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Introduction

In today's busy world, we rarely take the time to admire the surrounding environment. Even today, many individuals remain oblivious to the fact that most objects are strictly related to a mathematical equation. One of the predominant shapes that never ceases to fascinates the human eye and stimulate interest is the spiral. It is the main feature shared among a pineapple, shells, the Milky Way and a mountain goat's horn.

Spirals are one of Nature's most brilliant creations and surprisingly, it is found almost everywhere. For example, the horns that adorn a mountain goat's head display an exellent example of natural helixes found in the animal kingdom. Examples can be derived from the field of astronomy as well, since we observe that the Andromeda galaxy is the nearest spiral galaxy found beyond our Milky Way. As for the world of biology, we often overlook the double helix structure of the DNA molecule as an overwhelming example of spirals. In the domain of art, it is surprising to realize that the Solomon R. Guggenheim Museum in New York city, designed by Frank Lloyd Weight, represents a spiral too. Finally, we are all aware of the fact that sea shells demonstrate a perfect example of spirals being used in the marine world.

In the following pages, we will therefore elaborate on the subject of spirals found in molluscan species. We will be using the computer program Maple exclusively, thus enabling us to represent various mathematical formulas while helping the reader visualize the situation with graphs. We will also include short biographies of two influential mathematicians who contributed to the progress of science and discovered interesting facts about spirals as found in shells.

Mollusks

Mollusks, like spirals, are everywhere. Wherever one goes mollusks are there. From the Pre-Cambrian era to the present mollusks have flourished. After six hundred million years, mollusks have succeeded where many species have failed. While there exist over one hundred thousand different species of mollusk, and new ones are discovered every day, they are essentially united spirals.

A mollusk can be described as a soft visceral mass covered by a fleshy mantle and often encased in a calcified exoskeleton. They, of course, receive their name from the latin word "mollis" describing their soft body. The phylum Mollusca is the second largest phylum in the world only surpassed by the phylum Arthropoda (containing insects and crustaceans). The phylum mollusca can be subdivide into classes defined by the specific anatomy of the various mollusks. Five of these classes will be briefly mentioned although there exist several others, as well as some which contain only extinct species.

The most abundant mollusks are snails. These are in fact gastropods (stomach-foot) and they, as their name implies, walk or slither around on their stomach. Of all the classes Gastropoda is the only one containing species which have adapted to land, fresh water and sea water. They eat by using a small rasping tongue to scrape food off surfaces. Like all mollusks they have no brain but a network of ganglia (nerve conglomerations). Their shells vary in size, color and in shape. However the spiral is always apparent.

The second largest class is Pelycipoda (hatchet-foot) or bivalves named for their two valves. The animals in this class, unlike all the others, are filter feeders. They do not have a radula, which is one of the unique characteristic features of a mollusk. The pelycipods feed by filtering water through a siphon and extracting food from the water. This form of nourishement limits them to inhabiting bodies of water.

Another class of mollusk is Cephalopoda (head-foot). These are the squids and octopuses. This is the most anatomically advanced class of the phylum. With their lightning-fast reflexes and their advanced vision they are among the most advanced invertebrates. While they do not have a brain, they have evolved a "semi-brain". This "semi-brain" consists of a bunch of ganglia. For the most part these creatures have no outer shell and move along with the help of a jet propulsion system. This allows them to hunt for larger pray and more active pray. There remains one family of cephalopods which has kept its shell: the nautiluses. These relics navigate through the seas controlling their buoyancy by removing or filling chambers with gas. During the Ordovician period the ancesters of this magnificent animal (such as ammonites) ruled the oceans.

The other classes are more obscure and less abundant. The scaphopods (tooth shells) resemble tiny elephant tusks. They bury themselves partly in the sandy or muddy bottom and produce a current of water by waving "arms." They are then able to retrieve the proper nutrients from the passing water flow. The class amphineura can be divided up into two subclasses: polyplacophora and aplacophora. These are the chitons and the semi-segmented wormlike mollusks. These are quite ancient species with a very basic anatomy. A chitons shell resembles that of a pillbug divided into eight plates while aplacophores are shelless.

Its important to note the formation of a shell. A shell is formed of calcium carbonate crystals combined with other substances, all extracted directly from the water. The shell is secreted in liquid form by special glands on the mollusk's mantle. The shell begins as a microscopic covering of the planktonic mollusk. The shell grows by forming whorls around a central axis thereby forming a spiral. While all shells spiral, they all do so to different degrees. The spiraling pattern of a mollusk's shell is no coincidence. It is the product of years of evolution; trial and error over millions of years. This shape has proved to be sturdy and compact; very important qualities when it comes to survival. Although a shell may be small it remains strong enough to protect the animal and large enough to house an animal by winding its body around the spiral.

Mollusks have managed to adapt to their environment very successfully. They can be found everywhere in the world; from the abyssal depths of our oceans to the heights of the Hymilayas. Mollusks have existed for millions of years and will continue to exist long after our species.

Spirals

A spiral is an abstract mathematical structure designed by the human mind to interpret the world around us. A mollusk's shell knows not that it is supposed to grow according to a mathematical formula or equation, it only knows to grow. The mollusk has been programmed by evolution. The spiral shell has proven itself over millions of years. Although the mollusk does not know the formula for a spiral, we are capable of defining it mathematically.

There exist a number of different spirals named for their discoverer as well as their formula. We shall be discussing a few of the most common spirals: Archimedes' spiral, Logarithmic spiral (Bernoulli, equiangular spiral). A spiral can be generically described mathematically as a rotating point on a line passing through the origin, where the point is approaching the origin at a velocity.

Archimedes' spiral is the combination of uniform angular motion with uniform linear motion. In this way, all Archimedes' spirals are similar. This means that they differ only in scale. Furthermore, their coils are all equidistant. This particular spiral has the polar (r,theta) formula of .

Here is an example of Archimedes' spiral:

Archimedes spiral with polar formula where . (Note that angle represents theta.)

> restart:with(plots):with(plottools):

> r:=5*angle;

> plot([r,angle,angle=0..20],coords=polar);

As for logarithmic spirals or Bernoulli's spiral, they can be described as a point starting from the origin and moving away from it on a linear path with an acceleration; the line follows a circular path which advances at constant angular velocity. This spiral is the same on every scale. lead-in statement... "The logarithmic spiral is the only form of coil that increases in size without changing shape. Thus, any organism that grows by coiling ( a good way to achieve strength of shell by compactness ), and reaches an advantageous shape worth preserving over a large range of size, must take the form of a logarithmic spiral." This lead Bernoulli to write "eadem mutata resurgo" ("I shall arise the same though changed") on his tombstone. In polar form, it has the equation .

Here is an example of a logarithmic spiral:

Bernoulli's spiral or Logarithmic spiral with polar formula where .

> restart:with(plots):with(plottools):

> r:=10*exp(1/5*angle);

> plot([r,angle,angle=0..20],coords=polar);

What is interesting about Bernoulli's spiral is that it all has to do with proportions. This first was discovered by the Greeks who discussed the divine proportion. This proportion arises in geometry when we divide a line segment into such that the ratio of the smaller side to the larger side will be equal to the ratio of the larger side to the entire segment or a:b=(a+b), where a<b. This can be rewritten as . If one solves for the zeros of the function (let y=0) two ratios will result:

> solve(x/1=(x+1)/x,x);

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