What is this figure? ... How we can built it? ...
The Welcome page of the book: Mathematics, the New Golden Age by Keith Devlin present this figure. Also the Patricia Cori Website offer it.
So, I like a mathematician I was captured to discover what is this figure and how we can draw it.
Yes. Also after study mathematics and do examinations in three Universities, there are for me still some Mathematics, to discover.
Generally, actual University courses are designed not to cover all the beauty of Mathematics, but the fundamentals and students (me included) were and are more interested to finish, than to enjoy the degree, and this behavior is valid for any subject and any University.
There are still another point I want to comment. The fundamental problem with knowledge is that we forgot, and this depends of the level of frequency in our DNA.
To answer the question I will give you three beauty answers, all equivalent touching several fashionable matters.
The previous spiral is the The Golden Spiral, connected with the Golden Section figuring out "De Divina Proportione".
The Fibonacci sequence, 1, 1, 2, 3, 5, 8, 11 ... was motivated by its author following the rabbits generations.
At the beginning we have 1 rabbit and its partner, which number is still 1. And they are two rabbits. (2). If this couple have a baby rabbit we will have (3 rabbits). If both couples have a new baby rabbit we will have (5) rabbits, and so on.
This sequence may be represented by the recurring sequence,
a_{1}=1, a_{2}=1, a_{n}=a_{n-1}+a_{n-2.}
We can also deduce a formula for this sequence.
where,
which is called "The Golden Mean".
Now, if we built squares harmonically, forming a spiral ... according to the Leonardo Pisano sequence, we will get something like that,
and therefore we will have, at infinitum, "The Golden Spiral".
In my modest opinion this is a valid "first proof" and well motivated.
It is logical, and here reside a great beauty noted by another Leonardo, Leonardo Da Vinci, that we can built a rectangle inside another, such that each vertex of the rectangle divide the side of the square in the ratio, 1: φ, like in the figure.
This is called "The Golden Rectangle", and it leads immediately to the famous Da Vinci proportione (Vitruvio_aureo)
Where Da Vinci, shows the ratio between a man with arms expanded is in the relation with its height, 1: φ.
Inside the Da Vinci picture we have a Golden Rectangle. Do you see? ... and from point A, we can draw the Golden Spiral.
This is proof number 2.
For an excellent third proof, I remain you the excellent book by **Harold Scott MacDonald "Donald" Coxeter** (originally quoted like H. S. M. Coxeter)** An Introduction to Geometry**, solve this issue in a very elegant and well seasoned mode.
You can read the Coxeter point of view in the book, available in DJVU, PDF (2 pages for screen) and only the subject commented here.
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