Recently, someone asked me how many of the following names I recognized: Tom Brady, Andrei Markov, Adele and Isaac Schoenberg. I’d like to pose the question to you. My guess is you will recognize Tom Brady and Adele.
Actually, my quiz is a fraud. It’s simply my awkward way of announcing that April is National Mathematics Awareness Month and to hammer the point that although mathematics and mathematical discoveries never make headlines like winning a Super Bowl or a Grammy, the two unknown persons on the list may play a more important role in our daily lives.
Let me first tell you about mystery person No. 2 on the list: Andrei Markov. Markov was a Russian mathematician who in 1906 worked out a mathematical theory, called Markov Chains, for describing how many physical systems evolve over time. Such a system may be anything from a baseball game, a frog jumping from lily pad to lily pad, the evolution of a biological population like bacteria or a virus or even a person surfing the Internet, clicking from one Web page to another.
Although Markov did not have any specific application in mind for his theory, Markov Chains are used today by engineers and scientists the world over. When it decided upon a strategy for ranking Web pages, Google imagined a person starting at some Web page, then moving from page to page. This can lead to dead ends at pages that have no outgoing links or around endless clicks of interconnected pages. This type of random walk is called a Markov Chain. Using Markov’s theory, it’s possible to find the fraction of time the surfer will spend at each page.
Suppose a person surfs the Web an infinite number of times. Of course, no real human can, but we can imagine it mathematically, and it’s possible to compute the fraction of times our imaginary person will spend at each page. This fraction is what Google calls the PageRank of the Web page. A Web page will have a high PageRank if it has links from other pages of high rank. If you enter the keyword “mathematics” in the Google search engine, the Web pages that show up will be ones with the highest PageRank for that keyword. Markov, who died in 1922, would be amazed at how his discovery is being applied today.
So, who is No. 4 on your don’t-know list?
In the 1940s while a professor of mathematics at the University of Pennsylvania, Romanian/American mathematician Isaac Schoenberg developed a theory of splines. Splines are special kinds of curves and surfaces that have a nice appearance and can be manipulated easily with a computer. However, the use of splines requires many arithmetic calculations, so Schoenberg’s theory was more or less on the shelf.
However, with today’s computers, splines are back with a vengeance with applications ranging from the design of automobile bodies and parts, to the images you see in animated movies, to computer-aided fabrication of prosthetic limbs, to the milling of dental restorations, such as inlays, crowns and bridges.
Nowadays, in the new digital age of dentistry, when your dentist recommends a new crown for a decayed or broken tooth, he or she begins by removing the decay and reshaping the tooth so it can receive the crown, then takes photographs of the reshaped tooth and surrounding area, then sends the images to a computer, whereupon the computer digitizes them and computes the shape of a crown that will fit perfectly over the prepared tooth. This new crown is, of course, inside the computer in the form of a mathematical spline, which means it is made up of maybe 1,000 cubic polynomial equations, each equation describing a tiny region of the tooth.
Although we never give it a second thought, mathematics is crucial in today’s society and becoming more so as time passes. Each time we use a mobile phone or make ATM and Internet transactions, the information transmitted is encrypted using mathematical algorithms. Even photos and digital music, which we email and download, are compressed mathematically for fast communication.
Jerry Farlow is professor emeritus of mathematics at the University of Maine.