Recommended reading: On Growth and Form
by D'Arcy Wentworth Thompson (Author),
John Tyler Bonner (Editor)
available at Amazon
The Fabulous Fibonacci Numbers
by Alfred S. Posamentier and Ingmar Lehmann
available at Amazon
AL's Sacred Geometry Corner by Alan S. Glassman Third in our popular series of articles of interest to anyone involved in the building professions. (View Part 1 of the Series or view Part 2 of the Series). The relevance of geometry is obvious to those involved in the planning, design and construction of our buildings and infrastructure. Sacred Geometry is still little understood and here, Part 3 continues to demystify this fascinating topic.
The Basics - Part 3: Meaning in Shapes and Forms --
Nature Displays Phi, Fibonacci, and Five
Leonardo Bigollo Fibonacci, also known as Leonard da Piza (Pisano), traveled to Algiers as a young man and brought back to Europe the use of Arabic numerals to replace the cumbersome Roman numeral system. From geometers in Algiers he also learned about various additive series of numbers which are generated by the Golden Ratio of phi.
There are many sequences of whole numbers that are called additive series (progressive numbers that are generated by adding the first and second numbers to arrive at the third number), but only one such sequence is classified by what is now known as the Fibonacci Series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, and so on. The interesting thing is that when we calculate the ratio between each number and the one just before it in the series, the higher we go the closer we arrive to the numeric value of phi. For instance, 5/3 = 1.666666 ; 55/34 = 1.617647; and 987/610 = 1.6180327.
Generating a Fibonacci Series Logarithmic Spiral
From the Fibonacci succession of numbers, we can geometrically produce a particular kind of spiral. One way to do this is by nesting ever increasing sizes of 90 degree rotated rectangles divided with squares according to the phi ratio, and then connecting the points where the squares divide the sides of the rectangles into Golden Ratios with curved lines that trace one-quarter of a circle. The resultant areas of the squares are in the sequence of the Fibonacci Series of numbers, as shown below:
Fibonacci Spirals in Nature
Now, it just so happens that we see such a Fibonacci Spiral repeated time and time again in Nature. In fact, while symmetry is often seen in our world (for example, we have two symmetrically placed arms, legs, eyes, ears, nostrils, etc.), there are actually more examples of the asymmetry of the Fibonacci Series. The problem is that we've forgotten to look for them.
The Fibonacci Series is represented in the structure of many living organisms and physical processes, such as the distribution of leaves on a plant, spiral seashells, all flowers having 5 petals, the bottom pads on the foot of a cat, the planetary orbit of Venus as seen from Earth, the distribution of seeds on a cactus and on a sunflower, changes in the radiation of energy, the ratio of male to female bees in a hive, the DNA molecule, the surface pattern of pine cones, multiple reflections of light through mirrors, the cochlea of the human ear, the breeding pattern of rabbits, spiral galaxies, the path of certain predatory birds flying down to snatch their prey, and much more. Here are pictures of some of these examples:
Nautilus Sea Shell in Cross-Section
(PhotoSources: www.iadb.org and
Orbit of the Planet Venus Viewed Over Time from Earth
DNA Spiral with Phi Relationships
Human Ear Showing Cochlea Spiral
Galaxy with Fibonacci Spiral
Human Arm and Hand Showing Fibonacci Sequence of Joints
The Number 5 in Nature
You will recall from last month's AL's Corner that the number 5 had a direct relationship to phi. We showed that the ratio between a diagonal and a side of a regular pentagon equals phi and that by connecting all the pentagon's corners we can inscribe a pentagram star within the pentagon shape. In the figure below, you'll notice that inside the pentagram is another, smaller pentagon, and within that, too, we can inscribe a still smaller pentagram. This can be done indefinitely going up the scale in size (macrocosm) and down the scale in size (microcosm).
A fascinating characteristic of the lengths of the lines used to make the pentagon/pentagram figure(s) is that, as we proceed from larger to the next smaller line (a to b, b to c, c to d, etc.), the ratio of line lengths to each other is exactly the value of phi.
Visually, this figure begins to resemble a flower. That is to be expected, for in Nature we see the number 5 and the subsequent Fibonacci Series played out in many different kinds of flowers. Perhaps the most beautiful is the rose.
Rose with Petals in a Fibonacci Series Spiral
All edible fruit-bearing plants have flowers with their number of petals based on 5, while daisies have a number of petals always equal to one of the numbers in the Fibonacci Series.
Daisy with Fibonacci Series Number of Petals
(Photo Source: www.bloggers.it)
The internal shape of edible fruit itself is often based upon the number 5 and the Fibonacci Series. For instance, an apple core has 5 seed pod segments that make almost a perfect pentagram.
(Photo Source: blog.sciencenews.org/)
(Photo Source: www.pollinator.com)
Inventor Copies Nature's Efficiency
In an article by Kelpie Wilson, Environmental Editor for Truthout, which appeared online Monday, October 29, 2007, he discussed the annual Bioneers conference held earlier that month in San Raphael, California. In the article, Wilson highlights an invention he saw that was made in the image of the Fibonacci Spiral. He says:
Inventor Jay Harman presented a family of designs for fans and impellers based on the natural spirals found in seashells and blossoms. My favorite was a mixer for giant municipal water tanks. When water is stored for long periods, it can stagnate and become unhealthy. Harman's mixer is tiny, barely bigger than my fist. Turning in the middle of the tank, nothing much happens at first, but over time it sets up a natural vortex in the tank that keeps the water circulating and fresh. Harman said even if you stop the mixer, the vortex will keep on spinning for days.
Inventor Jay Harman's water tank mixer,
based on the natural spirals found in seashells and flower blossoms.
(Photo: Pax Scientific)
Next month, we'll discuss the contributions of Pythagoras as we continue with Part 4 of AL's Sacred Geometry Corner -- The Basics.