Plus Blog

November 20, 2014

Our images of the week show the amazing snow art created by Simon Beck.

Koch snowflake

Snow art. Image © Simon Beck.

Koch snowflake

The Von Koch snowflake.

Koch snowflake

Simon Becks' book, Snow Art.

Beck creates those beautiful geometrical shapes by walking through the snow. The shape you see above is based on the Von Koch snowflake. To create the full shape, you start with an equilateral triangle and replace the middle third of each side by a "spike" consisting of two sides of a smaller equilateral triangle. Now do the same for each of the twelve straight-line segments of the resulting shape and repeat, ad infinitum. The Von Koch snowflake is an example of a fractal, a mathematical shape that is infinitely intricate and self-similar: it exhibits the same structures over and over again as you zoom in on smaller and smaller pieces. You can find out more about fractals here.

Beck can't produce the Von Koch snowflake exactly, of course, because it involves an infinite process. But he has created an amazingly good approximation, which comes from using the first few iterations of the process. He gets the shape down in the snow by counting steps to measure distances and using a compass when changing direction to make sure he gets the correct angles between straight-line segments. "In the case of the Koch shape, I found I was soon able to judge the 60 degree angles and do it quicker to an acceptable level of accuracy without using the compass," he says.

If you like the pictures, you can see more of them in Beck's new book, Snow Art, which you can purchase from his website. Here are a few more samples:

Koch snowflake

Image © Simon Beck.

Snow art

Image © Simon Beck.

Snow art

Image © Simon Beck.

Snow art

Image © Simon Beck.

You can see previous images of the week here.

November 19, 2014
A Hubble image

An image taken by the Hubble Space Telescope, courtesy NASA, ESA and E. Sabbi (ESA/STScI).

How important are experiments in science? Scientists use experiments to check whether a theory's predictions match up with reality, so without them you can't pick out bad theories.

In theoretical physics, however, there are many theories that cannot be tested. Not only because our experimental tools are nowhere near good enough, but also because there's some fundamental reason that stops us exploring some of the predictions those theories make. Examples are string theory, M theory and the various multiverse theories. Should we pursue them anyway, or dismiss them as speculation?

This debate, featuring one of our favourite theoretical physicists, David Tong (among others), explores this question and asks whether physics has strayed too far from experiment. It's been produced by the Institute of Art and Ideas in London.

November 17, 2014

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

  \[ f(x) = \frac{x^2}{4a} \]    

which has its focus at the point $F=(0,a).$ (Use the left-hand slider to change the value of $a.$) You can roll the parabola along using the right-hand slider and see the catenary the focus traces out. Its equation is

  \[ y = a\cosh {\frac{x}{a}.} \]    

Nice!

November 12, 2014
Book cover

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.

November 5, 2014

Our image of the week is all about chemistry!

Image created by Alexander Bujotzek, Peter Deuflhard and Marcus Weber. © Zuse Institute, Berlin, Germany.

This image reveals the shape of enzymes and how they bind together. It is produced using mathematical methods. Created by Alexander Bujotzek, Peter Deuflhard, and MarcusWeber. © Zuse Institute Berlin, Germany.

The picture is one of the images that appear in the book 50 visions of mathematics, which celebrates the 50th anniversary of the Institute of Mathematics and its Applications.

You can see previous images of the week here.

November 3, 2014

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 these Plus articles.

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