Bellezza regolare
dedicate yourselves a star as a present
Beauty by the rule.
The poster on which you have found this Web address may seem of striking beauty to you, the product of a brilliant, creative mind, like for instance mine. Alas, it isn´t. It is God´s creation, if you choose to believe in him or her. If not, it is the product of the purest science of all, mathematics, very simple mathematics for that matter.
The rule which generates this image and trillions more is
Z(n+1) = Z(n) Z(n) + Z(0)
This does not look like a promising rule. It is a rule on how to generate a next number from the previous one. Start with any number, multiply it by itself and add the number from which your started
1 -> 1*1+1=3 -> 3*3+1=10 -> 10*10+1=101 -> 10202 .... and off we go into infinity
But will we go off to infinity always? Let´s try with -1:
-1 -> (-1)(-1)+(-1)=0 -> 0*0+(-1)=(-1) -> 0 ... and we keep oscillating between -1 and 0 for infinity for ever and ever and ever ...
For which starting numbers do these series go to infinity, for which do they oscillate and for which do they converge to a fixed and finite value?
Mr. Mandelbrot back in the 80´s of the last millenium asked himself the same question. Not just for all the numbers on the infinite line from minus infinity to plus infinity, but for all numbers in the plane. Each point is denoted by two numbers there, the horizontal and vertical distancesfrom the center of the universe. Mathematicians call it Cartesian coordinates, but you don't have to worry about it.
Mandelbrot, very much to his dismay, did not come up with an answer, not an easy one, nor a complicated one, either. Back in the 80´s they already had computers. So instead of thinking he did what kids do, he just tried. For every point in the plane he simply tried on the computer whether the series would explode or stay calm. Being a trained mathematician and not a kid, he did think a little bit and found out that once the series would arrive at a value of 2 or more, the series was doomed to explode. For every number in the plane he kept trying until the series would exceed 2. For some he had to give up, e.g. the -1 above, for the others he counted the number of steps before the series exceeded the value of two. The number of repetitions of the calculation he used to decide on the color of that point in the plane. No more than that.
He was hoping to see a simple result, given that the rule was that simple, at least for a trained mathematician , get an idea in which direction his mathematical proof of a simple criterion had to go. Alas, he was unlucky. He got a picture like the one in front of you, the apparent result of chaos. Beautiful chaos, but still chaos resulting from order, a very simple order.
Well, really, he was lucky. The unexpected and only mathematically dissatisfying result was beautiful enough to get him famous. He would not have gotten famous, had he proven that the series would stay limited for all starting values say with a modulus or a square root between -2 and 0. A mention in some mathematical and otherwise obscure chronicles, at the most.
These pictures are not just beautiful. They make us think. How come that one addition and one multiplication can create such a chaos? Why does this chaos look so beautiful to us? Is there a simple mathematical rule behind any kind of beauty? Is beauty objective? Is there a simple rule for any chaos beautiful or not, e.g. my office? Why should we continue trying to be creative, if anyhow, whatever our mind can possibly conceive will never ever be able to compete with the result of a simple multiplication and addition. So let's stop writing, painting, composing and singing. Relax and have a beer.
I have no answers for you, as Mandelbrot didn't. You will have to find out for yourself. Just look at this picture and meditate. If you like, you may download a Mandelbrot generator program for your computer, if you own one, or your smart phone or tablet or iWhatever or toaster and spend hours on it. With a pinch of your fingers or a click on the mouse you can zoom in on any part of the picture and find ever new self similar structures and sometimes total surprises.
All these pictures are for free. No royalties. God does not ask royalties. You may use them as your screen desktop, on your home page, or print them on a canvas like the one in front of you, on the sheets of your bed, on your T-shirt, your coffee mug or on your car.
If you are not experienced with using computers and online printing companies, drop me a line in the form below. I can help you with the production. It won´t cost you more than the production cost at the company you choose. God not asking royalties, neither do I. You may choose to donate what you saved to a charity of your choice or of my choice.
While you won't be the author, really, it's God, maths or whoever or whatever; neither you nor me nor Mandelbrot, it is you who chose your definitely unique design in the trillions of Mandelbrot sets.
The poster on which you have found this Web address may seem of striking beauty to you, the product of a brilliant, creative mind, like for instance mine. Alas, it isn´t. It is God´s creation, if you choose to believe in him or her. If not, it is the product of the purest science of all, mathematics, very simple mathematics for that matter.
The rule which generates this image and trillions more is
Z(n+1) = Z(n) Z(n) + Z(0)
This does not look like a promising rule. It is a rule on how to generate a next number from the previous one. Start with any number, multiply it by itself and add the number from which your started
1 -> 1*1+1=3 -> 3*3+1=10 -> 10*10+1=101 -> 10202 .... and off we go into infinity
But will we go off to infinity always? Let´s try with -1:
-1 -> (-1)(-1)+(-1)=0 -> 0*0+(-1)=(-1) -> 0 ... and we keep oscillating between -1 and 0 for infinity for ever and ever and ever ...
For which starting numbers do these series go to infinity, for which do they oscillate and for which do they converge to a fixed and finite value?
Mr. Mandelbrot back in the 80´s of the last millenium asked himself the same question. Not just for all the numbers on the infinite line from minus infinity to plus infinity, but for all numbers in the plane. Each point is denoted by two numbers there, the horizontal and vertical distancesfrom the center of the universe. Mathematicians call it Cartesian coordinates, but you don't have to worry about it.
Mandelbrot, very much to his dismay, did not come up with an answer, not an easy one, nor a complicated one, either. Back in the 80´s they already had computers. So instead of thinking he did what kids do, he just tried. For every point in the plane he simply tried on the computer whether the series would explode or stay calm. Being a trained mathematician and not a kid, he did think a little bit and found out that once the series would arrive at a value of 2 or more, the series was doomed to explode. For every number in the plane he kept trying until the series would exceed 2. For some he had to give up, e.g. the -1 above, for the others he counted the number of steps before the series exceeded the value of two. The number of repetitions of the calculation he used to decide on the color of that point in the plane. No more than that.
He was hoping to see a simple result, given that the rule was that simple, at least for a trained mathematician , get an idea in which direction his mathematical proof of a simple criterion had to go. Alas, he was unlucky. He got a picture like the one in front of you, the apparent result of chaos. Beautiful chaos, but still chaos resulting from order, a very simple order.
Well, really, he was lucky. The unexpected and only mathematically dissatisfying result was beautiful enough to get him famous. He would not have gotten famous, had he proven that the series would stay limited for all starting values say with a modulus or a square root between -2 and 0. A mention in some mathematical and otherwise obscure chronicles, at the most.
These pictures are not just beautiful. They make us think. How come that one addition and one multiplication can create such a chaos? Why does this chaos look so beautiful to us? Is there a simple mathematical rule behind any kind of beauty? Is beauty objective? Is there a simple rule for any chaos beautiful or not, e.g. my office? Why should we continue trying to be creative, if anyhow, whatever our mind can possibly conceive will never ever be able to compete with the result of a simple multiplication and addition. So let's stop writing, painting, composing and singing. Relax and have a beer.
I have no answers for you, as Mandelbrot didn't. You will have to find out for yourself. Just look at this picture and meditate. If you like, you may download a Mandelbrot generator program for your computer, if you own one, or your smart phone or tablet or iWhatever or toaster and spend hours on it. With a pinch of your fingers or a click on the mouse you can zoom in on any part of the picture and find ever new self similar structures and sometimes total surprises.
All these pictures are for free. No royalties. God does not ask royalties. You may use them as your screen desktop, on your home page, or print them on a canvas like the one in front of you, on the sheets of your bed, on your T-shirt, your coffee mug or on your car.
If you are not experienced with using computers and online printing companies, drop me a line in the form below. I can help you with the production. It won´t cost you more than the production cost at the company you choose. God not asking royalties, neither do I. You may choose to donate what you saved to a charity of your choice or of my choice.
While you won't be the author, really, it's God, maths or whoever or whatever; neither you nor me nor Mandelbrot, it is you who chose your definitely unique design in the trillions of Mandelbrot sets.
