Can lines that are not parallel or perpendicular to each other fill up $\mathbb{R}^3$? As title says, can a union of lines that are not paralell or perpendicular to each other be $\mathbb{R}^3$? The number of lines does not matter. It may be countable or uncountable.

There is an injection from the set of points in $\Bbb R^3$ to the set of directions contained in the (open) first octant, for instance, via space-filling curves. (Fill the space with a curve, this gives a bijection from $\Bbb R^3$ to $\Bbb R$. Now inject that real line into the space $\{(\theta, \phi) | 0 < \theta, \phi < \pi/2\}$, consisting of directions for the lines in spherical coordinates.)
No two such directions are orthogonal, and every point gets a line through it with a unique direction, so it turns out you can fill the space with lines neither parallel nor perpendicular.
