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.