Those strangely intermediate folks who are at once unfamiliar with compass-and-straightedge constructions and yet not intimidated by their spectre may find the Wikipedia page a reasonable starting place. But what I'm proposing today is something different, which I'm a bit startled hasn't been discussed more prominently: three-dimensional constructions. I've had this in my virtual back pocket for a while; this seems as good an opportunity as any to pull it out for a looksee, and figure out if anyone else has encountered anything like this.
The idea here is to extend to three dimensions what ordinary compass-and-straightedge constructions do in two dimensions. The first thing is to define the tools and rules for their use. For instance, in two dimensions, the tools are a compass and straightedge (like a ruler, but with only one edge and no markings), and with them, one may:
- 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).
It turns out that these are impossible, and can be proved to be so, using some notions from field theory. That has not, of course, stopped people from submitting reams upon reams of alleged constructions of one of these three objects, all of which (you may be assured) are somewhere bogus.
But enough of that for now. In three dimensions, the canvas is not a flat plane, as it is in two dimensions, but all of space. And we introduce a new tool, which I will call a flatiron, which permits you to draw planes. The flatiron rules are as follows; in addition to the above, one may:
- Draw the unique plane containing any three non-collinear points.
- Draw a sphere with one point as the center, and any other point on its surface.
- Draw an arbitrary point on a plane or a sphere, or off it.
- Draw the line at the intersection of two planes (if they intersect).
- Draw the circle (or point) at the intersection of two spheres (if they intersect).
- Draw the circle (or point) at the intersection of a plane and a sphere (if they intersect).
- Draw the point (or two) at the intersection of a line or circle with a plane or sphere (if they intersect).
- Draw spheres of radius PQ around both P and Q.
- Draw the circle C at the intersection of spheres P and Q.
- Draw R, an arbitrary point on circle C.
- Draw a sphere of radius PR around R.
- Draw S, one of the two points of intersection between circle C and sphere R. PQRS is then a regular tetrahedron.
So, a couple of questions, one easy, and one not so easy:
- Suppose that indeed, the cube root of two is not constructible in three dimensions. What about the fourth root of two? In which dimensions might that be constructible?
- Is it possible to construct all five of the regular polyhedra? In addition to the tetrahedron and cube, these include the regular octahedron (eight faces), dodecahedron (twelve faces), and icosahedron (twenty faces).