1.10. Some historical notes 31 an angle of 2π/n radians. Using the other side of this angle construct an- other angle of 2π/n radians. Continue constructing these angles until you have gone completely around the point with n angles. Now, with O as the center, use the compass to construct a circle of some radius. The points of intersection of the circle with the rays that form the sides of the angles will be the vertices of the regular polygon of n sides. Suppose it were possible to construct with straightedge and compass alone a regular 18-sided polygon. Then it would be possible to construct an angle of 2π/18 = π/9 radians. But then it would be possible to trisect a π/3 angle, which we have seen to be impossible. Hence, a regular 18-sided polygon cannot be constructed. In Book IV of Euclid there are descriptions of how to construct regular polygons of 3, 4, 5, 6, and 15 sides. So, it is possible to construct, using Euclid, angles of 2π/3 radians = 120◦, 2π/4 radians = 90◦, 360/5 radians = 72◦, 2π/6 radians = 60◦, and 2π/15 radians = 24◦. Since any angle can be bisected by straightedge and compass alone, it follows that many additional regular polygons can be constructed. This was known in the Greece of Euclid. Nothing more was discovered until 1796, when a 19-year-old German mathematics student showed that a regular 17-sided polygon is constructible. After announcing his discovery, this student continued his work in mathe- matics with phenomenal success. In 1801, at the age of 24, Carl Friedrich Gauss published his Disquisitiones Arithmeticae. In the final article of that work, he not only showed that a regular 17-sided polygon could be con- structed, but he described every regular polygon that is constructible in terms of the Fermat primes, a certain collection of prime numbers. Carl Friedrich Gauss was born on April 30, 1777. Neither his mother nor his father were educated and there was nothing in his family’s background to predict the enormous stature held by Gauss in his lifetime and today. He is acknowledged to be one of the best mathematicians, if not the best, to have ever lived. Gauss’s father was a poor laborer who shifted from one job to another: gardener, canal tender, bricklayer. In none of these tasks did he show any distinction. Gauss’s mother seems to have had a feeling and respect for learning she had ambition for her son and marveled at his accomplishments. Her brother had intellectual inclinations, but his education was cut short by economic considerations. Gauss often remarked in later life that his uncle was a lost “genius”. Youthful precocity is no guarantee of mature success. Albert Einstein was an example of the opposite phenomenon, as he didn’t learn to talk until he was 4 years old. Many know of people who, as young children, were dazzling and stood head-and-shoulders above the mental accomplishments
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