Often greeted by a roll of the eyes, or a fear-filled quickening of the pulse, mathematics and statistics have a reputation for being dry, dull or simply too hard by many outside the scientific community (and some inside). Here, International Innovation aims to prove the hypothesis that these disciplines are in fact interesting by looking at some of the more unusual places mathematicians and statisticians can find themselves

THE UNREASONABLE EFFECTIVENESS OF MATHEMATICS IN THE REAL WORLD

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FEATURE

EUGENE WIGNER ONCE spoke of the ‘unreasonable effectiveness of mathematics in the natural sciences’. It is quite peculiar, as Wigner pointed out, that both the dimensions of a circle and the Gaussian distribution of a population may be expressed as functions of pi; but is it not infinitely more perplexing that we should be able to express music, sport and the like as mathematical constructs?

The supreme applicability of mathematics and statistics to the world around us is unquestionable. In this overview, we offer some real-world examples of the surprising topics and questions mathematicians and statisticians are elucidating.

MUSICThere was perhaps none better qualified than polymath Gottfried Leibniz to marry the seemingly disparate disciplines of mathematics and music: “Music is a hidden exercise in arithmetic, of a mind unconscious of dealing with numbers,” he famously wrote.

On first inspection, the relationship between mathematics and music is perhaps an obvious one. Both disciplines employ a language of symbols, each symbol denoting a unique value; and each has rules, failure to adhere to which makes for a disagreeable product. However, one might very well say the same of the written word, or practically any other information-bearing abstraction, so what makes music so mathematical?

It is said that Pythagoras first recognised harmony as a branch of mathematics as he walked past a blacksmith. The famed philosopher and mathematician noted the relationship between the mass and sound of the falling hammers – a two-fold increase or decrease in hammer mass was associated with a frequency change we now term an ‘octave’. Relating this observation to conventional instruments, the first octave ranges from C1, or around 32.7 Hz, to C2, around 65.4 Hz; a twofold difference in frequency. This phenomenon hearkens back to Wigner’s allusions to mathematics as a system intrinsic to nature.

Another clear link can be found between mathematical and musical abilities. While it is a myth that playing Mozart’s Requiem to a newborn will inspire genius, playing an instrument as a child is still believed to cultivate one’s mathematical faculties. Furthermore, a pleasing proof or equation evokes the same neural activity as agreeable music. Unfortunately, insight as to the reason behind this inextricable link is lacking; whilst musical composition may be supported by mathematical calculation, nobody knows why we find this rational music so appealing.

MARITIME SALVAGEEquivalent to finding a needle in a haystack, locating a ship lost at sea can be a considerable challenge. However, with the use of Bayesian search theory, a form of game theory, this challenge is greatly diminished.

Take the example of the USS Scorpion, a nuclear submarine of the US Navy lost at sea during the Cold War. The ship was known to have sunk somewhere between the Azores Islands and Norfolk, Virginia, a 2,600 mile stretch across the north Atlantic Ocean. After the Navy had spent five fruitless months in search of the vessel, they tasked naval scientist Dr John Craven to tackle the problem. Staggeringly, in just five days he found the ship.

Craven began by collecting audio recordings from remote listening stations in the north Atlantic, and analysing these recordings for sounds indicative of the ship’s wreckage. This step reduced the search area to a circular region of 20 miles in diameter. Craven then had salvage experts and Navy personnel place bets on where the USS Scorpion might be, providing a number of possible scenarios, each associated with a probability density function.

Information obtained from the undersea audio recordings represented the ‘prior distribution’ while odds derived from the bets served as the ‘likelihood function’. Assembling the original Bayesian equation, the probability of a location based on scenario was determined, and graphically represented as a contour map of probability. The lost submarine was found just 220 yards from the location of greatest probability, an astounding demonstration of probabilistic reasoning that justifies the continued use of Bayesian search theory for search and rescue missions.

AFRICAN CULTUREEthnomathematics is a niche specialism of culture’s relation to mathematics, and the forte of cyberneticist Dr Ron Eglash. Eglash is best known for his work on fractals, self-similar shapes whose form recurs at every level of scale and which account for the morphology of many natural structures. It puzzled Eglash to find these geometric figures manifest in the layout of a traditional Tanzanian village, whose indigenous people could not possibly know of non-Euclidean geometry.

Eglash later identified further instances of fractal geometry in African culture, from hairstyles and art to architecture and geomancy. He

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noted their likeness to the work of Western computer engineers, albeit on a grander scale. When asked, natives would often explain their use of fractals as being aesthetically motivated – not derived from calculation, but intuition. Considering the ubiquity of fractal geometry in nature – the dendritic arborisation of neurons or the fronds of a fern, for instance – indigenous man’s use of fractals is likely an example of biomimicry.

Fractals are also formed in the construction of Tanzanian straw fences. Eglash quizzed a local fencemaker on his designs, finding them to be purely utilitarian and unintentionally fractalic. As wind travels faster a few feet higher than at ground level, the top of the fence is where the most straw is incorporated into its structure, tightly woven to block the wind and dust. Further down the fence – where wind speed is slower – longer strands of straw, more loosely woven, are used to save on resources. Eglash plotted the log of straw length versus the log of fence height to calculate the scaling exponent. Quite miraculously, the calculated value matched almost exactly that which describes the relationship between wind speed and height in a Western engineering handbook.

SPORT SCORESOne might proclaim ‘may the best man win’ before a sporting competition, but to what extent are scores truly determined by skill, as opposed to sheer luck? A scoring system, ideally, ought to differentiate between the two, favouring skill as the game-winning asset. In sports that feature multiple rounds of play, such as squash, snooker or tennis, the more skilful player has a better chance of exhibiting his or her skills than if only a single point were played.

If a player maintains a constant performance aptitude, and thus probability, p, of winning each point (and therefore 1 – p of losing each point), and n games must be won before match victory, the player’s chance of winning n points before losing n points may be calculated. Where opponents are equally matched, so p is close to ½, 12+s, here s is a tiny positive value signifying skill. The probability of n wins before n 12+2snл.

In a hypothetical scenario, where p remains ½ and s is zero, and the probability of winning the match is also ½, you need not reach for the calculator when approximating the outcome of such a simple head-to-head. But if one player has a slight advantage (ie. s is slightly positive for this player) over his or her competitor, this advantage is inflated by the square root of n as the number of points increases

12), becomes increasingly evident as more points are played. Hence, in tennis a match between Novak Djokovic and Roger Federer often requires five sets instead of three, so as to raise Q to significantly above ½. This allows the small skill difference between the players to influence the scores, rather than permitting chance to dictate winner and loser.

DOOMSDAY PREDICTIONSMany have predicted the end of days and all have been wrong. However, when an esteemed mathematician declares the end to be nigh, one is more inclined to take note. One such academic, John Leslie, founded his prediction on probability.

If we consider the timeline of our species’ existence, from advent to extinction, and approximate the number of humans born so far, we can determine probabilistically where we fall on that timeline, and thus how much longer we have. The theory posits that humans are born in a random order, and chances dictate that any given human is born roughly in the middle.

The Copernican principal, denoting with N the total number of humans to be born in the past and future, suggests that each of us is equally likely (as likely as the remaining N – 1 of the population) to exist at any ordinal position, n, in the total population, N. We may therefore assume, before determining our position absolutely, that our fractional position, f = n/N, is uniformly distributed about the interval of 0 to 1. Statistics tells us that there exists a 95 per cent chance that f falls within the interval of 0.05 and 1, or f >0.05, meaning we can be 95 per cent confident that we are among the last 95 per cent of all humans ever to exist.

Leslie estimated there to have been 60 billion humans in existence thus far and, therefore, a 95 per cent chance that N will equal less than (20 x 60 billion) 1.2 trillion. Translating this into time remaining requires the use of some labile variables, such as life expectancy, and so cannot be considered terribly accurate. For the sake of conclusion, assuming the Gaia hypothesis kicks in at a population of 10 billion, stabilising human numbers, and life expectancy remains at an average of 80 years, the remaining 1,140 billion Homo sapiens will be born over the next 9,120 years – almost enough time to get to grips with calculus.

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