Editorial, February 2017

Editorial, February 2017

Happy Valentine’s Day from all on the Editorial Board! Many of us celebrate love and romance on 14 February, but just who was this mysterious saint? According to the History website (www.history.com), at least three Saints Valentine are associated with this date as Christian martyrs, including Valentinus of Rome who died on 14 February around AD 270. However, not until AD 496 did a Papal declaration formally recognise this date as St Valentine’s Day. Its association with love began in the 14th Century and gradually developed into an occasion for exchanging poems, flowers and symbolic gifts such as the key to one’s heart. In this issue of Mathematics Today, we celebrate St Valentine’s Day with a symbolic gift for you: Möbius hearts!

Our Editorial Officer, Rebecca Waters, has designed, developed and documented this amusing and romantic project for you to enjoy. Her inspiration comes from Colin Wright’s excellent article on topology in last August’s issue and a highly informative and entertaining YouTube video clip on the subject by maths communicator Matt Parker. The insert comprises four strips, two of which are marked with the letters A, B and C. You can use these two strips with the instructions on the next page to create a pair of interlocking Möbius hearts, and then make another pair for a sweetheart, relative, friend or pet by repeating this procedure for the two unmarked strips. All you need are sticky tape and scissors. Follow the instructions carefully though, as half-twisting both strips the same way produces a hat and a mask, while omitting either half twist results in a giant square!

Figure 1: Mathematical hearts. Plots are generated using Mathcad and Surfer (https://imaginary.org/program/surfer)

The extensive online resource Wolfram Mathworld demonstrates other ways to construct mathematical hearts, including those displayed in Figure 1. The first curve, a cardioid, is defined in polar coordinates by the simple equation r=1-\sin\theta and forms the central region of the infinitely beautiful Mandelbrot set (type ‘fractal zoom’ into your browser and be amazed). The second curve, a delightful shape with two cusps, is defined parametrically by the equations

    \begin{align*} x&=16\sin^{3}t \quad \textrm{and}\\ y&=13\cos t-5\cos \left( 2t \right)-2\cos \left( 3t \right)-\cos \left( 4t \right). \end{align*}

Better still, the three-dimensional surface is defined by the implicit sextic equation

    \[{{\left( {{x}^{2}}+9{{y}^{2}}/4+{{z}^{2}}-1 \right)}^{3}}-{{x}^{2}}{{z}^{3}}-9{{y}^{2}}{{z}^{3}}/80=0\]

and was included in the Institute’s golden anniversary book 50 Visions of Mathematics.

This February marks 100 years since the death of French mathematician Jean Gaston Darboux (1842–1917), who made important contributions to differential geometry and analysis. These include orthogonal systems of surfaces, partial differential equations and the Darboux integral, which is based on the convergence of upper and lower bounds for areas below curves. He also conducted influential research into kinematics and dynamics, and some of his doctoral students subsequently became great mathematicians, including compatriots Borel, Cartan, Goursat and Picard.

Interestingly, the cardioid in Figure 1 is generated by the path of a point on a circle that rolls around another circle of equal radius. Darboux investigated the harder problem of surfaces rolling on other surfaces and published this work in 1896 within the final book of an extremely impressive four-volume treatise on infinitesimal geometry that is now freely accessible on the Internet Archive website (https://archive.org). He received several international honours for his contributions, including election to the Académie des Sciences in Paris and the Royal Society in London, which awarded him with the Sylvester Medal in 1916.

Among other news, September’s issue of the European Mathematical Society Newsletter printed a transcript of a fabulous interview with Abel Laureate Sir Andrew Wiles. Similarly noteworthy, Physics Nobel Laureate Peter Higgs published an essay on Maxwell’s Equations: the Tip of an Iceberg in the Autumn Newsletter of the James Clerk Maxwell Foundation. Both articles are freely available online and well worth reading. Furthermore, December saw the opening of the much anticipated Mathematics: The Winton Gallery at the Science Museum in London.

This issue of Mathematics Today contains a wide variety of topics that you are welcome to read after successfully constructing your Möbius hearts, including the use of software for education and lasers for astrophysics, along with some personal opinions.

The IMA is grateful to Gordon Blunt for writing many Reflections on Council articles, the last of which is in this issue. He recently retired from Council and Guy Marshall has kindly agreed to write these articles in future. Finally, let us know if you discover any other interesting topological puzzles!

David F. Percy CMath CSci FIMA
University of Salford

Reproduced from Mathematics Today, February 2017

Download the article, Editorial February 2017 (pdf)

Image credit: st. valentine by Konstantinos Papakonstantinou / Flickr / CC BY-SA 2.0

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