THE FIBONACCI SEQUENCE
- Grady McMurtry
- December 12, 2020
God does ALL things decently and in an orderly manner. (1 Cor. 14:40) In creation, He places His “fingerprints” all over it to prove that He is the only One responsible. We may prove His existence and His continuing involvement in human history by using many different avenues of information. Whether we look at the Laws of science, the natural processes, the physical evidence, His control of history (notably in the dates and historical events of His Feasts - “The Feasts of the Old Testament” available in our Bookstore), logic, or mathematics we see Him as The Creator!
Two mathematical proofs of God’s existence are the Fibonacci Sequence of numbers, and The Golden Mean/Ratio. The mathematical value for the Ratio is accepted as 1.618. The Greek letter phi in lower case symbolizes the Golden Mean Ratio (φ).
The Sequence is the series of numbers generated by adding the sum of the two preceding numbers starting from 0 and 1 and going on forever. Initially, the series looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, …. [0+1=1; 1+1=2; 1+2=3; 2+3=5; 3+5=8; 5+8=13; …]
The Golden Mean Ratio is generated by dividing each number in the Sequence by the previous number in the Sequence. For example: 89/55=1.618; 987/610=1.618; and 28657/17711=1.618. The reciprocal would look like this: 55/89=0.618; 610/987=0.618; and 17711/28657=0.618. These Ratios cannot be random artifacts. These numbers are obvious proof of God’s direct involvement!
Although the Sequence was known by the Ancient Greek mathematicians, and by mathematicians in India as early as the second century, it was Leonardo Bigollo Pisano (c. 1170 – c. 1250) from the Pisa region of Italy that made it truly famous. His nickname is Fibonacci (“Son of the Bonacci Family”). He is considered the most talented Western mathematician of the Middle Ages. In his book, Liber Abaci [Book of Calculation] (1202), he was responsible for changing the mathematics of Europe from the awkward Roman numeric system to the streamlined Hindu numeric system.
The introduction of his numeric system revolutionized commercial bookkeeping and banking, the conversion of weights, money, and measurements. Fibonacci wanted to solve the problem of how rabbit populations would grow under ideal conditions. The Fibonacci Sequence described the result accurately. Fibonacci Numbers are found so often throughout nature, mathematics, and manmade objects that a quarterly journal is dedicated to their study.
The modern application of Fibonacci numbers includes computer algorithms, search engines, and graphs used for interconnecting parallel and distributed systems. It is in biological and architectural systems, however, that we find our greatest interest. In science we find the Sequence to be ubiquitous.
In Biology, the Sequence may be seen in the branching of trees, the arrangement of leaves on a stem, the uncurling of spiral fern leaves follows the Golden Mean Ratio, the arrangement of bracts on a pinecone, the arrangements of petals and pistils on flowering plants, spiral shells (Chambered Nautilus and snails), the spirals of hurricanes and tornadoes, the spirals of galaxies, the curl of a ram’s horn, and the spiral tail of a Seahorse.
The arrangement of seeds on a Sunflower are formed by two spirals, each going in the opposite direction to the other. The seeds form a Golden Mean Ratio. The same is true of the seed arrangement in species of Daisies: the combination of counterclockwise and clockwise spirals are counts of 21 and 34. The needles of pine trees are arranged in fascicles of 2, 3 or 5 - all are Fibonacci numbers. The bracts of pineapples are arranged in three spirals that number 5, 8, and 13 - Fibonacci numbers. Romanesco broccoli is an edible flower bud that would look like a fractal to some observers; however, the spiraling follows the rules for Fibonacci numbers.
The Golden Mean Ratio is seen in millions of places in nature.
The Golden Mean or the Golden Ratio is a Divine Proportion the beauty of which is seen as a Golden Spiral in living and non-living things. These Golden Spirals reveal God’s handiwork and interest in beauty, function, and order. We see it in the cochlea of the human ear, whirlpools, and the DNA molecule.
To make a visual representation of a Golden Spiral, we have to start with a Golden Rectangle. The Rectangle is formed by using the ratio for phi. Within the rectangle you can section off a perfect square. The resulting rectangle remaining has the same proportions as the original rectangle, but smaller. In the new smaller rectangle, you can section off another perfect square, and the remaining rectangle will have the same proportions as the original rectangles. Dividing the Golden Rectangle into perfect squares creates the blueprint for the Golden Spiral, a visual representation of the Golden Ratio. As the squares continue to get smaller, the Golden Spiral curve fills each square in the same ratio of space. The spiral can continue inward and outward, retaining the same proportions.
Perhaps the most famous of these Spirals in nature would be the shell of the Chambered Nautilus. As the shell gets larger, it retains its identifying form. Since the body of the organism grows in the path of a spiral that is equiangular and logarithmic, its form never changes.
The most pleasing human faces have the Golden Ratio shape. Truly it is a “fingerprint” of the Great Creator and Artist of the universe.
How do we see the function of this form in nature?
We may see it in the designed arrangement of petals around a flower or the arrangement of leaves around a plant stem. For examples with petal arrangement: a Lily has 3 petals, Yellow Violet 5, Delphinium 8, Mayweed 13, Aster 21, Pyrethrum 34, Helenium 55, and Michaelmas Daisy 89.
A similar observation may be made with the spiral arrangement of leaves as we look straight down the stem and note the arc of the stem from one leaf base to the next leaf base and note the fraction of the stem circumference which is inscribed. In an Elm the arc is 1/2 the circumference; in Beech and Hazel, 1/3; Apricot and Oak, 2/5; in Pear and Poplar, 3/8; in Almond and Pussy Willow, 5/13; and in some pines either 5/21 or 13/34.
What is the purpose of these arcs? These arc ratios assure that each leaf will receive a maximum exposure to sunlight and air without shading or crowding any other leaves. This is clearly not coincidence.
Of course, the information to make these spirals and position these petals and leaves is stored in the plant’s DNA. Is it any wonder then that the DNA molecule is based on the Golden Ratio! The DNA molecule is 21 angstroms in width and the length of one full turn in its spiral helix is 34 angstroms, both of them Fibonacci numbers. The DNA molecule, the most compact and dense information storage device in the universe, may be described literally as one long series of attached Golden Rectangles.
How do we see the function of this form in beauty?
This ratio has been found to be remarkably pleasing to the human eye. Ancient Greek sculptors like Phideas, Renaissance painters like Leonardo da Vinci, Impressionist painters like Monet - all used the Golden Ratio in their works of art. Indeed, all painters wanting to use dynamic symmetry which gives their work “life” use the Ratio.
We see it used widely in architecture and office supplies. The front of the Parthenon in Athens is constructed using the Golden ratio. What governs the shape of credit cards, playing cards, postcards, light switch plates, and writing pads? Why are 3-by-5 and 5-by-8 cards 3-by-5 and 5-by-8? They follow the Fibonacci Ratio.
Why should we study the Fibonacci Sequences? They are just one of many of the “Fingerprints” of God that He has inscribed on His creation so that we may know that He is God! “For since the creation of the world His invisible attributes, His eternal power and divine nature, have been clearly seen, being understood through what has been made, so that they [we] are without excuse.” Romans 1:20(NAS95)