Why can’t the Cube be Doubled or the Circle be Squared?
It took over two thousand years for algebra to expose the limitations of the straight edge and compass
Most of us have either painful or pleasurable memories of using a straight edge and compass at school. Mine were always in little tin boxes, and these simple instruments were our main connection with ancient Greek geometry. For those minutes we used them, we would be constructing shapes in almost exactly the way the Greeks did.
At age 10, my teacher set the class a problem. He asked us if we could use only a straight edge and compass to construct a cube that had double the volume of a unit cube. He made it clear that we were only allowed to use a straight edge and compass — NOT a ruler — so we had no way of taking actual measurements.
Myself and a couple of other more math-nerdy classmates spent a long time in trial and error trying to find ways to do this — a few times we thought we’d found it but our teacher would immediately show that it was invalid. After we were exhausted with that problem, he set us another: given a unit circle, can we construct a square of the same area?
We were wasting our time of course, because ten years later as a Pure Mathematics undergraduate, I would discover proofs that it is impossible to double the cube or square the circle using just a compass and straight edge using a finite number of steps. What amazed me about these proofs was that they used pure unadulterated algebra, which I had previously considered a separate mathematical field to geometry. And of course, I felt a bit sorry for the Greeks, who obsessed about these questions but did not have the toolkit to solve them— a toolkit which was only fully completed by the mathematician Ferdinand von Lindemann in 1882.
In this article I want to talk about the steps involved in proving that the cube cannot be doubled and the circle cannot be squared. I will only be offering proofs of the simple parts in this article just to keep it a manageable length, but I will provide directions to proofs of other results as we go.
1. Rings and Fields
We start our journey in the hotbed of abstract algebraic development that was heralded in by mathematicians such as Euler and Gauss but really hit its peak in the mid- to late-19th century with Kronecker’s general definition of an abelian group. Around this time, Richard Dedekind introduced a definition of an algebraic structure called a ring.
If we look at how a ring is defined, we can see that it is a special case of a group and that it is borne out of an attempt to understand how our most common number systems fit within group theory — something which was newly developed at the time.
Definition 1.1: A ring is a set R together with two binary operators on that set + (addition) and ｘ(multiplication), such that:
- R is an abelian group under the addition operator. This means addition is associative, so (a + b) + c = a + (b + c) for any a, b, c in R. It is also commutative, so a + b = b + a for any a, b in R. There is an additive identity, usually denoted as 0, where a + 0 = a for any a in R. And for any a in R there is an additive inverse, usually denoted as ﹣a, where a + (﹣a) = 0.
- R under the multiplication operator is associative and R contains an identity under multiplication, usually denoted as 1.
- Multiplication is left and right distributive when it acts on addition. That is a ｘ (b + c) = (aｘb) + (aｘc) and (a + b)ｘc = (aｘc) + (bｘc) for any a, b, c in R.
Note that multiplication does not need to be commutative (aｘb =bｘa), but in many of the most common rings multiplication is commutative, and these are known as commutative rings.
Example 1.2: The most common ring you’ve probably worked with is the set of integers ℤ under standard addition and multiplication. This is a commutative ring. Similarly, the set of all polynomials with integer coefficients —ℤ[x] — is a commutative ring. An example of a non-commutative ring is the set of all 2 x 2 matrices with real number entries under standard matrix addition and multiplication operators.
Of course a ring seems quite weak as a structure because there is no requirement for a multiplicative inverse. So if we add this requirement, we can generate a much more powerful algebraic structure.
Definition 1.3: A field is a commutative ring F which has a multiplicative identity 1 which is different from the additive identity 0, and where a multiplicative inverse exists for every element of F except 0.
Example 1.4 : Sound familiar? It should, because adding this extra condition allows us to differentiate different number sets in terms of their algebraic ‘tightness’ so to speak. We can now see the difference between ℤ and, say, ℚ (the set of all rational numbers) or ℝ (the set of all real numbers). All three are rings, but only the latter two are fields. Also, don’t be fooled into thinking that fields must be infinite. The smallest field — the Galois Field GF— has only two elements. You’d probably know it better as the binary field. It consists of a 1 and a 0, where the additive operation is OR (XOR to be more precise) and the multiplicative operation is AND.
2. Interesting facts and results about fields
Here are a few interesting facts and results about fields — mostly selected to help me get back to my original point about Greek geometry. If you are keen to explore this area and find important proofs then I would recommend this book.
Fact 2.1: Fields can be extended to larger fields, where the degree of the extension is equivalent to the dimension of the larger field as a vector space over the smaller field.
Examples 2.2: All complex numbers is a field extension of all real numbers. The degree of the extension is 2, because all complex numbers can be formed as duples of real numbers.
Fact 2.3: In a chain of finite degree field extensions, the degrees of the extensions are multiplicative. That is, if J, K and L are a chain of field extensions, where L has degree s over K and K has degree t over J, then L has degree st over J.
Common early uses of fields and subfields related to the understanding of polynomial roots.
Fact 2.4.1: If J is a subfield of K then an element ⍺ of K is said to be algebraic over J if it is the root of a polynomial in J[x] (the set of all finite-degree polynomials with all their coefficients in J). Otherwise ⍺ is said to be transcendental over J.
Fact 2.4.2: A polynomial f in J[x] which is of the lowest possible degree such that ⍺ in K is a root of f is known as an irreducible polynomial over J. The degree of such a polynomial is equivalent to the degree of the field extension of K over J.
Examples 2.5: ∛2 is algebraic over ℚ and its irreducible polynomial is x³﹣2, so therefore ∛2 lies in a field extension of degree 3 over ℚ by Fact 2.4.2.
3. So what has all this got to do with compass and straight edge constructions?
Well, we know that there is a limit to what can be done with a compass and straight edge. So let’s first define those limits.
Definition 3.1: Given two points in cartesian space, we can define construction rules as follows:
- Any straight line through both points is a valid construction.
- Any circle centered on one point but intersecting another point is a valid construction.
- Any intersection points of valid constructions are valid constructions.
There is a lot you can do with these rules. With a finite number of steps you can construct parallel lines, bisect angles (see the example below) or find the midpoint of lines for example. You may remember being taught how to do these things at school.
Definition 3.2: Using these rules, we can call a number q constructible if, starting with a unit line in cartesian space, you can construct a line of length q in a finite number of valid construction steps with a compass and straight edge.
Fact 3.3: You can prove that the set of all constructible numbers forms a field under standard addition and multiplication (see here), and a simple corollary of this is that all rational numbers are constructible.
Now this is where our above facts about fields come in handy. First we notice that any straight line constructed through two points with co-ordinates in ℚ is a polynomial of degree 1 over ℚ, and similarly any circle constructed with a center in ℚ and intersecting another point in ℚ is a polynomial of degree 2 over ℚ. From here it is possible to prove two important facts:
Fact 3.4: For any two lines with coefficients in ℚ, their intersection points are also in ℚ. (This should be obvious).
Fact 3.5: For a line and a circle or for two circles with coefficients in ℚ, their intersection points must lie in a field extension of degree at most 2 over ℚ. (This is shown by substituting one of the equations into the other).
Fact 3.6: It follows from the Facts 3.4 and 3.5 above that any constructible number must lie in a field of degree 2ᵏ over ℚ for some finite positive integer k, since each construction step must extend ℚ by degree 1 or 2 (Fact 3.5) and since field extensions are multiplicative (Fact 2.3).
(Note that a corollary of this is that all constructible numbers are algebraic over ℚ.)
Now we can prove that what the Greeks were trying to do was indeed impossible:
Fact 3.7: Doubling the unit cube is impossible in a finite number of steps using a compass and straight edge. To double the unit cube we need to know that ∛2 is constructible. But we observed in Examples 2.6 that this lies in a field extension of degree 3 over ℚ, so therefore it cannot be constructible by Fact 3.6.
Fact 3.8: Squaring a circle of unit radius is impossible in a finite number of steps using a compass and straight edge. To square the circle we need to prove that √π is constructible, which is equivalent to proving that π is constructible. But Lindemann proved in 1882 that π is transcendental over ℚ — meaning that there is no finite degree polynomial with coefficients in ℚ for which π is a root. Therefore π lies in a field extension of infinite degree over ℚ, and consequently by Fact 3.6 π cannot be constructible.
I personally find this link between abstract algebra and ancient Greek geometry both beautiful and inspiring. I hope you do too. If you are interested in playing around with it, you can try and prove some of the other impossible constructions. For example, while it was shown above how to bisect any angle, try to prove that it is impossible to trisect a 60 degree angle. (Hint: look for an equation that can express cos3θ in terms of cosθ and then see what follows from the logic in this article).