Weird Maths: At the Edge of Infinity and Beyond

by David Darling

The known digits of pi are part of the frozen reality of the mathematical universe, and so they cannot be random. But what about the digits lying beyond those that have been computed? Assuming pi is normal in base 10 they remain essentially statistically random to us.

The evolution of any given pattern in the Game of Life is entirely deterministic yet unpredictable: the outcome only becomes known when every step along the way has been calculated.

In maths, things can be unpredictable even if they’re not random. But, until the turn of the twentieth century, most physicists held the belief that even if we couldn’t know every detail of what happens in the physical universe, we could, in principle, know as much as we wanted. If we had enough information then, using the equations of Newton and Maxwell, we could figure out how events would unfold, to whatever level of accuracy we chose. The dawn of quantum mechanics, however, saw that idea fly out the window.

It may well be that quantum randomness is here to stay. However, there have been physicists, and Einstein was famously one of them, who couldn’t stomach the idea (to paraphrase Einstein) that God plays dice with the universe. These opponents of quantum orthodoxy favour the view that, behind the apparent quixotic behaviour of things at the ultra-small level, there are ‘hidden variables’ – factors that determine when particles decay and suchlike, if only we could learn what they are and be able to measure them.

On the other hand, the string 0001100110, which was generated randomly, has the maximum amount of information possible for its length. The reason for this is that one way of quantifying information is the amount by which the data can be compressed. A truly random string can’t be written in any shorter way while retaining all of its information.

This puzzling result, that the coastline of an island, or a country or a continent, doesn’t have a well-defined length, was first talked about by the English mathematician and physicist Lewis Fry Richardson several years before Mandelbrot expanded on the idea.

There are many different forms of the fractal dimension, but one of the easiest to grasp is the box-counting dimension, also known as the Minkowski–Bouligand dimension. To calculate it in the case of the coastline of Great Britain, we’d cover a map of the coastline with a transparent grid of small squares and count the number of boxes that overlap the coast. Then we’d halve the size of the boxes and count again. If this was done with a straight line, the number of boxes would simply double, or go up by a factor of 21, where the exponent (1) is the box-counting dimension. If it was done with a square, the number of boxes would quadruple, or go up by a factor of 22 giving a dimension of 2, and in the case of a cube (using a three-dimensional grid) the number of boxes would increase by a factor of 8, = 23, because a cube is three-dimensional.

Fractals such as the Koch snowflake and Sierpiński sieve are self-similar, which means that they’re made up of successively smaller copies of themselves. In nature, most fractals aren’t exactly self-similar. However, they’re statistically self-similar and so we can still work out their fractal dimension by applying the box method as before.

In the real world, there are only three spatial dimensions but time is also sometimes considered to be the ‘fourth dimension’. It’s no surprise, then, that fractals can exist in time as well as in space. An economic example is the stock market.

Another example of a time-based fractal is something we’ve already come across – the changing length of the coastline of an island, such as Great Britain. At any given moment, the coastline is a purely spatial fractal, the measured length of which depends on the magnification factor. But over time, as mentioned earlier, there are additional variations because of continual erosion (and deposition), the coming and going of tides and even of individual waves, and the almost imperceptible rise or fall of whole land masses due to tectonic activity.

The simplest possible Julia set is when c = 0 because then the rule for getting new values of z becomes simply ‘multiply z by itself’. What happens to a complex number z when it’s iterated in this way? If z starts off inside the unit circle (a circle of radius 1) centred on 0, it will rapidly spiral in towards 0. If z is outside this circle, it will rapidly spiral out to infinity. The Julia set is therefore the boundary of the unit circle, the basic of attraction is everywhere inside the unit circle, and the attractor is the point 0. Imagine the Julia set with c = 0 to be like a steel ball placed exactly between two magnets; the ball will stay put (on the Julia set, although in practice z can move about unpredictably as long as it stays on the Julia set) but if it were placed even slightly differently it would rapidly be attracted to a magnet. In this case, one of the magnets represents 0 and the other infinity.

The Mandelbrot set is the set of all values of c for which the Julia set is connected. It’s one of the most recognisable, yet counterintuitive, fractals.

Fractals are related to another phenomenon in mathematics, known as chaos. Both arise from iterated functions, or rules that are cycled through over and over. Each iteration takes the state of the previous iteration as an input to produce the next state. In the case of fractals, the iterations generate a repeating or somewhat repeating pattern to which there’s no end no matter how much we zoom in. The distinguishing features of chaos are complexity that lacks any repeating pattern and an extreme sensitivity to initial conditions, or the starting state of the system.

weather gives a familiar example. Nowadays we can easily forecast the weather in the short term, over a few days or a week, and get it right much of the time. But there are no reliable forecasts for longer timescales, such as a month. That is because of chaos.

This gave rise to a principle that Lorenz called the butterfly effect: a reference to the fact that if a butterfly flapped its wings today it might lead to a hurricane a month later.

The Feigenbaum constant emerges from the process we’ve just considered, but what makes it fundamental to chaos theory is that it can be found in all similar chaotic systems. No matter what the equation, as long as it satisfies some basic conditions, it will have cycles that double in length according to the Feigenbaum constant.

For a physical analogy of this, imagine a ping-pong ball and an ocean. If the ping-pong ball is released above the ocean it will rapidly fall until it reaches the water. If it’s released below the surface it will rapidly float up. But once it’s on the ocean’s surface, its motion becomes unpredictable and chaotic. Likewise, if a point is not on a strange attractor it will rapidly move towards it. Once it’s on the strange attractor, though, it moves around in a chaotic manner. Fractals are fascinating to explore and among the most visually stunning objects in maths.

In cosmology, the distribution of matter across the universe is like a fractal and its structure may descend below the atomic and nuclear level down as far as the shortest length to which any physical meaning has been ascribed, the so-called Planck length, a mere 1.6×10-35 metre, or about one hundred million trillionth the width of a proton.

But there’s something about the sounds produced by professional drummers that distinguishes them from the perfectly steady, impeccably accurate beats of their synthetic counterparts. That ‘something’ is the slight variations in timing and loudness – the little deviations from perfection – which, research has shown, are fractal in nature.

What became known as Gödel’s incompleteness theorems meant that there’d always be some mathematical truths that were unprovable.

The gist of this is that something can be calculated or evaluated by human beings (ignoring the little matter of resource limitations) only if it’s computable by a Turing machine or a device that’s equivalent to a Turing machine. For something to be computable, it means that a Turing machine, given the program as input (encoded into binary), can run before ultimately terminating with the answer (similarly encoded) as output. A key implication of the Church–Turing thesis is that a general solution to the Entscheidungsproblem is impossible.

Conway’s game is played on an (in theory) infinite square grid of cells that can be coloured either black or white. Some starting pattern of black cells is laid down and then allowed to evolve according to two rules: 1. A black cell remains black if exactly 2 or 3 of its eight neighbours are black. 2. A white cell turns black if exactly 3 of its neighbours are black. That’s all there is to it. Yet, despite the fact that a child could play it, Life has all the capability of a universal Turing machine – and therefore of any computer that has ever been built.

Polynomial algorithms may or may not be practical in reality, but exponential algorithms are certainly not practical when dealing with large inputs. Fortunately, there’s also a wide range of algorithms between the two, and it’s often the case that algorithms that are nearly polynomial work well enough in practice.

In principle, everything an NTM can do, a DTM could also do, given enough time. But that ‘enough time’ is the catch. An NTM could do in polynomial time what would take a DTM exponential time. Too bad, we can’t ever actually build one. What these computers of the imagination let us do, however, is get to grips with one of the great unsolved problems of computer science and of mathematics as a whole: the so-called P versus NP problem.

P and NP are the names given to two sets of problems having different classes of complexity. Problems in set P (‘polynomial’) are those that can be solved by an algorithm running in polynomial time on an ordinary (deterministic) Turing machine. Problems in set NP (‘non-deterministic polynomial’) are those that we know how to solve in polynomial time if we had access to an NTM. (Factoring large numbers is one such problem. An NTM can search through the binary tree for the ‘right’ factor rapidly, in polynomial time, whereas a DTM has to search every single branch, taking exponential time.)What this means is that every problem in P is also in NP as an NTM can do anything an ordinary Turing machine can do in the same amount of time. It

One million dollars is a handsome prize, but how could anyone ever claim it when it means proving (or disproving) that all NP problems are P? A small ray of hope is offered by certain problems in NP that are said to be NP-complete. NP-complete problems are remarkable in that if a polynomial algorithm could be found that would run on an ordinary Turing machine and solve any one of these problems, then it would follow that there’s a polynomial algorithm for every single problem in NP. In this case, P = NP would be true.

Only in 1972 was it shown that the Travelling Salesman problem is NP-complete (meaning that a polynomial algorithm for this problem would prove P = NP), which explained why generations of mathematicians, latterly even using computers, had trouble finding optimal solutions for complicated routes.

The Travelling Salesman problem may be easy to grasp but it’s no easier to solve than any other NP-complete problem: they’re all as hard as each other. Mathematicians are tantalised by the fact that finding a polynomial algorithm for any NP-complete problem would prove that P = NP. Such a proof would have serious implications, including that there’d be a polynomial algorithm for cracking RSA – the method of encryption, described later on, that we rely upon on a daily basis, for example for banking. But in all likelihood none exists.

A qubit, which can be as simple as an electron with an unknown spin, has two properties due to quantum effects that an ordinary bit in a computer does not. First, it may be in a superposition of states. This means a qubit may represent both a 1 and a 0 at the same time, and only resolves into one or the other when it’s observed. Another way of interpreting this is that the quantum computer, along with the rest of the universe, splits into two copies of itself, one of which has the bit 1 and the other the bit 0.

Two entangled qubits, although separated in space, are linked by what has been called ‘spooky action at a distance’ so that measuring one instantaneously affects the measurement for the other.

Quantum computers are computationally equivalent to Turing machines. But, as we’ve seen, there’s a difference between being able to compute something at all (given enough time) and being able to compute it efficiently. Anything a quantum computer can do (or will be able to do) is also achievable on a classical paper-tape Turing machine if we’re prepared to wait a few geological eras or more. Efficiency is a different matter altogether.

One problem that has been solved in polynomial time with quantum computers, which was previously thought to have no such solution (assuming P = NP is false), is the factoring of large numbers.

Quantum computers, like most new upstart technologies, bring both hope and headaches. Among the latter is the possibility of cracking codes that were previously thought to be highly secure, mainly because, despite decades of research, no one has found any polynomial-time method of cracking them.

In any event, before the security of our data is threatened by any developments on the P versus NP problem or more efficient algorithms, quantum mechanics may come to our rescue. The field of quantum cryptography may result in a cipher that is completely unbreakable, no matter what decryption techniques are brought to bear on it.

An example of a truly unbreakable cipher was found as long ago as 1886 and is known as the one-time pad. The key is a random sequence of letters that is as long as the message. The message is combined with the key by converting letters into numbers (A = 1, B = 2, and so on), adding the corresponding numbers for each letter of the message and key, subtracting 26 if the sum is greater than 26, and converting back to letters. This has been proven to be completely unbreakable.

But, assuming aliens ever found one of the Golden Records and managed to play the music as intended, the question is whether they’d recognise it for what it is. And, similarly, if alien music somehow reached our ears, would we appreciate it as being musical?

they saw a perfect marriage of music and mathematics in the heavens. At the centre of physical space, according to their cosmology, was a great fire. Around this, carried on transparent celestial spheres and moving in circular paths were 10 objects, in order from the centre: a counter-Earth, Earth itself, the Moon, the Sun, the five known planets or ‘wandering stars’ (Mercury, Venus, Mars, Jupiter, and Saturn), and, finally, the fixed stars. The separations between these spheres, they taught, corresponded to the harmonic lengths of strings, so that the movement of the spheres gave rise to a sound (inaudible to human ears) known as the ‘harmony of the spheres’.

From an early age, our brains become accustomed to the music that’s pervasive around us, just as they adapt to the local language, the tastes of our home food, and the ways of the people with whom we grow up. Music from other cultures may sound unusual and surprising, and, yet, for the most part, is still pleasing to the ear. The different scales, intervals, rhythms, and structures of musical pieces from other parts of the world may take some getting used to but we almost always recognise them as being musical. This is because they too are based on acoustic patterns that can be reduced to relatively simple mathematical relationships that govern such elements as melody, harmony, and tempo.

Mathematicians who’ve applied information theory to the songs see in them a complexity of syntax and hierarchy of structure previously not encountered outside of human language. But whatever the whales are doing they’re not having regular conversations because the songs, while subtly and continually changing, are too repetitive. Think of them perhaps as being like jazz or the blues, where riffing and improvisation is allowed, even encouraged, but within well-defined guidelines.

If we want an objective definition of music we must turn to the science of acoustics and the laws of mathematics, and ultimately reduce sounds, and combinations of sounds, to numbers.

The Bach contributions last 12 minutes 23 seconds, or roughly one seventh of the playing time of the whole record, reflecting the belief of those who assembled the collection that the highly structured nature of Bach’s pieces, including his clever and complex use of counterpoint to interweave multiple melodic lines, would appeal to both the intellect and aesthetics of any advanced beings who came across the spacecraft.

If mathematics is universal, then so, with many variations, may be the fundamentals of music, including similar scales and methods of tuning. There’s a certain inevitability about the development of equal temperament, for example, that may be repeated wherever intelligent beings want to be able to play a variety of different instruments and harmonise them in many different keys.

It’s often assumed that the first message we receive from ‘out there’ will be scientific or mathematical in content. But what better way to extend a greeting than by sending a really good piece of music, one that has not just a logical basis but is full of the passion and emotions of its creators.

Cataldi’s big prime is the 19th Mersenne number (M19) and the 7th Mersenne prime. It took almost a century and a half for a larger prime to be found, by the Swiss mathematician Leonhard Euler in 1732. Almost a century and a half after that, in 1876, the record-holder became Édouard Lucas, who showed that the 127th Mersenne number (M127), with a value of roughly 1.7 trillion trillion trillion, is also prime.

Another unsolved conjecture has to do with pairs of prime numbers that differ by just 2, such as 3 and 5, and 11 and 13. These are called twin primes and the so-called twin prime conjecture is that there are infinitely many of them. To date, though, no one has been able to show that this is true beyond doubt.

the prime number theorem and is widely regarded as one of the greatest achievements in number theory. In a nutshell, it says that for any number N that is large enough, the number of primes less than N is roughly equal to N divided by the natural logarithm of N.

Further important progress towards refining the distribution formula was made by the Russian mathematician Pafnuty Chebyshev between 1848 and 1850. But the biggest breakthrough of all came through the efforts of the German Bernhard Riemann who, in 1859, published an eight-page memoir, his only writing on the subject, titled ‘On the Number of Primes Less Than a Given Magnitude’. In it he put forward a suggestion, subsequently called the Riemann Hypothesis, which has teased and tormented mathematicians ever since in their attempts to prove

What the Riemann Hypothesis claims is that the deviation of the distribution of primes from n/log n is no greater than what the law of averages predicts.

The Riemann Hypothesis, it turns out, has a subtle but direct connection with the subatomic universe.

So-called random matrix theory can be used to carry out calculations on the energy levels of particles inside heavy atomic nuclei. Dyson recalled his surprise at seeing the same equations pop up in a field to do with the distribution of prime numbers: