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

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

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

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

How is the solution to: $\displaystyle x^2 + x - 1 \:=\:0$ related to this? The solution to this equation is: .$\displaystyle x \;=\;\frac{-1\pm\sqrt{5}}{2}$

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

How is this ratio used to form the Golden Rectangle? Construct a rectangle with width 1 and length $\displaystyle \phi.$

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