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 December 12, 2014 The number 12 is very flexible. You can write it as 2 x 6 or as 3 x 4. Or, even better, as 2 x 2 x 3. Which brings us to those very special numbers called the primes: those numbers that are not divisible by any other number apart from 1 and themselves. Every number can be decomposed into a product of primes, for example 12 = 2 x 2 x 3. The primes are the atoms of number theory, they are also connected to one of the hardest open problems in maths, and even to the weird world of quantum physics. If you love the primes, or at least feel curious about them, then try Marcus du Sautoy's book The music of the primes: why an unsolved problem in mathematics matters. It's a gentle introduction to the primes, their importance in the modern world, and all the mathematicians they have taunted over the centuries. To get a feel for the subject, read du Sautoy's Plus articles The prime numb lottery and The music of the primes, or watch his lecture below. Return to the Plus Advent Calendar December 12, 2014 Welcome to the 117th Carnival of Mathematics! What treats are in this months Carnival? As is tradition we start with a fact about this Carnival's number: 117. It's divisible by 3. That might not be so exciting to everyone, but to someone such as myself, as a mathematician who is notoriously bad at numbers and arithmetic (my friends think it's hilarious to ask me to divide the bill at a restaurant), being able to tell this quickly about any integer gives me a feeling of vast power over these numbers. You can tell if a number is divisible by 3 if the sum of the number's digits is divisible by 3: in our case, 1+1+7=9. Neat! This trick gives me a great sense of power over the integers as it allows me to rule out vast swathes of them (well, a third) as not being prime, at a glance. Combine that with our instant knowledge that any even numbers, or those ending in 5 or 0 aren't prime, and you've really got a good of chance of being able to tell if any random number someone tells you is prime. There's other neat division rules too, for 7 and 11 and so on. But the really interesting thing is why they work… Can you explain why the division rule for three works? Give us your best explanations in the comments below! Now to the Carnival. We had many great submissions, thank you to everyone who sent one in. The one that had me hooked for the longest was the wonderful Parable of the Polygons by Vi Hart and Nicky Case, submitted by Katie. This brilliant interactive post explores the maths of diversity based on the work of Nobel Prize-winning game theorist Thomas Schelling. I also really enjoyed Kevin Houston's fascinating and detailed article The Beatles' magical mystery chord, delving into the mathematics and music behind the opening chord to a hard days night. It's made me want to catch the next talk Kevin gives on the maths of autotune! A very mathematical snowflake For a real treat, why not spend a long lunch hour watching Terry Tao's talk The Cosmic Distance Ladder. This forms part of one of Mike Lawler's blog post about the lecture, where he discuses how the radius of the Earth and the radius of the Moon's orbit were measured, and how Tao's lecture was an incredible opportunity to show some fun and exciting maths to children. Lawler said in his submission: "Amazingly this lecture by Tao on youtube has fewer than 2,000 views right now. I wish that more people knew about it (kids especially) as it is such an amazing opportunity to learn from one of the greatest mathematicians living today." We heartily agree! (And you can find out more on Tao's work here.) I, and any other people madly manufacturing paper snowflakes for decorating school halls and homes, must thank Mr Reid for his excellent post A public service announcement regarding paper snowflakes. Really, if you're in the middle of cutting a million and one of those suckers out you need to read this, it will make it all much better! And here are some more treats to while away your paper-snowflake-cutting-time… Is truncating the Y-axis misleading?, submitted by Katie. Some book suggestions for your letter to Santa, submitted by Shecky R. And why not make someone a Sunglass Gyroid for Christmas, thanks to Andrea Hawksley. And of course, the answer to all your other gift dilemmas is answered behind the doors of our Advent Calendar. Mark Dominus and his daughter learn that algebra is awesome . Mathistopheles' take on the maths of the game Pointless. You can also read more about Pointless in John Haigh's article for Plus. Meanwhile, on another channel, the Mathematical Ninja is taking on the wall game on Only Connect. An exploration of an interesting function, the Gudermannian-function from the Thinking Machine. That's it for this Carnival, we hope you enjoyed the show! You can find the last Carnival at CavMaths and the details for all past and future Carnivals, as well as how to be involved at Aperiodical. Don't forget to submit ideas for January's Carnival, hosted by Andrew at AndrewT.net. (Hint for the division rule: A number with digits abcd=ax1000+bx100+cx10+d = ax(1+999)+bx(1+99)+c(1+9)+d…) 1 comments December 11, 2014 If you've ever redecorated a bathroom, you'll know that there are only so many ways in which you can tile a flat plane. But once you move into the curved world of hyperbolic geometry, possibilities become endless and the most amazing fractal structures ensue. Beautiful images such as these form the heart Indra's pearls: the vision of Felix Klein by Caroline Series, David Wright and David Mumford. Image by David Wright For a feel of the book read our short introduction to the maths behind their beautiful images. In the book itself, as well as pages of stunning images, you'll find instructions for how to create the images yourself, as well as detailed explanation of the mathematics involved. One of the most compelling ideas is the way these contained images can represent the infinite. In many eastern philosophies, especially Buddhist, this idea of the infinite appearing from copies within copies is pervasive: "In a single atom, great and small lands, as many as atoms." This concept was so exactly reflected in the mathematics of their pictures that it inspired the title of the book, taken from the ancient Buddhist myth of Indra's Web: In the heaven of the great god Indra is said to be a vast and shimmering net, finer than a spider's web, stretching to the outermost reaches of space. Strung at each intersection of its diaphanous threads is a reflecting pearl. Since the net is infinite in extent, the pearls are infinite in number. In the glistening surface of each pearl are reflected all the other pearls, even those in the furthest corners of the heavens. In each reflection, again are reflected all the infinitely many other pearls, so that by this process, reflections of reflections continue without end. Return to the Plus Advent Calendar December 10, 2014 Why do diamonds sparkle? Why is the shower the best place to sing? Where is the 4th dimension in Dali's paintings? If you don't know the answers to these questions, or perhaps didn't even know you didn't know these interesting facts, then John D. Barrow's book will help. Its title, 100 essential things you didn't know you didn't know about maths and the arts, explains what's in it and you can a taster for Barrow's writing in this Plus article. You can be sure it's good though, after all this is Barrow's 22nd book! If you like your science and maths visual, then we also recommend Barrow's book Cosmic imagery, which explores some key images in the history of science (find out more in our visual podcast). If, on the other hand, you like your arts and crafts to be mathematical, then check out Crafting by concepts and Origami polyhedra design. Return to the Plus Advent Calendar December 9, 2014 If you've got maths friends then you might have noticed that a large subset of them are also Simpsons fans. And there's a good reason for that: as Simon Singh's book The Simpsons and their mathematical secrets explains, a large number of Simpsons writers are mathematicians. The book explores the maths that has been hidden (and not so hidden) throughout the series and will make a great present for maths/Simpsons nerds. Return to the Plus Advent Calendar December 8, 2014 In June this year five mathematicians turned into millionaires when they were awarded the Breakthrough Prize set up by Facebook founder Mark Zuckerberg and internet entrepreneur Yuri Milner. Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Terry Tao and Richard Taylor each received £1.8m. That's more than the £930,000 that's awarded for the Nobel prize and should make up for the fact neither of the five is likely to ever to receive a Nobel: there isn't one for mathematics. Back in November the five winners gave a lectures at the Breakthrough Prize Symposium, which have been filmed for you to watch. We especially recommend Terence Tao's talk on collaboration in maths (mathematicians seem to be getting more sociable): Richard Taylor's talk about some beautiful number theory (does get a little technical later but starts very accessibly): And Jacob Lurie's talk on analogy and abstractions in maths, involving some impossible shapes: Simon Donaldson and Maxim Kontsevich's talks are a little more technical, but still worth a look if you're a bit more advanced in maths. Just click on the links!