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).
You have to define how you use the compass and straightedge to define points that are not in the original plane. You can construct segments of the proper lengths to have the coordinates of the corners of the Platonic solids. How you translate those coordinates into a point in 3D is not clear to me. How do you get the $z$ axis given the $xy$ plane?
Arthur suggested we consider the compass to make spheres and the straightedge to be able to make planes. To construct the regular tetrahedron then you just construct an equilateral triangle in the plane and swing spheres from each corner with radius equal to the side of the triangle. The point where the three spheres intersect is the top point. Constructing a cube is easy as well. Construct a square, then swing spheres of size $\sqrt 2$ times the side from to diagonally opposite corners plus one of side $\sqrt 3$ times the side from the other two. The intersections get two of the top corners of the cube. The others can also be done because the corners are all at points that can be expressed with addition, subtraction, multiplication, division, and square roots.