/3 ↺5^ (4.0359^ + 6.33013iv) = (10.9641^ + 2.33013i^)
    • π^/3 ↺5^ = (10.9641^ + 2.33013i^) = 4i^

  • ^/3 ↺4i^ 5^ = (0.9641^ + 10.33013i^)
    • π^/3 ↺4i^ (0.9641^ + 10.33013i^)      = (5.9641v+ 1.66987i^)
    • π^/3 ↺4i^ (5.9641v + 1.66987i^) =    5^

  • ^/3 ↺(3v + 4i^) (5^ + 6iv) = (15.66025v + 2.0718i^)
    • ^/3 ↺(3v + 4i^) (15.66025v + 2.0718i^) = (1.66025^ + 15.9282i^)
    • ^/3 ↺(3v + 4i^) (1.66025^ + 15.9282i^) = (5^ + 6iv)

The Orthogonal rule in R2

The second rule of rotation is the Orthogonal rule which states that the rotation by a unitary is 90o. Unlike the return rule, the Orthogonal rule does not fully apply in R2.

Unitary rotation in R2 is not standard because there is no free axis of rotation in R2. This results in no rotation when rotating/multiplying by 1^ and 180o rotation when rotating/multiplying by 1v. However, rotation by i^ and iv are orthogonal.

Conclusion

Try these examples of R2 rotation and multiplication with our online calculator.

Next: Reversal, Remain and Zero Rules

Previous: Multiplication Table

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