Chasing the impossible dream

The challenge of making the impossible possible is what drives the advances of today’s rapidly changing society. The common motivational (and advertising) trope in the modern world that best symbolises this sentiment are the words “Nothing is impossible”. These are the opening words of David Richeson’s book Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity, a book that chronicles an intellectual journey involving some of the greatest mathematicians in history and unravels their combined struggles over centuries that led to a successful subversion of the aforementioned trope in the mathematical arena. The book highlights the strange power of impossibilities in mathematics and deftly demonstrates that they constitute a fundamental part of the fabric of the mathematical universe. For the mathematician, the impossible holds as much promise and lure as the possible and Tales of Impossibility recounts a fascinating pursuit that has attracted the attention of mathematically gifted savants and oddballs alike.

Richeson’s latest book resembles his earlier popular book Euler’s Gem: The Polyhedron Formula and the Birth of Topology¹ in tone and style, combining historical and mathematical threads to weave an enticing and coherent tale of intellectual conquest. For a book that is concerned with history and mathematics, two subjects stereotypically considered “dry” by most lay readers, Richeson pulls off a commendable feat of delivering an exciting account, both mathematically and historically.

A preambular chapter introduces the readers to the titular problems of impossibility which are the classical Greek geometric construction problems of antiquity—using a compass and straightedge to square the circle (i.e., constructing a square with the same area as a given circle), doubling the cube (constructing a cube with double the volume of a given cube), and trisecting an angle along with the closely related problem of using a compass and straightedge to construct a regular n-sided polygon.

Richeson follows this up with an enlivening discussion about the prospect of impossibility from a mathematical perspective. The book chooses to use a mathematical puzzle as an explanation tool, in particular, the famous 15-puzzle initially claimed as the creation of American puzzler Sam Lloyd but later known to have actually been discovered by Noyes Chapman. The more knowledgeable reader will recognise the puzzle to be connected to a problem on permutation groups, but the example still succeeds in getting across the fact that impossibilities are a dime-a-dozen in the world of mathematics and are at the heart of many deep results. Indeed, it is well-known that the whole field of non-Euclidean geometry spawned from the impossibility of proving Euclid’s fifth postulate from the other four, or that topology and more directly graph theory both owe their origins to an impossibility result, vis-a-vis the unsolvability of the Bridges of Königsberg problem.

Richeson reveals very early on that the four impossibility theorems can be reduced to algebraic statements about certain real numbers. Primarily, the questions boil down to the algebraic character of the numbers \sqrt[3]{2}, \cos(\theta/3),\cos(2\pi/n) and \pi and if one can show that these numbers are not “constructible”, then it will follow that one cannot double a cube, trisect a general angle, construct a regular n-gon, or square a circle respectively. Of course, the book takes a while to get to a precise understanding of the notion of constructibility, and the journey that follows is enjoyable.

The tale begins in earnest in ancient Greece in the fifth century BCE which offers the earliest time and place of origin for the problems. The book has a lot to say about the mathematics of the Greek period, particularly the limitations they had in being restricted to the use of natural numbers. In the geometric framework of the early Pythagoreans, what mattered was the ratios of lengths of line segments or of areas of geometric figures, and thus they treated all quantities as being commensurable, that is as being multiples of a fundamental unit, and were largely confined to the use of rational numbers. Irrational magnitudes came to be acknowledged by the time of Plato and Euclid, the latter of whom provides a delightful proof of the reductio ad absurdum type for the irrationality of \sqrt{2} in Book X of his Elements. This was of course along the same lines as an earlier proof by Aristotle.

Most people exposed to constructions using a straightedge and compass wonder what is so special about these specific tools and whether they are just an arbitrary choice on the part of the early Greek mathematicians. But the book furnishes a proper historical and analytical rationale for this choice as it discusses the mathematical understanding of the Greeks and demonstrates that the choice of the compass and straightedge make perfect sense when seen in the light of the mathematical advances that followed. Chapters ten and twelve of the book consider variants of these geometric questions. What would be the consequences, for example, if one were allowed to use a marked straightedge and compass or, alternatively, only a compass or only a straightedge? And the answers are surprising. A chapter on the construction of some classical curves also takes a look at the many ingenious instruments that were designed to produce curves such as the quadratrix, conchoid, limaçon of Pascal, spirals, carpenter’s square curve, etc. The trammel of Nicomedes to construct parts of the conchoid, in particular, can be seen as a precursor to the later works of mathematicians such as Chebyshev on linkage mechanisms, and attempts of Kempe to prove a one-one correspondence between algebraic plane curves and linkages.

However, as far as the choice of straightedge and compass is concerned, the fact that there are many equivalent sets of tools giving rise to the same set of constructible objects is the key observation, as is made clear in the ensuing chapters of the book. In modern parlance, this essentially means that this set of constructible lengths (numbers) have a neat algebraic characterization and that a number is constructible precisely if it can be obtained from the integers using any combination of the four arithmetic operations and square roots. Ergo, the set of constructible numbers form a well-defined set known as a Euclidean field, and one that can also be described in many equivalent ways.

Mathematical advances are spurred on not just by climactic joyous triumphs but also by the preceding frustrating failures.

This was the momentous idea pioneered by Descartes in the first half of the seventeenth century, who amalgamated the geometric and the algebraic elements and set into motion a chain of mathematical revelations regarding the essential nature of the impossibility problems. The book informs us about Descartes’s geometry and, in particular, his use of a compass and straightedge to build a working model of arithmetic for line segments that allowed him to perform operations of addition, multiplication, division and subtraction with these objects. This led him to conjecture about the potential objects of construction determined by a ruler and a compass. Descartes’ observations in the first half of the 17th century were sandwiched between two significant events. The first was the previous work of the Italian school, who benefited from the arrival of algebra into Europe, chiefly that of the self-taught mathematician Tartaglia (Niccoló Fontana), Gerolamo Cardano and the prodigious Ludovico Ferrari, on solutions of cubic and quartic equations, and the second was the eventual understanding and acceptance of the complex numbers and their properties following Descartes, both of which provided crucial pieces to solving the puzzle. The book sheds light on this turbulent episode in chapters thirteen to eighteen. Descartes was also influenced by the work of François Viète, the amateur French mathematician who made vital contributions and the book spends some time delving into the rudiments of his “new algebra”. That Viète was a lawyer by profession makes his accomplishments all the more striking.

As can be expected, Gauss had quite a bit to say about all this. At the age of nineteen, Gauss showed that one could construct a 17-sided regular polygon (heptadecagon). Later on, in 1801, Gauss proved a more general theorem regarding the construction of n-sided regular polygons. The proof relied on the properties of complex numbers, algebra and number theory and was an early exhibit of the harmonious interdependency between these different spheres of mathematical study. It would be instructive at this point to recall an enlightening essay, “Is there beauty in mathematical theories?”, where the mathematician Robert P. Langlands while treading similar ground and discussing exactly the same work of Gauss, remarks, “I have found over the course of the years, without making any particular effort, that the more one penetrates Gauss’s writings the closer one comes to mathematics”. One cannot help but agree.

This remarkable result of Gauss, which was later improved upon by the undeservedly obscure French mathematician Pierre Wantzel, established a surprising link between the constructibility of regular polygons and the primality of the famous Fermat numbers. The powerful Gauss-Wantzel theorem states that only those regular n-sided polygons with n of the form

n = 2^jp_1… p_k, where j>0 and p_is for i=1… k are distinct Fermat primes, are constructible.

Nevertheless, Gauss’ proof did not involve an explicit method for the construction of the heptadecagon. Several possible ways of constructing the 17-gon were furnished in later years. The methods of explicitly constructing regular polygons with an even number of sides are relatively elementary. It is the construction of regular polygons with the number of sides equal to a Fermat prime that poses difficulties. It is also interesting to note that although infinitely many constructible polygons exist, there are only 31 known with an odd number of sides, resulting from the combinations of the products of the five known Fermat primes viz. 3, 5, 17, 257 and 65537. Among these, a rather involved construction of a regular 65537-gon was provided by Johann Gustav Hermes in a 200-page manuscript in 1894 [4].

Gauss’ theorem, as it was stated, only covered the sufficiency part. It is therefore a credit to the brilliance of the young engineer and mathematician Pierre Wantzel that he delivered what would be the pièce de résistance with his proofs of three of the four impossibility theorems in a seven-page article in 1837, the most important page of which Richeson dubs the “pagina mirabilis”—the miraculous page. In this paper, Wantzel proves the necessary condition, by exploiting the theory of cyclotomic polynomials.

This left the other impossibility problem of squaring the circle as the last fort to be felled. As mentioned earlier, this is intricately connected with the distinctive quality of \pi. The book treats the early history of this most famous mathematical constant at length, and leads us on a globetrotting expedition as we initially traverse the diverse mathematical landscapes of the Egyptian and Mesopotamian civilizations, as also of India and China. Richeson details the early attempts of Archimedes to estimate its value but at the same time is careful to note that the toil was not limited to the Greeks. Indeed, noteworthy progress was made in China and India. This is, for instance, evidenced by the works of the Hindu mathematician Mādhava from the Kerala School of Mathematics and Astronomy. Mādhava’s work on estimating the value of \pi is built on his work on infinite series and is cited in the Mahajyānayana prakāra (“Methods for the great sines”), attributed to Mādhava himself, and also finds mention in the treatise Tantrasaṃgraha by Nīlakaṇṭha and in a couple of later commentaries, notably Jyeṣṭhadeva’s Yuktibhāṣā. While the book mentions the Śulbasūtras in relation to the problem of squaring the circle, it however fails to mention Āryabhaṭa who gave an early approximation of \pi accurately up to three decimal places with a method different to that of the Greeks.

The \pi voyage picked up pace when it was first shown to be an irrational number by Johann Lambert and later by Adrian-Marie Legendre, finally peaking with a proof of its transcendentality deduced by the German mathematician Ferdinand von Lindemann in 1873, who largely built on the earlier ground-breaking work of Charles Hermite. Hermite had in the same year shown in two different ways that the other well-known mathematical constant e is a transcendental quantity. With this, the fate of the last of the impossibility theorems was sealed. Although there is a brief mention of the consequent events, such as David Hilbert’s seventh problem which serves as a more general result on transcendental numbers and its subsequent resolution by Alexander Gelfond and Theodor Schneider, Richeson resists the temptation to go any further and sticks to a more tractable scope.

Given that the book tries to encapsulate close to 25 centuries of mathematical saga, involving some of the most well-known mathematicians of the time whose works serve as the foundations of the modern mathematical edifice, it packs in quite a bit of historical detail. However, the exposition expertly blends the historical material with just the right amount of mathematical content, and a favourable trait is the way in which both those aspects are interwoven cleverly.

Rather than bombard the reader with a melee of facts, Richeson opts to unravel the story at a leisurely pace without losing sight of the underlying grand theme he seeks to develop. That the book chooses to deal with the history around a chosen set of problems, rather than the history of a subject itself, works to its advantage and keeps the historical exposition tight.

The narrative is also infused with a strong human element and paints an impressive picture of the enormous intellectual labour and unrelenting perseverance shown by the many mathematicians whose life stories make up the long mathematical odyssey that the book documents. The book also refrains from reducing these stories to a mere series of anecdotes and opts to present an objective version of historical events, choosing instead to trust the reader’s intelligence.

Interspersed between the principal chapters are the “tangents” which follow each chapter, and serve as engaging interludes. Some of these side excursions are of the “did-you-know?” type with a strong historical flavour. Did you know that the Indiana House of Representatives legislated a bill on the value of \pi, or that Leonardo da Vinci grappled with trying to square the circle, or about the \tau revolution trying to replace \pi by the more convenient 2\pi?

Some other “tangents” are of the do-it-yourself kind which include geometric constructions using toothpicks, trisecting an angle using a tomahawk², and doubling the cube using origami, and methods to compute \pi at home. These rather interactive sojourns furnish a hands-on experience for the enthusiastic reader looking for a better understanding of the nitty-gritty of the problems. One peculiar and amusing “tangent” concerns mathematical “cranks”—people who are convinced that the impossible remains possible and attempt to prove the same—and even provides tips to spot them! All in all, these eclectic detours complement the main theme well and allow for a deeper appreciation of the central ideas. The richly annotated endnotes and bibliography are a welcome addition to whet the appetite of the reader intending to dig deeper into the mystery and history of these problems.

The historical account is also a beautiful testament to an often-ignored truth in society, that no great mathematical breakthrough happens in a vacuum, but is rather the culmination of painstaking efforts of many mathematical minds over numerous years. In this context, one must note that although the usual suspects such as Euclid, Archimedes, Euler, Descartes and Gauss find abundant mention in the book, Richeson also brings to the fore the stories of many lesser-known mathematicians of their time, often unfairly forgotten or overlooked.

Notable among these are the debatable legend of Hippasus from ancient Greece whose discovery of incommensurable magnitudes was met with stiff opposition and punishment from the early Pythagoreans, the enterprising undertaking of the Renaissance era geometer Viète who never trained as a professional mathematician, and none more so than the tragically short and uncelebrated life of Pierre Wantzel, who provided a satisfactory conclusion to three of the four main impossibility problems, but was surprisingly ignored and unknown for almost a century despite the prominence of the problems in the mathematical community.

Richeson pays tribute to the unheralded feat of Wantzel by devoting an entire chapter to sketch his biography and also dwells on the disappointing lack of attention given to his achievements. In contrast, as Richeson observes, Lindemann, who successfully settled the remaining impossibility theorem of squaring the circle had his shot at fame although his findings were largely an extension of that of Hermite, who himself seemed content to stay out of the limelight. The following quote attributed to Hermite sums up his disposition:

We are servants rather than masters in mathematics.

As for the mathematical side of things, a valuable feature is that the book quite effectively functions as an introduction in proof-writing for the prospective math major trying to grasp the intricacies of analytical reasoning and logic. This quality is all the more enhanced due to the historical context provided for the mathematical details, a didactic method now finding increasing traction in higher mathematics education [1].

Although the book does not include any strong mathematical proofs or assume any background on the part of the reader, there is a clear intent to ensure that the reader gets the “big picture” and is able to recognise how distinct results from different mathematical domains come together, leading to the resolution of the four geometric construction problems, thus showcasing the elegant confluence of the currents of abstract algebra, Euclidean geometry and number theory.

The book does not shy away from presenting the reader with dozens of messy yet beautiful Euclidean geometry figures where necessary, to illustrate the nuts and bolts of the constructions being discussed. This is particularly evident in the chapter on “Neusis constructions” where an appreciation for geometric figures is invoked. Indeed, the reader is sure to find sitting down with a pen and paper, to work through some of the algebraic examples in the book and a compass and ruler for the geometry bits, to be a rewarding endeavour. On the other hand, Richeson also does not mind dispensing with technical expressions when dealing with the formulae of solutions to certain cubic equations or the complex number formulae to avoid interrupting the reader’s flow.

… showcases the elegant confluence of the currents of abstract algebra, Euclidean geometry and number theory.

Another heartening fact is that the book places enough emphasis on exploring the unsuccessful attempts as much as the eventually successful ones, quite rightly sending a message that mathematical advances are spurred on not just by climactic joyous triumphs but also by the preceding frustrating failures. All in all, the book admirably captures the trials and tribulations of the many geniuses whose work on the impossibility problems has left a long-lasting mathematical legacy. Again, we hearken back to Langlands who quotes

Two striking qualities of mathematical concepts are that they are simultaneously pregnant with possibilities for their own development, and are of permanent validity.

Ultimately, Tales of Impossibility is a mathematical epic with an inspiring human story at its heart and comes close to providing a glimpse of that often-sought attribute of beauty in the mathematical realm. \blacksquare

References

1. David Bressoud – Calculus Reordered: A History of the Big Ideas Princeton Press, 2019.
2. David S. Richeson – Euler’s Gem: The Polyhedron Formula and the Birth of Topology Princeton University Press, 2008.
3. Robert P. Langlands – Is there beauty in mathematical theories? Parts 1 and 2, Bhāvanā https://bhavana.org.in/langlands-beauty-in-mathematical-theories-part1/, https://bhavana.org.in/is-there-beauty-in-mathematical-theories-part2/
4. Johann Gustav Hermes (1894). “Uber die Teilung des Kreises in 65537 gleiche Teile”. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German). Göttingen. 3: 170–186. http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002496585
5. M.L. Wantzel (1837). Recherches sur les moyens de reconnaítre si un Problème de Géométrie peut se résoudre avec la règle et le compas, Journal de Mathématiques Pures et Appliquées, 2: 366–372.

Footnotes

1. Winner of the 2010 Euler Book Prize awarded by the Mathematical Association of America. ↩
2. A type of single-handed axe native to the many Indigenous peoples of North America, traditionally resembling a hatchet with a straight shaft. ↩