## Plus Blog

November 26, 2010

The dramatic curved surfaces of some of the iconic buildings created in the last decade, such as 30 St Mary's Axe (AKA the Gherkin) in London, are only logistically and economically possible thanks to mathematics. Curved panels of glass or other material are expensive to manufacture and to fit. Surprisingly, the curved surface of the Gherkin has been created almost entirely out of flat panels of glass — the only curved piece is the cap on the very top of the building. And simple geometry is all that is required to understand how.

One way of approximating a curved surface using flat panels is using the concept of geodesic domes and surfaces. A geodesic is just a line between two points that follows the shortest possible distance — on the earth the geodesic lines are great circles, such as the lines of longitude or the routes aircraft use for long distances. A geodesic dome is created from a lattice of geodesics that intersect to cover the curved surface with triangles. The mathematician Buckminster Fuller perfected the mathematical ideas behind geodesic domes and hoped that their properties — greater strength and space for minimum weight — might be the future of housing.

To try to build a sphere out of flat panels, such as a geodesic sphere, you first need to imagine an icosahedron (a polyhedron made up of 20 faces that are equilateral triangles) sitting just inside your sphere, so that the points of the icosahedron just touch the sphere's surface. An icosahedron, with its relatively large flat sides, isn't going to fool anyone into thinking it's curved. You need to use smaller flat panels and a lot more of them.

Divide each edge of the icosahedron in half, and join the points, dividing each of the icosahedron's faces into 4 smaller triangles. Projecting the vertices of these triangles onto the sphere (pushing them out a little til they two just touch the sphere's surface) now gives you a polyhedron with 80 triangular faces (which are no longer equilateral triangles) that gives a much more convincing approximation of the curved surface of the sphere. You can carry on in this way, dividing the edges in half and creating more triangular faces, until the surface made up of flat triangles is as close to a curved surface as you would like.

You can read more about geodesic domes on Wikipedia and about Buckminster Fuller on the MacTutor History of Mathematics Archive. And you can read more about the Gherkin, geodesics, engineering and architecture on Plus.

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November 26, 2010

Worried you missed Pi day? Never fear! Thanks to the kind people of Wolfram we now have a bevy of mathematical dates to celebrate — six in November alone! November 23, or 11/23 for people in the US, is Fibonacci day as 1,1,2,3 is the start of the Fibonacci sequence. And Twilight fans will be excited that recent months have not only included a Vampire day on 5th of October (since 10052010 = 5001 × 2010), but also that 25th of September was a Cullen day (as 25 and 9 are Cullen numbers)!

Today — 26th of November, or for Americans like the people at Wolfram, 11/26 — is a Eulerian day. The Eulerian number, A(n,k), is the number of ways you can rearrange the integers {1,...,n} so that your permutation is made up of exactly k increasing runs of numbers. And 11 and 26 are the 4th and 5th Eulerian numbers when k is 2 — reason enough to celebrate for Plus! We're off to the pub!

You find out all the mathematical dates we have to celebrate in the Wolfram Blog: Happy Vampire Day.

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November 5, 2010

Suppose you're trying to decide which university to go to. You find out that last year the university you're interested in admitted 30% of male applicants but only 21.3% of female applicants. Looks like a clear case of gender bias, so you're tempted to go somewhere else. But then you look at the figures again, this time broken up by department. The university only has two departments, maths and English. The English department admitted 40% of male applicants and 42% of female applicants. The maths department admitted 10% of male applicants and 11% of female applicants. So if you look at the figures by department, if anything there's bias in favour of women. What's going on?

This is an example of Simpson's paradox, which arises when you look at percentages without giving the actual numbers involved. Suppose the English department admits quite a high proportion of applicants, while the maths department is more choosy and admits only a small proportion. Now suppose that most of the male applicants apply to the English department. Then this drives up the overall percentage of successful male applicants, as English is easier to get into. Similarly, if most women apply to the maths department, then this lowers the overall percentage of successful women applicants, because maths is harder to get into. So it can happen that, although neither department discriminates against women, the overall percentage of successful female applicants is lower than that for males.

Let's go back to the example: suppose that 100 men apply to the English department, so that means that 40 of them got in (40%). Suppose that only 50 women applied to the English department, so 21 of them got in (42%). Suppose the maths department had only 50 male applicants, so 5 got in (10%), and 100 female applicants, of which 11 got in (11%). Then the overall proportion of male applicants who were successful is 45/150 corresponding to 30%. For the women the overall proportion is 32/150 corresponding to 21.3%. Mystery solved.

This paradox isn't just a theoretical curiosity. In 1973 The University of California at Berkeley was sued for sex bias on the basis of figures that were an illustration of Simpson's paradox. It turned out that on the whole women had applied to more competitive departments and that's how the seemingly biased figures arose.

You can find out more about Simpson's paradox on Plus.

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November 5, 2010

Image: L. Shyamal.

Mathematical language can heighten the imagery of a poem, and mathematical structure can deepen its effect. This lovely blog by JoAnne Growney lets you feast on an international menu of poems made rich by maths.

Here's an example of a Fib, that's a poem in which the number of syllables in each line follow the Fibonacci sequence, which appears on the blog. It was written by Athena Kildegaard.

seek
beauty—
all else is
false hope or blind faith.
What can be seen or heard or known
by pressing hard against this world—that is beautiful.

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November 3, 2010

You're unlikely to ever run into a black hole, but here's what it "looks" and "sounds" like when two black holes run into each other. The movie below, produced by Frans Pretorius at Princeton University, shows a simulation of the gravitational waves generated when two black holes collide and form a third. Gravitational waves are ripples in the fabric of spacetime, resulting from events involving massive objects which distort spacetime. The waves were predicted by Einstein's theory of gravity. No-one has directly observed gravitational waves yet, but there's indirect evidence for their existence. And if you could see these ripples, this is what they would look like.

"[Collisions like this are] expected to happen in the Universe for any two black holes that are sufficiently close to form a bound, orbiting system," says Pretorius. "Their orbital motion causes the gravitational waves to be radiated outwards. However, gravitational waves carry energy, which comes at the expense of the orbital energy, which is why the black holes spiral in and merge into a single large black hole. "

Another research group, based at MIT and led by Scott A. Hughes, has turned the gravitational waves generated by a black hole collision into sound waves. Click here to listen.

You can find out more about gravitational waves in our interview with Bangalore Sathyaprakash and about how the gravitational waves are turned into sound on the MIT website.

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November 3, 2010

Ever wondered how your cat can be dead and alive at the same time, or whether there could be a version of you with tentacles editing a magazine in a parallel universe? (Rest assured, Plus is currently tentacle-less and in this universe.)

Science Brainwaves, a student-led organisation from the University of Sheffield, with the aim of bringing all things science to the masses, present this year’s Christmas Lecture. Dr Paul Stevenson will discuss weird physics — the brain contorting conundrums that give us lasers, phone apps, MRI scanners and light switches.

The talk will take place on the 17th December at 6pm in the grade II listed St George’s Lecture Theatre in the heart of the city, followed by a festive reception where the University’s physics department will provide demos and other activities for the public to get involved in. The event is free to attend and is suitable especially for families with children from 8+. To register for tickets visit www.sciencebrainwaves.com/events.

To get in the mood, you can read more about weird physics on Plus.

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