Monk's 'Startling' Math Discovery

April 1, 2001 -- An astonishing instance of mathematical anticipation has recently come to light, showing that a monk discovered a complex formula centuries earlier than previously thought.

The Mandelbrot set is one of those dizzying fractal shapes thathave come to symbolize the modern science of chaos theory. Like all suchshapes, it is indefinitely convoluted and has arcs giving rise to smaller arcs and flares branching into smaller flares.

Oriented vertically, it can be viewed as a star, and it is this way of seeing the set that leads to what is probably one of the most arresting discoveries in the history of mathematics.

In his wonderfully multitudinous Web site, The Apothecary’s Drawer (see Web link, left), Roy Girvan, an Englishscience writer and Web designer, tells the story of a retired mathematics professor traveling in Germany who was shocked to see in a medieval religious manuscript a nativity scene with a Star of Bethlehem in the unmistakable shape of the Mandelbrot set.

Bill Gates in the Dead Sea Scrolls

The professor, Bob Schipke, confessed to being stunned. “It was like finding a picture of Bill Gates in the Dead Sea Scrolls. The title page named the copyist as Udo of Aachen, and I just had to find out more about this guy.”

After investigation, Schipke discovered that this 13th century monk, previously known only for his poetry and essays, had discovered the simple secret for the generation of the Mandelbrot set 700 years before Benoit Mandelbrot!

Happily, some of Udo’s notebooks have survived, and Schipke foundthat the genius monk wrote knowledgeably about the notion of probability. He also discovered that the monk had discovered a method for estimating thevalue of pi that had previously been thought to have been first employed by 18th century naturalist Comte de Buffon.

But Udo of Aachen’s most staggering feat is the construction of theMandelbrot set, which involves the multiplication of complex numbers.

Theseare numbers of the form a + bi, where a and b are what we normally think ofas real numbers, and i, a so- called imaginary number, is the square rootof -1.

Spiritual, Profane and Complex

Incredibly these numbers would not become a part of mathematicalpractice for another 500 years, but Udo of Aachen wrote of theintertwined spiritual and profane parts of all entities, including numbers,and through some scrim of theological reasoning, he concluded that theproduct of two purely profane numbers was a negative spiritual number, justas i² = -1.

In this way Udo derived rules for dealing with profane/spiritualnumbers that were essentially identical to those used today for manipulatingcomplex numbers.

Using these rules, more theology (a number multiplied byitself is somehow the analogue of spiritual reflection), and repeatedlysubstituting complex numbers into the expression z² + c to determine their“fate” eventually led Udo of Aachen to the Mandelbrot set.

Prof. Schipke, with the help of historian Antje Eberhardt atthe University of Munich, wrote up his findings on Udo in the March 1999issue of the Harvard Journal of Historical Mathematics.

Few Comparable Finds

But despite the prestigeof the publication, the discovery has not received the attention itwarrants, and this may be one reason that Ray Girvan has so beautifully toldthe story on his web site and is certainly why I am publicizing it here.

Not since the Piltdown-Sokal Theorem was discovered to have firstbeen proved in 1907 by a provincial Russian electrical engineer, and not in1979 by Piltdown and Sokal as originally had been thought, has the date of afirst proof been so misjudged.

There is only one case of misattribution inthe history of mathematics that is of comparable magnitude. The famoustriangular array of numbers discovered by Pascal in the 17th centurywas anticipated five centuries before him by the Chinese mathematician ChiaHsien.

Even this example, however, is trumped by the 700 year differentialbetween Udo’s and Mandelbrot’s discovery.

Professor of mathematics at Temple University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on appears on the first day of every month.