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The Monk Who Discovered the Mandelbrot Fractal
Some years ago, during the COVID work-from-home era, I spiralled and fell down the rabbit hole of fractals, starting with a one-line Mandelbrot function, and developing into a whole family of complexity indices in NeuroKit (and some groundwork trying to make sense of them). What kept me there was a kind of vertigo: the Mandelbrot set (the mother of all fractals) comes out of one line of arithmetic: take a (complex) number c, start from zero, and repeat z → z² + c. Some values of c send the sequence off to infinity; others keep it circling forever, and the set is the collection of those trapped values. Its area is finite, and yet its boundary is infinitely detailed. Infinite complexity inside a bounded space.
It was around that time that I came across the story of Udo of Aachen...
The copyist of Aachen
The story starts when Bob Schipke, a retired professor of combinatorics, was visiting Aachen cathedral, where Charlemagne is buried. In the manuscript of a 13th-century Christmas carol, he noticed that something was off about the Star of Bethlehem in a tiny illuminated nativity scene. Up close, the gilded star was not a star at all: it looked like the Mandelbrot set. It was, in his words, "like finding a picture of Bill Gates in the Dead Sea Scrolls". The colophon named the copyist: Udo of Aachen, an obscure 13th-century Benedictine monk.
Schipke followed the trail into the ecclesiastical archives of Bavaria, and found the Codex Udolphus, signed by Udo himself and so badly faded that it had to be photographed in infrared. A local historian had come across it in the 19th century, taken it for obscure theology, and filed it away again. It had three chapters.
The first, Astragali ("Dice"), had been read as a sermon against gambling. It was actually probability theory - rules for adding and multiplying chances, applied to card and dice games. The second, Fortuna et Orbis ("Luck and a Circle"), estimated π by scattering sticks on a ruled surface and counting how many fell across the lines - Buffon's needle, five centuries before Buffon.
The third and longest, Salus ("Salvation"), is where it all came together. Udo wanted a method to determine who would reach heaven. In line with the theology of his time, he saw each soul as made of two parts, a material, profane part and a spiritual part, and he gave each part a number. Then he followed the soul through a lifetime - seventy years, the span the Bible gives a human life - during which, every year, the two parts act upon each other and reshape the soul.
His rule for one year had three steps. The material part feeds on itself and is worn down by the spirit: it becomes its own square, minus the square of the spiritual part. The spiritual part is born of their encounter: it becomes twice the material part multiplied by the spiritual part. And the soul returns to its own nature: the two numbers it was born with are added back, each to its own part.
Take a soul born with no material part and one unit of spirit, which we can write (0, 1), material first. In its first year, its material part becomes 0² − 1² + 0 = −1, and its spiritual part 2 × 0 × 1 + 1 = 1: the soul is now (−1, 1). A year later it is (0, −1), then (−1, 1) again, and it swings between these two for the rest of its life - wavering, but never straying. Its neighbour, born with one unit of each, (1, 1), follows the very same rule and runs away: (1, 3), then (−7, 7), then (1, −97), and long before its seventieth year its numbers are too large to write down.
This is what Udo saw. Once a soul strays far enough from zero (beyond a distance of 2, we know today), it never comes back, and its numbers race off towards infinity: that soul is lost to the outer darkness. The others stay confined, however long they live, and are drawn to God. And there is no simple rule telling the two apart. More spirit is not better: a soul born (0, 2), with twice the spirit of the first, is (−4, 2) after a single year, already too far to ever return. Nor does matter alone condemn: a soul of pure matter, (−1, 0), wavers between −1 and 0 forever. The only way to know a soul's fate is to work through its years.
So Udo worked through them all. He laid out a grid of possible souls, 120 by 120 of them, in a frame he called the columbarium (the dovecote, for its rows of niches): each soul's material part set how far across the frame its niche lay, and its spiritual part how far up. He followed every soul through its seventy years, marked the niches of those that stayed, and watched a shape emerge. He named it the Divinitas: the Godhead. The whole grid takes about 300,000 of those yearly steps (most souls are lost within a few years, and only the saved need all seventy), a bit less than a hundred a day for the nine years it took him. A true monk's work.
God or devil?
As a reward, the abbot gave Udo an assistant, a monk named Thelonius. They disagreed at once. For Udo, the shape was God, but for Thelonius, it was the Devil: the souls escaping to infinity were the ones flying free to heaven, and those trapped inside had fallen into the pit. The dispute ended in blows in the refectory, and the abbot forbade them to continue, on pain of excommunication. On the last page of the codex, Udo writes that he must put down his dice and his numbers, having glimpsed a realm of heavenly complexity whose door is now closed. But he hid clues to his work in the poems and manuscripts of his later life - a golden star over a nativity scene, for instance - and waited seven hundred years for someone to notice...
It is Mandelbrot!
For indeed, Bob Schipke realized that the figure was the Mandelbrot set, and that Udo's method was nothing but the recipe for drawing it. The two parts of the soul are the two halves of a complex number a + bi: the material part is its real part a, the spiritual part its imaginary part b. Udo's yearly rule is exactly what squaring such a number does - since i² = −1, (a + bi)² = (a² − b²) + 2abi - and returning the soul to its own nature is adding c: z → z² + c, once a year for seventy years. The souls drawn to God, those whose numbers stay bounded forever, are exactly the Mandelbrot set; our wavering soul (0, 1) is the number i, which sits right on its edge. Without knowing it, Udo had invented complex numbers, and drawn their most famous fractal seven centuries before Benoit Mandelbrot.
I loved everything about this story, a mathematical discovery born of a question about salvation, as well as the image at its heart: in Udo's scheme, the frontier between the saved and the damned is the boundary of the Mandelbrot set - infinitely folded, impossible to draw. Souls far from it settle their fate in a few years; the closer one sits to the edge, the longer it takes to know which way it will go, and for some, seventy years are not enough.
Udo of Aachen never existed
And neither did Bob Schipke, nor the Codex Udolphus, nor Thelonius. The story was written by Ray Girvan, a British technical writer, meant as a hoax.
When I discovered the truth, I was amused, and somewhat disappointed. It was so good I wished it was true. But the story stayed with me, because it could have happened, because infinitely complex things like fractals can emerge from the simplest of mechanisms, and someone could have stumbled upon this one centuries ago.
The original article, Ray Girvan's "The Mandelbrot Monk" (1999), is preserved on the Internet Archive; the three manuscript images in this post are from it. See also the Wikipedia entry on Udo of Aachen, and Robert Munafo's Mu-Ency entry.
