# The Golden Ratio: What is it?

Consider the above picture (called the golden rectangle). There is a rectangle that when we separate it into a square and another rectanlge, the rectangle formed is similar to the original rectangle. Let $a$ be the longer side of the rectangle, and $b$ the shorter side. Then, because of the rectangles' similarity, we have the proportion:

$\frac{a}{b} = \frac{b}{a-b}$

We will name this value $\phi = \frac{a}{b}$, the greek letter phi, and call it the Golden Ratio, just as the ancient Greeks did. At this point, this may just seem like some arbitrary geometrical figure. But $\phi$ has a lot of hidden properties, which we will uncover. First, let's try to find a numeric value for $\phi$: Taking the reciprocal of both sides of the equation above gives

$\frac{b}{a} = \frac{a-b}{b} = \frac{a}{b} - 1$

or, by definition of $\phi$,

$\frac{1}{\phi} = \phi -1$ $\rightarrow 0 = \phi^2 - \phi - 1$ $\rightarrow \phi = \frac{1+\sqrt{5}}{2} \approx 1.618$

Now we know the actual value of the ratio of the sides of the rectangle first pictured above. The following forms of equations involving $\phi$ will be very useful to us in the future:

$\phi^2 = 1 + \phi$ $\phi = 1 + \frac{1}{\phi}$

You may be wondering right now why I took only the positive root when I applied the quadratic formula to find $\phi$ above. In the geometric sense, I wanted a ratio, and ratios are always positive. The negative root results in a negative value for $\phi$. This value is called the conjugate of $\phi$ and has many similar properties to $\phi$. But we will normally restrict ourselves to the positive value of $\phi$. Tune in tomorrow for more mathematical information about the Golden Ratio.

Here is the next segment in the series.

Note by Bob Krueger
6 years, 6 months ago

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So, you're planning on doing posts everyday? Nice introductory post!

It should be noted that the golden-rectangle is said to be the most visually aesthetic rectangle.

- 6 years, 6 months ago

Thanks! Every one or two days, that's correct. And about the golden rectangle being aesthetically pleasing, we will definitely be delving deeper into that concept! A quick estimation shows that this will probably continue through February.

- 6 years, 6 months ago

Okay then! Good luck!

- 6 years, 6 months ago

Nice, looking forward to more entries.

- 6 years, 6 months ago

Good job! Keep it up, you still have a LOT to go through ;)

- 6 years, 6 months ago

Thanks! I've already got the next six days written up.

- 6 years, 6 months ago

nice work bro

- 6 years, 6 months ago

Why are the two rectangles similar??

- 6 years, 4 months ago

It's gr8...

- 6 years, 6 months ago

- 6 years, 6 months ago

nice work!

- 6 years, 6 months ago

WOW! EXCELLENT.

- 6 years, 6 months ago

It's great

- 6 years, 6 months ago