Polar Roses and Snakes on Cartesian Planes: The Mathematical Essence of Nature


Nature is scribed in the language of Mathematics: the science of numbers, quantitative information and space. Many of the hypnotizing shapes and aesthetically-pleasing patterns and instances of symmetry we encounter in the natural world are but mere overlays of some of man’s most well-known graphs atop nature’s favorite colors. The greatest part of this, though, is the functionality behind the form: These mathematical correspondences we find in the frailest flowers and the burliest trees exist not to impress onlookers but rather to yield maximum efficiency.

When speaking of math in nature, everything seems to turn to gold: the golden ratio (or the golden mean), the golden rectangle, the golden triangle, the golden angle and the golden spiral, to name a few. All of these terms are associated with the omnipresent Fibonacci sequence, beginning with two consecutive 1s and continued by summing the previous two numbers together to produce the next number: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. The golden ratio, calculated to be approximately 1.618, can be derived in a few ways — namely from an infinite series of fractions or square roots (see table?). There also exists a more geometrical manner of understanding the golden ratio: A given line segment of arbitrary length can be divided into two segments — a larger segment, A, and a smaller segment, B — such that the original line segment’s length is to the length of A as the length of A is to B (i.e., the ratio of lengths of the whole segment to A is proportional to that of A to B). This is precisely the manner in which the golden rectangle is constructed, and from that, the golden spiral. A similar geometric setup is used to calculate the golden angle: The circumference of one circle can be sectioned into two arcs such that the proportion between the smaller and larger arcs is equal to that between the larger arc and the entire circumference. The angle subtended by the smaller arc measures approximately 137.5 degrees — the golden angle. The correlation between these golden geometries and the Fibonacci sequence is quite a beautiful phenomenon: The ratios of consecutive Fibonacci numbers match up rather precisely with the approximation of the golden ratio after each iteration of its fractional derivation. These numbers and proportions are “golden” and “divine” because they are innate archetypes found around every corner of the natural world that allow nature to function with the utmost productivity.

One of the most charming examples of these golden mathematical overtones found in nature can be observed in the common sunflower. Its mesmerizing center — called the flower head or, more scientifically, pseudanthium — is dotted with disk florets that eventually mature into sunflower seeds, the “fruit” of the sunflower. These disk florets are arranged in a tight network of spirals — the quintessence of the golden angle. The florets are oriented at about 137.5 degrees from one another and the rotation, or turn ratio, is undeniably close to phi, the golden ratio. Here’s yet another gold medal on the sunflower: The number of leftward and rightward spirals are generally consecutive Fibonacci numbers, say 34 in one direction and 55 in the other. Some of the larger sunflowers sport 89 and 144 spirals of disk florets, respectively. The question remains: How, exactly, is this efficient? This neatly-packed alignment is the only way to arrange the disk flowers atop the flower head without leaving spaces between florets. Changing the angle by less than a degree either way would leave noticeable gaps, thus causing space to be wasted. The more disk florets on the sunflower, the more seeds the flower can produce and spread for floral reproduction. Beautifully efficient — that’s nature for you.

If you thought that was it, you were wholly mistaken. We haven’t even touched upon stems yet. Spirals — such as the golden spiral mentioned earlier — play a major role in phyllotaxis, the arrangement of leaves on plant stems. While the list of arrangement classifications covers a wide range of patterns, Fibonacci numbers still find their way into the mix. The fractions that describe the angle of windings of repeating spirals of leaves often involve consecutive Fibonacci numbers as numerators and denominators: 1/3 angle for beech and hazel, 2/3 for oak and apricot, 3/8 for sunflower and pear, 5/13 for willow and almond, etc. Such Fibonacci arrangements allow for optimal access to sunlight, dew moisture and rainwater for each individual leaf.

There are a slew of other mathematical concepts and elements that appear in nature, such as trigonometric waves — as the title of this article suggests — as well as other geometrical shapes and symmetries. Many flowers, for example, produce predictable numbers and formations of petals that follow numerical sequences (such as the Fibonacci sequence mentioned earlier) and the graphs of sine and cosine functions, respectively. The visual that first comes to mind of sinusoidal graphs is one of ever-oscillating waves that roll over and sweep under the x-axis of the Cartesian (rectangular) coordinate system, apparent in the lateral undulatory movement of snakes as they slither about the ground. Lateral undulation is the most common mode of locomotion for many fish, reptiles and amphibians, useful both terrestrially and aquatically. The snake’s body flexes left and right, and these flexed areas propagate in waves resembling sinusoids that push posteriorly against various contact points along the surface, causing a reactionary force that thrusts the snake into an overall forward motion. Undulatory locomotion — motion distinguished by wave-like movements — is one of the most effective types of motion as it allows for the movement of limbless creatures. However, take those familiar waves and transpose them into the polar coordinate system — in which points are defined not by perpendicular distances but rather by a radial distance and an accompanying angle — and they transform into gorgeous petal-like graphs reminiscent of some of nature’s beloved flowers. These mathematical roses, also known as rhodonea curves, have a specific number of petals based on the coefficient of the angle in their equations. Odd-numbered coefficients yield a petal count equal to that coefficient, while even-numbered coefficients yield a petal count double the coefficient. Actinomorphic flowers display a radial symmetry about their centers similar to that of polar roses about the coordinate system’s origin. There has been a shift from radial to bilateral symmetry (as in zygomorphic flowers) in the continually-changing evolutionary progression. This kind of symmetry can be analogized as symmetry about one of the axes of a rectangular coordinate system, much like in mirror images. Regardless, floral symmetry in general is essential for attracting insects and instigating pollination.

These few instances hardly scratch the surface of mathematical appearances in nature. The hypnotizing phenomenon of fractals — intricate patterns that remain mathematically similar at different scales — are found in all kinds of places, from snowflakes and frozen tree branches to river deltas and clouds — even in the Romanesco broccoli sitting on the kitchen counter. Simpler patterns are so common in nature that it’s easy to overlook them. Take honeycombs, for example: The waxen honeycombs of beehives are always equilateral hexagons so as to result in the most compact fit and economic use of both wax and labor. Even the bees follow the proven truths of mathematics! Math is everywhere, an inherent dialect that the entire universe — as chaotically beautiful as it may appear — seems to speak and that we, as humans, are just beginning to learn.

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