Gauss and Regular Polygons: Gaussian Periods

Introduction

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

Introduction

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

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

Introduction

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

Introduction

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.

π(PI) and the AGM: Gauss-Brent-Salamin Formula

After a heavy dose of elliptic integral theory in the previous posts we can now prove the celebrated AGM formula for $\pi$ given independently by Gauss, Richard P. Brent and Eugene Salamin. So here we go

π(PI) and the AGM: Evaluating Elliptic Integrals contd.

Complete Elliptic Integrals of Second Kind

After getting familiar with the AGM sequences and Landen Transformation, it is time to apply these concepts to evaluate elliptic integrals. Here we are going to focus on elliptic integrals of the second kind. To be more specific we are going to deal with $J(a, b)$ defined by
$$J(a, b) = \int_{0}^{\pi / 2}\sqrt{a^{2}\cos^{2}\theta + b^{2}\sin^{2}\theta}\,d\theta$$
Our strategy (well actually Landen's and Legendre's) here will be to analyze the defining integral under the Landen transformation
$$\tan(\phi - \theta) = \frac{b}{a}\tan\theta$$

π(PI) and the AGM: Evaluating Elliptic Integrals

Complete Elliptic Integrals of First Kind

In my earlier post I described the method for calculating complete elliptic integrals of first kind namely $K(k)$ and $I(a, b)$. To summarize the results we have
$$K(k) = \int_{0}^{\pi / 2}\frac{d\theta}{\sqrt{1 - k^{2}\sin^{2}\theta}} = \frac{\pi}{2M(1, \sqrt{1 - k^{2}})} = \frac{\pi}{2M(1, k^{\prime})}$$ $$I(a, b) = \int_{0}^{\pi / 2}\frac{d\theta}{\sqrt{a^{2}\cos^{2}\theta + b^{2}\sin^{2}\theta}} = \frac{\pi}{2M(a, b)}$$ where $M(a, b)$ denotes the Arithmetic-Geometric Mean of two numbers $a$ and $b$ and $k^{\prime}$ is the complementary modulus related to $k$ by the following relation $$k^{2} + k^{\prime 2} = 1$$

π(PI) and the AGM: Legendre's Identity

While studying elliptic integrals (refer to previous post for an introduction to elliptic integrals) Legendre discovered a remarkable identity connecting the elliptic integrals of the first and second kinds. This identity at the same time connects these integrals to the mathematical constant $\pi$. This relation to $\pi$ was exploited by Gauss to derive a formula for $\pi$ based on AGM (which is the main topic of this series of posts).

π(PI) and the AGM: Introduction to Elliptic Integrals

Introduction

In my last post I had described the adventures of Gauss with AGM (in this post AGM means Arithmetic-Geometric Mean, see the linked post). Gauss had established the deep connection between Elliptic Integrals and AGM and used his results in this field to ultimately derive a formula for calculating $\pi$ using AGM, but since it involved extraction of square roots it was not considered of much value in pre-computer era.

Arithmetic-Geometric Mean of Gauss

Prelude

Contrary to the popular belief that mathematics is the most dreaded subject, many people in their younger years are struck by many mathematical curiosities. Some of them use these curiosities as puzzles for friends, others try to find the reason behind it and are satisfied once they find the reason. But the great heroes of mathematics are those who, being intrigued by a mathematical curiosity, develop the idea in a systematic manner and connect it to other ideas of existing mathematical knowledge.