Mathematics helps us discover nature
Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the
arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the
skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and
the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the
golden ratio operating as a universal law. Zeising wrote in 1854:
The Golden Ratio is a universal law in which is contained the ground-principle of all formative striving for beauty
and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal,
all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical;
which finds its fullest realization, however, in the human form.
Many flowers offer a beautiful confirmation of the Fibonacci mystique. A daisy has a central core consisting of tiny florets arranged in opposing spirals. There are usually 21 going to the left and 34 to the right. A mountain aster may have 13 spirals to the left and 21 to the right.Pine cones are also constructed in a spiral fashion, small ones having commonly with 8 spirals one way and 13 the other. The most interesting is the pineapple - built from adjacent hexagons, three kinds of spirals appear in three dimensions. There are 8 to the right, 13 to the left, and 21 vertically - a Fibonacci triple.
Plants do not know about this sequence - they just grow in the most efficient ways. Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and fir cones also show the numbers, as do daisies and sunflowers. Sunflowers can contain the number 89, or even 144. Many other plants, such as succulents, also show the numbers. Some coniferous trees show these numbers in the bumps on their trunks. And palm trees show the numbers in the rings on their trunks.
Both the Fibonacci Sequence and the Golden Ratio appear in natural forms ranging from the geometry
of the DNA molecule (and the human body) to the physiology of plants and animals. In the past few years, science
has taken
a quantum leap in knowledge concerning the universal appearance and fundamental importance of Fibonacci mathematics.
Some of history's greatest minds, from Pythagoras to Isaac Newton, have held Golden Section and the Fibonacci
sequence in the highest esteem and reverence.
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Fibonacci numbers and the Pascal triangle
Rasko Jovanovic provides this website on Fibonacci numbers and
Pascal's triangle. Information is provided on the history of Pascal's
triangle; the relationships between Pascal's triangle and Fibonacci
numbers, Lucas numbers, Catalan numbers, Fibonacci polynomials, and
Chebyshev polynomials; the Titius-Bode law; and Golden sections and
their relationships with Fibonacci numbers and Lucas numbers. The site
also includes a calculator for Fibonacci and other numbers. This site is
available in English, German and Serbian.
Fibonacci in Nature:
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Johannes Kepler (1571-1630):
"Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel."
Stan Gris:
"The Fibonacci numbers are Nature's numbering
system. They appear everywhere in Nature, from the leaf arrangement in plants,
to the pattern of the florets of a flower, the bracts of a pinecone, or the scales
of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every
living thing, including a single cell, a grain of wheat, a hive of bees, and even all
of mankind."
Frederick A Hottes:
"All human senses, including hearing, touch, taste, vision and pain receptors, have not only spiral physiology, but also response curves that are logarithmic (having a fibonacci structure). Cellular action membrane potentials, which are important for muscles and nervous system, have a voltage equal to the log of the ratio of the ion concentration outside the cell to that of inside the cell. The brain and nervous systems are made from the same type of cellular building units and look similar microscopically, so the response curve of the central nervous system is probably also logarithmic."
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