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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
4 years 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.

Mursalin Habib - 4 years ago

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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.

Bob Krueger - 4 years ago

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Okay then! Good luck!

Mursalin Habib - 4 years ago

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Nice, looking forward to more entries.

Yash Talekar - 4 years ago

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Why are the two rectangles similar??

Satvik Golechha - 3 years, 10 months ago

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nice work bro

Hariman Malhotra - 4 years ago

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

Daniel Liu - 4 years ago

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

Bob Krueger - 4 years ago

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It's great

Säyàñtàñ Tirtha - 4 years ago

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WOW! EXCELLENT.

Nemesio Ditchon - 4 years ago

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nice work!

Sean Gyen Medrano - 4 years ago

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I read first about this in THE DA VINCI CODE by Dan brown

Abhishek Singh - 4 years ago

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It's gr8...

Shiven Bholwani - 4 years ago

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