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Tags  →  mathematics

Many children must have sat through lessons with a teacher who knew their stuff but couldn't teach, or (even worse) with a poor teacher with no real knowledge of their subject. Here is someone very different...

Recently Bill Gates nominated Salman Khan as one of Time Magazine's The 100 Most Influential People in the World, writing:

"Like a lot of great innovators, Salman Khan didn't set out to change the world. He was just trying to help his teenage cousin with her algebra from across the country. But from a closet turned office in his Silicon Valley apartment, Sal, 35, has produced an amazing library of online lectures on math, science and a host of other subjects. In the process, he has turned the classroom — and the world of education — on its head.

"The aspiration of is to give every kid a chance at a free, world-class education. The site has over 3,000 short lessons that allow kids to learn at their own pace. Practice exercises send students back to the pertinent video when they're having trouble. And there's a detailed dashboard for teachers who use Khan Academy in their classrooms.

"Early pilot programs in California classrooms show terrific promise. I've used Khan Academy with my kids, and I'm amazed at the breadth of Sal's subject expertise and his ability to make complicated topics understandable."

It's not just kids that will appreciate the free video lessons of this wonderful teacher. I'm not bad at maths, but I am still enjoying going back and watching his way of putting over topics that have faded from my memory over the years - even some very basic ones.

Whether you're interested in maths, science, finance & economics or humanities (or if your kids are interested or need to be), do check this out. It has to be one of the greatest educational resources in the world - and it doesn't cost a bean.

One of many works of fractal, abstract and 3D art by Iwona Fido, a.k.a. Fiery-Fire , whose other work is well worth exploring.

"Torus knot fun" by zipper

I couldn't resist snaffling this from ToeTagJaneDoe (Sofie)'s fine pages (now on Categorian).

It took a while to track down the original, but I eventually discovered it on this very fine mathematical art page, which is worth visiting in its own right.

A great page for all kinds of mathematical art.

From the page:

In "Nature by Numbers," filmmaker Cristóbal Vila presents a series of animations illustrating various mathematic principles, beginning with a breathtaking animation of the Fibonacci sequence. Then it moves on to the Golden and Angle Ratios, the Delaunay Triangulation and Voronoi Tessellations. This would be math-class gold, and it's awfully sweet even if math class is years behind you.

If you are fascinated by this brilliant animation (which I found on Ian's pages), here is a quick guide to finding out more (should you need or want it):

The Fibonacci Sequence is formed by starting with 0,1 and adding the two numbers together to get a new number (giving 0,1,1). If you keep doing this using the last 2 numbers in the sequence then you get 0,1,1,2,3,5,8,13,21... Fibonacci thought up this sequence (in the year 1202) as described in Fibonacci's Rabbits, which is well worth reading, as is all the other information that you will find there.

Two numbers "a" and "b" are said to be in the Golden Ratio if the ratio of "a" to "b" is the same as the ratio of "a+b" to "a". You can't write this ratio exactly as a decimal number, but it is approximately 1.618.

The Golden Ratio, referred to as phi (greek letter), is approximated ever more closely by the ratio of any two successive numbers in the Fibonacci Sequence, as the sequence is taken further and further. From the sequence above you can see that 13/8 is an approximation of the ratio, but 21/13 is a better approximation (and so it goes on).

Spookily, you can start a sequence with any two numbers, e.g. (235,1) and extend it using the same rules as for the Fibonacci Sequence, and the ratio between the last two numbers will still get closer and closer to the Golden Ratio as the sequence extends. In other words, it is the rules, rather than the starting numbers, that matter.

The Golden Section is a line segment divided according to the Golden Ratio.

The Golden Angle is what you get if you divide a circle's circumference according to the Golden Section, draw two lines from the centre of the circle to where the circumference is split, and take the smaller angle between those two lines. It is about 137.5 degrees. You will see it appearing in the animation.

I can't tell you much about Delaunay Triangulations and Voronoi Tessellations (not having met them before), except that they are reflected in the way in which Nature packs things together. This amazing video has certainly inspired me to try to find out more.

If you like this...

[A great post on the Fibonacci series from laydgray]
[... and you might like the mind-stretchers tag at the top of this post]

"Jewel of the Smiles", a beautiful entry in the Benoît Mandelbrot Fractal Art Contest 2007 by Joe Zazulak
(Joe's link is well worth following up)

In the introduction to the contest, Benoît Mandelbrot writes:

"What distinguishes fractal geometry within mathematics is an exceptional and uncanny characteristic. Its first steps are not tedious, hard, and unrewarding, but playful and extraordinarily easy, and provide rich reward in terms of stunning graphics. To the mathematician, they bring a bounty of very difficult conjectures that no one can solve. To the artist, they provide backbones around which imagination can play at will. To everyone, a few steps in about any direction bring extraordinary pleasure. Nothing is more serious than play. Let's all play."

[The amazing Mandelbrot Set]