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Please tell me about the Golden Ratio (or Golden Mean), the Golden Rectangle, and the relation between the Fibonacci Sequence and the Golden Ratio.

The Golden Ratio

The golden ratio is a special number approximately equal to 1.6180339887498948482. We use the Greek letter Phi to refer to this ratio. Like Pi, the digits of the Golden Ratio go on forever without repeating. It is often better to use its exact value:
        1 + sqrt{5}
The Golden Rectangle

A Golden Rectangle is a rectangle in which the ratio of the length to the width is the Golden Ratio. In other words, if one side of a Golden Rectangle is 2 ft. long, the other side will be approximately equal to 2 * (1.62) = 3.24.

Now that you know a little about the Golden Ratio and the Golden Rectangle, let's look a little deeper. Take a line segment and label its two endpoints A and C. Now put a point B between A and C so that the ratio of the short part of the segment (AB) to the long part (BC) equals the ratio of the long part (BC) to the entire segment (AC):

The ratio of the lengths of the two parts of this segment is the Golden Ratio. In an equation, we have

          AB     BC
         ---- = ---- .  
          BC     AC
Now we're ready for the definition of the Golden Ratio. The Golden Ratio is the ratio of BC to AB. If we set the value of AB to be 1, and use x to represent the length of BC, then
            1     x
            - = ----- .  
            x   1 + x
If we solve this equation for x, we'll find that it is the value given above, (1+sqrt{5})/2, which is about 1.62.

If you have a Golden Rectangle and you cut a square off it so that what remains is a rectangle, that remaining rectangle will also be a Golden Rectangle. You can keep cutting these squares off and getting smaller and smaller Golden Rectangles.

Fibonacci Sequence

In the Fibonacci Sequence (0, 1, 1, 2, 3, 5, 8, 13, ...), each term is the sum of the two previous terms (for instance, 2+3=5, 3+5=8, ...). As you go farther and farther to the right in this sequence, the ratio of a term to the one before it will get closer and closer to the Golden Ratio.

With the Fibonacci Sequence you can do the opposite of what we described above for the Golden Rectangle. Start with a square and add a square of the same size to form a new rectangle. Continue adding squares whose sides are the length of the longer side of the rectangle; the longer side will always be a successive Fibonacci number. Eventually the large rectangle formed will look like a Golden Rectangle - the longer you continue, the closer it will be.

See "The Relation of the Golden Ratio and the Fibonacci Sequence" in the Dr. Math archives.

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