Constructible Geometry: Easy as Squaring the Circle

To square the circle is to achieve the impossible.

If you’ve ever heard of the phrase “squaring the circle”, you’ll know that it means something along the lines of “attempting to do the impossible”. But have you ever wondered where this phrase came from, and whether there are other such impossible constructions which, at first, might seem plausible?

The phrase, in fact, comes from the Ancient Greeks. The emphatic geometers pursued the fun-time activity of trying to solve the problem of squaring the circle – constructing a square with the same area as a unit circle, using only a compass and a straightedge* (with a finite number of steps of course). As is the usual fashion, many tried, and failed, for centuries, leaving a trail of false proofs in their wake. It was finally proved impossible in 1882, when π was proven to be transcendental.

A theoretically squared circle

Of course, this didn’t stop people from trying to do it anyway. In the words of the Hon. Malcolm Turnbull:

“The laws of mathematics are very commendable, but the only law that applies in Australia is the law of Australia.”

Apparently, this also applies in the US.  In 1894, amateur mathematician Edwin J. Goodwin claimed he had found a way to square the circle. Of course, the proof was phoney, and was essentially equivalent to redefining π as equal to 3.2. In order to assert his mathematical “genius”, Goodwin proposed the Indiana Pi Bill, which would allow this redefinition of π to be taught in schools without having to credit Goodwin. The bill passed unopposed in the state house. Fortunately, Professor C. A. Waldo, a real mathematician, happened to be in the legislature the day the bill was being voted on, and he promptly put a stop to it. 

Some other famous examples of impossible constructions are trisecting an angle and doubling the cube. All of these examples beg the question – which geometries are constructible?

Constructible Polygons

It’s pretty easy to draw a square, right? You just draw a line, measure it, and then draw 3 more lines of the same length that meet up at right angles. This is, of course, assuming that you have a ruler, and can make perfect right angles no less. How do you do it using only a compass and, say, the spine of a book? Well, let’s start with something easier, like a hexagon. A hexagon can be constructed by drawing a circle, drawing two circles of the same radius along a diameter of that circle, and joining the points of intersection, as shown in the gif below.

Constructing a regular hexagon

So if you can make a hexagon, you could make a dodecagon by bisecting the sides, and so on. Once we know an n-gon, we can just as easily construct a 2mn-gon. Using 5 axioms of constructible geometry, it is possible to create a number of regular polygons. These axioms are:

  • Creating the line through two existing points;
  • Creating the circle through one point with centre another point;
  • Creating the point which is the intersection of two existing, non-parallel lines;
  • Creating the one or two points in the intersection of a line and a circle (if they intersect);
  • Creating the one or two points in the intersection of two circles (if they intersect).

Given that it is pretty trivial to create a polygon with an even number of sides once we know any other polygon, it makes sense that the odd-numbered constructible polygons are few and far between in comparison. The first few odd-numbered constructible polygons are 3, 5, 15, 17, 51, 85, 255, 257, … In 1796 Gauss found a construction for a 17-gon, and promptly generalised to a sufficient condition for constructible polygons, which was later proven in full by Pierre Wantzel in 1837. The Gauss-Wantzel Theorem is:

A regular n-gon can be constructed with compass and straightedge if and only if n is the product of a power of 2 and any number of distinct Fermat primes (including none).

i.e., an n-gon is constructible iff  n = 2kp1p2pt, where k and t are non-negative integers, and the pi‘s (when t> 0) are distinct Fermat primes. (A Fermat prime is a prime of the form {\displaystyle 2^{(2^{m})}+1.}) The first 5 known Fermat primes are 3, 5, 17, 257, 65537. A full construction for a 257-gon can be found here, and the beginning of a construction of a 65537-gon can be found here. The concept can of course be generalised to include constructions for a number of markings on the ruler, which allows for more complex constructions, including the trisection of an angle.

Constructible Numbers

Instead of looking at integer-valued polygons, we could investigate constructible numbers in the real sense. A number r is said to be constructible if, given a line segment of unit length, there exists a way to construct a line segment with length |r|, in a finite number of steps. For example,  √2 is constructible by creating a line perpendicular to the unit line which ends at the unit circle, and joining their endpoints to create a right-angled isosceles triangle. We can think about this more easily by using the Cartesian coordinate system, with the original line segment joining (0,0) and (1,0). In this system, the coordinates of constructible points are constructible numbers.

Constructing the square root of 2

In algebraic terms, a number is constructible if and only if it can be obtained using the four basic arithmetic operations and the extraction of square roots, but of no higher-order roots, from constructible numbers, which always include 0 and 1. In the language of field theory, the constructible numbers form the quadratic closure of the rational numbers: the smallest field extension that is closed under square roots. This effectively transforms geometric questions about compass and straightedge constructions into algebra, and leads to the solutions of many famous mathematical problems, which defied centuries of attack. In particular, this algebraic system can be used to disprove the squaring of the circle and the doubling of the cube.

 

*A straightedge is basically a ruler but with no markings, i.e. anything with a straight edge – who knew?

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