July 22, 2024
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Axioms – Rotation

Introduction

The axioms of rotation do not have an equivalent in Axioms for Real Numbers. As a fundamental operation, rotation requires axioms. This has led to the rules of rotation that follow.

A rotation moves a point represented by Opposite Value(s) to a location represented by other Opposite Value(s) by a circular movement around  an axis which is represented by Opposite Value(s). At least one point remains fixed during rotation.

Rules of Rotation

The following are the rules of rotation:

Return Rule

The first rule of rotation, the Return Rule, 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.  These rotations routes are called Return Rings. A point can move to all other axes as long as it is not rotating on its own axis.

Orthogonal Rule

The second rule of rotation, the Orthogonal rule, states that multiplication by a unitary is the equivalent of rotation around the unitary’s axis by 90o. Rotation by an Opposite Value with a ^ sign is in a counterclockwise direction and rotation by an Opposite Value with a v sign is in a clockwise direction. The sign used is the sign of the unitary.

R1

Unitary rotation is not standard as there is no axis perpendicular to x-axis upon which to rotate. Rotation is around a single point.

Imagine multiplication by the unitary 1^ as a rotation around a dimensionless point on the z-axis by 90^o. As a result, the point located at 1^ rotates counterclockwise to i^ on an imaginary y-axis. As the point i^ does not exist in R1, it cannot rotate by 90^o and multiplication by the unitary 1^ does not result in any movement of a point.

However, multiplication by the unitary 1v is also imagined as a rotation around a dimensionless point on the z-axis by 90vo.

As a result the point located at 1^ rotates clockwise to iv on an imaginary y-axis. Although the point iv does not exist in R1 and it cannot be rotated by 90vo, multiplication by the unitary 1v results in the movement of a point. It moves a point at 1v to 1^ and a point at 1^ to 1v. It rotates by 180vo which is flipping. The availability of only one axis warps rotations in R1.

R2

Unitary rotation is not standard as there is no axis perpendicular to the x-y plane in R2. Rotation is around a single point.

Unitary rotation by i^ and iv follow the orthogonal rule. Given i^ and iv follow the orthogonal rule, then it can be deduced that no rotation occurs when multiplying by the unitary 1^ and 180o rotation occurs when multiplying by unitary 1v.

R3

Unitary rotation is standard in R3, so the Orthogonal Rule fully applies. For example: j^*1^ = π^/2 ↺z (1^,0,0) = i^.

Reversal Rule

The third rule of rotation, the Reversal Rule, states that a rotation reverses by multiplying the result of the first rotation by the Operator of the first rotation with the sign reversed.

In other words, a 90o? degree turn in one direction reverses by turning back in the other direction. In the cases where a point rotates around its own axis, no rotation actually takes place, but the Reversal Rule still applies. 

R1 and R2

The Reversal Rule does not apply because a perpendicular axis of rotation is not available. This results in no rotation when multiplying by the unitary 1^ and 180o rotation when multiplying by the unitary 1v. So 1^*1^ = 1^; 1v*1^=1v

R3

The Reversal Rule fully applies in R3. For example:

  • j^*i^ = 1v; jv*1v=i^
  • 60^o ↺(4^+ 5i^ + 6j^) (1^ + 2i^ + 3j^) =  (1.627^ +  1.447i^ +  3.043j^)
  • 60vo(4^+ 5i^ + 6j^)  (1.627^ +  1.447i^ +  3.043j^) = (1^ + 2i^ + 3j^)

Remain Rule

The fourth rule of rotation, the Remain Rule, states that a point remains in the same place when rotated about its own axis.

Zero Rule

The  fifth rule of rotation, the Zero rule for R3, states :    

  • (a): a??*b?? + a??*b?-? = 0. For example:
    • 1v*iv + 1v*i^ = j^ + jv = 0
  • (b): a??*b?? + b??*a?? = 0 where a?? and b?? are not the same Opposite Type and sign. For example:
    •  jv*i^ + i^*jv = 1^+ 1v = 0
  • (c) a? + b?(b?.a?) = 0 where a?? and b?? are not the same Opposite Type and sign. For example:
    • 1v + jv(jv*1v) = 1v + jv*i^ =  1v + 1^ = 0

Next: General Axioms

Previous: Properties of Equality

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