Ioana Bouros

If you ask a random person when and where do they believe mathematics as we know it today first originated from, most would probably answer the same: Ancient Greece. However, the Ancient Greek mathematicians were more geometers than arithmeticians, obsessed with the outer world and the symmetries and rules that govern it.

One of the issues they faced back then was that units of measure were unreliable, and like the perfectionists that they were, the wise of old would not be satisfied with crooked lines or anything but exact measures of angles. Therefore, it became very important to be able to construct geometric representations using only a compass and a straightedge (a ruler without a measure of distance).

In fact, there is a theorem which states that any common geometric figure in a plane (we call such representations Euclidean) can be drawn using only these two simple tools. Here we assume that the compass is a “collapsible” one, i.e. it cannot be used to translate distances, but rather to simply draw circles once we fix the radius, and reverting to its original form after the circle is drawn.

However, using both of these tools is in fact excessive. In 1672 Georg Mohr proved that you can draw any common geometric figure using just the collapsible compass (the straight edge can be used at the end for aesthetic reasons to more easily visualise what the construction accomplishes, but it never plays an active role in its creation). Mohr’s discovery was unearthed again in the 20^th century by Lorenzo Mascheroni, and since he discovered it independently from Mohr in 1797 and made the result popular, the theorem bears both of their names.

The proof of the result is purely constructive: you only need to show the basic shapes that comprise all geometric figures, which you can construct with a compass and straightedge, can be done in fact using only the former. There are five such basic constructions:

1.   Creating the line through two existing points A and B

2.   Creating the circle through one point (B) with centre at another point (A)

3.   Creating the point (X) which is the intersection of two existing, non-parallel lines (AB and DC)

4.   Creating the one or two points (P, Q) in the intersection of a line AB and a circle C (if they intersect)

5.   Creating the one or two points (P, Q) in the intersection of two circles (if they intersect)

Constructions 1, 2 and 5 are quite straightforward (if you assume for the first that you use the straightedge to actually visualise the line); but it is shapes 3 and 4 that require active drawing and we will see how to do each of them step-by-step. We begin by looking at some preliminary constructions that will be required in our proofs.

Shape 1

Construct the reflection of a point C with respect to a line AB.

Shape 2

Extend a line segment AB to ABC.

Shape 3

Construct the inverse of a point on a circle.

The inverse I of a point P with respect to a circle of radius r is a point on the line connecting the centre of the circle O and the original point such that IP / r  = r / PO (see below diagram for clarification) 

Shape 4

Determine the centre of a circle that passes through 3 points.

We will now use these 4 shapes to draw the last two remaining basic constructions:

Construction 3: Creating the intersection of two existing, non-parallel lines

Construction 4: Creating the one or two points in the intersection of a line and a circle (if they intersect)

Summary

With these two last constructions, we have shown how to create the 5 basic operations that can be done with the compass and straightedge drawing, using only the compass.

Since we already know from Euclid’s work that one can build any Euclidean shape from these 5 basic operations, this concludes the proof of the Mohr-Mascheroni theorem: that any shape can be drawn only using the compass. 

As a way to end this piece, I suggest you try your hand at solving one of the more famous construction problems – Napoleon’s Problem (it is still unclear if he was the one to solve or come up with it). It states:

Try to divide a circle into four equal arcs given its centre (or in other words, construct a square inscribed in the given circle).

 A solution for the problem can be viewed here.