Espiral Aurea

please comment

Interesting interpretation of Fibonacci.

 

Cheers.

Is this a golden mean spiral ... is that the same as Fibonacci series .... well, that gloden mean does fullfill x(n+2) = x(n)+ x(n+1) .. so golden mean is a case of fibonacci, right? Interesting.

Here are 3 different Spirals.

1 This is the commonly named Fibonacci Spiral. It is consisting by 90 degrees Arches, starting with two which radius are = 1; followed by other r = 2, next: r = 3, r = 5, r = 8, etc... (Fibonacci Series).

2 The number 2 is consisting by 90 degrees Arches were ever the next is 1.618 times bigger. All rectangles are Golden ( 1 x 1.618 ).

3 This one doesn't have circular Arches. All the red line is a Spiral that gradually grows it's curvature a way to increase it's size 1.618 every 90 degrees.I think nobody has done this before, so this is my contribution!.

4 This shows the difference between 2 and 3. (Blue shows Arches and Red show a continue Spiral).The same in the main picture.

I hope this can help. Regards, Juan

Ok, the two first one are continuos by tangent but not by radius adn the third one is differential spiral continuous to second order and possibly higher. I once played with a similar problem with Mathematica. The idea was to create a mean line for a wing profile with a certain angle and radius at leading edge and two further points with tangent&radius conditions. I didn't get it the way I wanted - just did not look right so eventually I found Eppler had done it in a more beautifull way.

Thanks Juan Gomez, I really appreciate your graphical explanation.

And now for something COMPLETELY different! I don't understand how it works but the underlying sense of order is fantastic.

Sounds very complicated & I didn't know how it works either, but its definitely very interesting & original. Looks like some sort of spiral stairs to me :-)

I think you already did a very good definition of it: "...AN underlying sense of order THAT is fantastic." Just that!

I think this is just interesting and it is not necessary to understand it to make art or beauty. It have been for centuries an attempt to explain what is beauty by mathematics. It is much more complex than these spirals...