By moving the concept of geometric construction into three dimensions, could one trace the 3D wireframe of any of the five platonic solids using only a compass and straightedge?
If not, what additional tools would be required?
I imagine the construction taking place in a "void" of sorts, without the luxury of a preexisting plane. No one $xy$, $xz$ or $yz$ plane is visualized.
Rules copied from TheNullHypodermic:
Draw a line between any two distinct points.
Draw a circle with one point as the center, and any other point on its circumference.
Draw an arbitrary point on a line or a circle, or off it.
Draw the point at the intersection of two lines (if they intersect).
Draw the point (or two) at the intersection of two circles (if they intersect).
Draw the point (or two) at the intersection of a line and a circle (if they intersect).