In the last couple of weeks I have paid a number of visits to dentists' offices. Extractions, moldings, cleaning and a number of other procedures led me to the discovery of a curious connection between dentistry and mathematics.
In the late 1800s Austrian dentist Adolph Zsigmondy came up with a simple system that replaced prior schemes involving latin names. This scheme involved three parameters: upper or lower, left or right, and the numbers one though eight. For example UL3 would be the third tooth from the middle in the upper left jaw. In reality, Zsigmondy used graphical symbols instead of the letters, but this is inconsequential. This beautiful coordinate system was simultaneously discovered by the Ohio dentist Corydon Palmer, and an ugly priority fight ensued. The result was the Zsigmondy-Palmer system. The symmetry of the human jaw is clearly reflected in the choice of parametrization, and similar teeth have the same numbers - canines are threes, molars are sixes, sevens and eights, incisors are ones and twos, and so on. The mathematical elegance of this system had a devastating effect on European dentistry. Dentists mesmerized by the intrinsic beauty of Zsigmody-Palmer encoding went on to study mathematics and left the population in the hands of blacksmiths and street vendors performing the crudest forms of dental procedures. Simultaneously, mathematical sciences blossomed with group theory, representation theory and combinatorics, all of which had roots in these simple observations. Former dentists-turned-mathematicians include Arthur Cayley and Felix Klein. Evariste Galois was never a dentist but he was shot by one.
The British Isles have been particularly devastated by what I would not hesitate to call a Copernican revolution, and to this day Brits suffer dearly for their mathematical contributions. This is partly due to Alan Turing who modified Zsigmondy-Palmer into modern FDR (Federation Dentaire Internationale) system. FDR designates each tooth by two digits - the first digit with value one through four denotes the jaw's quadrant and the second digit (one through eight) is the tooth number (counted from the center). Four quadrants with a bite of 8 teeth in each. However, Turing's slurred speech after a painful extraction led to "bite" becoming "byte", a unit of digital information used today. One can clearly see that modern coding theory, cryptography and digital signal processing are descendants of this basic ideas. And while FDR was officially adopted in 1970, this was only because Turing's work was declassified long after his untimely death and it took some time to make it into the mainstream. Meanwhile these ideas led to the victory in WWII, while at the same time lack of dental care condemned the Brits to soft and overcooked food.
On the other side of the Atlantic, the US was watching these developments with trepidation and decided that to become the dental superpower - the explicit goal of several consecutive presidents - it had to dial down the mathematical intricacies of the dental encoding. The resulting Universal system - used only in the US - is mathematically very simple. It was proposed by J. Perreidt in 1882, but the basic idea was pioneered by Carl Friedrich Gauss a century earlier. Sometime in 1783, a 6-year old Carl was being examined by the school nurse, who was methodically counting his teeth while looking for cavities. She started at the left, counting 16 teeth, and then went down going in the opposite direction all the way to 32. Just before the nurse found a cavity and his brain shut down in terror of what followed, the idle mind of Carl Friedrich quickly noted that the numbers of the opposite teeth add up to 33. When he returned to the classroom, his bored teacher, who was aching to read personals in the Sudetische Beobachter, told the class to add up all the numbers from one to hundred. At this point Gauss, who was still shaken by his dental experience, was thankful that he did not have 100 teeth. However, he did not miss the fact that if he did have 100 teeth then the numbers of the opposite teeth would add to 101. In seconds he multiplied 50 by 101, handed the answer to the teacher and instantaneously became the most famous mathematician in the history of mankind.
So this is the Universal system - a simple count from left to right in the upper jaw, which continues from right to left in the lower jaw. This simplicity keeps American dentists glued to their profession. Indeed, over time the US became a dental superpower, while mathematics was often left for foreign talent. However, American teeth are so white and shiny that if their owners could smile a little more, the reflected light would stave of the global warming. To further keep their dental primacy, the US adopted a special scheme to encoding baby teeth. Instead of numbers, only the letters A through T are used. Even the brightest kid would get lost in such an intellectual desert and not be lured into sciences while poking around in their mouth.
And what about Gauss? In spite of simplicity, there was enough math in the Universal system for Gauss to remain interested. His observation that if teeth were counted at the top and bottom in the same direction led to the discovery of modular arithmetic and further solidified his reputation. Throughout his life he took good care of his teeth, and during visits to dental offices he was always on the lookout for mathematical talent. Sophie Germain and Bernhard Riemann were dental assistants that he took under his wings and trained as mathematicians. Sadly, this alone set back European dentistry by half a century and prolonged the use of pliers and low-speed drills.
