April 21, 2018
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Counting on mathematics, now more than ever

By Jerry Farlow, Special to the BDN

Recently, someone asked me how many of the following four names I recognized: Jeremy Linn, Andrei Markov, Meryl Streep and Isaac Schoenberg, and I’d like to pose the same question to you.

My guess is you will recognize two of the four. Actually, the quiz is a fraud. It’s just my awkward way of announcing that April is National Mathematics Awareness Month and to make a basic point that although mathematics never makes headlines like an NBA star or winning an Oscar, it might play a more important role in our daily lives, and that’s where the two unrecognizable names on the list come in.

Andrei 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 might be anything from a baseball game, a frog jumping from lily pad to lily pad, the evolution of a biological population of bacteria 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, today Markov Chains are used by engineers and scientists the world over. When Google decided upon a strategy for ranking websites, it 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 and 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, and 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 websites 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.

And who is the other mystery mathematician? In the 1940s while professor of Mathematics at the University of Pennsylvania, Romanian-American Isaac Schoenberg developed a theory of what are called splines. Splines are special kinds of curves and surfaces that have a nice appearance and can be manipulated easily inside a computer. However, the use of splines requires many arithmetic calculations so Schoenberg’s theory was more or less on the shelf for years.

Now, with today’s ultra-fast computers, splines are back with a vengeance with applications ranging from the design of automobile bodies and parts, to images in animated movies, to computer-aided fabrication of prosthetic limbs, to the milling of dental restorations, such as inlays, crowns and bridges.

In the digital age of dentistry, when a new crown for a decayed or broken tooth is needed, the dentist removes the decay and reshapes 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. Each of these 1,000 equations requires 16 numbers to describe it, hence the computer requires 16,000 numbers to describe the crown. This virtual crown is then displayed on a screen where the dentist or dental assistant fine-tunes it with a computer mouse or similar device so it will not rub against neighboring teeth and have the desired characteristics, all the while the computer is number-crunching the 16,000 numbers that defines the crown.

After the crown is deemed satisfactory, the dentist then clicks a button on the computer screen and the 16,000 numbers that mathematically describe the virtual crown are sent electronically to a milling machine where after a few minutes a block of porcelain is converted to the desired crown, whereupon the dentist cements it in place in your mouth.

Today, we are in a golden age of mathematics. If you want to see just a few ways mathematics is affecting our lives, go to www.ams.org/mathmoments, a site run by the American Mathematical Society.

Jerry Farlow is a professor of mathematics at the University of Maine in Orono.

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