In a kind of ongoing project, I am working on some of the often quite tricky problems from Roger Penrose's opus magnum for the (more or less) general audience, The Road To Reality, writing up the results of my efforts as concise as I can. The lengthy formulations (as well as the titles) of the problems are obviously mine: I prefer to have everything as self-contained as possible. In some cases I even modified the problems a bit, to make them more accessible. The stars denote the difficulties of the problems according to Penrose; the maximum *** (‘not to be undertaken lightly’) actually means, as far as I can tell: really hard, resembling more a little research project than an exercise in a book for the general audience, and certainly requiring some good math/physics background.
If you are interested, have a look. For questions, feel free to contact nicolaspott@web.de.
An impressive collection of existing solutions can be found here.
| Chapter | Solutions done | Chapter content |
|---|---|---|
| Chapter 2 An ancient theorem and a modern question |
8/8 | A broad panorama of Euclidean, hyperbolic and elliptical geometry. As it turns out, there is somewhat more to say about the nature of abstract mathematical space, and its accompanying notions of points, lines, angles and so forth, than one might naively think at first sight. Spoiler: The ‘ancient theorem’ ist the Pythagorean theorem, valid only in Euclidean geometry. The ‘modern question’ is the question whether the large-scale-geometry of physical space is Euclidean or not. |
| Chapter 3 Kinds of number in the physical world |
4/4 | Starting from the Pythagorean theorem proved in the last chapter, we see it's almost inevitable to postulate the existence of ‘real numbers’ (ironically in some sense much less ‘real’ than the well-known natural and rational numbers) to describe the lengths of ideal mathematical objects like rectangular triangles. We learn that the ancient Greek already knew quite a lot about those real numbers, representing them as continued fractions instead of today's decimal expansions. It follows a lucid introductory discussion about what kind of numbers are best suited to describe physical reality as we understand it today. |
| Chapter 4 Magical complex numbers |
6/6 | The process of expanding the ‘allowed’ kind of numbers that led us, in the previous chapter, from the natural numbers to the reals finds its logical continuation in the introduction of the complex numbers, which allows us to solve equations like x2 = -1. In this chapter, we learn some basics about these complex numbers and their geometrical representation in the complex plane, and also make a first acquaintance with three of their more ‘magical’ properties, like their relevance for detemining the real (!) solutions of cubic equations, or the circle of convergence for power series, or for generating the enticingly beautiful Mandelbrot set with its endless fractal structures. |
| Chapter 5 Geometry of logarithms, powers, and roots |
15/15 | This chapter broadens our geometric perspective on the complex numbers and introduces some important new functions. In the first and easy part we learn about the geometric interpretation of complex addition (as translation) and multiplication (as simultaneous rotation and expansion/contraction). In the second and more challenging part we see how the exponential function can be defined as a power series on the complex plane, and how from this one can define the natural logarithm with it's several ‘magical’ properties like intrinsic multi-valueness and the multiplication-to-addition property. Related to this topic is one of the most important basic insights about complex numbers, the Euler formula eiφ = cosφ+i sinφ, which however in this text is only stated but not proved. Anyway, with the help of exponentiation and natural logarithm, one can finally define the generel power function wz = ez log w, and from this the logarithm to an arbitrary base as well as the z-th root w1/z, which completes our basic assortment of complex functions. The chapter closes with an interesting physical application of the n-th roots of unity: for any given n, these n complex numbers form a certain symmetry group that can also be used to describe the so-called multiplicative quantum numbers like parity. |
| Chapter 6 Real-number calculus |
10/10 | Calculus, or mathematical analysis, is probably the single most important mathematical prerequisite for almost all of modern physics and engineering. Developed independently by Newton and Leibnitz in the 17th century, and later made formally precise by Cauchy in the 19th century, it is concerned with functions and their continuity, diffentiation and integration. Calculus provides a huge amount of calculational techniques that can be used to compute e.g. slopes and curvatures, areas and volumes. In this chapter, where only functions between real numbers are considered, we are introduced to these topics on a desciptive level (the elaborate formal development using the ε-δ-technique is skipped), and learn about some of the important rules that in many cases allow for the more or less mechanical calculation of derivatives and (to a lesser extent) integrals. Additionally, Penrose presents an eluminating discussion how the concept of intuively ‘nice’ functions (not glued together, perfectly smooth) might be made mathematically more precise in terms of analytic functions (i.e. functions everywhere in their domain locally identical to some power series). |
| Chapter 7 Complex-number calculus |
8/8 | A rather dense chapter that contains on only 12 pages the basics of complex analysis, generalizing last chapter's real-number calculus to the world of complex numbers. As it turns out, the differentiability of a complex function is a much stronger notion than its counterpart in real analysis; to be differentiable, a complex function must obey the so-called Cauchy-Riemann equations. These equations then establish various unique properties of complex functions. Most importantly, any complex differentiable function is also analytic (see Chapter 6) and has a certain ‘rigidity’ that couples, by Cauchy's integral formula, its value at a given point non-locally to specific contour integrals around this point (for this reason, complex-analytic functions are also called holomorphic). Furthermore, the residue theorem allows to calculate arbitrary contour integrals by just studying the behaviour of the function to be integrated at its singularities inside the contour; this theorem can be applied to evaluate, in a truly magical and often surprisingly simple way, various difficult definite integrals and infinite series involving only real numbers. An other aspect of the above-mentioned ‘rigidity’ is the possibilty of analytic continuation which can be seen as a powerful method to uniquely extrapolate a complex function beyond its original domain of definition. |
| Chapter 8 Riemann surfaces and complex mappings |
8/8 | This chapter deals, so to say, with the ‘higher geometry’ of complex numbers. A prominent example for this kind of geometry are the Riemann surfaces, i.e. 1-dimensional, often quite bizarre looking complex manifolds that occur as the natural domains of definition of complex-analytic (or holomorphic) functions, especially for those functions that, like log z, can be defined on the ordinary complex plane only by introducing a discontinuous branch cut. The study of Riemann surfaces leads to an enormous variety of new insights. To start with, it turns out that Riemann surfaces must obey, as manifolds, a conformal geometry, i.e. transformations from one coordinate patch to the other are necessarily conformal mappings, which means mappings that locally preserve angles and shapes, but not always size. This notion is quite important, since in particular all complex-analytic functions are conformal. Particular fruitful is the study of compact Riemann surfaces, i.e. Riemann surfaces of finite size. The simplest example for this kind of objects is the Riemann sphere, which is equivalent to the complex plane with a single point representing ‘infinity’ added. We learn that more complicated compact Riemann surfaces (like the torus) can be topologically classified by their genus (number of handles) and additionally, if one is interested more specifically in their equivalence up to comformal mappings, one or more complex numbers called moduli, characterizing the conformal structure of the surfaces. Finally, to help us appreciate the impressive power of conformal mappings, we make the acquaintance of the Riemann mapping theorem which asserts basically that any open, simply connected region of the complex plane can be mapped conformally to the interior of the unit circle. |
| Chapter 9 Fourier decomposition and hyperfunctions |
14/14 | A real periodic function f(t) with a period T can usually be expressed as an infinite sum of it's ‘pure tones’, i.e. of sine and cosine functions of increasingly smaller periods T, T/2, T/3 and so on, the Fourier series of this function. In this chapter, complex analysis is used to study this phenomenon. It turns out, for example, that the Fourier series is just a special case of the Laurent series of a complex function, an important generalization of the usual Taylor series well known from real analysis, with an annulus instead of a disc as region of convergence. Another surprising aspect is that the Fourier series of a function f(t) can be seen as the split of a corresponding complex function F(z) on the unit circle into a ‘positive frequency’ part F+(z) and a ‘negative frequency’ part F-(z), both of them analytic on opposing hemispheres of the Riemann sphere (see Chapter 8), where the uniqueness of the split is a result of the analyticity constraints on these functions. Furthermore, it is demonstrated that the concept of the Fourier decomposition is even more powerful than one might think at first sight: It mostly works even in the limiting case of an infinite period T, i.e. non-periodic functions, the infinite sum of the Fourier series then becoming an integral from minus to plus infinity over the Fourier transform of f(t), and also for many non-continuous functions like e.g. the square-wave function. Finally, the exotic concept of hyperfunctions is introduced, which is motivated by the frequency splitting on the unit circle as mentioned above. These hyperfunctions are defined as equivalence classes of pairs of functions, where the two functions forming the pair are defined on two disjunct but adjacent complex domains (like opposing hemispheres of the Riemann sphere). A hyperfunction embodies the ‘jump’ between these two functions at the border between the disjunct regions. It turns out that hyperfunctions can be seen as a generalization of the usual concept of a function, including all ‘normal’ functions, but also several exotic objects like the Dirac delta function which is of considerable importance in the mathematics of quantum mechanics. |