So it is a spiral point but I didn't zoom in closely enough?

No, the standard spiral remains the same under any zoom. However your spiral rotates rather slowly in comparison with moving away and as it makes one rotation ($\theta$ increases by $2\pi$) the exponent increases by $5 \times 2\pi/\sqrt{5}\approx 14$ and the radius increases $e^{14}\approx 1.2 \cdot 10^6$ times. If the initial distance was 1 mm, then after one rotation it becomes 1.2 km.

Try plotting $x'=a x- y$, $y'=x+ ay$ for $a=.001, .1, .5, 1, 1.5, 2$ to observe that at for some $a$ you just cannot observe rotation.