146 // Why Gauss Became a Mathematician

hose, and so on, he uses the hose to put out the fire with the minimum expenditure of energy.

Later, the mathematician wakes up and smells smoke. He goes into the hallway and sees a (third) fire. He notices the fire hose on the wall, and thinks for a moment . . . Then he says, `OK,

........................................... a solution exists!' ­ and goes back to bed.

Why Gauss Became a Mathematician

Carl Friedrich Gauss.

Carl Friedrich Gauss was born in Brunswick in 1777 and died

¨ in Gottingen in 1855. His parents were uneducated manual workers, but he became one of the greatest mathematicians ever; many consider him the best. He was precocious ­ he is said to have pointed out a mistake in his father's financial calculations when he was three. At the age of nineteen he had to decide whether to study mathematics or languages, and the decision was made for him when he discovered how to construct a regular 17-sided polygon using the traditional Euclidean tools of an unmarked ruler and a compass.

This may not sound like much, but it was totally unprece- dented, and the discovery led to a new branch of number theory. Euclid's Elements contains constructions for regular polygons (all Why Gauss Became a Mathematician // 147

sides equal length, all angles equal) with 3, 4, 5, 6 and 15 sides, and the ancient Greeks knew that the number of sides could be doubled as often as you wish. Up to 100, the number of sides in a constructible (regular) polygon ­ as far as the Greeks knew ­ must be

2; 3; 4; 5; 6; 8; 10; 12; 15; 16; 20; 24; 30; 32; 40;

48; 60; 64; 80; 96

For more than two thousand years, everyone assumed that no other polygons were constructible. In particular, Euclid does not tell us how to construct 7-gons or 9-gons, and the reason is that he had no idea how this might be done. Gauss's discovery was a bombshell, adding 17, 34 and 68 to the list. Even more amazingly, his methods prove that other numbers, such as 7, 9, 11 and 13, are impossible. (The polygons do exist, but you can't construct them by Euclidean methods.)

Gauss's construction depends on two simple facts about the number 17: it is prime, and it is one greater than a power of 2. The whole problem pretty much reduces to finding which prime numbers correspond to constructible polygons, and powers of 2 come into the story because every Euclidean construction boils down to taking a series of square roots ­ which in particular implies that the lengths of any lines that feature in the construction must satisfy algebraic equations whose degree is a power of two. The key equation for the 17-gon is

x16 þ x15 þ x14 þ x13 þ x12 þ x11 þ x10 þ x9 þ x8 þ x7 þ x6

þ x5 þ x4 þ x3 þ x2 þ x þ 1 ¼ 0

where x is a complex number. The 16 solutions, together with the number 1, form the vertices of a regular 17-gon in the complex plane. Since 16 is a power of 2, Gauss realised that he was in with a chance. He did some clever calculations, and proved that the 148 // Why Gauss Became a Mathematician

17-gon can be constructed provided you can construct a line whose length is

"

1 pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi

À 1 þ 17 þ 34 À 2 17 þ

16

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#

pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi ffi

68 þ 12 17 À 16 34 þ 2 17 À 2ð1 À 17Þð 34 À 2 17Þ

Since you can always construct square roots, this effectively solves the problem, and Gauss didn't bother to describe the precise steps needed ­ the formula itself does that. Later, other mathematicians wrote down explicit constructions. Ulrich von Huguenin published the first in 1803, and H.W. Richmond found a simpler one in 1893.

Richmond's method for constructing a regular 17-gon. Take two

perpendicular radii, AOP0 and BOC, of a circle. Make 4OB ¼ 1

OJ

and angle 4OJP0 ¼ 1. Find F Such that angle EJF is 458. Draw a

OJE

circle with FP0 as diameter, meeting OB at K. Draw the circle

with centre E through K, cutting AP0 in G and H. Draw HP3 and

GP5 perpendicular to AP0. Then P0, P3 and P5 are respectively

the 0th, 3rd and 5th vertices of a regular 17-gon, and the other

vertices are now easily constructed.

Gauss's method proves that a regular n-gon can be con- structed whenever n is a prime of the form 2k þ 1. Primes like this are called Fermat primes, because Fermat investigated them. In Why Gauss Became a Mathematician // 149

particular he noticed that k must itself be a power of 2 if 2k þ 1 is going to be prime. The values k = 1, 2, 4, 8 and 16 yield the Fermat primes 3, 5, 17, 257 and 65,537. However, 232 þ 1 ¼ 4;294;967;297 ¼ 64166;700;417 is not prime. Gauss was aware that the regular n-gon is constructible if and only if n is a power of 2, or a power of 2 multiplied by distinct Fermat primes. But he didn't give a complete proof ­ probably because to him it was obvious.

His results prove that it is impossible to construct regular 7-, 11- or 13-gons by Euclidean methods, because these are prime but not of Fermat type. The analogous equation for the 7-gon, for instance, is x6 þ x5 þ x4 þ x3 þ x2 þ x þ 1 ¼ 0, and that has degree 6, which is not a power of 2. The 9-gon is not constructible because 9 is not a product of distinct Fermat primes ­ it is 363, and 3 is a Fermat prime, but the same prime occurs twice here.

The Fermat primes just listed are the only known ones. If there is another, it must be absolutely gigantic: in the current state of knowledge the first candidate is 233;554;432 þ 1, where 33;554;432 ¼ 225 . Although we're still not sure exactly which regular polygons are constructible, the only obstacle is the possible existence of very large Fermat primes. A useful website for Fermat primes is mathworld.wolfram.com/FermatNumber.html

In 1832 Friedrich Julius Richelot published a construction for the regular 257-gon. Johann Gustav Hermes of Lingen University devoted ten years to the 65,537-gon, and his unpublished work

¨ can be found at the University of Gottingen, but it probably contains errors.

With more general construction techniques, other numbers are possible. If you use a gadget for trisecting angles, then the 9-gon is easy. The 7-gon turns out to be possible too, but that's

........................................... nowhere near as obvious.

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