The Golden Ratio seems to get its name from the Golden Rectangle , a rectangle whose sides are in the proportion of the Golden Ratio. Whether or not the Golden Ratio or the Golden Rectangle are of aesthetic significance, the ratio does turn out to have considerable significance in problems of natural symmetry.

If the Golden Ratio turns up in examples of five-fold symmetry, it may well be because the number itself is fundamentally related to the number five. It happens, then, that below zero, we find the same Fibonacci Numbers, but they alternate as positive and negative.

Now, the Fibonacci Numbers turn up in nature.

• The table at right illustrates an interesting way in which the Fibonacci Numbers occur naturally in relation to the Golden Ratio.
• Because of this, any power of the Golden Ratio can be ultimately reduced to the sum of an integer and an interger multiple of the Golden Ratio.
• While we see that the Fibonacci series emerges naturally in the evaluation of the powers of the Golden Ratio, this does not necessarily make it clear why the ratio of the members of the Fibonacci series should approach the Golden Ratio as a limit.

This is why ratios of Fibonacci numbers approximate the Golden Ratio, they are all solutions to the unique continued fraction for the Golden Ratio!

Thus, while the Golden Ratio may not be as important as other mathematical constants, it does have its claim to fame and does have its own unique properties. This is a much more significant and mysterious number than the Golden Ratio -- not bad as a kind of cousin.