Seventeen Parts of A Circle
Ways were found to cut a circle into three and five equal sectors early in Euclid period. However ways had not been found to cut a circle into seven, nine and eleven equal sectors.
It was not until the 19th century that the great Germany mathematician Guass proved that for an odd number n, a ruler and a compass can be used to divide a circle into n equal sectors only when n is a Fermat prime number or the product of different Fermat prime numbers. He himself make a regular seventeen-sided polygon with a compass and a ruler.
The ways to cut a circle into 17 parts:
(1) make a circle and draw two diameters vertically with each other. Get two points in the circle P and B;
(2) make OJ=1/4OB, and make ∠OJE=1/4∠OJPO, ∠FJE=45°
(3) Draw a circle with line FPO as diameter, it intersects the line OB at K, draw a circle with E as center, line EK as radius. It intersects line OPO at N5 and N3.
(4) Draw a parallel line across N5 and N3. It intersects Circle O at P5 and P3, and then equally depart line P5P3, get the point P4;
(5) Line P3P4 is just one line of the sixteen-polygon. Using this line’s length to cut the Circle O can get every point.
Guass found the ways to draw a sixteen-polygon by using of the ruler and compass in his 18 years old. From then on he began his splendent mathematic career and become the greatest mathematician in first-half period of 19th century.