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 that have come to symbolize the modern science of chaos theory. Like all such shapes, 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 English science 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 found that the genius monk wrote knowledgeably about the notion of probability. He also discovered that the monk had discovered a method for estimating the value 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 the Mandelbrot set, which involves the multiplication of complex numbers.
These are numbers of the form a + bi, where a and b are what we normally think of as real numbers, and i, a so- called imaginary number, is the square root of -1.
Spiritual, Profane and Complex
Incredibly these numbers would not become a part of mathematical practice for another 500 years, but Udo of Aachen wrote of the intertwined spiritual and profane parts of all entities, including numbers, and through some scrim of theological reasoning, he concluded that the product of two purely profane numbers was a negative spiritual number, just as i² = -1.
In this way Udo derived rules for dealing with profane/spiritual numbers that were essentially identical to those used today for manipulating complex numbers.
Using these rules, more theology (a number multiplied by itself is somehow the analogue of spiritual reflection), and repeatedly substituting 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 at the University of Munich, wrote up his findings on Udo in the March 1999 issue of the Harvard Journal of Historical Mathematics.
Few Comparable Finds
But despite the prestige of the publication, the discovery has not received the attention it warrants, and this may be one reason that Ray Girvan has so beautifully told the story on his web site and is certainly why I am publicizing it here.