https://users.math.yale.edu/public_html/People/frame/Fractals/MandelSet/MandelMonk/MandelMonk.html
THE MANDELBROT MONK
by Ray Girvan
Mandelbrot set: modern image Until recently, Udo of
Aachen occupied a sideline in the history books as
a minor poet, copyist and theological essayist.
Even his birth and death dates of this mediaeval
Benedictine monk are unknown, though he probably
lived from around 1200-1270 AD. [*1] A new study of
his work, however, has led to his recognition as an
outstandingly original and talented mathematician.
While Udo himself is little-known, one of his works
is far more familiar. This 13th century German monk
was the author of a poem called Fortuna Imperatrix
Mundi (Luck, Empress of the World) in the
collection of mediaeval underground verses now
known as the Carmina Burana. [*2] Orchestrated by
composer Carl Orff in 1937, Udo's poem is now
widespread as the choral work, O Fortuna, which has
been used by the media many times, from incidental
music to the film Excalibur to the backing for
after-shave lotion advertisements.
The first clue to Udo's undiscovered skills was
found by mathematician Bob Schipke, a retired
professor of combinatorics. On a holiday visit to
Aachen cathedral, the burial place of Charlemagne,
Schipke saw something that amazed him. In a tiny
nativity scene illuminating the manuscript of a
13th century carol, O froehliche Weihnacht, he
noticed that the Star of Bethlehem looked odd. On
examining it in detail, he saw that the gilded
image seemed to be a representation of the
Mandelbrot set, one of the icons of the computer
age. [*3]
O froehliche Weihnacht Discovered in 1976 by IBM
researcher Benoit Mandelbrot, the Mandelbrot set is
the most famous fractal (a mathematical object with
the property of infinite detail). Only the advent
of fast computers made feasible the repeated
calculations involved - or so it was thought. [*4]
"I was stunned," Schipke says. "It was like finding
a picture of Bill Gates in the Dead Sea Scrolls.
The colophon [the title page] named the copyist as
Udo of Aachen, and I just had to find out more
about this guy."
Schipke visited Bavaria, where the poems, Cantiones
profanae (now the Carmina Burana), were discovered
in 1837. Written by wandering scholars and monks in
the 13th century, they were collected as an
anthology in the Benedictine monastery at Beuron,
near Munich, and Schipke began his search there.
With the help of historian Dr Antje Eberhardt at
the University of Munich, Schipke gained access to
ecclesiastical archives, where he found a document
called the Codex Udolphus. Written in illuminated
Latin, with informal marginalia in Greek, the Codex
bore the signature of Udo himself.
"Although it had been discovered in the 19th
century, it had promptly been filed away again,"
Schipke says. "The local historian who found it was
clearly no mathematician, and dismissed it as
obscure theology. But it yielded several major
surprises."
In a recent paper, Schipke and Eberhardt report on
Udo's discoveries. [*5] The first chapter,
Astragali (Dice) was originally thought to be a
discourse on the evils of gambling. It turned out
to be Udo's research into what we now would call
probability theory. He derived simple rules to add
and multiply probabilities, and thus devised
strategies for several card and dice games.
The second part, Fortuna et Orbis (Luck and a
Circle) describes Udo's determination of the value
of pi by scattering equal sticks on a ruled
surface, and counting what proportion lie across
the lines. This was an anticipation of the Buffon's
Needle technique, named after the 18th century
mathematician normally credited with its discovery.
[*6] This is a very laborious method, but Udo
managed to get a respectable - but very lucky -
approximation of 866/275 (3.1418...) and had enough
confidence in it to dispute the value of pi=3
implied in the Bible. [*7] (I say 'lucky' because
Buffon's Method converges extremely badly, and it's
well possible that Udo achieved this good result by
choosing his stopping point judiciously - perhaps
influenced by the 3.1418 quoted by his
contemporary, Leonard of Pisa, otherwise known as
Fibonacci).
Schipke continues: "What was interesting at this
point was that we looked back at the words of O
Fortuna, and suddenly they fell into place. Verse
two - Luck / like the moon / changeable in state /
We are cast down / like straws upon a ploughed
field / Our fates measuring / the eternal circle -
is very clearly an allusion to the Buffon's Needle
method." [*8]
More was to come. In the final and longest chapter,
Salus (Salvation), Schipke uncovered the most
radical work. Udo had, it seemed, investigated the
Mandelbrot set, seven centuries before Mandelbrot.
O froehliche Weihnacht (detail) Initially, Udo's
aim was to devise a method for determining who
would reach heaven. He assumed each person's soul
was composed of independent parts he called
"profanus" (profane) and "animi" (spiritual), and
represented these parts by a pair of numbers. Then
he devised rules for drawing and manipulating these
number pairs. In effect, he devised the rules for
complex arithmetic, the spiritual and profane parts
corresponding to the real and imaginary numbers of
modern mathematics.
In Salus, Udo describes how he used these numbers:
"Each person's soul undergoes trials through each
of the threescore years and ten of allotted life,
[encompassing?] its own nature and diminished or
elevated in stature by others [it] encounters,
wavering between good and evil until [it is] either
cast into outer darkness or drawn forever to God."
When Schipke saw the translation, at once he saw it
for what it was: an allegorical description of the
iterative process for calculating the Mandelbrot.
In mathematical terms, Udo's system was to start
with a complex number z, then iterate it up to 70
times by the rule z -> z*z + c, until z either
diverged or was caught in an orbit. [*4]
Below the description was drawn the first crude
plot of the Mandelbrot, which Udo called the
"Divinitas" ("Godhead"). He set it out in a 120x120
frame he termed a "columbarium" (i.e. a dovecote,
which has a similar grid of niches) and records
that it took him nine years to calculate, even with
the newly imported technique of algorism',
calculation with Arabic numerals rather than
abacus.
"It tends to be taken for granted," Schipke says,
"That the Mandelbrot is too calculation-intensive
to be done without computers. What we have to
remember is the sheer devotion of the monastic
life. This was a labour of faith, and Udo was
prepared to work for years. Some slowly-converging
pixels must have taken weeks."
Why did the work of this gifted mathematician go
unnoticed for so long? Schipke blames, in part,
specialisation. "When the Codex was unearthed in
1879, only a non-mathematician got to see it, and
he didn't know what he was looking at. It's a
common enough story. Take Hildegard of Bingen,
whose accounts of her visions were taken as pure
mysticism, but neurologist Oliver Sacks instantly
recognised them as accurate descriptions of
migraine symptoms. Likewise, literary critics
dismissed Edgar Allan Poe's final work, Eureka, as
alcoholic ravings. But now scientists are finding
valid insights in it, such as Poe's correct
solution of the Olbers paradox in astronomy, or his
coining of the classic Einsteinian phrase, 'Space
and duration are one'." [*9, *10]
"But there were also contemporary reasons why Udo's
knowledge didn't make it into the mainstream. His
basic belief - that salvation and damnation could
be determined in advance - was heretical, and his
use of Arabic numerals was thought a bit of a black
art. And there was the disagreement with
Thelonius."
Codex Udolphus Despite the borderline nature of his
work, Udo impressed his abbot at the monastery of
Sankt Umbertus near Aachen. Life for a 13th century
monk wasn't necessarily austere: the scurrilous
Cantiones profanae poems record the delights of
sex, eating, drinking and gambling. In a footnote
to Astragali, Udo writes: "My enumeration of the
ways [of dice] helped my lord abbot to win
thirty-two florins and a fine new cloak from the
Burgermeister at Irrendorf, and he has promised me
a helper for my work".
But Udo and his helper, Thelonius, ran into instant
disagreement. Udo had always interpreted the
Mandelbrot as signifying God. Thelonius took the
opposite view: that it represented the Devil.
Numbers that escaped to infinity, he argued, were
souls flying free to heaven, and those caught in an
orbit had fallen into the pit of Hell. Like many
theological collaborations, they had a schism on
their hands.
Udo noted that their differences brought all work
to a halt, and finally the two were reprimanded by
the abbot for coming to blows in the refectory.
"Sadly I write," says Udo on the last page of the
Codex Udolphus, "that on pain of excommunication I
must lay down my dice and my numbers. I have seen
into a realm of heavenly complexity, and my heart
is heavy that the door is now closed."
Bob Schipke comments: "It's a pity that personal
differences ended research that could have moved
mathematics forward by centuries. But fortunately,
Udo couldn't leave the subject alone. By dropping
clues into the Cantiones profanae and the
manuscripts he illuminated later in his life, he
ensured that we were able to recover his work and
give him the recognition that he deserves."
References:
[1] "The Benedictine Order: a Historical
Miscellany", edited by Rose M Wolanski,
Springer-Verlag, 1965.
[2] "Carmina Burana, Frequently Asked Questions",
by Charles Cave. http://www.classical.net/music/
comp.lst/works/orff-cb/carmina.html
[3] "O froehliche Weihnacht", ms. circa 1250 AD,
Aachener Dombibliothek, acquisition nr. GM801-237,
Blatt 1a. Photograph by Bob Schipke.
[4] "Chaos: making a new science", James Gleick,
Abacus Books, 1989.
See also the sci-fractals FAQ, maintained by
Michael C. Taylor and Jean-Pierre Louvet. (ftp://
rtfm.mit.edu/pub/usenet/news.answers/sci/
fractals-faq).
[5] 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.
[6] "Buffon's Needle, an Analysis and Simulation"
by George Reese. (http://www.mste.uiuc.edu/reese/
buffon/buffon.html).
[7] 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).
[8] Lyrics, translated by William Mann, to Orff's
"Carmina Burana (Cantiones profanae)", EMI
recording SAN 162, 1965.
[9] Oliver W Sacks, "Migraine: Evolution of a
Common Disorder", University of California Press,
1970.
See also: "Hildegard of Bingen": website by Sabina
Flanagan, University of Adelaide ( http://
www.uni-mainz.de/~horst/hildegard/documents/
flanagan.html).
[10] "Edgar Allan Poe's Eureka: I Have Found It!"
by David Grantz; at The Poe Decoder, Poe analysis
site by Christoffer Nilsson. (http://
www.poedecoder.com/).
The Mandelbrot Monk has been cited at:
El monje de Mandelbrot: Oct/Dec 1999 issue of
ContactoS, an educational e-journal from the
Universidad Autonoma Metropolitana, Mexico;
Netsurfer Digest 07.07, March 15th 2001;
Cool Math Site of the Week, March 18th 2001,
Canadian Mathematical Society's KaBoL project;
Newsletter of the Mew Zealand Mathematical Society,
#82, August 2001;
ABCNews.com: Monk's 'Startling' Math Discovery,
John Allen Paulos, "Who's Counting column, April 27
2001.
(c) Ray Girvan (ray@raygirvan.co.uk), April 1st 1999.
My sincere thanks to the late Bob Schipke for
permission to reproduce his work.
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