Pages: 62

No chapter abstract available.

Pages: 19

Mathematics didn't begin with the Greeks, but Chapter 1 of this book does. We start with Pythagoras and his school in Croton, probably the beginning of mathematics as a formal system rather than a calculating tool. We spend some time discussing Euclid's Elements, paying particular attention to his definitions and axioms. We describe some of Euclid's constructions using straight-edge and compass. Then we dwell on Apollonius's Conics, in particular some elements of ellipses which we will need later in the book; we describe the life and sad end of Hypatia on the way. Following that some description of the contribution made by Hindu and Islamic mathematicians of the middle ages, especially regarding trigonometry and spherical geometry, before Fermat and Descartes and the introduction of coordinate geometry.

Pages: 10

The Elements by Euclid was the definitive word on Geometry for over 2000 years, however right from the beginning mathematicians were uneasy with Euclid's fifth axiom: given a line l and a point p not on l, there is one line through p parallel to l. Many tried to prove this as a consequence of the other axioms, and finally in the 19th century it was realised that this axiom is not necessarily true: it is perfectly consistent to imagine a geometry with no parallel lines, or infinitely many. These non-Euclidean geometries were put forward by Bolyai, Lobachevsky and Gauss, and in this chapter we tell the somewhat tragic story of their discovery. We also introduce a model of hyperbolic geometry, the disc model due to Poincaré, and we investigate how we would cover this hyperbolic space with polygons showing several hyperbolic tilings and also a brief mention of hyperbolic space.

Pages: 16

The ancients were concerned with symmetric curves like circles and ellipses, but as we enter the Renaissance mathematicians wanted to study curves in general. The first notion to attach to a curve's shape is its curvature, and building on Oresme's definition of the curvature of a circle being the reciprocal of its radius, we are able to fit circles to curves, osculating circles, and use them to define its curvature. Then Euler used this idea to investigate the curvature of a surface by intersecting planes with the surface, and finally Gauss took the product of those principal curvatures to define the Gauss curvature. He went further, introducing the line element and then the Gauss Bonnet theorem which relates the angle sum in a geodesic triangle to the curvature contained in it. This inspired Riemann to extend to higher dimensions using the metric, which in turn lead to Einstein's notion of spacetime and general relativity.

Pages: 15

The curves we have seen so far all have the same property: if you zoom in really far on a portion of a curve it will start to look like a line. Mandelbrot in the 20th century wanted to extend geometry in a way that would capture the complex shapes we observe in nature; as he said “mountains are not cones”. He introduced the notion of a fractal, and we first describe fractals generated in a geometrical way, by removing portions of shapes and repeatedly iterating this process infinitely many times. Historically we see examples of this idea in the Pappus chain and Apollonian gasket, but the main modern example is the Sierpinski gasket or carpet. We use this to define a new way to calculate the dimension of these objects and show they have non-integer dimension. Then we use a non-standard way of understanding complex numbers to define and explore the most famous fractal of all: the Mandelbrot set.

Pages: 62

No chapter abstract available.

Pages: 15

When most people are asked to think of mathematics they think of equations, however the development of symbolic maths was very slow in coming. We go way back to the beginnings of number systems as tally marks like on the Ishango bone, through early Egyptian and Babylonian numerals, then to Greek, Hindu and Islamic/Arabic number systems. We show how early problem solving by Chinese, Babylonian and Indian mathematicians were what we would recognise today as linear, quadratic and cubic problems. Then we see the pre-cursor of modern algebra in Diophantus and al-Khwārizmī, before looking at the story of the solution of the cubic with Cardano and Tartaglia in Renaissance Italy. We finish with Fermat and Descartes introducing the symbolic algebra we would recognise today, called analytic geometry, and how they interpreted linear and quadratic equations as lines and conic sections like ellipses and hyperbolae. We also reference Fermat's last theorem to point out the necessity of extending the notion of number to include real numbers.

Pages: 13

Mathematicians were very uneasy working with complex numbers, to the extend where most simply ignored them altogether. Bombelli realised the necessity of negative numbers under the square root sign, and developed rules for their use, but we describe a more geometrical way of understanding complex numbers as simply points in the plane that can be multiplied. Then we study two main results: first we look to Euler and his use of power series to define the exponential function, leading to understanding e^it as a curve in the complex plane which we show is a circle which then leads to Euler's formula, often called the most beautiful equation in maths. Then we look at the fundamental theorem of algebra and some early attempts to prove it including by Gauss, before describing an elegant proof of the theorem using the notion of winding number and thinking carefully about functions and what continuous means.

Pages: 17

At the start of the 19th century algebra meant solving equations, but by the end it had become a deep and abstract study of structure and form. We consider what it means to solve an equation, and point out how the quintic has no formula for its roots. The main character we dwell on in the development is Galois, and we tell the tragic story of his life as well as introduce the reader to the notion of a group via some definitions and examples such as rearranging lists of symbols. We describe the fundamental group via a novel approach: long walks around the city of Dublin with an invisible thread that gets snagged on the spire. This then extends naturally to the fundamental group on the sphere and torus, which is followed by a more in-depth description of the life and contribution of Emmy Noether, probably the most accomplished female mathematician.

Pages: 15

As Hamilton was walking by a canal in Dublin one day he was struck by a moment of inspiration and carved on the stone of a bridge nearby his famous quaternion equations; this was the first time the necessity of spaces of dimensions greater than three was demonstrated. We introduce notions of vectors and linear combinations and demonstrate these ideas via the technique of k-means clustering. Then we use the setting of trying to count the loops in a graph to introduce the method of Gaussian elimination and the notion of a basis, before describing the work of Cayley establishing matrices (plural of matrix) as mathematical objects with rules and symbols. We make the point that vectors, matrices and the techniques of Linear Algebra are a crucial part of the modern developments of artificial intelligence, AI, and machine learning, via a convolutional neural network used to classify images of cats and dogs.

Pages: 68

No chapter abstract available.

Pages: 17

Calculus is the greatest idea mathematics ever had, but the path to our modern notion of calculus was a long and winding one. We start with Archimedes and the method of exhaustion, and early attempts to calculate the digits of pi. Then we see in the early modern age how Fermat used a sleight of hand to calculate the area under a curve, leading to the work of Newton and Leibniz and their power series methods to do the same. The other important problem they attacked was finding the tangent to a curve, and we show their different methods, but especially how Leibniz introduced much of the notation and terminology we use today. We describe the controversy around calculus and how this was resolved by Cauchy and others in the definition of a limit, and we dwell on historical questions like: who invented calculus?

Pages: 20

Since the very beginning people have wanted to understand the motions of the stars and planets, and we describe some Greek models such as nested spheres and epicycles. We consider the ideas of Kepler, not all of which worked out, and his three laws which were then given a firm mathematical basis by Newton in his Principia. After a little digression into Newton's laws and their use to describe mechanical systems such as springs, we describe how Laplace extended Newton's methods to consider the perturbations of the orbits of planets due to other planets via differential equations and the osculating elements. Laplace showed the elements are slowly oscillating over time, and Mary Somerville presented his work in a more digestible form which helped lead to the prediction of the existence of Neptune. However, this clockwork universe was soon undermined by the work of Poincaré who described chaos in the solar system, specifically via periodic orbits in the three-body problem, and then Einstein and his theory of relativity which said there is no such thing as force but instead everything moves due to spacetime curvature.

Pages: 16

There are many classical problems which we can describe as optimisation problems, such as: of all the rectangles with a fixed perimeter, which one encloses the largest area? Fermat was the first to use what we now call the derivative to solve problems like this, and this was quickly expanded by Leibniz and the Bernoullis to functions of several variables and partial derivatives, their motivation being the caustic formed by families of curves. We then describe how many physical phenomena can be phrased as optimisation problems, such as reflection and refraction, with the brachistochrone problem and resolution leading to the development of the calculus of variations by Euler and Lagrange. This has many uses in mechanics and we consider the three-body problem and the existence of integrals and Noether's theorem, but also important geometrical problems such as shortest paths, known as geodesics, and the connection again to General Relativity.

Pages: 13

Nature is change, and the way we capture this change is with the derivative. Equations involving partial derivatives are called partial differential equations (PDE's) and they have lots of applications. We start with the vibrating string as investigated by d’Alembert and Bernoulli which lead to the method of separation of variables, then the Laplace equation to study the gravity around spheroids and then Fourier's investigation of heat transfer which lead to Fourier series. We see there are three main classes of PDE: hyperbolic (wave equation), elliptic (Laplace equation) and parabolic (heat equation). We then dwell on Sophie Germain and her investigation of the vibration of Chlani plates before finishing with a sketch of gravitational waves as predicted, and then detected, via Einstein's field equations and the wave equation.

Pages: 64

No chapter abstract available.

Pages: 15

Topology is the most recent branch of mathematics studied in the book, although there are some early manifestations via the bridges of Konigsberg and the Euler characteristic of Platonic solids. We prove the alternating sum V-E+F is equal to 2 using Thurston's flowing charges approach, but then point out how other polyhedral and surfaces such as the torus have different Euler characteristic. We make the connection to geometry via the Gauss Bonnet theorem and discuss how the total Gauss curvature is fixed, then develop more abstract notions of when two sets are the same from a topological point of view. We demonstrate this new way of thinking by showing that the set of oriented lines in the plane is the same as a cylinder, and finish with a key idea in topology: the use of invariants to distinguish and identify different objects.

Pages: 17

Often in Mathematics there are intuitive ideas that are made formal and precise and some of the elements of this chapter are a good example. We see three different expressions of the notion of the degree of a map: the winding number which we then connect to the Jordan curve theorem, the rotation number which we use the classify all closed plane curves using a result of Whitney, and the index of a zero of a vector field due to Poincaré. We explain how a vector field can be projected onto a sphere and then how the sum of the indices of the zeroes of a vector field adds to two, also known as the hairy ball theorem, and more generally how the sum of the indices is the Euler characteristic. We can then use this to consider higher dimensional spaces such as the 3-sphere and we see via the Poincaré-Hopf theorem that the Euler characteristic of odd-dimensional spheres is 0, but for even-dimensional spheres it is 2.

Pages: 14

How many holes does an object have? We use some examples to demonstrate our use of the word “hole” is not precise enough and instead we turn to Poincaré's seminal work in counting the number of holes using cycles and boundaries on manifolds. In contrast to the other chapters in this book, we do some detailed calculations of the Betti numbers for the sphere and torus, and show how these can then be interpreted as the number of holes of different dimension. Also, we see how the alternating sum of Betti numbers is the Euler characteristic, giving us more invariants to use in distinguishing manifolds. Finally, we sketch the rephrasing of homology in terms of groups due to Noether and others via sets of chains, cycles and boundaries.

Pages: 16

Given a set of objects, such as surfaces, we would like to be able to classify them according to their topology, to put them into boxes where all the elements of one box are the same as each other but different to the elements of the other boxes. This was completed for surfaces with boundary, and we first explain notions of orientability and connected sum then use plane models and word representation to show how such a thing could be proved. Along the way we meet the Mobius strip, Klein bottle and projective plane. Then we extend to higher dimensions, and reach the Poincaré conjecture: that all 3-d closed manifolds that are simply connected must be spheres. We explain Thurston's geometrization and Hamilton's Ricci flow, before describing how Perelman stunned the world with his famous proof.