The Fibonacci Sequence

My painting Maler/Bilder is composed of panels joined to a wood grid on the back. The ratio of the height to the length uses something known as the divine proportion. In fact, the panels beginning with the smallest are all squares and representative of a sequence of gnomic growth called the Fibonacci Numbers. This growth sequence is found in nature in the nautilus shell, pine cones, sunflowers, spiral galaxies, etc. You can see a diagram of the sequence, at right.

Leonardo of Pisa, known as Fibonacci, was a 13th-century scholar who created the series of numbers given his name. The ratio created by any number in Fibonacci Sequence to the next larger number approximates a proportion also known as the divine proportion or the golden section. The divine proportion is a ratio approximately 1:1.6. As a rectangle, it would be the ratio of height to length.

In classical Greece, this ratio was the basis for many design elements found in art and architecture. The Parthenon’s facade has a height-to-length relationship using this ratio.

Fibonacci found that a sequence of numbers, if carried on indefinitely, would approach this same ratio and that it would become more exact the further one carried the sequence. As the sequence progresses, each new number is the sum of the previous two numbers. Thus, the sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc.

Applying this sequence to geometry and the area of a rectangle, start with a square of any size and call it #1. Using a grid based upon this unit, start building. Add an equal square next to #1 (1+1). This square is "1" unit, too.

Now combine these two squares to create the sides and size of the next square. The next square’s side is equal to 1+1 or "2." Adding the length of "2" with the previous square "1." The next square’s side equals 3.

As this sequence increases in size, the proportion of the rectangle becomes closer or more refined to equal the ratio of "golden section." In nature, the growth of plants and animals is very similar. Think of seeds on a pine cone, sunflower or the familiar and exotic nautilus shell.

Cole Carothers earned his master of fine arts degree from American University and a certificate from Ecole des Beaux-Arts (Paris). His work is part of the permanent collections of the Cincinnati Art Museum, the RSM Company, and the University of Kentucky Art Museum.

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