Cool and Calculating

In several previous articles, I've shared how important and pervasive fractal geometry is in the natural world all around us. We take for granted that the discovery of this intriguing branch of mathematics is a purely modern phenomenon, made possible only by the advent of powerful enough computers. These computers were necessary in order to produce the first stunning image of what we know of as the Mandelbrot Set (see thumbnail picture at right). But in the relatively recent past, information has come to light that questions the modernity of this mathematical wonder. In fact, the painstaking, diligent search conducted by an obscure monk over 700 years ago shows that the knowledge of fractal geometry is by no means new to our time. In fact, this personage who was so devoted to the ascetic life was on the threshold of unlocking the mathematics that describe the phenomena of our natural world.

You see, this unassuming 13th century German monk, whose name was Udo of Aachen, became very interested in devising a method for determining who would reach heaven. His findings were detailed in a work entitled Salus (Salvation). He began his quest with the understanding that man was composed of two parts he called profanus (profane) and animi (spiritual). But our good monk Udo was also fascinated with probabilities and mathematics, so he strived to develop a mathematical model for showing the progression of souls towards or away from the celestial realm. Therefore, he proceeded to represent these two parts of humanity via pairs of numbers. As he visualized the lives of people, he saw them either approaching Infinity or moving away through numerous trials in the course of their lifetimes. His task, then, was to find a way of representing this via pairs of numbers.

Accentuating the Positive, Eliminating the Negative?

As Udo worked on this problem, he developed rules for drawing and manipulating these pairs of numbers. In so doing, and without actually knowing it, he devised the rules for complex arithmetic. The spiritual and profane parts of humanity corresponded to what we now know of as the real and imaginary numbers of modern mathematics. The result of Udo's work with these numbers was an equation surprisingly similar to the one now used to plot the familiar image we know of today as the Mandelbrot Set. Udo's monastic life enabled him to devote the many, many hours, days, weeks, months, even years, that it took to calculate the path of each "soul", as represented by number pairs on a grid, and determine whether the soul was going to advance or descend. The result of this painstaking work was an image he called the Divinitas, or "Godhead". The illustration shown in Udo's Codex Udolphus bears a striking resemblance to a crudely drawn Mandelbrot Set image.

Lost Tragically in a Schism Chasm

So why did it take over 700 years for us in the modern world to learn of Udo's incredible discoveries? Well, as it turns out, Udo worked with a colleague by the name of Thelonius, and the two of them got into a heated disagreement over what the unique image represented. Udo was of the opinion that it represented the Divine, but Thelonius was adamant in his position that the unusual mathematical depiction represented the opposite. Thelonius asserted that all the points (souls) that escaped into infinity were the ones ascending, while those that did not were those trapped below. At the height of this intense dispute, the monastery abbot was obliged to step in and halt it. He reprimanded the two men, and then forbade any further work concerning the Divinitas. With great sadness, Udo ceased his pursuit, but not before leaving clues throughout his works. These clues enabled contemporary researchers to discover the astonishing feat achieved by this diligent monk hundreds of years ago.

For Further Reading:

  • "The Benedictine Order: a Historical Miscellany", edited by Rose M Wolanski, Springer-Verlag, 1965.
  • "Carmina Burana, Frequently Asked Questions", by Charles Cave.
  • "O froehliche Weihnacht", ms. circa 1250 AD, Aachener Dombibliothek, acquisition nr. GM801-237, Blatt 1a. Photograph by Bob Schipke.
  • "Chaos: making a new science", James Gleick, Abacus Books, 1989.
  • Schipke, R.J. and Eberhardt, A. "The forgotten genius of Udo von Aachen", Harvard Journal of Historical Mathematics, 32, 3 (March 1999), pp 34-77.
  • "Buffon's Needle, an Analysis and Simulation" by George Reese.
  • II Chronicles, iv, 2: "Also he made a molten sea of ten cubits from brim to brim, round in compass ... and a line of thirty cubits did compass it round about" (Authorized King James Version).
  • Lyrics, translated by William Mann, to Orff's "Carmina Burana (Cantiones profanae)", EMI recording SAN 162, 1965.
  • Oliver W Sacks, "Migraine: Evolution of a Common Disorder", University of California Press, 1970.
  • "Edgar Allan Poe's Eureka: I Have Found It!" by David Grantz; at The Poe Decoder, Poe analysis site by Christoffer Nilsson.
  • Girvan, Ray, The Mandelbrot Monk, April 1, 1999.
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Image credit: LariAnn Garner