Structure of the Ear and the Leaning Tower of Pisa

Now what could the Leaning Tower of Pisa possibly have to do with structure of the ear?  Here’s the story of how Pisa influenced an explanation of how the cochlea is constructed and how it relates to other spirals that occur in nature.

First The Background

In the late 10th and 11th centuries, the Republic of Pisa was a Tuscan powerhouse nation.  At that time, Pisa dominated Mediterranean and Italian trade and was  one of the five main maritime republics of Italy. These maritime Republics were the Republic of Venice, the Republic of Genoa, the Republic of Pisa, the Republic of Ragusa, and the Republic of Amalfi.  From the 10th to the 13th centuries they built fleets of ships both for their own protection and to support extensive trade networks across the Mediterranean, giving them the capability to an essential supply and support role in the Crusades.

Guglielmo Bonacci, a wealthy Italian merchant and, by some accounts, the consul for Pisa, directed a trading post in Bugia (now Algeria), which was at the time a port in the Almohad dynasty‘s sultanate in North Africa. Leonardo Bonacci (1170-1250), Bonacci’s son, often traveled to Bugia with his father and was exposed to Arabic culture at a young age. During these trips, young Leonardo would meet with merchants and other businessmen to learn their methods of arithmetic and accounting.  As he learned the system, it became obvious that the Hindu–Arabic numeral system was less cumbersome than the old Roman Numeral system used in the Mediterranean area for accounting and mathematics. As Leonardo’s knowledge increased in the use of this numbering system he completed a book in 1202,  Liber Abaci (Book of Abacus or Book of Calculation) which revived and popularized Hindu–Arabic numerals in Europe.

What’s a Fibonacci Number?

Leonardo became known by many names, the most popular of which was known as Leonardo Fibonacci but also known as Leonardo of Pisa and Leonardo Pisano Bigollo.  No matter which name is used, according to the records of the time, he was the most talented mathematician of the Middle Ages.

Leonardo’s pivotal book Liber Abaci also posed and solved a problem involving the growth of a population of rabbits based upon idealized assumptions. The solution was a sequence of numbers later known as Fibonacci numbers. Fibonacci’s work contains the earliest known description of this mathematical sequence outside of India, where the sequence had been noted as early as the sixth century. The Fibonacci sequence of numbers uses a different rule for determining the next number from the sum of the previous two and it is interesting as these numeric relationships occur throughout nature and art. 

Of special interest, according to the mathematicians, is what occurs as result of the ratios of successive numbers.  Mathematically, if you sum the squares of any series of Fibonacci numbers, they will equal the last Fibonacci number used in the series times the next Fibonacci number.  This property of the numbers results in the Fibonacci spiral,  thus our relationship to the cochlear structure.  The sequence is based on the following progression and properties of the Fibonacci series: 1² + 1² + . . . + F(n)² = F(n) x F(n+1).

 How does this relate to the Cochlea and the Pinna?

 The sequence is based on the following progression and properties of the Fibonacci series: 1² + 1² + . . . + F(n)² = F(n) x F(n+1).  When one number in the sequence is divided by the number preceding it, the result is very close to 1.618, called the “Golden Ratio and a rectangle whose sides is equal to the golden ratio is known as a “golden rectangle”.

As indicated by Meisner (2012) the sequence also describes the “golden spiral,” which is the result of  a “golden rectangle”(see above) that is subdivided into smaller and smaller golden rectangles with the result being a predictable spiral.   Meisner uses the Fibonacci example:  1² + 1² + 2² + 3² + 5² = 5 x 8.  This relationship is depicted presented in the golden rectangle (above); notice how the relationship of the numbers in the squares predict the spiral and looks somewhat as we see in a pinna shape.    

While these relationships exist in nature,  Batts (2007) indicates that another example of a biological structure in the mammalian body which is very close to a “golden spiral” is the cochlea. It is, however, not a perfect golden spiral and there is individual variation between and within species.  Hayha (2012) states that this bony structure (the cochlea), filled with fluid, has a logarithmic spiral shape with a fixed angle of α=73°43´ containing the golden ratio.   There have even been attempts to explain the spiral of the pinna as well as hearing physiology using the Fibonacci sequence. One report indicates that Bell Telephone labs have confirmed the Fibonacci sequence and the shapes of both the cochlea and the pinna, but the report is not available to the public.  It appears that though this is an interesting exercise, there is not much that it tells us about auditory physiology other than an interesting complex of golden ratios, rectangles and spirals. 

So, what does the Leaning Tower of Pisa have to do with the Structure of the Ear?

Turns out……probably not too much.  We see these logarithmic spirals many other places as well as the ear such as, the nautilus shell, certain snails, the horns of goats, spider webs, even the galaxies. But before we say “why do I care” we may want to instead consider this sequence a nice mathematical relationship that is interesting; who knows if the creator of the hearing system and other natural shapes conducted these calculations to obtain the exact, proper spirals and rectangles? As for Fibonacci, he became a guest of Emperor Frederick II, who enjoyed mathematics and science and, in 1240, the Republic of Pisa honored him by granting him a salary for the rest of his years. 

References:

Batts, S., (2007). Fibonacci numbers, the cochlea, and poetry.  Science Blogs.  Retrieved July 7, 2015:  http://scienceblogs.com/retrospectacle/2007/10/12/fibonacci-numbers-the-cochlea/

Hayha, Y. (2012).  Fibonacci Numbers a thing of beauty.  Islamic Research Foundation International.   Retrieved July 7, 2015: http://www.irfi.org/articles/articles_251_300/fibonacci_numbers.htm

Meisner, G., (2012). Spirals and the golden ratio. The Golden Number.  Retrieved July 7, 2015:  http://www.goldennumber.net/spirals/

Images:

Macha, V., (2014). Write a C program to generate the first n terms of the sequence. | Fibonacci Sequence program in C.  Silly coldes.  Retrieved July 7, 2015:   http://www.sillycodes.com/2014/05/write-c-program-to-generate-first-n.html