Showing that the distributive law holds in twizzle arithmetic is equivalent to proving the trigonometric addition formulae.

\begin{eqnarray} r_1 cis(\theta_1) \times r_2 cis(\theta_2) &=&r_1r_2 cis(\theta_1+\theta_2)\\ &=& r_1r_2(\cos\theta_1 + i \sin\theta_1)(\cos\theta_2 + i\sin\theta_2)\\ &=& r_1r_2((\cos\theta_1\cos\theta_2-sin\theta_1\sin\theta_2) + i(\sin\theta_1\cos\theta_2+\cos\theta_1\cos\theta_2))\\ \end{eqnarray}

That's not the focus of this question though. What we're
really looking for is some experimentation with the animation
leading to an understanding of the behaviour of (z-i) , (z+i) , and (z-i)(z+i) as z moves in a loop-like path.

It's quite useful to look at the case where the beige
a -twizzle is zero in the
second animation.