Euclidean GeometryRuler and Compass Construction
You might have noticed that Euclid’s five axioms don’t contain anything about measuring distances or angles. Up to now, this has been a key part of geometry, for example to calculate areas and volumes.
However, at the times of Thales or Euclid, there wasn’t a universal framework of units like we have today. Distances were often measured using body parts, for example finger widths, or arm lengths. These are not very accurate and they vary for different people.
To measure longer distances, architects or surveyors used knotted cords: long pieces of string that contained many knots at equal intervals. But these were also not perfectly accurate, and different string had the knots placed at slightly different distances.
Greek mathematicians didn’t want to deal with these approximations. They were much more interested in the underlying laws of geometry, than in their practical applications.
That’s why they came up with a much more idealised version of our universe: one in which points can have no size and lines can have no width. Of course, it is
Euclid’ axioms basically tell us what’s possible in his version of geometry. It turns out that we just need two very simple tools to be able to sketch this on paper:
A straight-edge is like a ruler but without any markings. You can use it to connect two points (as in Axiom 1), or to extend a line segment (as in Axiom 2).
A compass allows you to draw a circle of a given size around a point (as in Axiom 3).
Axioms 4 and 5 are about comparing properties of shapes, rather than drawing anything. Therefore they don’t need specific tools.
You can imagine that Greek mathematicians were thinking about Geometry on the beach, and drawing different shapes in the sand: using long planks as straight-edge and pieces of string as compass.
Even though these tools look very primitive, you can draw a great number of shapes with them. This became almost like a puzzle game for mathematicians: trying to find ways to “construct” different geometric shapes using just a straight-edge and compass.
To begin, draw a line segment anywhere in a box on the right. With the
Next, draw two circles that have one of the endpoints of the line segments as center, and go through the other endpoint. With the
We already have two vertices of the triangle, and the third one is the intersection of the two circles. Use the line tool again to draw the two missing sides and complete the triangle.
Now these two sides and these two sides are each
Midpoints and Perpendicular Bisectors
COMING SOON – Constructing Midpoints and Perpendicular Bisectors
COMING SOON – Constructing Angle Bisectors
In the next chapter, we will see even more shapes that can be constructed like this. However, there is a limit to Euclidean geometry: some constructions are simply impossible using just straight-edge and compass.
According to legend, the city of Delos in ancient Greece was once faced with a terrible plague. The oracle in Delphi told them that this was a punishment from the gods, and the plague would go away if they built a new altar for their temple that was exactly twice the volume of the existing one.
Note that doubling the volume is not the same as doubling an edge of the cube. In fact, if the
This still sounds pretty simple, but doubling the cube is actually impossible in
Trisecting the angle
We already know how to bisect angles. However it is impossible to similarly split an angle into three equal parts.
Doubling the cube
Given the edge of a cube, it is impossible to construct the edge of another cube that has exactly twice the volume.
Squaring the circle
Given a circle, it is impossible to construct a square that has exactly the same area.
Note that these problems can all be solved quite easily with algebra, or using marked rulers and protractors. But they are impossible if you are just allowed to use straight-edge and compass.