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).