Fact: The shell of a nautilus does NOT exhibit the golden spiral!
In mathematics the golden ratio and the golden spiral describe a specific type of geometric relationship. It states that two numbers are in ‘golden ratio’ when their sum creates the same relationship as the larger of the two original numbers. To express this mathematical concept graphically many authors have used examples of spiral patterns found in nature, including the very delicate spiral found on the cross section of the chambered nautilus. While the shell of the nautilus does have a beautiful spiral, which on first glance appears to follow the rules of the golden spiral, in actuality the spiral produced is slightly tighter than the golden spiral.
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The nautilus is a marine cephalopod related to squids and octopuses. They are pelagic meaning they live in the middle of the ocean, neither next to a coast nor near the ocean floor. Unlike the soft body of an octopus, the nautilus is the only cephalopod that produces a shell to protect its body, much like snails and other gastropods. This shell contains mother of pearl and has been highly sought after in the centuries past to create items for aristocracy, such as this chalice. When the shell is cut in two down the centerline, it reveals a series of chambers that show how the animal has grown successively larger, and build the shell up over time. The outline of this interior shows a spiral and this has been reported by many sources over the years as exhibiting the “golden spiral”.
Golden Ratio, Golden Spiral and Fibonacci Numbers
At the beginning of the 11th century Italian Mathematician Fibonacci penned the following sequence in his book Liber Abaci.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
Called Fibonacci numbers, this sequence shows a recurring relationship where each new number is the sum of the two numbers immediately before it. This number sequence is closely related another relationship called the Golden Ratio where two numbers once added have a relation that is the same as the larger number to the smaller number.
For example, in X + Y (where X is the larger number) Y is related to X, in the same way as X is related to X + Y.
This relationship is called ϕ, which comes out to roughly 1.68. Meaning that X is 1.68 times larger than Y, and X+Y is 1.68 times larger than X.
When ϕ is turned into a spiral (that grows at the rate of ϕ )the specific geometric pattern produced is called a Golden Spiral which is a type of logarithmic growth.
Math and Biology together
The geometry based on these mathematical sequences can be seen in certain patters found in nature. This can be seen in structures as MASSIVE as The Whirlpool Galaxy and are still present all the way down to the sub-molecular level in atoms of Cobalt Niobate. One example that has been cited along with these others is that of the shell of the nautilus.
Although written about in both biological and mathematical literature for decades, Dr. Clement Falbo of Sonoma State University demonstrated that the spiral of the nautilus although similar, does not actually follow the golden spiral. It instead uses a lower ratio closer to 1.3 or 1.4 than that of the golden spiral’s 1.68. This is discussed in detail in The Golden Ratio: a Contrary Viewpoint.
Nautilus shells are many things. They are the homes for the animals that create them, and they serve as ornaments and precious materials for craftsman that manipulate them. But one thing that these beautiful shells are not is perfect Golden Spirals!
You may also want to know about:
- The Math behind ϕ and the Nautilus (http://www.goldennumber.net/nautilus-spiral-golden-ratio/)