- Snapshots of modern mathematics
- Diderot Mathematical Forum 2013: “Mathematics of Planet Earth”
- Pierre de Fermat and Andrew Wiles in Czech Republic stamps
- Stefan Banach (March 30, 1892 – August 8, 1945)
- Diderot Mathematical Forums
- Guessing the Numbers
- What is mathematics for Ehrhard Behrends
- What is mathematics for Krzysztof Ciesielski
- The Three Ducks Trick
- What is mathematics for Franka Brueckler
Plus Magazine Monthly Column – January 2017
Plus http://plus.maths.org is a free online magazine about mathematics aimed at a general audience. It is part of the Millennium Mathematics Project, based at the University of Cambridge and our aim is to open a door on the world of maths for everyone. We run articles, videos and podcasts on all aspect of mathematics, from pure maths and theoretical physics to mathematical aspects of art, medicine, cosmology, sport and more. Plus has a news section, covering news from the world of maths as well as the maths behind the mainstream news, reviews of books, plays and films, as well as puzzles for you to sharpen your wits.
One of our most popular articles (and videos) over recent months was an interview with Andrew Wiles, recorded by Plus editor Rachel Thomas at the Heidelberg Laureate Forum in September 2016.
As you’d expect, Wiles talked beautifully about his most famous achievement, the proof of Fermat’s Last Theorem. “It’s just fantastic,” he said when asked what it felt like to finally complete the proof. “This is what we live for, these moments that create illumination and excitement.” Most mathematicians will be familiar with this feeling, even though for most of us it probably doesn’t concern results that are quite as momentous as Fermat’s last theorem.
What most fascinated our non-mathematician readers, however, were Wiles’ comments on the process of doing maths, especially on what to do when you are stuck. “It’s part of the process and you have to accept [and] learn to enjoy that process,” he said. “Yes, you don’t understand [something at the moment] but you have faith that over time you will understand — you have to go through this.”
Perseverance is the only way, said Wiles, and something that’s normally considered a deficiency, a bad memory, actually helps. “I really think it’s bad to have too good a memory if you want to be a mathematician. You need a slightly bad memory because you need to forget the way you approached [a problem] the previous time because it’s a bit like evolution, DNA. You need to make a little mistake in the way you did it before so that you do something slightly different and then that’s what actually enables you to get round [the problem].”
It was interesting to see our readers’ strong positive reaction to these comments. Perhaps it illustrates a common misconception about maths: that ordinary mortals with imperfect minds, bad memories and a propensity for getting stuck have no access to it. It would be great if the process of doing mathematics, with all its trials and tribulations, were talked about more, in schools and in public. That, and of course the enormous amount of creativity that’s involved in doing maths. “You listen to Bach or Beethoven, that’s not a series of numbers, there’s something else there,” said Wiles. “It’s the same with us. There’s something very, very creative [in mathematics] that we get very passionate about.”
From the beauty of maths we move swiftly on to the beauty of nature. Everybody loves a spiral and spirals are common in nature. We’ve all admired the spirals that occur in seashells, we can find spirals in plants, and even in the arms of galaxies or weather patterns. In terms of mathematics, these examples bring to mind the golden ratio and its associated logarithmic spiral, another example of the supposed ubiquity of this most irrational of all irrational numbers.
But there’s also another mathematical concept linked to spirals: cellular automata. Spirals are not always the result of slow growth but can occur spontaneously in biological or chemical systems. A famous example from chemistry is the Belousov-Zhabotinsky (BZ) reaction: when several chemicals are mixed together in a petri dish, the resulting solution forms changing spiral patterns, as seen in this video:
In biology a particular slime mould, called dictyostelium discoideum, gives rise to similar patterns.
In both these examples there is nothing to direct the global formation of spirals. The systems consist of individual units that only interact with their nearest neighbour. Which is why cellular automata offer themselves as models for these processes.
In his article Spontaneous Spirals Wim Hordijk presented an interesting description of such a model, called the ISCAM model for spiral waves. It starts with a two-dimensional grid of cells. Imagine that each cell is a container that contains a certain amount of a substance. At regular time intervals a certain proportion of the substance disappears from the cell, and a fixed amount is replenished. The proportion that disappears depends on the amount of substance that disappeared from the cell at the previous time step, but it also depends on the proportion that disappeared from the eight neighbouring cells at the previous time step (see the article for the precise rule). In this way, the amount of the substance in each cell varies over time in a way that depends on its eight neighbours and the past.
For the right parameter choices ISCAM is capable of generating spontaneous spiral waves very similar to the ones observed in our two examples, the BZ reaction and dictyostelium. The figure below shows several selected time steps from one particular run of the ISCAM on a 100×100 grid, starting from a purely random initial configuration. The different shades of red represent the amount of the substance in each cell.
The one-minute movie below shows a similar run of the ISCAM in real time, using the same parameter values as in the figure above.
“Natural systems tend to produce many beautiful and often complicated patterns,” writes Hordijk in the article. “However, as the above example shows, the underlying mechanisms and mathematical principles can be surprisingly simple. Using a basic mathematical model, spontaneous spirals in spatial systems cannot only be accurately reproduced, but also studied and understood in more detail. This certainly makes for some interesting psychedelic science!”
Finally, we are thrilled with our animal-tastic front page, which currently features rabbits, a capuchin monkey and a penguin! The excuse for the rabbits is the one you’d expect: we’ve produced a new package of articles introducing the Fibonacci sequence and the golden ratio.
The monkey represents a recent news story reporting on a study which suggests that capuchin monkeys have a basic grasp of probabilities. In the study the monkeys were presented with two transparent jars, each containing a mix of peanuts, which they love, and monkey pellets, which they don’t like so much. The proportion of peanuts and pellets were different in each jar, and did not necessarily reflect absolute numbers: the jar containing more peanuts was not necessarily the jar containing the higher proportion of peanuts. A researcher then randomly picked an item from each jar and presented the items in closed hands to the monkeys. The researchers claim to have observed a significant trend in some monkeys to go for the hand containing the item from the jar that had a higher proportion of peanuts. This, the researchers say, suggests that monkeys not only understand proportions, but also what they mean for you chances of getting a peanut when one item is randomly selected.
The penguin was our contribution to penguin awareness day, which happened on the 20th of January. A few years ago we reported on a problem facing a captive breeding programme at Bristol Zoo. Penguins regularly rotate their unhatched eggs about their long axis, but nobody knew why. Without that knowledge, however, it’s difficult to correctly reproduce this behaviour in incubators. And without rotation, the chicks won’t hatch. A team of mathematicians came up with an answer to the rotation question: they modelled the fluid dynamics within the eggs and suggested that the rotation helps disperse nutrients and waste. The team also recommended what type of further research would be necessary to improve the model. If their recommendations were put into practice, then it’s safe to assume that there are currently penguins alive on this planet that owe their existence directly to mathematics. Who would have thought!