The Golden Ratio, Fibonacci, & Nature’s Spirals – 5-28-24

 

The real world is beautiful to our senses, mathematics is beautiful to our thinking, and each type of apprehension enhances the beauty of the other – the whole is greater than the sum of its parts.

 

The Pentagram, Golden Ratio, Fibonacci Numbers, and Spirals

The same series of numbers creates hundreds of spirals found in the Cosmos and in Earthly Nature – including the spirals of galaxies, pinecones, snail shells, and spiral staircases.

The homepage of our website, www.ContinuingCreation.org, begins with photographs of spiral of galaxies, pinecones, snail shells, and staircases. Here they are again:

 

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The Spiral Pattern is related to a special number called the Golden Ratio, and to a unique string of numbers called the Fibonacci Sequence.

 

The Golden Ratio, (or Golden Mean, or Golden Proportion)

The Golden Ratio is the number 1.6180339887498948482…  (The use of three dots conventionally indicates that the series goes on forever.).  Mathematicians named this number after the Greek letter “Phi,” (pronounced “Fie”) which the ancient Greeks wrote as a small oval with a vertical line drawn through it — ϕ.

The simplest way to calculate the Golden Ratio is to take any straight line and divide it into two segments in a certain way. Here’s a drawing of this “line-dividing method,” plus a restatement of the line concept using a bit of simple algebra: we define the golden ratio as the number we get when we divide a line into two parts so that:

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The “Golden Ratio” – An Ancient Greek Ideal of Beautiful Proportion

For centuries, artists and architects have used the Golden Ratio segments to define the proportions of their work because they seem to be aesthetically pleasing to the human eye and mind.  This use has been called the “divine proportion.”

A rectangle with its sides in this ‘golden’ ratio is thought of as aesthetically pleasing.  So, a painting 34 inches wide would often be made 20 7/8 inches high, because 34 / 21 = 1.619, which is about equal to Phi.

The architecture of the hundreds of thousands of buildings, old and new, including the Parthenon in Athens, Greece, incorporate the Golden Ratio.  In the Parthenon, the height of the lower columned façade is 1.618 times the height of the face (the “pediment”) above the columns.  

 

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For centuries, artists and architects have used the Golden Ratio segments to define the proportions of their work because they seem to be aesthetically pleasing to the human eye and mind.  This use has been called the “divine proportion.”

A rectangle with its sides in this ‘golden’ ratio is thought of as aesthetically pleasing.  So, a painting 34 inches wide would often be made 20 7/8 inches high, because 34 / 21 = 1.619, which is about equal to Phi.

The architecture of the hundreds of thousands of buildings, old and new, including the Parthenon in Athens, Greece, incorporate the Golden Ratio.  In the Parthenon, the height of the lower columned façade is 1.618 times the height of the face (the “pediment”) above the columns. 

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Note:  For more information about the intersection of art and mathematics, including beautiful illustrations, search online for the Wikipedia article, Mathematics and Art.

The Golden Ratio and the Fibonacci Sequence

In 1202, an Italian mathematician named Fibonacci (short for “Son of Bonacci”) wrote the Liber Abaci (“Book of Calculation”). In this book, Fibonacci described the idealized pattern of population growth using a sequence of numbers we now call the Fibonacci Sequence. In the Fibonacci Sequence, each number is the sum of the previous two numbers.  Here is the sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610… and so on.

We see that in this sequence, 8 + 13 = the next Fibonacci number after 13, which is 21.  Then, 13 + 21 yields the next number in the Fibonacci sequence, which is 34.  And so on.

Centuries later, in the late 1700’s, Johannes Kepler discovered that when you take any two adjacent numbers in the Fibonacci Sequence, and divide the bigger one by the smaller, you get an approximation of the Golden Ratio!  And if you do this using neighbor-numbers further and further out on the Sequence, the approximation of Phi becomes more and more exact.  

Thus, using Fibonacci numbers 13 / 8 = 1.625000. Using more distant Fibonacci numbers, e.g., 610 / 377 = 1.618037135 (awfully close to Phi, which is 1.618039887…).

Therefore, the Golden Mean (phi) is connected to the Fibonacci Number Sequence. This is one of the many relationships in mathematics that are amazing and almost magical when they are first discovered.

Spiral Patterns Come from The Golden Ratio, and Fibonacci Numbers

We can use the Fibonacci numbers to draw spirals if we use each number in the sequence to make a square. So, the “size-21-square” would be 21 inches (or centimeters, etc.) on a side. Next, we set these squares next to each other in a pattern like this:

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We add new squares by moving in a counterclockwise direction. We place the squares so that each square touches the smaller square we drew in just before it. The figure above shows how to do it.

Then, if we draw an arc through the opposite corners of the 3-square, and then through the opposite corners of the 5-square, the 8-square, the 13-square, and so on – we will have drawn a spiral. Specifically, a “Golden Ratio” or “Fibonacci” Spiral.

 

The Fibonacci Spiral appears over and over in the natural world – ranging from impossibly massive spiral galaxies to the small spirals of rose blossoms, pinecones, and snail shells.

 

In fact, in pinecones, sunflowers, and other plants, there are two Fibonacci Spirals – one clockwise and one counter-clockwise — interweaving with each other. This pattern is shown in the photograph of sunflower, below:

 

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Flower petals mostly occur in counts equal to Fibonacci numbers…There are daisy varieties with 8, 13, 21, and even 55 (all Fibonacci numbers) petals. Why? Because there is a little segment in the genetic code of the daisy that tells the emerging flower to generate flower petals using a Fibonacci-like counter 

 

Clearly, there is something about the forces, materials, and/or processes of Nature’s Continuing Creation that incorporates this spiral pattern. It even shows up in the crystals of certain aluminum alloys. 7  Is mathematics somehow behind Nature’s Continuing Creation?  

 

(First published online, with sources footnoted, at Mathematics Evolves in Nature’s Continuing Creation – Continuing Creation %.)