Golden Rectangle and Spirals (please help)

**ticktack** · March 10th 2008, 07:00 AM

Hi, I am really confused on the following

What is meant by dividing up a line segmant according to the Golden Mean or Golden Ration? What does the solution to the equation
0 = x^2 + x - 1 have to do with this?

How is this ratio used to form the Golden Rectangles? (Please explain how does one calculate and construct it)

How is the Equiangular spiral (logarithmic spiral) formed from the Golden Rectangles?

**roe_gamma** · March 10th 2008, 07:29 AM

Golden ratio - Wikipedia, the free encyclopedia

You get the golden ratio when you solve the quadratic by the quadratic formula.

To form the golden rectangle, choose a length n, for one of the pairs of sides, and multiply n by the golden ration for the other pair.

I'm not too sure about the spiral though. Sorry.

**ticktack** · March 13th 2008, 06:23 AM

thanxx but still confused.

**Soroban** · March 13th 2008, 01:29 PM

Hello, ticktack

What is meant by dividing a line segmant in the Golden Ratio?

Divide a line segment into two parts.
Let one part be 1, the other part be $a$ .

Code:

      : - - - -  a+1  - - - - :
      *---------*-------------*
           1           a

We want the point of division to be such that:
. . the ratio of the smaller part to the larger part
. . equals the ratio of the larger part to the entire segment.
[The larger part is the mean proportional of the smaller part and the whole segment.]

We have: . $\frac{1}{a}\:=\:\frac{a}{a+1}\quad\Rightarrow\quad a^2 - a - 1 \:=\:0$

The Quadratic Formula gives us: . $a \:=\:\frac{1 \pm\sqrt{5}}{2}$

Since length $a$ is positive: . $a \;=\;\frac{1 + \sqrt{5}}{2}$

This value is about 1.618033989... and is denoted $\phi$ (phi).

How is the solution to: $x^2 + x - 1 \:=\:0$ related to this?

The solution to this equation is: . $x \;=\;\frac{-1\pm\sqrt{5}}{2}$

And $\frac{\sqrt{5} - 1}{2}$ happens to be the reciprocal of $\phi.$

How is this ratio used to form the Golden Rectangle?

Construct a rectangle with width 1 and length $\phi.$

It is considered to be most "pleasing" rectangle.
It appears in many famous works including the Mona Lisa and the Parthenon.

An interesting bit of trivia . . .

If we cut off the square from the end of a Golden Rectangle,
. . the portion that remains is another Golden Rectangle.

Code:

      : - - - - φ  - - - - :
      *-----------*--------*
      | / / / / / :        |
      | / / / / / :        |
    1 | / / / / / :        | 1
      | / / / / / :        |
      | / / / / / :        |
      *-----------*--------*
            1

**earboth** · March 13th 2008, 02:04 PM

Originally Posted by ticktack

... how does one ... construct it

...

This is the simpliest construction of the golden ratio I know:

The legs of the right triangle have the ration $1 : \frac12$

The radius of the thin-lined circle is $r = \frac12$

Then the radius of the thick-lined circle is: $R = \frac{\sqrt{5}}{2}-\frac12$ which is one of the results Soroban has posted.