Just playin' around.
Same sequence as GoldenJ's worksheet: [url]http://www.geogebratube.org/student/m34491[/url]. With an eye toward the complex exponentials, I wrote the sequence this way: Let C0 be the center of the first circle, r0 the initial radius, k the scaling factor, α the rotation angle. Then to rotate any vector [b]v[/b] by α, premultiply by a rotation matrix R: [R] [b]v[/b], where [math]{\rm R}= \begin{bmatrix} \cos α & -\sin α\\ \sin α & \cos α \end{bmatrix}[/math] Here, rotations alternate direction. Two successive rotations cancel. But suppose they did not. Using complex vector products (rotation matrices), the final (composite) rotation can always be calculated in the same way. Now find the circle centers. They will be a linear combination of the given vectors. If n is even, the nth circle center is [math]C_n = C_0 + (k+1) ( (1 + k^2 +... k^n) {\rm r_0} + (k+k^3 +...k^n-1) {\rm [R] r_0})[/math] The nth radius has length [math](k^n)|{\rm r}|[/math] I made two sequences, and mashed them together with Sequence[Circle[Element[centers, i],abs([k^(i)) abs(r)], i , 1, n] But the questions about imaginaries are now arrived. Back to my KA application.