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In my last post I brought up the idea of being able to adopt different perspectives on the same topic, in the context of linear algebra. This certainly doesn’t always happen with every topic in mathematics, but it is very rewarding when it does. Partly it’s because the different perspectives can be complementary to each other, and partly because when seemingly unrelated conceptual domains converge it often becomes possible to uncover deep and surprising connections. In today’s post I’m going to tell you the story of one such surprising connection, which I learned about for the first time while teaching a course called “Topics in Geometry” in the summer of 2008 at my alma mater, McGill University in Montreal.
The idea of using various instruments to perform geometric constructions dates at least to the Ancient Egyptians, though it may be significantly older than that. However, it was the Ancient Greeks who not only formalized what these instruments could do – a compass can draw a circle with a given center and passing through a given point, while a straightedge can draw the line passing through two given points – but also put the art of geometric constructions on an axiomatic foundation. Euclid’s Elements, the culmination of these efforts, starts with 5 postulates and 5 common notions. The difference between them is that a postulate states the feasibility of a construction, such as drawing a line through two given points, while a common notion, or axiom, is a statement accepted without proof, such as “Things which equal the same thing also equal one another”). Euclid’s Elements is rightly considered the crowning jewel of Ancient Greek mathematics, and referred to as “the most successful textbook ever written”.
However, the Ancient Greeks left behind a number of geometric problems that they had not been able to solve using the two instruments. The status of two of them – doubling the cube (constructing the side of a cube having twice the volume of a given one) and trisecting the angle (constructing an angle whose measure was one-third the measure of a given one) – remained open for nearly two millennia. Then, in the 17th century, an important new development took place – Rene Descartes’ coordinate system made it possible to express geometric constructions in algebraic terms, and to understand algebraic quantities in a geometric way. Now it became possible to think of constructible numbers instead of just constructions (where a number r is constructible if a segment of length r can be constructed from a unit segment with the instruments). Building up on these developments, Pierre Wantzel’s brilliant 1837 article showed something very general: that the numbers constructible using a compass and straightedge were precisely the numbers that one could obtain by performing basic operations: addition, subtraction, multiplication, division, and square root extraction (that is, all the numbers you could get on an old pocket calculator). Then doubling the cube (which implies constructing the cube root of 2, the real solution of x3 = 2), and trisecting a 60° angle (which implies constructing the sine of 20°, the positive solution of 8x3 – 6x = 1), are both impossible with a compass and a straightedge, a straightforward corollary of Wantzel’s result.
Meanwhile, an algebraic question was the obsession of many mathematicians – which equations could and could not be solved by using only the basic operations and roots; not necessarily only square roots, but all integer roots. The Ancient Egyptians already knew how to solve quadratic equations; Cardano had shown how to solve cubic (degree 3) and quartic (degree 4) equations in his 1545 book Ars Magna. The status of the quintic (degree 5) equation was open, and it was only in 1823 that Niels Henrik Abel (eponymous with the Abel prize, an analogue of the Nobel prize for mathematics), extending the work of Paolo Ruffini, showed that a general solution using only these operations was impossible. But, if algebra was powerless to solve the quintic equation with a formula, then perhaps geometry could help?
Surprisingly, geometry can help, but only if we replace the straightedge by a ruler while keeping the compass. The difference between a straightedge and a ruler is that the latter has regularly spaced marks. But, taking two adjacent marks and the compass would allow us to construct all the other marks, so in fact, we only need a straightedge with two marks, also called a twice-notched straightedge. There are two extra operations we can do with this instrument: pivot it around a point or slide it along lines and circles, in both cases marking off the intersections with previously constructed figures. Although the Greeks had not solved the problems of doubling the cube and trisecting the angle with a compass and a straightedge (as we just saw, this is in fact impossible), Archimedes (the same one who cried Eureka! while running naked around Syracuse) already knew how to trisect an angle with a ruler and a compass, and Nicomedes, a contemporary of his, knew how to double the cube with it. However, the full power of the ruler and compass was not discovered until 2002 when Baragar showed that it can, in fact, be used to construct the solutions of the quintic x5 – 4x4 + 2x3 + 4x2 + 2x – 6 = 0, not solvable by radicals! In general, the ruler and compass combination can be used to construct solve any quadratic, cubic or quartic equation, as well as some quintics and a few sextics (polynomial equations of degree 6); which ones they are is still an open question, though I’m sure someone will solve it in the next couple of years.
However, there is another instrument, coming from a different cultural tradition, that is also interesting. Some would not even consider it an instrument at all – it’s our two hands! In other words, folding paper with our hands can be seen as a way of constructing lengths, and therefore numbers. There are some constructions that would require many steps if a compass and a straightedge (unmarked or marked) are used, but doable with only a handful of origami folds, including the solution to our two acquaintances, the problems of trisecting an angle and doubling a cube. However, the power of origami is also limited; the numbers that can be constructed with origami are precisely the ones that can be constructed with a compass and a marked straightedge (ruler), if only single folds are allowed. It’s unclear what happens if 2 simultaneous folds are allowed, but my intuition tells me there could be some interesting math there!
Before we finish this excursion, I’d like to mention one more famous geometric problem, also dating back to the Ancient Egyptians, that of squaring the circle (constructing a square of area equal to that of a given circle). Many great minds (including the famous philosopher Thomas Hobbes) have tried – and failed – to do so with a straightedge and compass. Since the area of a unit circle is π, squaring the circle is equivalent to constructing the number which is the square root of π. In 1882, Lindemann finally showed that π was transcendental (not the root of any polynomial with integer coefficients) and so established, among other things, the impossibility of squaring the circle. You can conclude from our earlier discussion that this is a problem where not only a (marked) ruler, but even origami, would not help, as they can only find the solutions of polynomial equations, while π cannot be the solution of any such equation or chain of equations. One moral of this story is that in mathematics, just like everywhere else, it’s important to not only master the instruments at our disposal, but also to understand their limitations. Maybe one day an instrument will be invented that will allow us to finally square the circle.