## What is Rotation in R1?

The Wave Number principles describe rotation as a fundamental operation. The rotation axioms support this. Rotation moves a point represented by an Opposite Value to a location represented by another Opposite Value. Rotation performs a circular movement around a single point axis represented by an Opposite Value. The single point at the centre of the rotation remains fixed during rotation.

Note that rotation around an axis in R1 is not possible. To rotate around an axis, the axis needs to be perpendicular to the location being rotated. There is only one axis in R1, so this is not possible.

## Syntax

Rotation requires an amount of rotation, a single point axis and a starting point. The syntax of the Rotation operation follows where ^{?} represents the point around which the rotation takes place. The default is the point 0.

(Amount of rotation) ↺^{?} (Opposite Value).

## Flipping

R1 has only one dimension which uses the x-axis. The standard rotation is the rotation from one side of the number line to the other. This has the effect of switching the Opposite Sign to the other Opposite Sign.

The ^{–} symbol represents this type of rotation, known as flipping. For example:^{–}1^{^} = 1^{v}.

The expression π^{^} ↺ 1^{^} = 1^{v} or π^{v} ↺ 1^{^} = 1^{v} also represents flipping. In other words a 180^{o} turn either counterclockwise or clockwise from 1^{^} moves a point to 1^{v}.

Similarly, ^{–}1^{v} = 1^{^} or π^{^} ↺ 1^{v} = 1^{^} or π^{v} ↺ 1^{v} = 1^{^}. In other words a 180^{o} turn either counterclockwise or clockwise from 1^{v} moves a point to 1^{^}.

## The Tee Rotation Prop

Rotation is a difficult subject to conceptualise. A physical prop helps to understand it. So, ideally the prop will fit between the index finger and thumb in order to allow for easy rotation. A wooden golf tee or something similar is suitable.

Scratch out points from 3^{v} through 0 to 3^{^} on the tee from left to right. Rotation requires an independent axis at right angles to the tee. R1 has only 1 dimension, so treat the z-axis as a dimensionless point at 0, the centre of the tee. This is the one point that remains fixed during rotation. Hold the tee between the thumb and index fingers with one finger in front of the tee and one behind as if the z-axis is from the front to the back. This allows the tee to rotate vertically.

Using an imaginary z-axis instead of an imaginary y-axis allows rotation of the tee so that the point 1^{^} on the x-axis can move to an imaginary point i^{^} on an imaginary y-axis. This is the equivalent to the real point i^{^} in R2. (Remember in R2 the x-axis goes from 1^{v} on the left to 1^{^} on the right, the y from i^{v} on the south pole to i^{^} on the north pole. This leaves the imaginary z-axis going from the near pole to the far pole. The R2 layout is different than for R3.)

## Rotating the Prop

Holding the tee at the centre between the thumb and index finger, the tee is parallel to the ground and point of the tee is facing the palm. Holding the tee like this, it is only possible to rotate it vertically so that 1^{^} moves to 1^{v} after a 180^{o^} counterclockwise turn. As there is only one dimension, the tee had to turn through an unavailable dimension.

If a Linelander living on a line in Lineland rotates 180^{o}, it would be as if it teleports from one side of Lineland to the other. It would just pop up there. It could not rotate by only 90^{o} as this would be to a dimension that did not exist in Lineland. But, if it pretended that it rotated by 90^{o^} it would end up at the imaginary i^{^} on the imaginary y-axis. It would be imaginary for a Linelander but not for a Flatlander who lived in 2 dimensions.

## Alternative Rotation Axes

Instead of using the imaginary z-axis point 0, use an imaginary y-axis point 0 as an alternative axis. This point remains fixed during rotation. Hold the tee parallel to the ground and in the middle with one finger on top of the tee and one below at an imaginary y-axis point 0. Now, the tee can only be rotated parallel to the ground. A 180^{o^} counterclockwise turn or clockwise turn moved the point 1^{^} to 1^{v }. This time, if it was rotated by 90^{o^}, it would end up at the imaginary j^{v} on the z-axis.

Instead of using the imaginary z-axis point 0 as the fixed point of rotation, you can use any other point on the x-axis. A 180^{o^} counterclockwise turn or clockwise turn around an imaginary axis at 1^{^} would move the point 4^{^}to 2^{v }. This is expressed as π^{^} ↺^{1^} 4^{^} = 2^{v} .

Rotation of a point around itself in R1 obeys the Remain rule of rotation and leaves the point at the same location. A 180^{o^} counterclockwise turn or clockwise turn around an imaginary axis at 1^{^} leaves the point 1^{^}at 1^{^ }. This is expressed as π^{^} ↺^{1^} 1^{^} = 1^{^} .

Rotation in R1 and R2 differ from higher dimensions in that they must use an imaginary axis to move a point.

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