Fibonacci

While the foundation of the Tower of Pisa was being being laid, a man was born nearby who developed math skills that might have helped the ill-fated architectural wonder.
Leonardo Pisano ("of Pisa") (c.1175-c.1250) was the son of Guglielmo Bonacci. Although known as "Leonardo Pisano" during his lietime, he signed his name "Bonacci" on his writings; an 1838 writer referred to him as "Fibonacci"—short for filius Bonacci ("son of Bonacci")—and the name stuck with his modern fans.

His father was a customs officer in Algeria, and between living there and traveling around the coast of the Mediterranean, Leonardo grew up exposed to education outside of the Greco-Roman/Western European tradition. He recognized the advantages of the Hindu-Arabic system of numbers over using Roman numerals, and worked to popularize it in Europe, starting with his 1202 work Liber Abaci ("Book of the Abacus" or "Book of Calculating"). In it, he presented to Europe the decimal system by which we all learn the four basic mathematical functions in school.

Calculating with the four functions in a decimal system.

The decimal system, with its "places" for ones and tens and hundreds, etc.,  was much "neater" than the system of Roman numerals and included a digit for "zero." Roman numerals had no "zero," and the words null or nihil were used to express a lack of something. The Roman tradition had great difficulty with the concept of "nothing" in math, because it seemed inappropriate to have a "something" that would indicate a "nothing."

If people have heard of Fibonacci in the present day, it is usually because of a particular sequence of numbers associated with him. In Chapter 12 of the Liber Abaci, he presents a math problem: how many rabbits are created in one year starting with one pair? After describing the progression in words, he shows the number progression as 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. Examples of this series existed prior to Fibonnaci, and it is likely that he was simply repeating something he had learned, but 19th century mathematician Edouard Lucas called this sequence the Fibonacci numbers. They have been found to relate to many phenomena found in nature. (A thorough discussion is impossible here, but look.)

A webpage with some simple representations of the Fibonacci sequence is here.

The Pisan Mistakes

According to legend, a rich widow named Berta di Bernardo left 60 coins in her will for the construction of a campanile, a bell tower, for St. Mary's Cathedral in Pisa. A 55-meter tower was planned, belvedere style so that people could walk up the outside and enjoy the view. Ground was broken August 9, 1173.

The architect/builder is unknown. Either he was simply undistinguished, or he was careful to keep his name out of the public eye once the project was begun in order to avoid the inevitable ridicule. Pisa was built at the junction of two rivers, the Arno and the (now dried and gone) Auser; when the builder dug a 3-meter deep hole for the foundation, he found only river clay and sand. This lack of bedrock did not deter him. He proceeded to build what was designed to be a 14,000-ton structure on top of this soft foundation; by the time he got to the third level in 1178, it had already started to list to one side.

They had no way to change the ground on which it stood, and they didn't want to give up the project, so they built the next floors with one wall higher than the other, so that the upper floors wouldn't slant.

A series of financial problems and wars with neighboring city-states (mostly Florence, but Pisa also lost a major conflict with Genoa) meant the campanile wasn't finished until 1319. Nothing they did could fix the increasing list. But then, nothing they did would have. Adding the bells? More leaning. Deciding to add an impressive 3.5 ton additional bell? More leaning. Even so, the tower was only leaning by five degrees in 1837, when Alessandro della Gherardesca decided to turn an embarrassment into a tourist attraction by exposing the famous flawed foundation. He dug around the base...below the water table. The space flooded, the clay and sand softened more, and over the course of a few days, the tower leaned another meter!

In 1934, the idea arose to drill holes in the foundation and force grout into them. It looked good on paper; in reality, it increased the tilt. Tinkering in 1966 and 1985 just made it worse, and the tower was closed to the public in 1990.

Years later, the combination of a steel corset around the base, concrete, and counterweights were supposed to be the ultimate fix. They weren't, although they did almost cause the whole tower to break apart from stress. Finally, someone decided to go to the root of the problem. They dug away at the foundation on the other side of the tower, hoping it would lean back the way it had come. A little over 800 years of tinkering since the problem first appeared, and not only was the tilt reduced, but the tower was stabilized and re-opened to the public. Once again, tourists can climb the tower and take photographs looking out over Pisa, rather than just ground-based pictures of them appearing to hold up the tower with their hands.