Maths.org

Looking for something to think about next time you gaze at your reflection when brushing your teeth? Then Sara Santos has some mathematical inspiration for your next daydream in her MMP public lecture, Through the looking glass... again and again!. If Alice took a magic trip inside a conic arrangement of mirrors, what would she find in this mathematical wonderland? You can take a look through a 3D kaleidoscope to see what happens to Alice's cubes and icosahedrons!

Sara Santos is Clothworkers' Fellow in mathematics at The Royal Institution of Great Britain (Ri) and is responsible for coordinating the UK-wide network of secondary Ri mathematics masterclasses. Sara will be speaking at 11am on Thursday 11 June 2009, at the Centre for Mathematical Sciences, Cambridge, just down the hall from Plus! Admission to the lecture is free but by ticket only — for tickets please contact Kerstin Enright, Millennium Mathematics Project, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA (01223 766839) or email: mmptalks@hermes.cam.ac.uk. You can also sign up for notifications of future MMP events at the MMP site.

And don't forget you can also see the London Mathematical Society's popular lectures on Monday 22 June in London and Tuesday 15 September in Birmingham. Come and see how physicists helped answer a hundred year old question about prime numbers and how random matrices and Riemann zeroes feature in a major Hollywood movie with Nina Smith. And Mark Miodownik will explain how fleas can jump over 100 times their own height, flies can walk on water and a hamster can survive falling from aircraft without a parachute.

Admission is free, but by ticket only. For more information and tickets, contact Lee-Anne Parker, London Mathematical Society, De Morgan House, 57-58 Russell Square, London,WC1B 4HS (email: leeanne.parker@lms.ac.uk), or visit the LMS website.

But if you can't wait til then, you can take a trip through the looking glass and discover the importance of scale to weightlifters and kitten on Plus!

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Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only? Read more.

2009-07-09 This enrichment event for Year 12 students will feature engaging mathematical content and an interesting insight into a range of intriguing applications of the subject. Lecture topics include 'Mathematics and the Movies', 'Avalanche!' and 'Particle Hunting at the Large Hadron Collider'. For full details please see the programme for the day at the Cambridgeshire Further Mathematics Centre website. Read more.

11th May 2009 Read more.

We had some extremely well-explained solutions to this problem. Read more.

The University of Cambridge today received a Gömböc. It was donated by its inventors Gábor Domokos and Péter Várkonyi. But what is a Gömböc and what is the University going to do with it?

A Gömböc (pronounce goemboets) is a three-dimensional body with one stable and one unstable equilibrium point. If you put it down on a horizontal surface, it will start wobbling around until it has safely reached the equilibrium position, a bit like a Weeble toy. In theory, you could balance it on the unstable equilibrium point, but in practice that's really hard because the slightest nudge will make it fall over, just like a pencil that is balancing on its tip. Unlike a Wheeble, whose self-righting ability is down to a weight in its bottom, the Gömböc is homogenous inside: its density is the same everywhere, ie there is no off-centre weight which forces it to take on a particular position. The Gömböc is also convex.

The question of whether a convex and homogenous body with one unstable and one stable equilibrium exists in three dimensions was first raised by the Russian mathematician Vladimir Arnold. Mathematicians knew before that in two dimensions there are no such shapes, and they also knew that every three-dimensional object must have at least two equilibria. Domokos and Várkonyi started working on the question and did not only prove that the Gömböc exists, but also built one. In fact, they're building many, from different materials, and they're selling them on the Gömböc website.

The Gömböc is not only beautiful and interesting, but also sheds some light on how a certain species of turtle, with a Gömböc-like shell, manages to get back on its feet after it has been toppled over. Gömböcs need to be engineered to the highest levels of precision, otherwise they won't work. The Gömböc that was today donated to the University of Cambridge can be admired at the Whipple Museum of the History of Science. Plus will interview its inventors next month and you'll be able to read the interview here soon.

You can see a Gömböc doing its thing on YouTube, though the video clip is in German.

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17th Apr 2009 Read more.