What is the equation of the Golden spiral

The Golden section squares are shown in red here, the axes in blue and all the points of the squares lie on the green lines, which pass through the origin (0,0).

Also, the blue (axes) lines and the green lines are each separated from the next by 45° exactly. The large rectangle ABDF is the same shape as CDFH, but is phi times as large. Also it has been rotated by a quarter turn. Similarly with CDFH and HJEF. This applies to all the golden rectangles in the diagram.

So to transform OE (on the x axis) to OC (on the y axis), and indeed any point on the spiral to another point on the spiral, we expand lengths by phi times for every rotation of 90°: that is, we change (r,theta) to (r Phi,theta+Pi/2) (where, as usual, we express angles in radian measure, not degrees).

So if we say E is at (1,0), then C is at (Phi,Pi/2), A is at (Phi2, Pi), and so on. Similarly, G is at (phi,-pi/2), and I is at (phi2, -pi) and so on because phi is 1/Phi.

The points on the spiral are therefore summarised by:

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