The Heptadecagon

As discussed in our last blog post, Gauss explored non-Euclidean geometry, but was extremely reluctant to publish his results. Today, we see the opposite; Gauss’ work with Euclidean geometry and his triumph as he makes the first advancement in 2,000 years, earning himself the title of “Prince of Mathematics.” That accomplishment is the proof of the constructibility of the heptadecagon (a 17-sided regular polygon) using only a compass and straightedge. It’s made all the more extraordinary by the fact that Gauss completed it at the age of just 19 (Dunham, 1990).

While at Göttingen University, Gauss read all the journals and mathematical work he could find. This collection was somewhat limited and he often would stumble upon proofs years after he had solved them himself, startled to find they were not original. Work he could find was often done by mathematicians such as A.G. Kastner, who Gauss considered mediocre. Meanwhile, he studied extraordinary classicists like Heyne, leading Gauss down the path of becoming a philologist. However, this all changed with his stunning discovery of the heptadecagon (Biographical, 1991).

While Gauss worked in many areas throughout his life, he considered mathematics the “queen of the sciences” and number theory the “queen of mathematics” (Dunham, 1990, pg. 240). At the time, number theory was a series of scattered results. In 1801, Gauss published Disquisitiones Arithmeticae. This was groundbreaking work as it systematically collected previous work in number theory, solved several outstanding problems, and set the standard of research for the next century and beyond. Gauss’ original work includes a proof of the law of quadratic reciprocity, the inception of modular congruence, the first examples of equivalence relations, further development of quadratic forms, and analysis of the cyclotomic equation. Such a collection instantly brought Gauss to the front of the mathematical world (Biographical, 1991).

Pages from Disquisitiones Arithmeticae (public domain).

Gauss culminated his book with work on the cyclotomic equation and lead it into connections with polygon construction, thus, combining number theory, geometry, and complex numbers. Polygon construction dates back to Book IV of Euclid’s Elements. Euclid showed how to construct regular triangles, squares, pentagons, hexagons, octagons, and pentadecagons using only a compass and straightedge. Although he never explicitly addressed it, it is also clear Euclid knew that doubling the number of sides of a constructible polygon results in another constructible polygon. Unclear, however, is if these were the only constructible polygons. No more had been discovered since Euclid (300 B.C.), so this was the assumption. That is, until out of nowhere, the young and unknown Gauss announced Euclid’s collection was unfinished (Dunham, 1996).

So how did he do it? Descartes proved early in the seventeenth century that the set of constructible numbers (i.e. numbers that can be constructed using only a compass and straightedge) includes positive integers, as well as any number resulting from a finite number of constructible numbers combined through addition, subtraction, multiplication, division, and square roots (Dunham, 1996). Gauss sought to prove cos(2π/17) was constructible. If he could do this, then the 17-gon was constructible. To see why, consider the following figure. We start with a unit circle with center O and construct segment OC with length cos(2π/17). Construct a line perpendicular to OC at C and label the intersection of the perpendicular and the circle as point B. Then, cos(<BOC)=OC/OB=cos(2π/17), implying <BOC=2π/17. Next, extend segment OC and label the intersection of OC and the circle as point A. If we then copy the chord AB seventeen times around the circle, we land exactly back at point A and have successfully constructed the 17-gon (Dunham, 1996).

The construction of a 17-gon (Dunham, 1996).

So, Gauss’ proof hinges on his ability to prove cos(2π/17) is indeed constructible. To do so, he ventures into complex numbers - a surprising move. If the geometric construction is, by definition, a concrete real-world representation, why make use of imaginary numbers? Perhaps this is why this proof took so long to come to light. Complex numbers had certainly been explored long before Gauss by some of the greatest mathematicians of their age, but imaginary numbers continued to live in the shadows. Gauss was the first to use them in a “really confident and scientific way” (Hardy & Wright, 2000, pg. 244). According to Gauss, this was because of their branding. He writes,

“If this subject has hitherto been considered from the wrong viewpoint and thus enveloped in mystery and surrounded by darkness, it is largely an unsuitable terminology which should be blamed. Had +1, -1, and √-1, instead of being called positive, negative, and imaginary (or worse still, impossible) unity, been given the names say, of direct, inverse, and lateral unity, there would hardly have been any scope for such obscurity” (Merino, 2006, pg. 5).

While Gauss never got his wish for direct, inverse, and lateral unity, he did coin the term “complex numbers.” His continued work with them resulted in four proofs of the Fundamental Theorem of Algebra which says the field of complex numbers is algebraically closed. The first of these earned Gauss his doctorate from the University of Helmstedt (McElroy, 2005). Uncharacteristically, he published it before satisfying his standards of rigor and thus it contained a gap that was not patched until 1920 (Jackson, 2017; Cain, 2005).

Back in Disquisitiones Arithmeticae, Gauss recognized that the number cos(2π/17) is a component of the complex number z = cos(2π/17) + i sin(2π/17) and thus one of the seventeenth roots of unity. Continuing down the path of complex numbers, he is able to prove the following:

As this expression is composed of only positive integers added, subtracted, multiplied, divided, and square-rooted, it is constructible. Thus, by our previous logic, the 17-gon must be constructible (Dunham, 1996).

A regular 17-gon (Kuh, 2013).

Gauss actually went beyond proving the 17-gon was possible; he proved it was possible to construct a polygon with a number of sides equivalent to a Fermat prime. That is, a prime of the form 22^n+1 for some non-negative integer n. Known to Fermat and Gauss were the first five Fermat primes, 3, 5, 17, 257, and 65,537, corresponding to n values of 0, 1, 2, 3, and 4. In fact, these remain the only five Fermat primes known today (Kuh, 2013). Hence, Gauss proved the constructibility of the regular 17-gon, the 257-gon, and the 65,537-gon.

Gauss did not actually construct each of these polygon; he simply proved it was possible. The constructions would not come until 1800, 1832, and 1894 by Erchinger, Richelot, and Hermes, respectively. They hold no practical use, but their possibility uncovered a mystery lurking in Euclidean geometry. The field was so familiar, Gauss made the first true innovation in Euclidean geometry in 2,000 years. He revelled in his discovery. So proud was he, that above all the original work he would create over his 77 year life, he asked for the 17-gon to be inscribed on his tombstone. Unformately, this was never done (Dunham, 1990). The comparison to Gauss’ work in non-Euclidean geometry is startling. While the 17-gon earned Gauss his name, his insight that Euclidean geometry was not the only geometry was arguably more revolutionary and impactful. However, one discovery he despised and the other he revered. His work with complex numbers though, brought him to the forefront of a field that would grow dramatically in the next century.

References

Biographical dictionary of mathematicians : Reference biographies from the dictionary of scientific biography (Vol. 2). (1991). New York, NY: Charles Scribner’s Sons.

Cain, H. (2005). C.F. Gauss's proofs of the fundamental theorem of algebra. Einstein Institute of Mathematics.

Dunham, W. (1990). Journey through genius. New York, NY: Penguin Books.

Dunham, W. (1996). 1996—A triple anniversary. Math Horizons,4(1), 8-13. Retrieved from http://www.jstor.org/stable/25678076

Hardy, G. H., & Wright, E. M. (2000). An introduction to the theory of numbers (4th ed.). Oxford: Oxford University Press.

Jackson, T. (Ed.). (2017). Mathematics: An illustrated history of numbers. New York, NY: Shelter Harbor Press.

Kuh, D. (2013). Constructible regular n-gons. Whitman College.

McElroy, T. (2005). A to Z of mathematicians (Notable Scientists). New York City, NY: Facts 
on File.

Maintaining the Status Quo

Carl Friedrich Gauss was constantly in pursuit of the next big idea; he is credited with more than 400 original ideas throughout his long life. However eager he was to uncover new truths though, he was hostile towards anything that disrupted established tradition (Biographical, 1991).

Gauss lived in a time of instability. A stalwart conservative, he was unsettled by the turmoil caused by the French Revolution and the Napoleonic period and was distrustful of the democratic revolutions happening in his homeland of Germany (McElroy, 2005). To Gauss, Napoleon epitomized all that was wrong with revolution. He remained a firm nationalist and royalist throughout his life (Biographical, 1991).

To understand why he took this position, it can help to look back towards his youth. Gauss grew up in a household that was financially uncertain. The child prodigy wished for a higher education, but was denied by his father. His talents were clear and at age 14, the Duke of Brunswick supplied Gauss with a stipend, making him financially independent of his parents. With this, he left home and began studies and research in the subject he loved; mathematics. The Duke provided Gauss with his “golden years of freedom” and became, in Gauss’ mind, the personification of an enlightened monarchy (Biographical, 1991, pg. 864). He would be honored with 75 official awards throughout his life, but often brushed them off. That is, unless a royal was present (Young, 1998).

Portrait of Charles William Ferdinand (1735-1806), the Duke of Brunswick during Gauss’ youth (public domain).

Then, tragedy struck. In 1806, the Duke lead the Prussian armies against Napoleon. In a humiliating defeat, the Duke was killed and Gauss’ anti-revolutionary beliefs were confirmed (Biographical, 1991).

This reluctance for radical change continued into Gauss’ academic life, as demonstrated in his dislike for the most revolutionary mathematical idea of the 19th century: non-Euclidean geometry (Biographical, 1991). His journey began at the University of Göttingen in 1795. As a student, he bore witness to the popular attempts at proving Euclid’s parallel postulate, but was quick to spot the inevitable misconceptions in each of them. None of them satisfied his infamously rigorous standards of proof. Gauss was so selective, much of his own work failed to meet his standards and went unpublished. Only his most polished work (work beyond criticism) was released. It wasn’t that he didn’t want the satisfaction that came with being first to discover, he just didn’t care if others knew of his preeminence. He had a great desire to be the first discoverer (going so far as to keep a dated record of all his results and those of others), but was satisfied for this to be known only to himself (Biographical, 1991).

As will be discussed in a later post, Gauss held jobs in astronomy and surveying. Thus, when working with geometry, he used the globe as his model. Here it was possible to create a series of vertical lines (longitudes) that each meet the equator at a right angle. However, every longitude intersects at the North and South Poles - a contradiction in Euclidean geometry. Gauss now had a concrete example showing the parallel postulate did not hold on curved surfaces (Development, 2002). Slowly, he ventured into the consequences of excluding it and cautiously moved towards the “new” geometry. However, he never dared make his thoughts known publicly. Mathematicians at the time believed Euclidean geometry was the “inevitable necessity of thought” (O’Connor & Robertson, 1996). Gauss, infamous in his dislike of controversy, was not going to be the one to disrupt this.

A model of the Earth lead Gauss to non-Euclidean geometry ([Latitude and Longitude], 2018).

As the years went on, his correspondence shows him clearly, but warily, growing in his belief that Euclid’s fifth postulate could not be proved. In 1816, he wrote a review of a book that contained explorations in non-Euclidean geometry and insinuated his agreement. His worst fears were confirmed when the idea was “besmirched with mud” by critics (Biographical, 1991, pg. 865). He never made the mistake again. He encouraged others who shared his thoughts, but only in private. He was committed to not publishing his own results (although he considered having them published after his death), but showed more willingness for someone else to take the fall.

In 1831, word came that János Bolyai was up to the challenge. Gauss wrote to his father, Wolfgang Bolyai, endorsing the work but (in Gauss’ off-putting style), declaring his own superiority. He wrote, “I regard this young geometer Bolyai as a genius of the first order” but, “to praise it would amount to praising myself. For the entire content of the work … coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years” (O’Connor & Robertson, 2004, pg. 1). After reading Gauss’ letter, János thought it a conspiracy to take credit for his work and quickly cut all ties with Gauss. It seems Gauss learned his lesson, because a decade later when Lobachevsky came on the scene, Gauss’ letter was full of praise and offered membership to the Göttingen Academy where he was working as an astronomer.

A portrait of János Bolyai (1802-1860), credited as one of the founders of non-Euclidean geometry (Márkos, 2012).

While Gauss was privately supportive (or at least, tried to be), in public he remained stubborn. Such a radical discovery was bound to be met with cynicism and a well-known mathematician like Gauss could have gone a long way towards bringing the idea into the mainstream if he had thrown support behind a like-minded mathematician. Instead, he refused to make his ideas known. Despite Gauss’ correspondence, analysis of Bolyai’s and Lobachevsky’s writings shows all three made their discoveries independently. It seems Gauss held a neutral (and sometimes negative) role in the development of non-Euclidean geometry. His public silence was taken as complicity in the general ridicule that prevented progress in the field. It would be decades before this was surpassed and came in part when it was revealed that the prince of mathematics had secretly been a non-Euclidean (Biographical, 1991).

While Gauss never published his work with non-Euclidean geometry, it does show up somewhat indirectly. His work as a surveyor led to work in differential geometry and in 1827 he published Theorema Egregium (Latin for “Remarkable Theorem”) which described what is now known as Gaussian curvature. In this, each surface is assigned a curvature number, the sign of which tells you the type of surface you are working with. A constant curvature of zero indicates the surface is in Euclidean geometry. If there is a constant positive curvature, the surface is a sphere and thus is in spherical geometry - a subset of elliptical geometry. If instead there is a constant negative curvature, the surface is “pseudospherical” and is in hyperbolic geometry (Mastin, 2010; Henderson & Taimina, 2017).

There are three types of constant Gaussian curvature (Mastin, 2010).

The main theorem of the paper states the curvature of a surface is not changed if it is bent without being stretched. Hence, it depends on how distances are measured, but not how they are embedded in space. For example, a cylinder has a Gaussian curvature of zero; equal to that of a flat surface. Thus, by “unrolling” a cylinder, it can be seen that it is isometric to a flat surface. Meanwhile, since a sphere has constant positive curvature and a plane has zero curvature, the two are not isometric. Therefore, any attempt to transform a sphere into a flat representation will distort measurements. This has clear connections to Gauss’ work as a surveyor, as this proves there is no perfect model of the Earth on a flat map (Mastin, 2010).

Gauss had a life full of uncertainty, both financially and politically. He linked Euclidean geometry with the past tradition he revered and feared the ridicule that would come with publishing his thoughts. Despite his reluctance to put his name on his work, he still left his mark. His work is seen in Gaussian curvature, a field mathematicians of the time did not connect with non-Euclidean geometry. More directly, the term “non-Euclidean geometry” was coined by Gauss himself.

References

Biographical dictionary of mathematicians : Reference biographies from the dictionary of
scientific biography (Vol. 2). (1991). New York, NY: Charles Scribner’s Sons.

Henderson, D. W., & Taimina, D. (2017, January 15). Non-Euclidean geometry. Retrieved from https://www.britannica.com/science/non-Euclidean-geometry

[Latitude and Longitude]. (2018, October 16). Retrieved from https://www.cemc.uwaterloo.ca/events/mathcircles/2018-19/Fall/Junior6_Oct16.pdf

Márkos, F. (2012). János Bolyai [Painting].

Mastin, L. (2010). 19th century mathematics - Gauss. Retrieved from https://www.storyofmathematics.com/19th_gauss.html

McElroy, T. (2005). A to Z of mathematicians (Notable Scientists). New York City, NY: Facts 
on File.

O'Connor, J. J., & Robertson, E. F. (1996, February). Non-Euclidean geometry. Retrieved from http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean_geometry.html

O'Connor, J. J., & Robertson, E. F. (2004, March). János Bolyai. Retrieved from http://www-history.mcs.st-and.ac.uk/Biographies/Bolyai.html

The development of non-Euclidean geometry. (2002, July 16). Retrieved from http://www.math.brown.edu/~banchoff/Beyond3d/chapter9/section03.html

Young, R. V. (Ed.). (1998). Notable mathematicians: From ancient times to the present. Detroit, MI: Gale Research.

The Archetypal Mathematician

Carl Friedrich Gauss (1777-1855), known as the Prince of Mathematics, is one of the most prolific mathematicians of all time. He joined the ranks of Sir Isaac Newton and Archimedes of Syracuse while living in Germany at a time where little mathematical progress was being made. His career included work in pure mathematics (arithmetic, number theory, geometry, algebra, and analysis), applied mathematics (probability, statistics, mechanics, and physics), as well as the mathematical sciences (astronomy, geodesy, magnetism, dioptics, actuarial science, and financial securities). He is credited with more than 300 publications, which represent just a fraction of the original ideas he developed (McElroy, 2005).

Portrait of Carl Friedrich Gauss by G. Biermann (1824-1908) (public domain).

Gauss had a hard start to life as he was born to a family on the edge of poverty. His mother was highly intelligent, but illiterate. In fact, after his birth, his mother never wrote down his birthday and could only recall that he was born on a Wednesday, eight days before Ascension Day (Bien, 2004). This missing knowledge about himself spurred Gauss to create an algorithm that would tell him the date of Easter in any given year (and in turn, Ascension Day and his birthday). He was ultimately successful in this endeavor (although he had to invent modular arithmetic to do it) and learned he was born April 30th (Gardner, 1981; Petrilli, 2012).

Despite Gauss’ mother’s illiteracy (or perhaps due to it), she was extremely supportive of her son’s education. Meanwhile, his father was practically minded and never valued his gifted son (McElroy, 2005). He worked as a gardener, laborer, foreman, assistant to a merchant, and treasurer of a small insurance fund in an effort to raise his family out of poverty. He did not understand the pull towards abstract mathematics and had a tenuous relationship with his son. Gauss described his father as “worthy of esteem,” but “domineering, uncouth, and unrefined” (Biographical, 1991, pg. 860).

Gauss’ talent was clear from a young age. It is said he learned to calculate before he could talk and began correcting mistakes in his father’s wage calculations by the age of 3 (something apparently unappreciated by his father). There is also the well-known story of Gauss’ arithmetic teacher who assigned him the task of adding together the first 100 integers. Rather than do this explicitly, Gauss found a pattern and grouped the integers into 50 pairs that each added to 101, quickly providing the answer of 5,050.

An illustration of Gauss’ method of adding the first 100 integers (Wilburne, 2014).

As for his personal life, Gauss married Johanna Osthoff in 1805 and with her had a son and a daughter. In 1809, Osthoff died while in childbirth and the child died soon after. The loss sparked a loneliness from which Gauss never recovered. He wrote that he had “closed the angel eyes in which for five years [he had] found a heaven” (Biographical, 1991, pg. 864). Less than a year later he married Minna Waldeck and had two sons and a daughter. However, the marriage was unhappy and Waldeck was often sick from tuberculosis (Young, 1998).

Gauss’ happiness in his first marriage is somewhat surprising as he spent most of his life as a recluse. Whether this is because there was no one at the time that he felt matched his intellect or because he was simply unwilling to work with others is unclear. He avoided teaching positions to the extent he could and never took anyone under his wing as one might have expected. The exceptions to this include correspondence with Eisenstein, Bernhard Riemann, and Sophie Germain. In general, he considered other mathematicians rivals and distractions. This even applied to his own family. The oldest son from his second marriage, Eugene, grew up as a prodigy like his father. Rather than mentor him, Gauss discouraged his son from a career in mathematics. Eugene took this to mean that his father feared he would soil their last name with subpar work (Biographical, 1991).

While Gauss was disinterested in people, he was especially fond of newspapers and magazines, a novelty in the 19th century. He was reportedly called the “newspaper tiger” in his university library for his penchant of staring down students who got to a new printing before he did (Young, 1998). While off putting for the people around him, this habit seems to have paid off for him. Motivated by his actuarial work, Gauss kept collections of statistics and observations from daily newspapers. This lead to financial speculations astute enough to create a net worth nearly 200 times his annual salary. Gauss had achieved the financial status his father had dreamt of, but without the “practical” career he had been told was necessary (Biographical, 1991).

Gauss lived in a time with no mathematical equals (a fact even they acknowledged). While the lack of collaborators and rivals may have discouraged most, Gauss was known for his solitary and reclusive attitude. Introspective and ambitious, Gauss represents the archetypal mathematician (McElroy, 2005). In our next few blog posts, we will dive deeper into the major accomplishments of this extraordinary mathematician.

References

Bien, R. (2004). Gauβ and beyond: The making of Easter algorithms. Archive for History of Exact Sciences, 58(5), 439-452.

Biographical dictionary of mathematicians : Reference biographies from the dictionary of scientific biography (Vol. 2). (1991). New York, NY: Charles Scribner’s Sons.

Gardner, M. (1981). Mathematical games. Scientific American, 244(2), 17-22.

McElroy, T. (2005). A to Z of mathematicians (Notable Scientists). New York, NY: Facts on 
File.

Petrilli, S. J. (2012). Servois' 1813 perpetual calendar, with an English translation - Gauss' calculation for the date of Easter. Retrieved from https://www.maa.org/press/periodicals/convergence/servois-1813-perpetual-calendar-with-an-english-translation-gauss-calculation-for-the-date-of-easter

Wilburne, J. M. (2014). [Sum of integers]. Retrieved from https://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/The-Story-of-Gauss/

Young, R. V. (Ed.). (1998). Notable mathematicians: From ancient times to the present. Detroit, MI: Gale Research.