R1 – Rotation – Multiplication Table

Orthogonal Rule

The Orthogonal rule, states that multiplication by a unitary is the equivalent of rotation around the unitary’s single point axis by 90^o. This allows the derivation of the R1 multiplication table that follows.

Multiplication Table

The unitary multiplication table of R1 shows the result of multiplying by 1^{^} or 1^v. Multiplying by 1^{^} equates to a 90^{^o} counterclockwise rotation around the point 0 whereas multiplying by 1^v equates to a 90^vo clockwise rotation around the point 0.

*	1^	1v
1^	1^	1v
1v	1v	1^

The only rotation possible in R1 is by 180^o which moves the point located at 1^{^} to 1^v and vice versa. This is flipping. The availability of only one axis warps rotations in R1.

1^{^} is a zero rotation.

1^v is a 180^o rotation which can be either clockwise or counterclockwise.

The Rotation axioms define unitary rotation as not standard in R1 because there is only one axis and no perpendicular axis upon which to rotate.

Interpreting Unitary Rotation

Imagine rotation by the unitary 1^{^} as a counterclockwise rotation around a dimensionless point at 0 on the z-axis by 90^{^o}. This results in the counterclockwise rotation of the point located at 1^{^} to i^{^} on an imaginary y-axis. As the point i^{^} does not exist in R1, a point cannot be rotated by 90^{^o} counterclockwise and stays where it is. As a result, the rotation rounds down and stays where it was.

Imagine rotation by the unitary 1^v as a clockwise rotation around a dimensionless point at 0 on the z-axis by 90^vo. This results in the clockwise rotation of the point located at 1^{^} to i^v on an imaginary y-axis. As the point i^v does not exist in R1, a point cannot be rotated by 90^vo clockwise. As a result, it moves on to the next available location at 1^v. In other words, it is as if the rotation rounds up to the nearest point on the R1 line.

In R1 and R2 multiplication tables, one of the multiplications by the unitaries 1^{^} and 1^v must result in 180^o rotation and the other in a 0^o. The axiom that 1^{^} results in 0^o rotation and 1^v in 180^o is arbitrary. It could have been the other way around.

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