The Universal Grand Design

Fibonacci’s Golden Mathematical Ratio

November 2, 2021 by

Observers of the natural world often notice the geometric spiral, the amazing design defined by the Fibonacci sequence. This spiral sequence forms the base of highly diverse structures throughout the natural world. Spirals are found in things as small as unicellular vorticella spiralis, but also stretch in size to spiraling cosmic galaxies, millions of light-years across. I believe this is no coincidence. It is the signature of the Master Designer – his “fingerprint” on creation.

The Pinwheel Galaxy is the stunning face-on example of a cosmic spiral:

Messier Pinwheel Galaxy. Image credit: George Barth.

Leonardo Fibonacci (surname means “son of Bonacci”) was an Italian mathematician of the thirteenth century credited with discovering the mathematical sequence named after him and its relationship to the golden ratio or the golden rectangle. When you take any two adjacent numbers in the Fibonacci sequence and divide the larger number by the smaller, you get a number close to 1.618, also known as the golden ratio. Herein lies the relationship between the sequence and the proportions of the golden rectangle.

2 NativeartspiralNative American Glyphs: To the Native Americans of the South West spirals were a symbol of water and wind, also life’s journey to the center.

Even hidden within art, architecture, and design this spiral and ratio appeals to the human eye. The spiral has profound significance in many indigenous and prehistoric cultures; it is inscribed in Native American petroglyphs and cave art. The architects of the Great Pyramid of Egypt used the ratio of the slant height to half the base dimension, which is exactly the golden ratio of 1.618! Balanced and beautiful, it has also influenced the works of the greatest artists of recent centuries. It is said to define the facial proportions in paintings like the famous Mona Lisa.

The extraordinary spiral sequence that Fibonacci described is defined by a simple function from natural numbers. One does not need calculus to understand it. It starts at the center, circling outward and magnifying by a very orderly numeric pattern. At its mathematical origin it starts like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc. Think how simple it is to add the two previous numbers in the sequence to find the next number in the sequence. (0 + 1 = 1; 1 + 1 = 2; 1 + 2 = 3; 2 + 3 = 5 . . .) Each number creates a square (sided by its own integer) that is rotated ninety degrees and added to the area of the previous rectangle, formed by the addition of the previous smaller squares – and presto, the golden rectangle is perpetuated. If you swing an arc, as with a compass, from the central corner of each square the ascending spiral grows.

3 graph a = 34, b = 21, a + b = 55 (34 / 21 = 1.618)

The spiral appears not only in art, music, engineering, and architecture, but also in the harmonies and the designs of nature. Some have described it as nature’s code and language, or the divine proportion.

I have coined it “the Fibo effect” and I love to discover it in yet new places in the natural world.

spiderwebIridescent spiderweb spiral

grape tendril Grapevine spiral tendril


Once you start looking, spirals are everywhere. In the gentle curves of the fern frond you can see the like spiral form of a curled elephant trunk or an octopus tentacle.

One of the most radiant examples of the Fibo effect is the sunflower. Its leaves spiral around the main stem for maximum exposure to sunlight, but its amazing compact seed configuration is full of complimenting and opposite spiral patterns. I love this example of the golden ratio.

5 sunflowercollage

The human body contains many examples of this spiral. The very strands of DNA genetic code form a helical spiral. The outer ear is patterned after the golden ratio; the cochlea is the spiral cavity of the inner ear that helps capture sound so we can hear the nuances of music. The whorl of hair on my grandson’s scalp is a spiral. Absolutely unique to every human being are fingerprints, the distinctive pattern of each person’s individuality. A trace of your own spiral whorl could convict or exonerate you.

6 hairwhorlHair whorl on author’s grandson

7 fingerprintAuthor’s fingerprint spiral

8 violinscrollcropinsunlightIt is no accident that the Fibonacci spiral appears on the scroll of a violin as well as on the spiral shaped F holes on its face. The spiral scroll is beautiful and functional. It is placed at the end with the tuning pegs to enrich the sounds the listener hears. Antonio Stradivari used the golden ratio and the Fibonacci spiral to create his masterpiece violins in the seventeenth and eighteenth centuries, worth millions today. Mozart’s sonatas are said to use the golden ratio. The violin is just one example of the spiral design melding math, beauty, tone, and resonance to create music that pleases. In Italian, bonaccia means “calm,” an interesting linguistic connection to the serene effect of the graceful spiral in sight and sound.

To me the most classic natural spirals are found in the marine world. The astounding beauty of the spiral takes thousands of variations in shells and fossils, each more beautiful than the last.

9 shellscollage Shell collage in order: Lightning Whelk, Florida Shell, Spiral shell, Ammonite fossils.

nautilusThe classic nautilus

spiral shell Even the mechanics of the human hand fit the spiral

11 snailsquareRight in my backyard with my grandsons I have discovered the Fibo effect – in snails, spiderwebs, fern fiddleheads, grapevine tendrils, curled grasses, and pinecones. It also can be seen in a butterfly proboscis, the strong and beautiful curve of a hen’s egg, and even in monarch caterpillars hanging in a jay.

a spiderweb Spiderwebs

b LeafspiralMG Grasses

c fiddleheadsFiddleheads

d jay Monarch caterpillar jay

e butterflyButterfly probiscus

f pinecone Pinecone spirals

Ancient cultures around the globe used the spiral as a symbol of spiritual significance and it is thought to symbolize origin, creation, and growth. The Celts carved spirals into stone megaliths in hundreds of burial chambers, possibly symbolizing reincarnation and spiritual immortality. In the Jewish tradition the spiral shofar (ram’s horn) was blown at ritual ceremonies, including the anointing of a king. In some Christian traditions the spiral labyrinth is symbolic of life’s journey from birth to death with the source and center being God.

13 labrynth Spiral labyrinth

Some of the most awe-inspiring spirals in nature are phenomena such as whirlpools, tornados, hurricanes, and typhoons. Powerful forces are at work like gravitational pulls, the centrifugal spin of the earth, and the resulting Coriolis effect. Look at any NASA stream of a spinning hurricane and the swirling Fibonacci spiral is apparent. But now this spiral is a thousand miles across, spawned by an intense internal low pressure, generating destructive windspeeds of over two hundred miles per hour. Oceanic currents and vortexes are no less powerful swirling across the oceans.


Then taking a cosmic leap, our own Milky Way galaxy is a Fibonacci spiral over one hundred thousand lightyears in diameter, made of hundreds of billions of stars! Our sun and its tiny earth lie on but one spiral branch known as the Orion Arm. Here is perhaps the greatest and most wonderful example of this divine design. Fibonacci is credited with describing this amazing sequence, mapping it with numbers and connecting its vast variety of manifestations in the world that we know. But who is the author of this incredible sequence? Who is the Master Designer, and why is this divine proportion woven into the tapestry of his master plan? I believe this spiral is his fingerprint and his signature on this, his wonderful creation!

14 JamesKurtzOriginalArtscanMWayGalaxy Artist’s conception of our Milky Way Galaxy. Airbrush painting by James Kurtz.

I consider myself an amateur observer, not a scientist, and any assumptions are my own. I included many original photographs, my own and from several friends, to showcase the spiral in just some of its myriad natural forms. Thank you, Georg Barth for your galaxy photograph and James Kurtz, for the original artwork. Thank you Mary Greenwood, Britta Wareham, Ivan Menz, Lisa Wipf, Pete Meier and John Henry Menz for contributing your photographs and enthusiastic support.


About the author

Mario Meier

Mario Meier

Mario Meier and his wife, Robin, are parents of four grown children, and grandparents of three.

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