Plus Blog

July 30, 2014

A perfect number is a natural number whose divisors add up to the number itself. The number 6 is a perfect example: the divisors of 6 are 1, 2 and 3 (we exclude 6 itself, that is, we only consider proper divisors) and

1+2+3 = 6.


If a non-perfect number were an animal, it might look something like this.

Hooray! People have known about perfect numbers for millennia and have always been fascinated by them. Saint Augustine (354–430) thought that the perfection of the number 6 is the reason why god chose to create the world in 6 days, taking a rest on the 7th. The Greek Nicomachus of Gerasa (60-120) thought that perfect numbers produce virtue, just measure, propriety and beauty. Numbers that are not perfect, for example numbers whose proper divisors add up to more than the number itself, Nichomachus found very disturbing. He accused them of producing excess, superfluity, exaggerations and abuse, and of being like animals with "ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands."

If you play around with numbers for a while you will see why people have always been so fond of perfect numbers: they are very rare. The next one after 6 is 28, then it's 496, and for the fourth perfect number we have to go all the way up to 8128. Throughout antiquity, and until well into the middle ages, those four were the only perfect numbers that were known. Today we still only know of 48 of them, even though there are fast computers to help us find them. The largest so far, discovered in January 2013, has over 34 million digits.

Will we ever find another one? We can't be sure — mathematicians believe that there are infinitely many perfect numbers, so the supply will never run out, but nobody has been able to prove this. It's one of the great mysteries of mathematics. You can find out more in Number mysteries.

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July 24, 2014

In the process of writing an article on curvature we got entirely distracted by making a geogebra worksheet showing the tangent, normal and osculating circle to any smooth function. The meaning and mathematics of all these terms is revealed in this article, but if you fancy getting your hands dirty yourself, have a play with the worksheet below. Please post your favourite function as a comment – the curvier and wigglier the better!

You can use this geogebra worksheet to see the tangent, normal and osculating circle of any smooth curve you choose - just change the equation f(x) in the left-hand panel.

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July 24, 2014
Sierpinski tetrahedron

Bat country. Image designed by Gwen Fisher, engineered by Paul Brown. © Gwen Fisher.

These lucky people are climbing around a 22 feet (6.7 metres) tall structure composed of 384 softball bats, 130 soft balls and a couple of thousand pounds of steel. The structure represents a Sierpinski tetrahedron: a fractal which has finite volume but infinite area. The image only shows an approximation of the fractal of course, as it would be impossible to make a full-on Sierpinski tetrahedon with its infinite intricacy, but it's beautiful anyway!

The picture, designed by Gwen Fisher and engineered by Paul Brown, is one of the images that appears in the book 50 visions of mathematics, which celebrates the 50th anniversary of the Institute of Mathematics and its Applications.

You can find out more about fractals here.

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July 18, 2014
The principles behind the Forth Bridge

IMage courtesy of the Department of Civil and Environmental Engineering, Imperial College London.

This is one of friend of Plus Ahmer Wadee's favourites images from the book 50 visions of mathematics. It is of a demonstration at Imperial College in 1887 of the mathematical principles behind (or should that be underneath?) the Forth Bridge. The bridge was the largest spanning bridge in the world at the time and the technique behind it was an innovation, essentially balancing the forces involved using cantilevers. The men on the chairs (Sir John Fowler and Benjamin Baker) represent the piers of the bridge and the load on the bridge, in this case Kaichi Watanabe, one of the first Japanese engineers to study in the UK, is supported by the tension (in the men's arms and in the ropes to the anchors) and compression in the structure.

So, what holds up the Forth Bridge? Why, maths of course!

You can read more about the maths of engineering in Constructing our lives on Plus

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July 16, 2014

If you love physics and making movies then this is for you. The Foundational Questions Institute (FQXi) is excited to present Show me the physics!, its first-ever video contest. Anyone can submit a video conveying the joys of physics to win a top prize of $10,000, and there are very attractive runner-up prizes too.

Whether you're a physicist or just a physics geek, and whether it's the geometry of space time (see top video on the right) or quantum immortality (see bottom video), all sorts of submissions are welcome. The aim is to enthuse non-physicists and provide a creative and visual space for the discussion and exchange of ideas. Examples of suitable topics are:

  • Unsolved physics mysteries
  • Physics experiments being carried out
  • Tales of physics discoveries
  • Accounts of how physics has improved our lives
  • Physicists, inventors, teachers, and others talking about their passion for physics
  • Fictional stories in which real physics plays a central role.

The closing date is August 8, 2014. See here here for rules and submission guidelines and here to see the current entries.

Good luck!

FQXi are our partners on the Information about information and Science fiction, science fact projects.

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July 11, 2014
Germany's 7-1 victory over Brazil in fractal form

Image by Lasse Rempe-Gillen – click on the image to see a larger version.

To mark Germany's historic win over Brazil in the World Cup semifinal this week, Lasse Rempe-Gillen (from the University of Liverpool) created this beautiful image. It shows the behaviour of a model that describes the phenomenon of phase-locking, something that can be seen in the synchronising flashes of fireflies or when a roaring stadium of football supporters gradually clap or stamp in unison. The image is related to recent research and you can read more in our news story Maths, metronomes and fireflies.

The grey parts of the image show where the model behaves chaotically – here even small changes in where you start can cause drastically different results in the model. The coloured parts of the image show where the model behaves in a more regular fashion where small differences won't dramatically change the results. This is because the model has attractors, special sets of conditions that create similar behaviour, either settling on a single outcome (called a fixed point) or running through a predictable cycle of outcomes. And in honour of the historic 7-1 score from the match, Rempe-Gillen's image has attractors of period 7 (with a repeating cycle of 7 points) and period 1 (a fixed point).

In contrast his image below has no periodic attractors, symbolising the other, goalless, semifinal between Argentina and Holland.

Holland v Argentina's 0-0 semifinal in fractal form

Image by Lasse Rempe-Gillen – click on the image to see a larger version.

You can read more about chaos, fractals and football on Plus!

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