### Straightedge and compass constructions

Classical Euclidean geometry is based on the familiar five postulates. The first two are equivalent to the assumption of the existence of a straightedge; the third gives the existence of a compass. The fourth essentially states that space is locally Euclidean, while the fifth, the infamous parallel postulate, assumes that space is globally Euclidean.

A quick digression: by now, most people are aware of the concept of non-Euclidean geometry as described by Bolyai and Lobachevsky, and independently by Gauss. These types of geometries make also the first four postulates. By simply abandoning the fifth postulate, we can arrive a geometries in which every pair of “lines” intersect (in other words, geometries in which “parallel lines” do not exist), or in which parallel lines are non-unique. (As a side note, one of the primary preoccupations of mathematicians is the existence-and-uniqueness of objects. With our obsession of classifying things, we ask often the question, “Does an object exist with properties X and Y?” and follow it up with, “Is there only one object that possesses properties X and Y?” One may have been subjected to this analysis as early as high school algebra when one is asked to classify whether a system of linear, algebraic equations have solutions, and whether the solution is a point or a line.) With even this relaxation, the space is still locally Euclidean (by the fourth postulate): a small portion of the surface will look “flat” when you blow it up (the same way how from our point of view, the Earth often looks flat). It is, however, possible to also relax the fourth postulate. The easiest example to imagine is by taking a piece of paper and rolling it up into a cone. At the vertex of the cone, going around one full turn is no longer going around 360 degrees. The “total angle” at the vertex will be smaller. So if we define a “right angle” as an angle that is exactly one quarter of the “total angle” at the point, the right angle at the vertex will be smaller! You can experiment with it yourself by drawing four rays from the vertex and then unfurling the piece of paper. Geometries with this kind of structures are often studied under the name of orbifolds.

In any case, let us return to the topic at hand.

Classical Euclidean geometry is filled with constructive proofs (the modern methods of mathematics–proof by contradiction, principle of strong induction, and existence proofs without construction–are only really popular after the 18th century). Say that you asked me to show that a triangle can be characterized by two of its angles and the side that sits between them (the ASA theorem). The classical constructive proof goes something like this:

1. First we construct a line segment of the given length.
1. On the sheet of paper, draw an arbitrary long line L using the straightedge.
2. Pick and mark a point on the line, call it P.
3. Using the compass, measure the length of the given side.
4. Draw a circle centered at P with the compass (whose radius is now the same as the length of the given side).
5. The circle intersects L at two points, pick and mark one of it Q.
6. PQ is the desired segment.
2. On P, we construct the first angle, and on Q, we construct the second angle, using the procedure
1. Given an angle A where A also denotes the vertex. We can copy it by first drawing a small circle of a fixed radius centered at A.
2. Draw a small circle O of the same radius centered at P.
3. The circle intersects the two sides of the angle A at points B and C. The circle intersects segment PQ at a point. Call it B’.
4. Measure the length BC. Draw a circle centered at B’ with that length. The circle intersects O at two points. Call them C’ and C”.
5. The angles B’PC’ and B’PC” are as desired.
3. Now extend the sides of the angles at P and Q so that they meet. This finishes the construction of a triangle.

The problem with this style of proofs is that their constructive nature makes it rather difficult to prove a statement is false, except in the case where one can do so by demonstrating the existence of a counter-example. Partly because of this, for two millennia, professional and amateur mathematicians alike faltered when trying to resolve the three “impossible” problems in constructive geometry: squaring the circle, doubling the cube, and trisecting the angle. It was not until the 19th century that these problems are proven to be impossible to construct by Wantzel, Lindemann, and Weierstrass, using tools from field theory, a subset of abstract algebra. (It is interesting to note that mathematics departments around the world are still occasionally receiving submissions from arm-chair mathematicians claiming to have found a construction for those three problems.)

The basic theory behind Wantzel, Lindemann, and Weierstrass’s demonstrations is a clever proof by exhaustion: to show that something cannot be done using the tools one is given, it is enough to enumerate all things that can be done using those tools, and showing that that something is not on the list. Of course, the total number of constructable things given a compass and a straightedge is infinite, so the list will get rather long. But we mathematicians tend not to let such minor roadblocks bother us too much. The question, then, is how to turn the unmanageable infinites to manageable finites. And the strategy boils down to distilling somehow the essence which is shared by all constructable things and only by all constructable things.

And this is where abstract algebra came in. After Descartes laid down his name-sake grid on the abstract plane, to every point there now attaches two numbers: its x- and y-coördinates. One then observes that straightedge and compass constructions only allow points, straight lines, and circles–and not any other curves. More importantly, a straight line can be characterized by just two points on it, and a circle can be characterized by its center and one point on the circumference. So in essence, the question of constructability becomes “which points in the plane can be constructed”.

Now points that are constructed in a geometric proof are all obtained by intersections of lines and circles. And here comes the analytic geometry! A circle, as one recall, is described by an equation of the type

$(x - x_0)^2 + (y - y_0)^2 = (x_1 - x_0)^2 + (y_1-y_0)^2$

where $(x_0,y_0)$ is the coördinate of the center of the circle, and $(x_1,y_1)$ the coöordinate of a point on the circumference. A line, on the other hand, is described by an equation in modified point-slope form

$(y_1 - y_0) (x-x_0) + (x_1-x_0)(y-y_0) = 0$

where $(x_0,y_0), (x_1,y_1)$ are the coördinates of two distinct points the line passes through. Furthermore, finding the point of intersection between two lines, or a line and a circle, or two circles, is the same as solving the system of equations of those geometrical figures.

How to we solve a system of linear and quadratic equations in two variables? By arithmetic manipulations alone (addition, subtraction, multiplication, and division), one can solve a linear equation for one of its variables x or y in terms of a simple arithmetic combination of the other unknown variable and the coefficients. Similarly, one can solve a quadratic equation for one of the variables x or y in terms of the other unknown and the coefficients, if one enlarge the set of allowed operations to include taking the square root. To give a concrete example, the equation for the circle can be re-written as

$x = x_0 \pm \sqrt{ (x_1-x_0)^2 + (y_1-y_0)^2 - (y-y_0)^2 }$

Combining the two, one sees the following fact

The solution to a system of linear or quadratic equations in two variables, should it exist, consists of points whose coördinates can be given as a combination of the coefficients of the equation using the operations of addition, subtraction, multiplication, division, and taking of square roots.

As it turns out, using abstract algebra, it is not too difficult to find out, given a set of numbers to start with, the collection of all numbers that can be constructed from the given set by addition, subtraction, multiplication, division, and taking square roots. And the constructions for those three impossible problems all require constructing numbers that do not belong to this collection. Hence their impossibility.

Now, for the most part, plane geometry and straightedge-compass constructions can be thought of as completely solved, in view of advances in abstract algebra. Let us turn our attention to non-Euclidean geometries.

As I mentioned earlier, in non-Euclidean geometries, the notion of compass and straightedge are still well-defined. (For the experts, as long as we assume a complete Riemannian surface…) The notion of a line/straightedge can be replaced by the notion of geodesics: on the surface of a sphere, they are the great circles. The notion of a circle/compass can be replaced by the set of points all at a fixed distance from a given point: on the surface of a sphere, where we choose the center of the circle to be the north pole, the circles are precisely the circles of constant latitudes. (For the experts, we can define a circle or radius r and point p by the action of the exponential map at point p on a circle of radius r in the tangent space.) And we are allowed to measure the geodesic distance between two points (a property of the compass).

So the question I want to ask is: what are the constructable points?

For a general Riemannian surface, the question is not particularly meaningful, since it will strongly depend on the local geometry. But let us say we are interested in the classical examples: the plane (which we already know about), the hyperbolic space (of constant negative curvature), and the sphere (of constant positive curvature). These are space-forms and symmetric spaces, so at every point on the surface the geometry is identical. Is there then an algebraic correspondence, as in the zero-curvature case, between constructable points and fields? Can any of the three (okay, two, I suppose, since doubling the cube is not a plane geometry problem, and so makes less sense in trying to generalize to curved surfaces) impossible problems be solved in a different type of geometry?

For the first question, I don’t have an answer yet. For the second, I strongly suspect the negative. I reason thus: one of the most important features of flat geometry is that it is scale invariant. This is no longer the case when space is curved. Like I mentioned before, assuming the fourth postulate of Euclid, space is locally Euclidean. That means that on a very small scale, space approaches that of a plane. But this also means that if something is generically constructable in curved geometry, it should also be, by scaling it down, constructable in flat geometry. So the argument that forbids angle trisection in Euclidean geometry should also imply the same in non-Euclidean geometry.

From a different heuristic point of view, we also see that Euclidean geometry is “special”. Classifying by curvature, the hyperbolic geometries have negative curvature, the spherical geometries have positive curvature, while Euclidean geometry has precisely zero curvature. In this trichotomy, the Euclidean condition is “closed” while the hyperbolic and spherical conditions are “open”. This tells us that the Euclidean condition is unstable: if we perturb Euclidean space a little bit, we end up with something that is no longer flat. Since a construction requires a delicate chain of steps, each of which depending on the geometry, it seems more likely that a perturbation will break this delicate chain, rendering something constructable in Euclidean geometry not constructable in non-Euclidean geometry, than that a perturbation will miraculously assemble this chain where no chain was possible before.

Anyway, this could be an interesting undergraduate research project.

Update July 1, 5:24pm EST: I forgot to mention a few more things to illustrate my question. By my argument above, it seems impossible to trisect an angle in spherical geometry. It is rather easy to see that trisecting and angle in spherical geometry is equivalent to trisecting a “line segment”. (Start from point P, by drawing two lines you can find its antipode Q. Draw two latitude circles at equal distances from P and Q. The two latitude circles each intersect with the two original lines, call the intersections P’, P” and Q’, Q”. Draw lines from P’ to Q” and Q’ to P”. Their intersection is a point on the equator relative to the poles P and Q. Now you can draw the equator relative to P, which is a great circle. I claim that it is obvious that the angle trisectors for an angle at P will trisect a corresponding segment on the equator. Similarly we can go backwards from a line segment, define its corresponding north pole, and find a corresponding angle there.) So while it is perfectly possible to trisect a line segment in Euclidean geometry, it is most likely impossible to do so in spherical geometry. (One can also see this fact from the construction used to trisect a line segment; it depends heavily on the parallel postulate and similarity of triangles.)

So I am pretty sure constructables in non-Euclidean geometry is a strict subset of the constructables in Euclidean geometry.

Another way to illustrate this. The sphere can be embedded into three dimensional space centered at the origin. “Lines” on the sphere are intersections of the sphere with a plane through the origin. “Circles” on the sphere are intersections of the sphere with an arbitrary plane. So again the “points” are solutions to a system of equations, one of which is quadratic, and the other two are linear. This suggest that the constructable points on a sphere is a strict subset of the constructable points in Euclidean space (not plane).

Similarly, the hyperbolic plane can be embedded into three dimensional Minkowski space in the usual way. Lines are again intersections with planes through the origin. But circles are now intersections with arbitrary space-like planes. Again, since the embedding of the hyperbolic plane is now a hyperquadric (in particular is given by a quadratic equation), we have the same sort of characterization that the constructable points on a hyperbolic plane is a strict subset of the constructable points in three dimensional Euclidean space.

In fact, using the argument above, one can show that the impossibility of trisecting an angle in Euclidean plane implies immediately the impossibility of trisecting an angle in spherical or hyperbolic geometry. (I will not ask about squaring the circle since the concept of a square is not exactly well-defined in non-Euclidean geometry.)

The main difficulty that remains, then, is how to characterize, algebraically in the ambient three dimensional space, a geometric operation on the non-Euclidean geometries. And I am almost certain that most of this can be deduced simply from trigonometric and hyper-trigonometric identities, which will tell us that, for example, trisecting a segment is a uniquely Euclidean thing.