Santiago Canez
The Mathematics
of Natural Patterns
Logarithmic Spirals
Many different patterns that occur in nature can be explained using mathematics. One well known mathematical pattern found in nature falls under the group of spirals called logarithmic spirals, also known as equiangular spirals. In nature, logarithmic spirals are found in nautilus shells, flowers, and other places. Even though these objects may not seem to exhibit mathematical concepts, aspects of these objects can be explained using logarithmic spirals. The pattern in which a nautilus shell grows and the arrangement of seeds in flowers are two well-known examples.
There are many different types of logarithmic spirals. All logarithmic spirals, however, follow the same general definition: a curve generated by the polar equation r = r0 kq ("Equiangular, Logarithmic, and Bernoulli spirals") where r0 is the initial radius, k is a constant, and q is the angle. This equation is a simple one but logarithmic spirals can be defined on simpler terms. One particular logarithmic spiral is associated with the numbers in the Fibonacci sequence and is often called the Fibonacci spiral. The Fibonacci sequence begins with the number 1. Every consecutive number is the sum of the previous two numbers, for example: 1, 1 (1+0), 2 (1+1), 3(1+2), 5 (2+3), and so on. The first seven numbers in the Fibonacci sequence are 1, 1, 2, 3, 5, 8, and 13. Using these numbers, one can easily construct a logarithmic spiral. If we start with a square of sides equal to a length of 1, then can construct a square next to it also with sides equal to length one, then construct a square above these two with sides equal to length 2, then continue constructing squares in a circular motion where each square has sides equal to the length of the sides of the previous two squares, we would get squares with sides whose lengths equaled Fibonacci numbers. If we start at the first square and draw an arc from one corner to the other, and then continue this arc in the same manner through all the squares, we would get a logarithmic spiral. Two websites that illustrate this are "Fibonacci Numbers and Nature" and "Golden Spiral".
Two places in nature where we see logarithmic spirals are nautilus shells and flowers. The nautilus shell follows a growth pattern that follows the idea of logarithmic spirals and Fibonacci numbers. Computer generated images of shells using logarithmic spirals match the shells found in nature. In flowers to this pattern is noticed. For example, the seeds of a sunflower are arranged in curves. These curves also match those constructed using logarithmic spirals. Many other examples of logarithmic spirals found in nature follow the same ideas as those mentioned above.
Many websites on the web are devoted to idea of logarithmic spirals and their occurrence in nature. One interesting webpage is "Fibonacci Numbers in Nature". This page deals not just with logarithmic spirals but also with other places in nature where Fibonacci numbers are found. This page helps you visualize the construction of a logarithmic spiral using Fibonacci numbers through an animation. Another interesting page is "Logarithmic Spirals". This page focuses more on the mathematics behind logarithmic spirals but it is also helpful because it helps to explain why logarithmic spirals occur in nature- "Any process which turns or twists at a constant rate but grows or moves with constant acceleration will generate a single logarithmic spiral."
Other websites offer computer generated images of logarithmic spirals. One such website is "Logarithmic Spirals (2)". This page offers three-dimensional images of shells generated using logarithmic spirals. These shells match those found in nature. This page also has a link to a program that lets you generate your own three-dimensional images using logarithmic spirals. The website "Equiangular, Logarithmic, and Bernoulli spirals" also has three-dimensional images of shells generated using logarithmic spirals.
Of the five websites I consulted, "Logarithmic Spirals" and "Equiangular, Logarithmic, and Bernoulli spirals" dealt more with the mathematics behind logarithmic spirals and "Fibonacci Numbers and Nature" and "Golden Spiral" dealt more with the relation between logarithmic spirals and Fibonacci numbers. "Logarithmic Spirals (2)" dealt with computer generated images. All of these websites, however, were useful in helping to understand logarithmic spirals and where and how they occur in nature.
Websites Used