Compass, ruler, origami, and squaring the circle

Dear readers,

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.

Top three most common misconceptions about mathematics

Dear readers,

In today’s post I will address the three most common misconceptions I encounter about mathematics from otherwise knowledgeable and often well-educated people. I strongly believe that understanding why these misconceptions are erroneous will be an important step towards eliminating the math anxiety (which, in contrast with this blog’s name, I could also call mathophobia) so prevalent in North America, as documented in a recent psychological study. To be fair, such anxiety seems to be prevalent elsewhere as well, but I have heard “Oh, I hate/don’t understand/suck at math!” in response to questions about my occupation far more frequently from North Americans than from people from other continents (which could also mean that North Americans are simply more outspoken about their negative feelings towards math; in the case of some individuals I met it almost gets to the point of wearing these negative feelings as a badge of honor). My hope is for this post to challenge the misconceptions and generate discussion.

First, it never ceases to amaze me how many people think mathematics is about numbers. Sure, you start learning math by learning how to count, and then go on to learn about the operations you can do with numbers. But by the end of primary school, or perhaps early middle school, the emphasis changes. By the time you get to algebra, geometry and trigonometry, the numbers take a back seat. Mathematics is no more about numbers than writing is about letters or music is about notes. What mathematics is really about is patterns. These patterns describe relationships between different mathematical objects, which could be numbers (for instance, prime numbers, the subject of an earlier post, serving as building blocks for all natural numbers), but could just as easily be more abstract (for instance, irreducible polynomials serving as the building blocks for all polynomials). Numbers, in fact, play a less and less significant role in mathematics as one learns more of it. It is common for mathematicians to be the least reliable people when it comes to splitting the bill at restaurants: a source of great shame for many of us.

Second, if I had a dollar for every time someone told me math is hard, I’d be a rich man now. Although many math textbooks are based on the idea of repeating the same exact exercise with different data, the real core of math is the concepts, and there are not that many concepts you need to know to understand math. Let’s take the example of solving an equation. There is a single concept – that of isolating the variable of interest – that underlies the entire problem. Sure, there are many techniques for doing this – adding the same quantity to both sides, dividing out common factors, combining like terms – but they are all subservient to the single unifying goal: isolating the variable. The exact same concept carries over to systems of equations with multiple unknowns. Unlike biology, mathematics has very few facts that need to be memorized; but it does have a set of core concepts that must be understood. Everything else is techniques that can be perfected through practice, much like writing or music. Not everyone may be able to become Gauss (or Tolstoy or Mozart), but everyone is capable of developing the basic skills to the point of moderate proficiency, which is all that is required for math, at least through the end of freshman year at universities. Incidentally, Albert Einstein famously said: “It’s not that I’m so smart, it’s just that I stay with problems longer.” That’s why math is not hard; it’s simply a skill like many others.

Finally, I am baffled by the North American obsession with getting the right answer when doing math, which plagued me throughout high school – a careless arithmetic mistake (of which I made quite a few) was then just as bad as using a completely erroneous approach. This is inappropriate for a discipline where process, much more than the result, is critical; if you reduce 1664 to 14 by “simplifying” the 6 on the top and bottom, you get the right answer… by the wrong method. I can only think of one reason for the popularity of the cult of right answers: grading is much easier when you only have to look at the answer (and as a grader, I did not always enjoy reading long, convoluted solutions to spot the mistakes in them). Just imagine if one tried teaching writing in this way: it wouldn’t matter what your thoughts were or how you expressed them, as long as there were no spelling mistakes. Math is about using the right approach, and the right approach means the right concept. Many of my friends, including myself, have written entire college-level math finals without even once opening the textbook (except to get the assigned problems), focusing instead on the concepts. We may not always have had enough time to derive everything from scratch in those finals to ace them, but we still remember these concepts now.

And if you were wondering whether getting the right answer matters in mathematical research, let me tell you that it doesn’t; it’s all about asking an interesting question. The mathematician and physicist Wolfgang Pauli once famously complained about a colleague’s work that it “wasn’t even wrong”. As for practical advice, one suggestion that I’ve repeatedly given to beginning mathematicians is to start with a small example. It’s surprising how concepts that might seem daunting at first encounter quickly become familiar once you play with them for a few minutes. There is really no reason to fear math if we approach it playfully. And the beautiful thing about math is that the same concept can usually be approached from so many different angles – one of my favorite branches of math, linear algebra, is the source of such great satisfaction to me because I can think about it geometrically (picturing planes or ellipsoids), algebraically (writing down equations or matrices), or abstractly (just using definitions and theorems). I truly believe that anyone can gain enough of an understanding and appreciation for math to enjoy it. And if it doesn’t work for you I’d much rather hear “I tried it, but I don’t really enjoy it” than any of the other negative statements about math that I commonly hear when I tell people what I do.

The path of love is never smooth

Dear readers,

In an earlier post I talked about how mathematical terms enrich the language that mathematicians speak in everyday life. But can mathematical terms, with their precise definitions, be used to convey feelings, so subjective by their nature? I think so, and in this post I’ll prove it by giving you an example, which also happens to be one of my all-time favorite songs. It is written by The Klein Four Group, which calls itself “the premier a cappella group in the world of higher mathematics”. Not surprisingly, it’s a love song that uses mathematical terms almost exclusively. I provide brief explanations for each term in bold to convey the main idea, though they are really superfluous; I highly recommend following along with the video.

The path of love is never smooth
[having derivatives of all orders makes you smooth]
But mine’s continuous for you
[it can be traced on paper without lifting your pen]
You’re the upper bound in the chains of my heart
[totally ordered elements in a partially ordered set]
You’re my Axiom of Choice, you know it’s true
[one of the key postulates of modern mathematics]

But lately our relation‘s not so well-defined
[a correspondence defined on pairs of objects]
And I just can’t function without you
[a function maps a possible input to a single output]
I’ll prove my proposition and I’m sure you’ll find
[every statement, or proposition, does need a proof]
We’re a finite simple group of order two
[structure formed by the numbers -1 and 1]

I’m losing my identity
[the element of a group that doesn’t change others]
I’m getting tensor every day
[a tensor is basically just a multidimensional array]
And without loss of generality
[a phrase used frequently in mathematical proofs…]
I will assume that you feel the same way
[… which is usually accompanied by the word assume]

Since every time I see you, you just quotient out
[divide an object into equivalence classes]
The faithful image that I map into
[a concept from representation and group theory]
But when we’re one-to-one you’ll see what I’m about
[in a one-to-one map, each output has one input]
‘Cause we’re a finite simple group of order two

Our equivalence was stable,
[equivalences partition a set of objects into groups]
A principal love bundle sitting deep inside
[a concept from topology and differential geometry]
But then you drove a wedge between our 2-forms
[a differential form used e.g. for double integration]
Now everything is so complexified
[turned from real numbers into complex numbers]

When we first met, we simply connected
[simply-connected means having a single piece with no hole]
My heart was open but too dense
[coming arbitrarily close to any element in a set]
Our system was already directed
[a concept from category theory I don’t understand]
To have a finite limit, in some sense
[infinite sums or products may have finite limits]

I’m living in the kernel of a rank-one map
[the set of all vectors that the map sends to zero]
From my domain, its image looks so blue,
[domain is the set of inputs; image, that of outputs]
‘Cause all I see is zeroes, it’s a cruel trap
But we’re a finite simple group of order two

I’m not the smoothest operator in my class,
[operators form classes; for smooth, see first line]
But we’re a mirror pair, me and you,
[two objects that are mirror images of each other]
So let’s apply forgetful functors to the past
[functors that drop certain properties of their inputs]
And be a finite simple group, a finite simple group,
Let’s be a finite simple group of order two
(Oughter: “Why not three?”)

I’ve proved my proposition now, as you can see,
So let’s both be associative and free
[a, b, and c are associative when (a*b)*c = a*(b*c)]
And by corollary, this shows you and I to be
[you know all about corollaries from this post!]
Purely inseparable.
[a concept from the study of roots of polynomials]

Q. E. D.
[Latin abbreviation, “which had to be proven”]

If you watch carefully, you will see the lead singer making the sign of a square with his fingers at the end, rather than that of a heart. That’s because the square symbol □ is used to denote the end of a proof.

In just over a month, many of you will need to surprise your valentines – and if this song helps provide you with the much-needed inspiration, I’ll be a happy blogger. Please share your story in the comments!

Hot spots, computers, and the academic life: an interview with Nilima Nigam

Dear readers,

I’m delighted to continue my interview series with someone who has been one of my greatest mentors, Nilima Nigam. I first met Nilima at McGill in 2004 when I took her Honours Numerical Analysis class. This was a very demanding course, where we not only solved problems and proved theorems, but also implemented numerical algorithms. It was worth every minute invested in it because I learned so much.

In 2005 I took Nilima’s Partial Differential Equations class. Just as the other class, Nilima’s philosophy was to allow us to learn by practicing as much as possible, rather than memorizing complicated proofs or accumulating abstract concepts with few examples. This was one of few classes where I experienced mathematical results being arrived at together by teacher and students, and it felt deeply empowering.

These two classes helped me finalize my decision to pursue more applied areas of mathematics, and I was very fortunate to have the chance to work with Nilima as a summer student that year. This resulted in three months of intense work culminating in a nice publication. Nilima also trained my team for the 2006 Mathematical Contest in Modeling though we ended up doing badly due to time mismanagement.

In 2008 Nilima left McGill for Simon Fraser University with her spouse, Paul Tupper, himselfa great teacher and applied mathematician. Although my interests shifted primarily to biological applications of mathematics, my general approach to research and teaching remains much inspired by Nilima’s and I also followed her example in exploring industrial mathematics after my PhD. Here is our conversation.

To find out more about Nilima’s work you can visit her homepage. Hope you enjoyed this interview!