- 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 – March 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 onto 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.
March is an important date in the mathematical calendar because it’s Abel Prize time. The Abel Prize is awarded annually to recognise “contributions to the field of mathematics that are of extraordinary depth and influence”. It was established in 2002 to celebrate the 200th anniversary of the birth of the Norwegian mathematician Niels Henrik Abel. Unlike the prestigious Fields Medal, which is only awarded to mathematicians under the age of forty, the Abel Prize is open to any age.
This year’s Abel Prize was awarded to Yves Meyer for the development of a powerful mathematical tool: wavelet theory. The theory enables you to break different types of information into simpler components which are easier to analyse, process and store — which is why it finds applications in a huge range of areas, from medical imaging to the detection of gravitational waves.
Before Meyer developed his results, mathematicians and engineers already had a tool for analysing and processing certain types of information. Fourier analysis, as it’s called, is best explained using sound as an example. The sound of the middle A on a tuning fork is represented by a perfect sine wave. Other sounds, like that of a violin playing the same note, are more complicated. However, it turns out that any periodic sound, in fact any type of periodic signal, can be decomposed into a sum of sine and cosine waves of different frequencies. This enables you to do a whole range of things. For example, it enables you to manipulate the different frequencies of a sound individually, or to “clean it up” by removing interfering noises. (You can read more about Fourier analysis on Plus.)
Fourier analysis is a versatile tool. It can also be used to analyse and process images and other types of information. However, it does come with a limitation: because the basic components — the sine and cosine waves — are periodic, Fourier analysis works best for repeating signals. However, most real-life phenomena, from the sound of human speech to seismic data, fall in the non-periodic category.
Wavelet theory works in a similar way to Fourier analysis, but the fundamental components a signal is broken into aren’t sine and cosine waves, but more localised oscillating functions called wavelets (find out more in this Plus article). It was Yves Meyer who came up with families of wavelets that exhibit the mathematical properties that are necessary for the processing and analysis of signals and developed the general mathematical framework of the theory. He’s been called a “visionary” by fellow mathematicians.
Love and Maths
As spring blooms, another thing that is on many people’s mind at this time of the year is love. If you are actively involved in the dating game, you’ve probably struggled with one of its central questions: how many people should you date before settling for something a little more serious? It’s a tricky question, and as with many tricky questions, mathematics has an answer of sorts: it’s 37%. Out of all the people you could possibly date, see about the first 37%, and then settle for the first person after that who’s better than the ones you saw before (or wait for the very last one if such a person doesn’t turn up).
Why is that a good strategy? You don’t want to go for the very first person who comes along, even if they are great, because someone better might turn up later. On the other hand, you don’t want to be too choosy: once you have rejected someone, you most likely won’t get them back. But why 37%? It’s a question of maximising probabilities. The basic idea is as follows: suppose that you see M people and then decide to settle down with the next person who is better than all those M people. It’s possible to work out the probability of ending up with the best person using that strategy, and that probability depends on M. Then choose the M for which this probability is highest – it turns out to be 37% of the total number of people. To see the mathematical details, read this Plus article.
The dating question belongs to the wider class of optimal stopping problems — loosely speaking, situations where you have to decide when is the right time to take a given action (go for a relationship) after having gathered some experience (dated some people) in order to maximise your pay-off (romantic happiness). Life abounds with these kind of problems, whether it’s selling a house and having to decide which offer to take, or deciding after how many runs of proofreading to hand in your essay. So even if you prefer to keep your romantic life well clear of mathematics, strategies like the 37% rule might help you with other tricky problems life decides to throw at you.
Now we’ve provided you with the maths you need to optimise your search for love, we can also offer you and your partner the opportunity of literally becoming one.
Our favourite exhibit at last weekend’s Maths Open Day, part of the Cambridge Science Festival, was the Face Fusion Photobooth, developed by the Cambridge Image Analysis Group. The software in the booth performs a face fusion: if you enter an image of your face and an image of someone else’s face, the software will merge the two, resulting in a strange combination of both.
The idea behind it is familiar from music, when different tracks of a song are mixed together. As we saw from the Abel prize story above, sound can be decomposed into its component frequencies; you can then adjust those frequencies separately using software called an equalizer to adjust the balance between the different frequency components of the sound.
In the case of the images, it’s not sound frequencies that are isolated, but structures within an image that display a particular level of detail. For example, the wrinkles on someone’s face display a particular level of detail (hopefully a very fine one). If you can isolate structures with that level of detail and adjust them, you can enhance or diminish the wrinkles. Alternatively, you could replace them with the wrinkles you’ve isolated on someone else’s face.
At the heart of the face fusion process lies a type of diffusion equation. The most famous diffusion equation is the heat equation, which describes how heat spreads (diffuses) through a material. For example when you hold one end of a metal rod held into a fire the heat will gradually spread towards your hand. When it comes to image analysis and image processing, we can use diffusion equations, not to spread heat around, but to spread information from the various pixels that make up the image. The diffusion equation used here is more complicated than the ordinary heat equation, and it preserves the larger features of the image, such as the overall shape of the face. It is known as the total variation (TV) flow and it is represented by a differential equation.
You might have missed your chance to fuse your face with your dearly beloved’s at the Open Day, but you can still find out more about the maths in this article.
Time for tea
And finally, it’s time to sit back and enjoy a cuppa. One of our favourite problems from our sister site NRICH are the teacups. Imagine you have four sets of cups and saucers. One set is red, one is blue, one is green and one is yellow. In each set there are four cups and four saucers, given you a total of sixteen cups and sixteen saucers all together. This means that there are a total of sixteen different cup-and-saucer combinations, which you can arrange in a square 4×4 grid.
Here’s the challenge. Can you arrange them so that:
- In any row there is only one cup of each colour
- In any row there is only one saucer of each colour
- In any column there is only one cup of each colour
- In any column there is only one saucer of each colour
- Any cup-and-saucer combination (for example red cup, blue saucer) appears only once in the grid.
We love this challenge because it’s a lot trickier than you might at first think. It also has an interesting history. You may have heard about Latin squares before: they are arrangements of symbols (for example numbers) in a square grid so that every symbol occurs exactly once in every row and every column of the grid. The solution to a sudoku puzzle forms a Latin square on a 9×9 grid in which every number from 1 to 9 occurs exactly once in each row and column.
Our tea cups are slightly more challenging. Once you have solved the problem, the grid of cups and saucers form a Graeco-Latin square. You can find out more about these, along with Euler’s problem with 36 officers, and more recent computer assisted proofs in the area, in this article. Enjoy!