"Eadem mutata resurgo. Although changed, I rise again the same"

-- Jacques Bernoulli,Spira Mirabilis

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From the double helix structure of our internal DNA database to our ancestors in the galaxies, nature's patterns that define our lives are intertwined with the ubiquitous spiral form. The spiritual, aesthetic, and mathematical beauty of spirals has enthralled men for ages. In most cases, precise definitions of the mechanisms that create real world spirals evade us. However, mathematicians have been able to approximate the spiral's form through the logarithmic spiral, the Fibonacci spiral, and other mathematical models that are described in the Useful References section shown below. This section provides information on some of nature's spiral patterns.

The horns of a Dall sheep are structures that differ everywhere in size but not in shape. The overall shape remains unchanged as the creature grows. The cross section of the horn's older portion becomes a template for the newer growth. The spiral is apparently formed because the top part of the horn grows at a faster rate than the lower part. The annular rings are evident because the horn grows only part of the year.

The logarithmic spiral models Dall sheep horns and many other patterns in nature where the size of a spiral increases while its shape is unaltered with each successive curve, a property known as self similarity. Mathematicians suggest that logarithmic spirals are also found (to name a few) in elephant tusks, unicellular foraminifera, galaxies, the flight patterns of peregrine falcons, nerves of the cornea, tropical cyclones, and mollusk shells. Romanesco broccoli is a cauliflower exhibiting a logarithmic spiral. This vegetable is also a visually striking example of an approximate fractal in nature.

The most popular appearance of a logarithmic spiral is in the shell of the chambered nautilus. As the mollusk within grows, it builds larger and larger habitation chambers within its shell while the remaining smaller compartments become gas-chambers that are used to regulate vertical movement in the water column. The shape of the shell remains the same as the creature grows.

Much has been written about the spiritual, aesthetic, and mathematical characteristics of the Fibonacci sequence. Much of it speculative at best. But, there is no doubt that some of nature's spirals can be described by the Fibonacci spiral which models the phyllotactic patterns in leaf and flower arrangements in many plants. Phyllotaxis and the Fibanocci sequence are described in the Useful References section below.

Phyllotaxis is the study of the successive arrangement of radial parts of a growing plant, such as the leaves on a stem or the seeds in the head of a sunflower. Look carefully at the arrangement of the little florets in the head of a daisy or a sunflower. You can clearly see the florets arranged in left and right handed spiral patterns emerging from the center. Below the photograph is a mathematically generated graph of the floret arrangement.

Each new floret on the head of a daisy or sunflower is positioned at a constant angle of 137.5 degrees with respect to its preceeding floret. Through succeeding generations of florets that are positioned according to this rule, a spiral phyllotactic pattern is defined. In the case of sunflowers and daisies, it is remarkable that the spirals which can be traced through this phyllotactic pattern result from integers of the Fibonacci sequence which is described in the Useful References. Mathematicians call this spiral model the Fibonacci spiral.

Fibonacci spirals permit the maximum number of seeds on a flower's seed head, packed uniformly, with no crowding at the center and no bald patches at the edges. On a sunflower seed head, the individual seeds grow and the center of the seed head continues to add new seeds, pushing those at the periphery outwards. Following the Fibonacci spiral, as a seed head grows, seeds will always be packed uniformly, and with maximum compactness. In addition to daisies and sunflowers, a Fibonacci spiral is seen in pine cones, fir cones, and many other seed arrangements.

The Fibonacci spiral is also seen in many arrangements of leaves that grow along the stems of plants. The basic idea is that the position of each new leaf growth is about 222.5 degrees away from the previous one, because it provides, on average, the maximum exposure space for all the leaves. This angle is called the golden angle which is derived from the Fibonacci sequence.

Why do these arrangements occur? In the case of leaf arrangement, some of the cases may be related to maximizing the space for each leaf, or the average amount of light falling on each one. Even a tiny advantage would come to dominate, over many generations. In the case of close-packed leaves in cabbages and succulents the correct arrangement may be crucial for availability of space.

Fibonacci numbers appear as a by-product of deeper physical processes. The spirals and leaf arrangements are not perfect. The plant is responding to physical constraints, not to a mathematical rule or model as is often implied in the literature.