The Fibonacci Sequence

1 • 1 • 2 • 3 • 5 • 8 • 13 • 21 • 34

1 • 1 • 2 • 3 • 5 • 8 • 13 • 21 • 34

I have been fascinated with The Fibonacci Sequence and The Golden Rectangle for some time. I finally got around to building a Fibonacci Gauge that was featured in WOOD Magazine.

The gauge maintains a constant proportion of 1:1.618 between the points. It is used to help determine visually appealing proportional dimensions. I am looking forward to using the gauge in future projects.

Follow the text below for some interesting history, a fun video from WOOD Magazine demonstrating the Fibonacci Gauge, and some online resources. This was a fun afternoon project which provided some much needed therapy and piece of mind!

MY FIBONACCI GAUGE

Constructed from thin cherry cut-offs from a prior project and finished with 2 coats of Tung Oil and 4 coats of lacquer rubbed out with wax. I found the solid brass binding posts in a little hardware store in Sisters, Oregon.

WHO WAS FIBONACCI?

Leonardo of Pisa, known as Fibonacci [pronounced fib-on-arch-ee], was the “greatest European mathematician of the middle ages”. His full name was Leonardo of Pisa, or Leonardo Pisano in Italian. He was born about 1175 AD in Pisa (Italy), the city with the famous Leaning Tower of Pisa. Pisa was an important commercial town in its day with links to many Mediterranean ports. Leonardo grew up with a North African education under the Moors. He traveled extensively around the Mediterranean coast meeting with many merchants learning their systems of doing arithmetic. He realized the many advantages of the “Hindu-Arabic” system over all the others. Fibonacci was one of the first people to introduce the Hindu-Arabic number system into Europe. This is the positional system we use today which is based on ten digits, a decimal point and a symbol for zero.

THE FIBONACCI SEQUENCE AND ITS RELATIONSHIP TO THE GOLDEN PROPORTION

The Golden Section, also called The Golden Ratio, The Golden Mean and The Divine Proportion was discovered by the Greek mathematician Pythagoras. Later, an Athenian architect using the Golden Section in building design came up with Phi, the number 1.618. Fibonacci made the next leap when he published a book in 1202 called “Liber Abaci”. He introduced a math problem where a pair of rabbits were placed in a field with the provision that they could not escape or die. At the age of 1 month the female gives birth to 2 new rabbits (1 male, 1 female). The female rabbit does this each month for 1 year. How many rabbits would there be at the end of the year? The answer to this question contains a series of numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55…..). This series of numbers is called the Fibonacci Series. If you look at the ratio that occurs after the number 3, you will see the number 1:618, which is the Golden Ratio.

Fibonacci devised a series of proportional relations [ 1 : 1 , 1 : 2 , 2 : 3 , 3 : 5 , 5 : 8 , 8 : 13 . . . ]. If you look closely, you will see the Fibonacci Sequence. This set of ratios, arrived at by adding the 2 previous numbers together to give the next number the new series, is been used in many aspects of life from architecture, finance, biology and engineering. Nestled in the Fibonacci series are the ratios 5:8 and 8:13 which are the classic “golden section” proportions.

WHAT DOES THIS HAVE TO DO WITH WOODWORKING?

The Golden Section or Fibonacci Numbers can be used to derive pleasing dimensions for any piece of furniture. For example, you have been commissioned to build a table for a client. You decide to use the Golden Rule to help determine construction dimensions that will be pleasing to the eye. The client requires the top to be 20 inches deep. To make the top into a Golden Rectangle, multiply 20 by 1.618 – the result is 32.36 inches. Rounding this to 32.5 yields a Golden rectangle measuring 20 inches x 32.5 inches. Interestingly, if you draw a square within your Golden Rectangle, the remaining rectangle will also be a Golden Rectangle. This principle can be used to scale all the other elements of the table. Ultimately, common sense and your eye should rule over the Golden Section. The Golden Section is a tool that you bring to the bench much like a finely tuned plane, razor sharp chisel or that special dovetail saw.

Construction of a Golden Rectangle

1. Construct a unit square.

2. Draw a line from the midpoint of one side to an opposite corner.

3. Use that line as the radius to draw an arc that defines the long dimension of the rectangle.

Some Interesting Online Fibonacci Resources

Fibonacci Gauge and How to Use It Woodworking Plan Featured in the November 2006 issue

YouTube - WOOD Magazine Fibonacci Gauge Demo

The Golden Mean Gauge

The Golden Number Grid

Fibonacci Numbers and The Golden Section in Art, Architecture and Music

UPDATE: 19 April 2008

For another review of proportional design check out Karson's well written blog entry Golden Mean, Golden Ratio in Constructing Items at Lumberjocks. Karson takes us thru a practical application of proportional design using The Golden Mean to dimension the parts for his Greene and Greene Bookcase entry for the Lumberjocks Bookcase Design Challenge.

For another review of proportional design check out Karson's well written blog entry Golden Mean, Golden Ratio in Constructing Items at Lumberjocks. Karson takes us thru a practical application of proportional design using The Golden Mean to dimension the parts for his Greene and Greene Bookcase entry for the Lumberjocks Bookcase Design Challenge.

Below is a short video clip posted on YouTube from WOOD Magazine demonstrating the Fibonacci Gauge in furniture design.

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