regular polygons with three, four, and six sides, and there is a slightly
more complex construction for a regular polygon with five sides. All these
were known to the ancient Greeks, but a classical construction of other
regular polygons proved elusive.^10 In the late 1920s, a manuscript attrib-
uted to Archimedes (who else?) was discovered that outlined a method of
constructing a regular heptagon by sliding a marked straightedge, but
almost two millennia would elapse from the era of Archimedes before
the four problems under discussion were finally resolved to the satisfac-
tion of the mathematical community.
The Mozart of Mathematics
Any list of the greatest mathematicians must include Carl Friedrich Gauss
(1777–1855), the Mozart of mathematics, whose mathematical talents were
evident at an extraordinarily young age. At the age of three, he was presum-
ably studying his father’s accounts, and correcting arithmetic errors if and
when they occurred. Just as Mozart is renowned for having composed mu-
sic at an exceptionally young age, Gauss is also known for demonstrating
genius at an early age. During an arithmetic lesson in elementary school,
the class was asked to add the numbers from 1 to 100. Gauss almost im-
mediately wrote “5050” on his slate, and exclaimed, “There it is!” The
teacher was stunned that a child could find the correct answer so quickly;
the technique Gauss employed is still known as “the Gauss trick” to math-
ematicians. Gauss realized that if one wrote down the sum
S 1 2 3 .. . 98 99 100and then wrote down the same sum in reversed order
S 100 99 98 .. . 3 2 1if one were to add the left sides one would get 2S, and if one were to add
the right side by thinking of it in terms of 100 pairs of numbers, each of
which summed to 101 (1 100, 2 99,... , 99 2, 1001), one would ob-
tain 2S 100 101 10,100, and so S 5050.^11
Even more incredible is the fact that when Gauss was given a table of
logarithms at age fourteen, he studied it for a while, and then wrote on
the page that the number of primes less than a given number N would
approach N divided by the natural logarithm of N as N approached infin-
ity. This result, one of the centerpieces of analytic number theory, was not
proved until the latter portion of the nineteenth century. Gauss did not
supply a proof, but even to be able to conjecture this at the age of fourteen
is simply extraordinary.^12
72 How Math Explains the World