Is there a known example of a 3D spiral that fills the space in a sphere-like manner, similarly to a spiral filling the plane as a concentrically circling curve? What is its name and its equation?
The 3D spiral examples I could find are either variants of a helix or a spiral on the surface of a sphere.
I suspect the answer is a spiral rotating around one (or more) axis.
Use the infinite band $B:=\bigl\{(\phi,z)|-\infty<\phi<\infty, \ -1\leq z\leq1\bigr\}$ as a billard table. Draw in $B$ the orbit $$\gamma:\ t\mapsto\gamma(t)=\bigl(2\pi t,z(t)\bigr)\qquad(t\geq0)$$ of a billard ball starting at $(0,0)$ and having irrational slope. Now spool $B$ in the obvious way around the unit sphere $S^2$ (i.e., such that it touches $S^2$ along the equator). This means that we map $B$ via via $$f:\quad (\phi,z)\mapsto (\sqrt{1-z^2} \cos\phi,\sqrt{1-z^2}\sin\phi,z)$$ onto $S^2$. As $f$ is area preserving it follows that $f(\gamma)$ will "cover" $S^2$ in a more or less uniform way. Now invent a monotone function $$r:\quad {\mathbb R}_{\geq0}\to{\mathbb R}_{\geq0},\qquad \phi\mapsto r(\phi)\ ,$$ such that $r(0)=0$, and $r(\phi)\to \infty$ as $\phi\to\infty$ ever so slowly that the "spiral" $$\sigma:\quad\phi\mapsto r(\phi)f\bigl(\gamma(\phi)\bigr)$$ asymptotically "fills space" with constant length per unit volume.