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This blog-post is an extract from my book Maths in 100 Key Breakthroughs
Newton by William Blake (1795)
Constructible Numbers
A sure route to mathematical fame is to resolve a problem that has stood open for centuries, defying the greatest minds of previous generations. In 1837, Pierre Wantzel’s seminal analysis of constructible numbers was enough to settle not just one, but an entire slew of the most famous problems in the subject, namely those relating to ruler-and-compass constructions.
As with so much in the history of mathematics, the topic had its origins in the empire of ancient Greece. The geometers of that period were interested not only in contemplating shapes in the abstract, but also in creating them physically. Initially, this was for artistic and architectural purposes, but later for the sheer challenge it posed. In time, mathematicians came to understand that the obstacles theyencountered in these ruler-and-compass constructions brought with them a great deal of mathematical insight. Nowhere was this more true than in the ancient enigma of squaring the circle, and what that revealed about the number (pi).
Classical problems
Greek geometers decided on a set of simple rules for building shapes, using only the simplest possible tools: a ruler and pair of compasses. The ruler is unmarked, so it can only be used for drawing straight lines, not for measuring length (therefore these are sometimes called straight-edge-and-compass constructions). The compass is used to draw circles, but it may only be set to a length that has already been constructed.
Today’s schoolchildren still learn how to use these devices to divide a segment of straight line into two equal halves and to bisect a given angle. These were two of the very first ruler-and-compass constructions. A more sophisticated technique allows a line to be trisected, that is, divided into three equal parts. What of trisecting an angle, though? Various approximate methods were discovered, which were accurate enough for most practical purposes, but no one could find a method which worked exactly. This proved a mystery, and gave the first hint that there was real depth beneath this question. But what does it mean if one task can be carried out by ruler and compass and another cannot?
The most famous of the ruler-and-compass problems, and indeed one of the most celebrated questions in mathematics, is that of squaring the circle. The question is this: given a circle, is it possible to create, by ruler and compass, a square which has exactly the same area? At the heart of this question lies the number (pi) (see page 54). The problem ultimately reduces to this: given a line 1 unit long, is it possible to construct by ruler and compass another line exactly (pi) units long?
Another classical problem was that of doubling the cube. This problem had its origins in a legend from around 430 BC. To overcome a terrible plague, the citizens of the island of Delos sought help from the Oracle of Apollo. They were instructed to build a new altar exactly twice the size as the original. At first they thought it should be easy: it could be done by doubling the length of each side. But that process leads to the volume of the altar increasing by a factor of 8 (since that is the number of smaller cubes that can fit inside the new one). To produce a cube whose volume is double that of the original, the sides need to be increased by a factor of ( sqrt[3]{2}) (that is the cube root of 2, just as 2 is itself the cube root of 8). The question of doubling the cube therefore reduces to this: given a line segment 1 unit long, is it possible to construct another exactly ( sqrt[3]{2}) units long?
Wantzel’s deconstruction
Working in the turbulent setting of France in the early 19th century, Pierre Wantzel turned these ancient questions over in his mind. He recognized that the form of many ruler-and-compass questions is the same. The key to them was this: given a line 1 unit long, which other lengths can be constructed? And which cannot? If a line of length (x) can be constructed, then Wantzel deemed (x) a constructible number. Setting aside the geometrical origins of these problems, he devoted himself to studying the algebra of constructible numbers. Some things were obvious: for example, if (a) and (b) are constructible, then so must be (a + b), (a – b), (a times b), and (a div b). But these operations do not exhaust the range of constructible numbers; Wantzel realized that it is also possible to construct square roots, such as (sqrt{a}).
His great triumph came in 1837, when he showed that everything constructible by ruler and compass must boil down to some combination of addition, subtraction, multiplication, division and square roots. Since (sqrt[3]{2}) is a cube root, and cannot be obtained via these algebraic operations, it followed immediately that the Delians’ ambition to double the cube was unattainable. A similar line of thought revealed the impossibility of trisecting an angle.
As for the greatest problem of all, squaring the circle, the final piece didn’t fall into place until 1882, when Ferdinand von Lindemann proved that (pi) is a transcendental number (see page 197). Then Wantzel’s work immediately implied the non-constructibility of (pi), and the impossibility of squaring the circle was finally established.
I’m pleased to present a new book: Maths in 100 Key Breakthroughs is published by Quercus and is now available as a softback or e-book. You can read an extract here, and buy it from any of the usual places.
As the title suggests, its hundred chapters, ordered chronologically, each deal with a major mathematical development (e.g. Aristotle’s analysis of logical syllogisms circa 350BC, the discovery of transcendental numbers in 1844, and the creation of Weaire-Phelan foam in 1993). My hope is that it should be accessible, attractive, and entertaining to people with little or no background in the subject – jargon and technical notation are kept to a minimum, and each chapter is accompanied by a beautiful full-page colour illustration.
My major concern was to avoid wrenching these breakthroughs out of context and artificially presenting them as stand-alone events. After all, mathematicians typically make advances by contemplating the insights of previous generations and answering questions posed by earlier thinkers. Without Kelvin’s conjecture (and perhaps without the work of Pappus and Thomas Hales on related geometrical questions) the discovery of Weaire-Phelan foam would have been less exciting. Equally, it often takes time and further insight for the significance of a breakthrough to become apparent: it was some years after their initial discovery that the deep importance of transcendental numbers was recognised.
So I hope that the book not only presents some wonderful discoveries, but also tells the back-stories, gives some sense of what the characters involved thought they were up to, and discuss why their work matters to us today.
A few weeks ago, I was excited to receive correspondence from a certain Kenneth Perko. The tale of the Perko pair is a wonderful mathematical story, and one I have told on numerous occasions, so let me do so again:
The Perko Pair I: the story so far
In the late 19th century, mathematicians and physicists started producing tables of knots. The idea here is that some knots are genuinely different from each other, while others can be deformed to match each other without cutting or gluing, making them essentially the same thing from a topologist’s point of view. The trouble is whether or not two knots are really the same is extremely hard to tell on first sight…. as we shall see.
The tables begin with the unknot, the only knot with no crossings. Then comes the trefoil or overhand knot, which (so long as we don’t distinguish between it and its mirror image) is the only knot with three crossings. There’s similarly one knot with four crossings, then two with five, and so on. In the late 19th century Peter Guthrie Tait and Charles Little got as far as listing the knots with 10 crossings.
Knot table up to 7 crossings, from Wikipedia
However the early knot tabulators (unsurprisingly) made a few errors, and it was not until 1976 that Dale Rolfsen put together a comprehensive list of the knots with up to 10 crossings, based on earlier efforts by John Conway, in turn building on the work of James Alexander in the 1920s. Of the ten-crossing knots they counted 166 separate varieties. (Today’s mathematicians have made it as far as 16 crossings.)
The surprise was that Rolfsen and Conway had also made a mistake! Even armed with 20th century techniques of algebraic topology, a duplication had slipped through. In 1973[1] Kenneth Perko had been studying a 19th century table of Little, and had realised that two ten-crossing knots (subsequently labelled by Rolfsen as 10161 and 10162) were actually the same thing in disguise.
This story has a very clear moral: telling knots apart is difficult. Really difficult. Even after 75 years of brainpower, mathematicians were still coming badly unstuck, and who knows how long it might have taken without Perko’s alertness. And of course, this just for knots with a paltry ten crossings. Imagine trying to decipher knots with thousands…
Now for the sequel:
The Perko Pair II: Perko strikes again
The above story has been told countless times, and is usually accompanied by a picture of the offending pair of knots, something like this:
Warning: Not the Perko pair!
The trouble is that this picture is itself wrong!
This error has infected, most likely among many other places, Wolfram Mathworld and Mathematics 1001 by myself. And guess who pointed this error out…
The explanation of the mistake is that two updated versions of Rolfsen’s 1976 table are now in circulation. In both, 10162 has been deleted as it should be. But one version (occurring for instance in the latest editions of his book) keeps his original numbering up to 10166 with a space between 10161 and 10163. In the other, see here for example, Rolfsen’s 10163 has been renumbered as 10162, and 10164 as 10163, etc., thus counting up to 10165.
So the knot numbered 10162 in the picture above was actually Rolfsen’s original 10163, and thus not equivalent to 10161! For the avoidance of doubt, here is the real Perko pair, with drawings provided by the man himself. Accept no imitations!
While we’re at it, let’s also put paid to the to the idea that Ken is an ‘amateur’ mathematician. Before going on to a career in law, he says “at Princeton I studied under the world’s top knot theory topologists (Fox, Milnor, Neuwirth, Stallings, Trotter and Tucker)”. I apologise for suggesting otherwise. At least I didn’t pronounce him dead unlike certain other popular maths authors… Also, he adds “That stuff about rope on the living room floor is pure internet nonsense. I did it with diagrams on a yellow legal pad. Ropes wouldn’t work anyway since the two knots are non-isotopic mirror images of each other.”
[1] Note the corrected date, usually given as 1974.
I have just heard the very sad news that Eric Jaligot has died.
I did not know Eric well, although our paths crossed several times and we once collaborated in a piece of work [pdf] on the model theory of groups, an area in which Eric was a leading expert. He was knowledgeable, thoughtful, kind and generous with his time, and often to be seen with a wry smile on his face. I’m sure he will be sorely missed by his many friends within the model-theory community and beyond.
A tribute page is hosted at the Institut Camille Jordan in Lyon, where Eric was based.
), all live around the edge of a giant circle. This is the city. We’ll imagine that its citizens are of two races: red and blue. Initially we populate the ring randomly, meaning that we go around each house in turn tossing a coin to decide the colour of the resident.
, and as with 
, for all 
with probability at least 
. (They conjecture that this can be further tightened to 
.)
…
such that for any 
(meaning “for all sufficiently large 
is greater than 
.
:
, which is to say people are happy so long as at least 30% of the neighbourhood are their own colour.
above this threshold, but below 0.5, the theorem above applies (though the value of 
will vary). For values below 
, the ring will be static or nearly so. More precisely, we prove that for any for any 
, the probability that a randomly chosen node changes colour at any point is less than 
.
. In this situation, common-sense suggests that we should see higher levels of segregation, and this time case common-sense gets it right, though the change is again more dramatic than one might have guessed:
.)
