Spiral Similarity is a combination of rotation and dilation.

It is an interesting geometric transformation for a variety of reasons. In the previous section, we found that any segment can be mapped onto any other segment using spiral similarity.

We wish to treat spiral similarity algebraically.

Let \( A = (r, \theta) \). Suppose we want to rotate it by an angle \( \theta_1 \) and also dilate it (multiply its length by) \( r_1\).

Clearly, after we perform the rotation and dilation, the resultant point is: $$ (r \times r_1, \theta + \theta_1) $$

This action of rotation and dilation can be encoded in an ordered pair of numbers: \( r_1, \theta_1 \).

The first number of this ordered pair says how much we should dilate (by what factor should we multiply the length). The second number gives the angle of rotation.

We will call this ordered pair of numbers an action. Hence \( (r_1, \theta_1) \) acts on \( (r, \theta ) \) to rotate and dilate it to the final position \( (r \times r_1, \theta + \theta_1) \)

A short hand way of writing this is $$ (r_1 \theta_1 ) \searrow (r, \theta) \to (r \times r_1, \theta + \theta_1) $$

Dual nature of points and actions

Notice that the action \( (r_1, \theta_1) \) can be regarded as a polar coordinate of a point. It is the position of a point that is \( r_1 \) distance away from origin and making an angle \( \theta_1 \) with the positive direction of x axis.

On the other hand the point that \( (r_1, \theta_1) \) is acting on, that \( (r, \theta) \) can be itself regarded as an action. It has the power of rotation by \(\theta \) and dilation by r.

We have arrived at the essence of our core idea: points can be regarded at once as objects waiting to be rotated and dilated OR actions that can rotate and dilate.

(This is a supplemental note to live lecture session of Complex Number module of Cheenta I.S.I., C.M.I. Entrance Program and Math Olympiad Program)