Maths and Physics Group
Equation of the month: The eigenvalue equation. Many systems can be understood or solved more easily if they can be written in terms of the eigenvectors and eigenvalues.
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Dec 13
Tony Gardiner - "Kissing and cuddling up. Sphere packing and applications in N dimensions" We all recall fitting six pennies round a central penny in 2D, and may have wondered about fitting unit spheres around a central unit sphere in 3D. In 2D, two touching unit circles each subtend pi/3 radians at the centre of the other circle, so six fit exactly round a central circle. In 3D we may use "solid angle" (4pi round a point). Integration shows that there is room for ~14.9 unit spheres around a given unit sphere. However, this "room" cannot all be used: Newton and Gregory argued whether 12 (Newton) or 13 (Gregory) was the best possible. The answer had to wait (until around 1950) - and a simple proof is still hard to find. The "six pennies round a given penny" image extends to fill the whole of 2D with the densest possible way to pack circles in 2D. But what about unit spheres in 3D? Do greengrocers know best when they pile up oranges? Kepler (1611) thought so. But the arrangement is far from unique, and its optimality was not settled until very recently (1997/2005/2014). If sphere packing in 3D is so hard, what hope is there for higher dimensions? And why would anyone care? Sphere packing in higher dimensions is linked to efficient codes that allow "error-correction". Such codes were essential in relaying classic images from space since the 1970s, and are now a routine part of internet communication and security. Our knowledge in dimension 4 and higher remains very poor. But something astonishing happens in dimensions 8 and 24. Jan 10
Video: Lecture by Timothy Palmer on chaos Feb 14
Paul Masham: TBD The schedule may change, so please check again closer to the date
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The Maths and Physics Group meets on the 2nd Wednesday of the month in the Lower Classroom at the Holy Trinity Parish Centre from 9.30 am until 10.45 am. Most meetings consist of one of our members giving a talk on a subject of their choice, but we are open to suggestions. There is also a WhatsApp group for people to ask each other questions, or to point out interesting news articles or books they've read.
New members are very welcome. You don't need to be Einstein be a member! You just need to be interested in maths or physics (or engineering, technology, astronomy ... ). Contact mathsphys@dorchesteru3a.org.uk for further information.
Previous Talks
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Experiment: Determining the wavelength of a laser
Polynomials and (Aero) Planes Radio Telescopes "Welcome to the Loxodrome" (!) The Wonderful World of Non-dimensional Groups Covert Radar Detection of Stealth Planes The Steam Engine The Concept of a Field Audio Compression and MP3 Lightning ! Galileo Trisecting the Angle Blockchain, BitCoin & NFTs Three talks on books:
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"Gödel for Dummies"
Bat Sonar – An Engineering Perspective Genetic Algorithms Do the Angles of a Triangle Add Up to Two Right Angles The Unreasonable Effectiveness of Mathematics Measuring the Earth Zipf’s Law and Benford's Law Short Introduction to Dirac and a Physics Experiment Summing Some Series and Quantum Physics The Role of Geometry in Understanding the World Catalan Numbers Mastering Masonry - the Physics of Ancient Buildings. Triangle Facts You Did Not Learn at School The 4 Colour Theorem The RF Mixer (Heterodyning) Physics of Nuclear Reactors Wind Turbine Maths |
Lastly, a bit of fun ...
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Easier
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Please make all the usual assumptions, and don't worry if you can't solve the problems! Solutions can be found in this pdf.
The questions are best viewed on a desktop as they may contain pictures that cannot be scaled easily for a mobile.
Two new problems will appear next month.
The questions are best viewed on a desktop as they may contain pictures that cannot be scaled easily for a mobile.
Two new problems will appear next month.
Page maintained by Dave Waymont