the Last Supper. There is a vast dodecahedron behind the central figures and its edges are in golden ratio to one another.
Over and above the works of these artists and architects, the psychology of the golden mean attracted the attention of Gustav Theodor Fechner (1801–1887), the pioneering German experimental psychologist. Fechner wanted to find out whether the Golden Section was correlated with the human ideas of what constituted beauty. His research concluded that there was a distinct preference for rectangles that were built on the golden mean.
Musicians and advanced music theorists like James Tenney (1934–2006) applied the numerical theories of the golden mean and the Fibonacci series to their musical compositions with striking results. Musicologist Roy Howat, who is also an excellent pianist, has found musical pieces that correspond to the golden mean.
The golden mean and the Fibonacci series are among the most intriguing mysteries of mathematics and numerology. They even overflow into the flora and fauna of the Earth’s biosphere, which are the subject of the next chapter. Nature, it seems, is the product of special numbers and special numerical sequences.
5
Numbers in Flora and Fauna
One of the strangest things about the power of numbers, as disclosed by both scientific mathematics and the mysteries of numerology, is their persistence throughout the biosphere. Animals, plants, fungi, and insects — in fact, all life forms — are surprisingly and persistently numerical. Why is there this persistent correlation between numbers and the multitude of organisms, from whales to bacteria, that fill the biosphere along with us? The component parts of most of these living things have inexplicable numerical sequences. They often line up with the Fibonacci series and with ф. What have life and mathematics and numerology to do with one another? It all seems to make numerology far more viable and significant. It raises the major question, yet again: is the mind of the Creator the mind of a mathematician?
Adolf Zeising (1810–1876) was a remarkable German psychologist, mathematician, and philosopher, who was convinced that he could trace the mathematical principles of ф and the golden ratio in plant stems and the arrangements of veins in their leaves. He pursued his research into animal skeleton formations and other zoological details such as the cardiovascular system, arteries, veins, neurones, and the lymph system. He also looked into the shapes and relative sizes of crystals, and he applied his favourite mathematical theories to chemical compounds. Wherever Zeising looked, he found indications of ф. For him, the golden ratio was a kind of universal law that he claimed he could detect in every phenomenon in the universe. Zeising didn’t just find significant number series in living organisms — he found them everywhere. In his view, the Fibonacci mystery permeated everything: organic and inorganic, acoustic and optical, forms and structures, movements and perspectives — and it found its ultimate perfection in human beings. Zeising can be thought of as the ultimate numerologist in terms of his belief in the power and ubiquity of numbers and number patterns. A great many examples are there to be examined.
Take the case of a hardy, drought-tolerant plant, usually called by the popular name of sneezewort (achillea ptarmica). It is also known as “fair-maid-of-France,” “white tansy,” “wild pellitory,” and “goose tongue.” It blooms through June until August and provides the researcher with a particularly clear example of the Fibonacci series of numbers. The sneezewort produces its growing points in the Fibonacci sequence, and these can clearly be observed: none to start with, then 1, then another, then 2, 3 … and so on.
Daisy petals
In addition to the Fibonacci numbers of growing points on plants, such as the sneezewort, the series occurs again when petals are counted.
Daisies can have as many as 89 petals, although 55 and 34 are more usual totals. All 3 of these totals are Fibonacci numbers. Buttercups are members of the ranunculaceae family — an interesting name that comes from the Latin for “little frog,” which has about 1,700 species worldwide. Buttercups produce 5 petals, “5” being a member of the Fibonacci series, and so do pinks, larkspurs, columbines, and wild roses. This fundamental link between plants and the Fibonacci numbers continues powerfully with many additional examples. The iris and the lily have 3 petals. Delphiniums have 8. Asters have 21 and corn marigolds have 13, as do cineraria. When the Lucas numbers are accepted as a parallel series, very similar in principle to the Fibonacci numbers, the fuchsia can be included with its 4 petals because 4 fits the Lucas series: 2,1,3,4,7,11.… Asters and chicory have 21 petals; plantains and pyrethrum have 34. Rose petal formations also comply with the Fibonacci numbers.
The technical botanical term phyllotaxis, also rendered as phyllotaxy, refers to the way that leaves are arranged on the stem of a plant. The word was coined in 1754 by a Swiss naturalist named Charles Bonnet (1720–1793). He derived it from 2 ancient Greek terms: phyllon, meaning a leaf, and taxis, meaning the way that things are arranged, set out, and displayed. A few years later, other naturalists discovered that each new leaf on a plant is set at a particular angle (137.5 degrees) from the one that preceded it. This is called the “angle of divergence.” It can also be described as the fraction of a circle differentiating a new leaf from its immediate predecessor. When the 137.5-degree angle is calculated, it turns out to be in the ratio of 1÷ф, and this leads straight back to our Fibonacci series and the golden mean: 1.618033 and its reciprocal 0.618033.
The spirals in a pinecone are designed according to Fibonacci numbers, and sunflowers are especially geared to the series. Some have 34 spirals, some have 55, and others have as many as 89.
Other Fibonacci phenomena can be seen when studying broccoli and cauliflowers. When the spirals are counted on a ripe cauliflower, the Fibonacci number “5” can be detected. There are 13 spirals (another Fibonacci number) on Romanesque broccoli. A slice through a banana reveals 3 sections; a slice through an apple shows 5. The Fibonacci numbers are there again.
The echinocactus grusonii cactus fits in well with the Lucas series and displays 29 ribs to the world. Best known by its popular name of the “golden ball,” or “golden barrel,” it grows in central Mexico.
These Lucas- and Fibonacci-based numbers in living things, however, are not confined to plant life such as cacti, cauliflowers, and broccoli — far from it. Other examples are found in natural spirals, such as those of snails.
Fibonacci and snail's shell
Theoretically, there is no limit to the size of this spiral. It could fill anything from a galaxy to a garden trowel. Each new square will have a side that is the same length as the 2 previous sides added to each other — the construction technique for the Fibonacci series. If we now consider the importance of ф and apply it to the spiral constructed from the Fibonacci-sided squares, it becomes clear that every quarter-turn of the spiral has rotated in such a way that each line drawn from the spiral to its centre will be approximately 1.618 times longer than its predecessor. The humble little garden snail is not alone in possessing a Fibonacci-based spiral shell: a great many other shells exhibit it both on land and sea. There has been some controversy between naturalists, mathematicians, and numerologists over the exactness of this coordination between spiral shell measurements and the Fibonacci series. Nautilus seashells seem to vary between about 1.9 and 1.6, nevertheless they seem to come close enough to ф to be worth serious consideration. But how powerfully, and how consistently, does nature conform to the Fibonacci series and the mysterious golden mean? And what does that conformity imply about mathematical structures and its mysterious first cousin, numerology?
The cochlea of the human ear also forms a Fibonacci spiral. From the complexity of these spirals to the simple but equally impressive zoological example that can be found in the human arm, the whole question becomes closer and more familiar. Begin by considering the humerus bone: just one single bone from shoulder to elbow. At the elbow, 2 bones begin: the ulna and the radius. The hand has 5 digits, and each digit possesses 3 small bones. The arm clearly illustrates the Fibonacci series: