We think mathematics to be a subject too coarse to have a connection to any spirit, let alone to have its own. But Mathematicians are probably the only people to have named a theorem ‘THE SOUL THEOREM’. The soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Every compact manifold is its own soul. In 1972, Cheeger and Gromoll proved the theorem by the generalization of a 1969 result of Gromoll and Wolfgang Meyer. The related soul conjecture was formulated by Gromoll and Cheeger in 1972 and proved by Grigori Perelman in 1994 with an astonishingly concise proof. The theorem states, If (M, g) is…

The classical three-body problem arose in an attempt to understand the effect of the Sun on the Moon’s Keplerian orbit around the Earth. It has attracted the attention of some of the best physicists and mathematicians and led to the discovery of ‘chaos’. We survey the three-body problem in its historical context and use it to introduce several ideas and techniques that have been developed to understand classical mechanical systems. The study of the three-body problem led to the discovery of the planet Neptune, it explains the location and stability of the Trojan asteroids and has furthered our understanding of…

Fluid Dynamics and the Navier-Stokes Equations The Navier-Stokes equations, developed by Claude-Louis Navier and George Gabriel Stokes in 1822, are equations which can be used to determine the velocity vector field that applies to a fluid, given some initial conditions. They arise from the application of Newton’s second law in combination with a fluid stress (due to viscosity) and a pressure term. For almost all real situations, they result in a system of nonlinear partial differential equations; however, with certain simplifications (such as 1-dimensional motion) they can sometimes be reduced to linear differential equations. Usually, however, they remain nonlinear, which…

The ratio, or proportion, determined by Phi (1.618 …) was known to the Greeks as the “dividing a line in the extreme and mean ratio” and to Renaissance artists as the “Divine Proportion”  It is also called the Golden Section, Golden Ratio and the Golden Mean. Just as pi is the ratio of the circumference of a circle to its diameter, phi is simply the ratio of the line segments that result when a line is divided in one very special and unique way. Divide a line so that: Definition: Phi can be defined by taking a stick and breaking it into two portions. If…

On a dark desert highway a tired driver passes one more hotel with a “No Vacancy” sign. But this time the hotel looks exceedingly large and so he goes in to see if there might nonetheless be a room for him:The clerk said, “No problem. Here’s what can be done – We’ll move those in a room to the next higher one. That will free up the first room and that’s where you can stay.” The mathematical paradox about infinite sets associated with Hilbert’s name envisages a hotel with a countable infinity of rooms, that is, rooms that can be…

The Möbius Band is an example of one-sided surface in the form of a single closed continuous curve with a twist. A simple Möbius Band can be created by joining the ends of a long, narrow strip of paper after giving it a half, 180°, twist, as in Figure 1. An example of a non-orientable surface, this unique band is named after August Ferdinand Möbius, a German mathematician and astronomer who discovered it in the process of studying polyhedra in September 1858. But history reveals that the true discoverer was Johann Benedict Listing, who came across this surface in July…

The Seven Bridges of Königsberg is a historical problem in mathematics. The negative resolution of the problem by Leonhard Euler led to the advent of graph theory and topology. The city of Königsberg in Prussia (now Kaliningrad, Russia) laid on either sides of the Pregel River and included two large islands—Kneiphof and Lomse—which were connected to each other, or to the two mainland portions of the city, by seven bridges. The problem was to to design a walk through the city crossing every bridge only once. Solutions involving either reaching an island or mainland bank other than via one of the bridges, or accessing any bridge without crossing to…

“In Mathematics there is no ignorabimus. We must know, we shall know.” -David Hilbert The incompleteness theorem, says roughly the following: under certain conditions in any language there exist true but unprovable statements. The Set of True Statements– We assume that we are given a subset T of the set L (where L is the alphabet of the language under consideration) which is called the set of “true statements” (or simply “truths”). In going right to the subset T we are omitting such intermediate steps as: firstly, specifying which words of all the possible ones in the alphabet L are…

“As far as the laws of mathematics refer to reality, they are not certain;and as far as they are certain, they do not refer to reality.“ -Albert Einstein In 1900, Henri Poincare put for a conjecture that colloquially states that“If it walks like a sphere and it quacks like a sphere, it is a sphere.” This statement, known as the Poincare conjecture, became one of the early questions in a field now called “low-dimensional topology” and proved itself to be extremely subtle and intractable. It attracted the attention of many great mathematicians and attempts to solve it lead to the…