Our good friend Julian Gilbey has just told us about an amazing fact: if you roll a parabola along a straight line then the shape its focus traces out as it goes is ... a catenary! That's the shape a chain makes when it hangs freely under gravity and also the shape that gives you the strongest arches (see here and here to learn more).
Just why the two curves are connected in this way is a mystery (at least to us) — you can do the maths to prove it, but there doesn't seem an intuitive reason.
Julian has also produced this beautiful applet to illustrate the result. It shows the graph of the parabola with equation
which has its focus at the point (Use the left-hand slider to change the value of ) You can roll the parabola along using the right-hand slider and see the catenary the focus traces out. Its equation is
Why do diamonds sparkle? Why is the shower the best place to sing? Where is the best place to stand to look at a statue? Where is the 4th dimension in Dali's paintings? Do you know the answers to these questions? Or perhaps, you didn't know you didn't know these interesting facts!
Never fear, John D. Barrow is here with all the answers to these and 96 other questions you didn't know you didn't know about maths and art. To celebrate the launch of his new book (his 22nd!) called 100 Essential Things You Didn't Know You Didn't Know About Maths and the Arts he's giving a talk at 6pm on Thursday 13 November for Gresham College. As well as find out more about the interplay of maths and art, you can also enjoy a free drink with Barrow after the talk!
You don't need to register for the event, it's first come first served. Find out more here. You can also read an extract of the book on Plus.
The tightrope walker Nik Wallenda's latest achievement is breath-taking. Without a safety net or harness Wallenda walked the gap between buildings on either side of the Chicago River before crossing between the two Marina City towers — blindfolded. His feat was televised, but with a ten second delay, in case he fell. Thankfully he didn't! You can see part of his walk in the movie on the right, but don't look if you're faint-hearted.
Like any tightrope walker, Wallenda carried a long pole to aid his balance. But why? What's the physics behind it? We rummaged in the Plus archive and found that a few years ago mathematician John D. Barrow had already come up with an answer. The key idea is something called the moment of inertia, which measures an object's resistance to being spun around an axis. The long pole increases the tightrope walker's moment of inertia by placing mass far away from the body's centre line (moment of inertia has units of mass times the square of distance).
As a result, any small wobbles about the equilibrium position happen more slowly. They have a longer time period of oscillation (the period of small oscillations about a stable equilibrium increases as the square root of the moment of inertia) and the walker has more time to respond and restore the equilibrium.
To convince yourself that this really works, you don't need to do a blindfolded tightrope walk 150m above the ground. Simply compare how easy it is to balance a one metre ruler on your finger compared with a ten centimetre ruler.
You can read more about the moment of inertia, as applied to space crafts and boomerangs, in thesePlus articles.
If you had a sloppy maths teacher at school you might have grown up with the idea that the number is equal to Now that is completely wrong. Writing those numbers out in decimal gives while . There’s a difference in the third decimal place after the decimal point!
How accurately do we need to know the value of π?
But so what? Who cares? Surely this small inaccuracy doesn't matter? Well, as the following extract from a longer article by Chris Budd shows, it really does.
The point is that is not any number. It lies at the heart of any technology that involves rotation or waves, and that is much of mechanical and electrical engineering. If rotating parts in, say, a typical jet engine are not manufactured to high tolerance, then the parts simply won’t rotate. This typically involves measurements correct to one part in 10,000 and, as these measurements involve , we require a value of to at least this order of accuracy to prevent errors. In medical imaging using CAT or MRI scanners, the scanning devices move on a ring which has to be manufactured to a tolerance of one part in 1,000,000, requiring an even more precise value of .
However, even this level of accuracy pales into insignificance when we look at modern electrical devices. In high frequency electronics, with frequencies in the order of 1GHz (typical for mobile phones or GPS applications), electrical engineers require a precision in the value used for in the order of one part in 1,000,000,000,000,000.
So, the modern world needs and it needs it accurately!
The chorus of people singing happy birthday to Maths Inspiration today could be deafening, particularly if the 100,000 teenagers who have been lucky enough to go along to these shows over the last 10 years join in!
Maths Inspiration, set up by Plus's good friend Rob Eastaway, is a fantastic project that runs interactive maths shows all over the UK. The shows are for 14-17 year olds and demonstrate how maths is useful beyond school. And they do more than that, they make maths cool! The shows are held in some of the most prestigious theatres in the country, including The Palace Theatre in London, The Crucible in Sheffield and The Bristol Hippodrome, and feature comedians, teachers, musicians, space scientists, stadium designers, medical professionals, and fitness instructors, who all also happen to be mathematicians. Each show features several talks on topics ranging from roller coasters to the maths used to convict criminals and is very interactive. Students are invited on stage to participate in demonstrations throughout the show.
And if you haven't had a chance to see one of their shows yet, you can still join the party. Their Autumn shows feature Rob Eastaway explaining what the competition between Coke and Pepsi can tell us about world peace, Hannah Fry examining how human connections can cause everything from the movement of crowds, to fashion, and can even help catch the odd burglar, and Ben Sparks will show why emotion, art and mathematics go hand in hand. You can book now at the Maths Inspiration website. And if you're not able to attend a show, you can still catch some of their shows on DVD.
From the Plus team and the 100,000 other lucky people to have seen your shows, Happy tenth birthday Maths Inspiration!