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Monday, June 9, 2014

The Golden Rectangle

We're fast approaching some really fascinating applications of this number 'phi' and the Fibonacci sequence. But before we get to looking at those applications, famous faces, buildings, etc., I'd like to introduce you to something new. We've talked briefly about the Golden Ratio, or the Golden Mean (the number 'phi', which is approximately 1.618...). But there is a very simple geometric shape that defines this ratio and which serves as a starting point for just about all of the geometric instances of Fibonacci.

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The shape is a rectangle and this very special rectangle is one of those shown to the left. Can you figure out which one it is? Which rectangle looks 'just right'?

You see, some rectangles are long and skinny, while others are so even in their dimensions they almost look like squared (the green rectangle toward the bottom middle, for example).

But there are certain rectangles that just seem to look 'right'. I don't know what it is, but they are pleasant to look at and just catch your eye. One that immediately jumps out at me is near the top middle of this collection. It's orange, and it lies just below a long skinny orange rectangle.

Now why does this one rectangle stand out to me? What is it about its dimensions that make it seem so 'perfect'? Ah, this is where Fibonacci comes in.

Remember the Golden Ratio, the ratio of consecutive numbers in the sequence? For example, we looked at the ratios 8/5, 13/8, 21/13, 34/21, etc. Those values are not the same, and yet as we work our way through the sequence, the ratios all approach this special value we call 'phi', which is approximately 1.618. It's known as the Golden Ratio (among other names).

Here is a rectangle whose dimensions reflect the Golden Ratio, along with several that do not. That one in the middle is pretty sexy, isn't it?
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Let's take a look quickly at how this 'golden' rectangle works. If you were to take a Golden Rectangle and cut from it a square whose length is the shorter side of the rectangle, it would look like this:
Photo Credit: Wikipedia

We now have a square, which measures b by b, and a rectangle. The thing is, we could then cut the new rectangle (A) into a square, again with side length equal to the shorter side of the rectangle, and that would leave another rectangle.

What's so interesting is that this pattern can continue forever. Each time you cut off a square, you end up with a new (smaller) rectangle. And that rectangle always has a length and width whose respective length ratio is phi.
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Now this is where it gets really cool, and we'll probably stop here for today. You can start way down inside the repeating pattern of rectangles and squares and think of that innermost rectangle(the bright pink one) as TWO squares that each measure 1x1. Next, the brown square right below it is then 2x2, and the blue square that follows is 3x3. See a pattern here?

The green square measures 5x5, the red one is 8x8 and now it shouldn't surprise you to learn that the orange square is 13x13, followed by the yellow square which of course now measures 21x21. In fact, you could continue this pattern forever, and eventually, the series of squares will all form into a spiral...a Fibonacci Spiral. More on that in my next post. For now, I will leave you with a drawing of the numbered squares that form the Fibonacci Spiral.
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