Gauss and Regular Polygons: Gaussian Periods



In order to solve the equation $ z^{n} - 1 = 0$ Gauss introduced some sums of the $ n^{th}$ roots of unity which he called periods, and using these periods he was able to reduce the solution of $ z^{n} - 1 = 0$ to a sequence of solutions of equations of lower degrees. The technique offered by Gauss is extremely beautiful and completely novel and it uses the symmetry between the various $ n^{th}$ roots of unity to achieve the final solution.

Gauss and Regular Polygons: Cyclotomic Polynomials



The word "Cyclotomy" literally means "cutting a circle". So the subtitle of the post suggests that the post is going to be about some polynomials which are related to cutting a circle. Cutting a circle actually refers to dividing a given circle into a number of arcs of same length. Supposing that we are able to divide a given circle into, say $ n$, arcs of equal length by means of points $ P_{0}, P_{1}, \ldots, P_{n - 1}$ then joining the adjacent points we obtain a regular polygon $ P_{0}P_{1}\ldots P_{n - 1}$ of $ n$ sides. Therefore cyclotomic polynomials are somehow related to the construction of regular polygons.

Gauss and Regular Polygons: Complex Numbers

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Introduction to Complex Numbers

Complex numbers are not really complex! In fact they are reasonably simple to understand and operate upon. The concept is definitely strange on a first look, but is damn powerful and has diverse ramifications in various branches of mathematics. Now, to illustrate the point that these numbers are really simple, we are gonna define them in terms of quantities already known.

Gauss and Regular Polygons: Euclidean Constructions Primer



What do we exactly mean by the term "Euclidean Constructions"? Informally the term refers to the geometrical constructions done using the ruler (also called straightedge) and compass. Such constructions are studied as part of high-school (7th to 10th grade) mathematics curriculum and I hope most readers are familiar with the construction of bisection of line segment, bisection of an angle and construction of equilateral triangles.

Gauss and Regular Polygons



After studying elliptic integrals and formulas for $\pi$, we shall now focus on one of the most beautiful gems discovered by Gauss at the age of 17. Gauss proved that the construction of a regular polygon of 17 sides is possible by using an unmarked ruler and a compass only (henceforth these will be known as Euclidean tools and such constructions will be called Euclidean constructions). This is quite remarkable because since 2000 years or so from the time of Euclid the only polygons which were constructible in such a fashion were having sides 3, 4, 5, 6, 8, 10, 12, 15. Gauss added a new number in this series namely 17 and generalized his results to add further numbers. Legend has it that Gauss was so excited by this discovery that he decided to make a career as a mathematician.