Fibonacci and the Golden Ratio

I had the opportunity recently to view a work of art that was based on the fascinating Fibonacci sequence. The Fibonacci sequence is named after Leonardo Pisa (or Pisano?) who was known as Fibonacci. Fibonacci’s 1202 book Liber Abaci introduced the sequence to Western European mathematics although the sequence had been described earlier in Indian mathematics.

By definition the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two. The first few numbers of the sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. As the numbers in this sequence get larger and larger, ratios of the larger number divided by the preceding smaller number get closer and closer to a number known as the golden ratio, 1.618 to 1.

The Greek mathematician Pythagoras (circa 580-500 BCE) is sometimes credited as the first to discover the golden ratio, though the Egyptians may have known about it more than a thousand years earlier. It can also be expressed as: (1+the square root of 5, all divided by 2) to 1.

This ratio produces an order of such seemingly great intelligence that it was considered sacred by those who understood it. There are countless examples of designs in nature that contain golden ratio proportions including nautilus shells, meadow daisies, pinecones, sunflowers, and the human body. A rectangle featuring the golden ratio, with a width of 1.618 and a height of 1 has been shown using psychological tests to be preferred by humans over all other rectangles. So although much variety can be found in nature, there is still a dominant proportional ratio that keeps showing up in many of nature’s designs; the golden ratio.

Mathematically, the golden ratio produces a logically ordered progressive relationship between two parts such that the proportion of the larger part to the smaller part is the same as the whole to the larger part. In other words, B is to A as A+B is to B. The golden ratio is the only type of ratio whereby the whole and the part relate to each other in the same harmonious way.

An important aspect of the golden ratio – one in which the mathematical and philosophical aspects are intrinsically connected – is that it has an integrated relationship with unity, which is mathematically represented by the number 1. The reciprocal is the opposite of a number, while the square is the self-multiplication of a number. The reciprocal of the golden ratio, 1/1.618 equals .618. The square of the golden ration, 1.618X1.618 equals 2.618.

Therefore the golden ratio minus its reciprocal (1.618-.618) equals 1 and the square of the golden ratio minus the golden ratio (2.618-1.618) also equals 1. In other words, the golden ratio stands in the center of its reciprocal and its square and is offset by a difference of unity, 1, in either direction. This is the only known number to have this property.

I, too, find something sacred about this harmonious ratio. It speaks to me of an intelligent designer, aka God, who created this world to have tremendous outward variety but also to have a profound inner unity to reflect his own unified, trinitarian nature. It’s also fascinating to me how the realms of mathematics and art can intersect on the basis of this golden ratio.

I got my information for this post from the book Classical Drawing Atelier, by Juliette Aristides, Watson-Guptill Publications, New York, 2006, pages 20-22 and from Wikipedia. Oh and by the way, the work of art that I refer to in the beginning of this post was a quilt. If you don’t think of quilting as an art form you haven’t been to any of the shows put on by the American Quilting Society.


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