Why does the function $r = \theta$ graph a spiral

Question

If $\theta$ denotes an angle (in radians I assume), and $r$ means the distance from the origin, then why does $r = \theta$ make a perfect spiral? I'm not that advanced in math (only in geometry and Algebra II), and I saw a 3Blue1Brown video that involved spirals and prime numbers, and I got curious. When I graphed $r=\theta$ in Desmos, I saw the spiral and asked "Why?"

Does anyone know why this happens?

Answer 1

So, if we graph $r = f( \theta )$, as we often do, what does this mean? This means, for each angle $\theta$ (where $\theta$ is a real number), its distance from the origin (or radius) $r$ is given by $f(\theta)$. This is often easiest to see if you take a particular function $f$ and graph a bunch of points $(r, \theta)$ with increasingly higher values of $\theta$. Animations are also nice. Some are in this video. You can play with some demos yourself on Wolfram here.

In this sense, what does $r = \theta$ represent? Basically, for any given angle $\theta$, its distance from the origin is equal in value to $\theta$.

• An angle of $\theta = 0$? The radius $r$ is $0$.
• An angle of $\theta = \pi/4$? The radius $r$ is $pi/4$.
• An angle of $\theta = \pi$? The radius $r$ is $\pi$.
• An angle of $\theta = 2\pi$? The radius $r$ is $2\pi$.
• An angle of $\theta = 10^{10}$? The radius $r$ is $10^{10}$.

All you're doing is graphing the points $(\theta,\theta)$, really (again, $r=\theta$). It's sort of like graphing $y=x$, which gives you the points $(x,x)$ -- now just with a polar flair.