Introduction
The definition of rotation has no equivalent in the Axioms for Real Numbers. As a fundamental operation, it necessitates its own definition:
A rotation moves a point, represented by Opposite Value(s), to a new location, also represented by Opposite Value(s), through a circular movement around an axis, which is similarly represented by Opposite Value(s). At least one point remains fixed during the rotation.
Rules of Rotation
The following are the rules of rotation:
Return Rule
The first rule of rotation, known as the Return Rule, states that continuous rotation of a point by the same angle will return the point to its original location when the total rotation equals a multiple of 360o. These rotational paths are referred to as Return Rings. A point can move to any other axis, provided it is not rotating around its own axis.
Orthogonal Rule
The second rule of rotation, the Orthogonal Rule, states that multiplication by a unitary is equivalent to a rotation around the unitary’s axis by 90o. Rotation by an Opposite Value with a ^ sign is in a counterclockwise direction, while 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 in R1 is non-standard since there is no orthogonal axis to the x-axis for rotation. Rotation is considered around a single, dimensionless point.
Consider multiplication by the unitary 1^as a rotation around a dimensionless point on the z-axis by 90^o. Consequently, the point located at 1^ would rotate counterclockwise to i^ on an imaginary y-axis. However, since i^ does not exist in R1, 1^ cannot undergo a 90^o rotation to i^, and thus, multiplication by the unitary 1^ does not result in any point movement.
In contrast, multiplication by the unitary 1v is conceptualized as a rotation around a dimensionless point on the z-axis by 90vo. As a result, the point at 1^would rotate clockwise to iv on an imaginary y-axis. Though iv does not exist in R1 and 1^ cannot be rotated by 90vo to it , multiplication by the unitary 1v does cause point movement. It effectively moves a point from 1v to 1^and from 1^ to 1v, corresponding to a 180o rotation, which is equivalent to flipping. The presence of only one axis distorts rotations in R1.
R2
Unitary rotation in R2 is non-standard since there is no axis orthogonal to the x-y plane. Rotation is considered around a single point.
Unitary rotation by i^ and iv adheres to the Orthogonal Rule. Given that i^ and iv follow the orthogonal rule, it can be deduced that no rotation occurs when multiplying by the unitary 1^ and 180o rotation occurs when multiplying by the unitary 1v.
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, therefore 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 an orthogonal 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, known as the Remain Rule, states that a point remains in the same location when rotated about its own axis.
Zero Rule
The fifth rule of rotation, known as the Zero rule for R3, states :
- a*b + a*–b = 0. For example:
- 1v*iv + 1v*i^ = j^ + jv = 0
- a*b + b*a = 0 provided a and b are not of the same Opposite Type and Opposite Sign. For example:
- jv*i^ + i^*jv = 1^+ 1v = 0
- a + b(b * a) = 0 provided a and b are unitaries and not of the same Opposite Type and Opposite Sign. For example:
- 1v + jv(jv*1v) = 1v + jv*i^ = 1v + 1^ = 0
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