Is my view of the golden ratio correct?

Question

It is when a number (1.618) is multiplied by one side of a shape and should equal the other side, right? Also, what is the golden ratio in radical form? Does this work only with rectangles?

Answer 1

Nearly correct. The simplest expression of the golden ratio 1.61803. . . in radicals is (1 + sqrt(5)) / 2.

If a rectangle has sides in the golden ratio, and you draw a line to divide it into a square and a rectangle, then the sides of the new rectangle will also be in the golden ratio.

Answer 2

Suppose you want to divide a line segment of length L like that

<----x*L---> <------ y*L ----------->
|--------------|-----------------------------|
<----------------L------------------------>

and in such way that y= x^2.

Then, x + y = 1 and x is the positive root of the equation x^2 + x - 1 =0, which implies x = (sqrt(5) - 1)/2. and y = (3 - sqrt(5))/2. Then, we say the segment L was divided according to the golden ratio.

Answer 3

2/(sqrt(5)-1)=goldenratio=gr=
.........................................1,618Round

if you have a line L an you want to cut it in two pieces using the golden ration you have the equation
......x..........x*gr
l-----------l----------------l
l--------------L------------l
L=x(1+gr)<=> x=L/(1+gr) it will give you the smallest part of the divided line

you can use it on nearly every thing for example the right-angled triangle

.....x
.X..x.........x
.....x...................x
.....xxxxxxxxxxxxxxxxx
................X*gr