Much is explained by looking at the polar equation of the spiral:
$$r=\exp(\theta\cot\alpha)$$
Here, $\alpha$ is the constant angle any tangent to the curve makes with the radius vector (a line segment joining the origin and the point of tangency). This explains the adjective equiangular (the verification of this property from the defining equation is left as an exercise). As an aside, insects flying towards a point light source like a candle or a light bulb follow the path of an equiangular spiral, since the usual strategy of an insect flying at the daytime to get their bearing is to fly at a constant angle from the sun's rays, and this strategy works against them when encountering man-made light.
Now, suppose we have an arithmetic progression of angles $\theta,\theta+\Delta\theta,\theta+2\Delta\theta,\dots$; if we get the corresponding values of the radius vector using the defining equation for the logarithmic spiral (geometrically speaking, this corresponds to a clockwise rotation by $0,\Delta\theta,2\Delta\theta,\dots$ radians), we get
$$\exp(\theta\cot\alpha),\exp((\theta+\Delta\theta)\cot\alpha),\exp((\theta+2\Delta\theta)\cot\alpha),\dots$$
which can be re-expressed as
$$\exp(\theta\cot\alpha),\exp(\theta\cot\alpha)\cdot\exp(\Delta\theta\cot\alpha),\exp(\theta\cot\alpha)\cdot\exp(\Delta\theta\cot\alpha)^2,\dots$$
which as you can see is a geometric progression; that is to say, the logarithms of the members of this sequence form an arithmetic progression. This is where the logarithmic adjective arises from.
From your description in the comments, it appears that you are taking $(r,\theta)$ coordinates and drawing them as though the $\theta$-axis is horizontal and the $r$-axis is vertical. [and for some reason you are plotting $\theta$ in degrees and you want your maximum $r$ to be 360; not sure what's up with that but it doesn't change the math.]
This is really only a change in perspective and there is no mathematical operation happening here in some sense, just exploring a biological limitation. We are creatures that are literally hard-wired to see everything in a Cartesian way. Even when you recognize a plot as polar you are seeing this from the Cartesian perspective; if you truly saw things from a polar perspective then whenever someone drew the secant curve you would react to it in the same way that I would if I saw someone draw bunch of lines, many of which overlapping.
Concretely this means that if you know $r=f(\theta)$, for instance with the Fibonacci spiral (a special case of the logarithmic spiral) you have $r=ae^{b\theta}$, then you can plot this on your square by simply forgetting your associations of "$r=$ radius" and "$\theta=$ angle", which might be easier if you replace them with $x$ and $y$. Then you will see that your square contains the Cartesian exponential curve $y=ae^{bx}$.
(Note: there is one interesting mathematical quirk which is that this change of coordinates is not a bijection: every point on the $\theta$-axis represents the same point, namely the center of the circle. If you try translating a curve that passes through the center of the circle you will see that strange things happen: in particular a continuous curve becomes apparently discontinuous. The "discontinuity" is just a biological problem: your eyes do not recognize nontrivial quotient spaces and so the points on the $\theta$-axis, although they are the same, appear to be different. But the mathematical framework you need to say that these two distinct pairs of numbers represent the same spatial point is quite nontrivial.)
FWIW: I don't see any particularly good reason to limit yourself to $0\leq\theta\leq 360$ just because you are studying circles, except that you now are quotient-ing by a more complex equivalence relation.
Best Answer
From Wolfram MathWorld:
Edit I searched for straightedge and compass constructions and found an old Popular Science article featuring the math behind the router. However, it doesn't show any spirals.