1. The problem is to describe a helicoid-like spiral surface extending from $z=0$ to $z=4\pi$ with radius $r$ from 0 to 1, wrapping twice around the $z$-axis. 2. A helicoid can be parametrized by cylindrical coordinates $(r, \theta, z)$ where $r$ is the radius, $\theta$ is the angle around the $z$-axis, and $z$ is the height. 3. Since the surface wraps twice around the $z$-axis as $z$ goes from 0 to $4\pi$, the angle $\theta$ changes from 0 to $4\pi$ as $z$ changes from 0 to $4\pi$. 4. We can express $\theta$ as a function of $z$: $$\theta = z$$ which means one full rotation ($2\pi$) corresponds to $z=2\pi$, so two rotations correspond to $z=4\pi$. 5. The radius $r$ varies from 0 to 1, so the parametric equations for the surface are: $$x = r \cos(z)$$ $$y = r \sin(z)$$ $$z = z$$ where $r \in [0,1]$ and $z \in [0,4\pi]$. 6. This parametrization describes a spiral surface that extends upward along the $z$-axis from 0 to $4\pi$, with radius expanding from 0 to 1, wrapping twice around the axis. Final parametric equations: $$x = r \cos(z), \quad y = r \sin(z), \quad z = z, \quad r \in [0,1], \quad z \in [0,4\pi]$$