A short sneak preview of the full-length ‘Mandelbulbs’ video currently in production. A Koch Snowflake is an example of a 2D fractal with infinite perimeter but finite area. Full details of the calculation in the final video… COMING SOON!
4.6692… – FEIGENBAUM’S CONSTANT
A new addition to the list of important mathematical numbers, Feigenbaum’s constant was only discovered in 1978 through the study of chaotic systems. Chaos in the mathematical sense is pretty much what you might expect – it describes something that is completely unpredictable. My favourite way to think of it is using magnets. If you have two magnets each with a North and South pole (those funky blue and red ones from school) and hold opposite ends near to each other, then they are attracted and will quickly move together. This is a nice stable system – we know what will happen every time. And if you slightly adjust how far apart you put the magnets, or at what angle you hold them near to each other, it won’t make a difference; they will still join up. Now, what happens if you try and hold two of the same ends near to each other? Well, for starters they’re now going to repel. But, what’s important here is that they move around unpredictably. Try it. If you force them towards each other they push back and can end up moving sideways, up, down, pretty much in any direction. And if you try to repeat the same movement you can’t. That’s chaos.
So, what does Feigenbaum’s constant have to do with all of this? The answer lies in fractals — a repeating pattern that continues to look the same despite zooming in further and further (see image below). The rate at which the image zooms in is Feigenbaum’s constant — cool, huh?
Feigenbaum’s constant is important because it describes the rate at which the simplest mathematical systems (called one-dimensional maps) descend into chaos. The really awesome thing is that you can create all kinds of different systems (following a few basic rules) and they will all go into chaos at the same rate given by this exact constant.
5 – FIVE
Apart from being a hit(?) boyband from the 90’s, five is also a number. It has the fun quirk of being the fifth number in the Fibonacci sequence: 1, 1, 2, 3, 5… where the next number is the sum of the two before it. It’s also the number of human senses (or at least the main ones) and the number of rings in the Olympic symbol. My favourite use of five, however, has to be the 5 Platonic Solids. These are the most, regular, symmetrical and beautiful shapes in all of maths – some people would say that they are so beautiful that they should be permanently inked onto your body… who are these crazy people? Hint: it’s me.
To be a Platonic Solid you need to be 3D, with each face the same shape, and at each corner the same number of faces must join together. Take the simplest example: the cube. We have 6 faces that are all squares and at each corner 3 squares join together. The smallest Platonic Solid is the Tetrahedron or triangle-based pyramid which has 4 faces that are all triangles. After the square comes the Octahedron which has 8 faces that are all triangles – basically two square-based pyramids stuck together so that the square face is inside the solid. The last two are the Dodecahedron with 12 pentagon-shaped sides and the Icosahedron which has 20 triangles all stuck together. These are the only 5 shapes that satisfy the very simple set of rules and they appear everywhere in nature, from the shape of viruses to the structure of molecules. In short, they’re grrrrreat (credit to Tony the Tiger).
6 – SIX
The number of impossible things that the Queen from Alice in Wonderland believes before breakfast and also the number of legs on every insect on earth. Insects are in fact the largest group of species we have on the planet, and even outnumber all of the other species combined… We also have 6 Quarks, which as well as being some of the fundamental particles that make up our universe, have the fantastic names of up, down, bottom, top, charm and strange. Sounds like a fun weekend…
Mathematically, six is what we call a perfect number: all of its factors (the numbers that divide it) add up to give six 1 + 2 + 3 = 6. It also happens to be the only number in existence where not only do all of its factors add up to give the number, they also all multiply together to give the number too: 1 x 2 x 3 = 6. Perfect numbers (the ones that equal the sum of their factors) are a little more common, the next three are: 28, 496, 8128. There are more of course, but they get a little tricky to find. Be my guest and see if you can work them out… sounds like another fun weekend to me.