top of page



When you read the following pages, you may say: "Wait a minute. He did this? All of it?" The answer is yes and no. The very first steps were on my own and entirely by myself, the Diploma Thesis, the Doctoral Thesis, and the Habiliation Thesis, as it should be. Then I morphed into a different working style, a style of collaboration with my doctoral students, with many colleagues in math, math education, and medicine around the world, with teams of teachers in Florida and Bremen, with my fantastic research staff at CeVis and MeVis, and Fraunhofer MEVIS.


Much of what I tried to do falls into the category of digital transformation and it seems typical for this age of change that innovations rarely come by single people, or lonesome wolves, as in the past, but rather by people who like to associate with people of different, complementing gifts and abilities. That is more what happend with me, what I wanted, what I created and had a passion for. I like to believe that the future is wiki-like collaboration, small wikis, large wikis, world-wide wikis, but wikis. Here I am not refering to the software wiki in particular. Rather I mean the attitude and spirit of collaboration which underlies the wiki culture. The true power of the digital age lies in unleashing the power of wikis. I like to think of myself as a tiny example of a local wiki enabler.




The Beginnings in Bonn

My initial work in mathematics around 1970-71 was in algebraic topology. I successfully was able to compute the cohomology ring for simplicial pairs on a computer. This was the first time that this had been done and a twofold challenge: I had to go back to the historical beginnings of algebraic topology (e.g. the legendary book by Seifert and Threfall from 1934 ) to develop algorithms that were implementable on a computer. The second challenge was implementation. The computer at hand at the University of Bonn at that time was an IBM 7090 and was considered to be one of the most powerful in Germany. However, looking back at it from the point of view of 2015 it had just a tiny memory of 128 KByte and a power of some 100 KFLOPS. It occupied an area as big as a single family home. Compared to an iPhone 6 with a memory of at least 32 GBYTE (which is some 250,000 times more) and a power of some 100 GFLOPS (which is 1,000,000 times faster) it was not much of a computer. In the course of the computation very large matrices had to be generated and matrix operations had to be performed, while the matrices did not fit in the memory at all. In fact they did not even fit onto the magnetic tape storage connected with the computer as memory extension. Today the field of computational algebraic topology is flourishing with the computer power at hand and today's users have no idea, what it meant to compute the cohomology ring of a decent manifold in 1970.


I left this area and transitioned into a combination of algebraic topology and nonlinear functional analysis for my dissertation in 1973. Christian Fenske and Friedrich Hirzebruch were the reviewers. Topological fixed point index theory, a generalization of Brouwer degree, and the Lefschetz fixed point theory was the area of work. Particular unstable fixed points were of special interest in a purely topological setting with no applications in mind.


From there, influenced by a one month Summer School in Montreal I became interested in what is known as fixed point theory with applications in the theory of functional differential equations. In fact the work in this area covered my Habilitation thesis in 1976. This was greatly influenced by Andrzej Granas and Roger Nussbaum. With Roger I discovered much later spurious solutions in the numerical approximation of functional differential equations (see below). Through the friendship with Roger I soon had learned enough about functional differential equations and aimed at transforming my general topological results about unstable fixed points into a framework which would be applicable at least to functional differential equations. This lead to a series of papers with the late Gilles Fournier about fixed point theory in cones.


Eugene L. Allgower, a mathematician from Colorado State University, made me aware of Emanuel Sperner and his famous result form 1929, which appropriately interpreted, provided a constructive proof of the celebrated Brouwer fixed point theorem, which has a generalization to infinite dimensions by the Schauder fixed point theorem, which in turn is one of the key elements in many existence proofs for solutions of ordinary and partial differential equations.  The idea thus was tempting to use this in an algorithmic numerical way: Differential equations, which have solutions based on Schauder, once numerically approximated would land in finite dimensions. There one should be able to use something like Sperner's Lemma to obtain a numerical approach to find a solution. This fascinated me beyond belief and my first doctoral student Michael Pruefer, while still in Bonn, worked on this idea and pushed it through. Then we looked at another famous result for solving differential equations: The global bifurcation theorem of Paul Rabinowitz. It guaranteed topological continua under certain conditions. The key element in Paul's proof was the homotopy property of the Leray-Schauder degree. We looked for a constructive finite dimensional version of that and provided an algorithmic and computational form of Paul's celebrated theorem. This led us into the domain of simplicial continuation methods, which Dietmar Saupe later used for the numerical computation of periodic solutions of functional differential equations. Finally, in domain we provided an entirely constructive, meaning algorithmic - computational setting for the Brouwer degree, which was a very unconvential approach to topological degree theory. In a way this was related to computational algebraic topology, which I had spearheaded in 1970. Later in 1978 I met E. Sperner, when I was invited to give the keynote on behalf of his 50th PhD anniversary at the University of Hamburg. He got his PhD at the age of 23!



From Bonn to Bremen

In 1977  I accepted a full professorship at the University of Bremen, which was trying to find its identity as a newly founded reform university. The early years in Bremen were a huge challenge compared to the prospering environment in Bonn. Bonn at the time was one of the world centers of mathematics with a generous funding situation and a unique mathematical environment for ambitious young researchers. To illustrate this point, in 1974, just one year after my dissertation, I had access to funding for several visting professors for several months and was able to organize an international conference during the summer with a substantial number of invited speakers. Among others I was able to bring my new mathematical friends from Italy, Massimo Furi, Mario Martelli and Alfonso Vignoli,  and  from the US, Roger Nussbaum, to Bonn. I had met all of them in Montreal a year earlier. Until my departure to Bremen I had access to funds which never appeared to dry up to invite many colleagues to Bonn as visiting professors and organize another international conference on Functional Differential Equations and Approximation of Fixed Points, which took place in 1978, while I was already in Bremen. The funding situation in Bonn was so excellent because of two multi-million DFG Sonderforschungsbereiche, one in Pure Mathematics and one in Applied Mathemtics. Even though quite young for such a role, I had my hands in writing the proposal for the latter one and was a member of its steering committee. Even when you assume that I may have looked promising to the ranking mathematicians in Bonn, it is still very remarkable, in particular for that time of the 1970s, where the typical German university was still petrified in hierarchical thinking and structures. Friedrich Hirzebruch had somehow infused this attitude and atmosphere, and in my mathematical neighborhood Eberhard Schock and Heinz Unger just gave us any freedom and support you might think of. This rubbed off on me big time and I cultivated the same attitude in my research centers first in CeVis and later in MeVis. In sharp contrast, I remember an incident after my habilitation. In Germany you had to leave the nest, so I applied for a professorship elsewhere. An offer came in from the University of Würzburg in Bavaria. After the colloquium lecture, which was used to introduce myself,   I was invited to a dinner with the faculty of the department, as it is the custom. Full professors sat at one table, associate professors at another table and the lower ranks yet at another table. 


I was a liberal, I wanted to be with people who hated hierachary and Bremen was a very outspoken liberal place at that time, to say the least. But Bremen had practically nothing in terms of funding and endowment for professors and drew a lot of criticism for its immature reform projects. But I wanted to be there, because I believed that the old German university system had to be replaced and I wanted to be a part of that transformation. Today the University of Bremen has an excellent reputation and I consider myself fortunate to have had the opportunity to actively participate in the transformation process over the last 30 years. However, the early years in Bremen were as hard as it gets, no resources, no reputation, and as many believed, but not us, no future. It still fills me with joy and satisfaction that we proved the mainstream opinion, according to which the University of Bremen would be a lost case, premature and very wrong. My strategy was simple, look for bright students and work with them in the Bonn style. And I did indeed find great students in Bremen. But maybe Bonn had also endowed me with an inexhaustible amount of self-esteem.


It didn't take long, a couple of years, and we aquired funding from the Stiftung Volkswagenwerk, and then, I believe we were the first in Bremen, we aquired even substantial funding from the Deutsche Forschungsgemeinschaft, where my Bonn experience in writing successful proposals came to use. And we, the late Peter Richter, Didi Hinrichsen, Ludwig Arnold and I founded the Institute for Dynamical Systems (IDS), which became an unusually fertile breading environement for young mathematicains, many  of which have obtained international fame, like Dietmar Salamon and Dieter Praetzel-Wolters. The IDS again was the launch pad for my own trajectory: CeVis, and then CeVis became the launch pad for MeVis, and that for Fraunhofer MEVIS and MeVis Medical Solutions AG. Around 2015 the two institutions combined employed some 300 researchers and software specialist. 


Sabbaticals in Salt Lake City

In 1976, while still in Bonn, I attended a NSF-CBMS Regional Meeting in Boulder Colorado on Isolated Invariant Sets and the Morse Index. It was there that I met Klaus Schmitt from the University of Utah, who inspired my interest more and more for differential equations. The conference featured Charles C. Conley's ground breaking work. Several years later Charly would visit Bremen a few times and we had unforgettable summer workshops on the tiny island of Helgoland.


Two sabbaticals 1979-80 and 1982-83 changed my mathematical path dramatically. Both were at the University of Utah in Salt Lake city sponsored by my friend Klaus Schmitt. The University of Utah had and has one thing that not too many universities have: a very strong mathematics department. But it also had something absolutely unique at the time of my sabbaticals: the University of Utah is the birthplace and home of computer graphics. "Almost every influential person in the modern computer-graphics community either passed through the University of Utah or came into contact with it in some way." says Robert Rivlin in his book on the history of computer graphics. 


Starting in 1979 Klaus Schmitt and I began a very fruitful collaboration for many years, where we merged our mutual expertise, his in ordinary and partial differential equations and mine in nonlinear functional analysis. And we started to use numerical and computer graphical experiments on the legendary Evans & Sutherland  PS/2 for our work on global bifurcation phenomena for nonlinear elliptic eigenvalue problems.

The legendary Evans & Sutherland PS/2 vector graphics system. Dials (above key board) were used to manipulate parameters of the graphics. 

The PS/2 was exclusive to Robert Barnhill, a pioneer in Computer Aided Geometric Design, and he generously allowed us to use his jewel machine. Remember, in 1979 computer graphics was practically not existent in mathematics or computer science departments around the world. Actually, I don't know of any other mathematics department at all.


Spurious Solutions and the Transition into Dynamical Systems

Being able to visualize our numerical experiments which were supposed to guide our new theory turned out to be crucial for a tremendous discovery. We discovered that the numerical solutions were of two kinds, those which were approximating the solutions of the partial differential equations, and those which strangely were not. These additional solutions were some kind of artifact of the numerical discretization. We knew that if the partial differential equations had solutions, then they had to have certain symmetries, and the strange solutions did violate these symmetries. That is how we discovered them on the screen of the PS/2. We called them "spurious" solutions and went on with our theory, because at that time we were not yet in numerical analysis. Actually, Klaus and I did not carry out these experiments ourselves, we just guided them. They were run by Dietmar Saupe who had joined me as a graduate student from Bremen. Regarding the discovery of spurious solutions we did not find any mention or explanation in the entire numerical literature. For many years we tried to understand what was going on. Eventually, and that had to do with my interest in chaos theory and in particular with the Poincare-Birkhoff theory of Chaos in Hamiltonian Systems caused by homoclinic structures, we found a complete answer, by embedding the numerical problem into the framework of dynamical systems. The emergence of spurious solutions turned out to be a rather peculiar homoclinic bifurcation in the associated dynamical system (A and B). And we found similar phenomena in functional differential equations

When I lectured about this on many numerical conferences I was expecting great interest, but that turned out not to happen. Rather I heard comments like: “A good numerical analyst would recognize spurious solutions and would not need a theory of them." I heard similar arguments many years later presenting our work on liver surgery planning, when some surgeons would say: “A good surgeon doesn't need the kind of risk analysis which you provide. He can draw on his experience." This changed dramatically after one of the greatest liver surgeons of all time, Koichi Tanaka from Kyoto University, adopted our methods and made them the standard in liver surgery.


Back to spurious solutions: After my presentations there never was a single question about our methods using homoclinic structures. In retrospect I should have known better. Homoclinic structures were Chinese to audiences of numerical analysts. Even at the University of Bonn, where we grew up as young mathematicians and were confident that we knew or at least had heard of the most relevant mathematics, we never came across the great ideas and achievements of Poincare and Birkhoff and dynamical systems as a field, and a basic or advanced course did not exist. Nevertheless, the Bonn values were to appreciate mathematical cross-disciplinary insights or building such connections. And that was in particular what Hirzebruch stood for. Numerical analysis as a field instead has fortress like fences and not much of a tradition to look beyond or to connect. Coming back home to Bremen in the spring of 1980 we managed to obtain access to a Tektronix 4014 storage tube computer terminal in the university's computer center and continued our experiments on homoclinic structures, based on clever algorithms by Hartmut Jürgens. One day I discovered a homoclinic bifurcation and I will never forget the sort of satisfaction, which I felt. The understanding of the mechanism for spurious solutions had yet to come (A and B). But our path had changed. From now on mathematical experiments became increasingly important. But we were still far away from having our own equipment. 


I found some of the generated pictures obtained on the Tektronix 4014 aesthetically very appealing and on a vacation to Hawaii I actually started to water color some (homoclinic structures and chaos in an area preserving diffeomorphism, December 1982):

Water colored phase portrait of area perserving diffeomorphism with homoclinic structures

Encounter with Fractals and Experiments

The second sabbatical in Salt Lake City from 1982-83 impacted my path even more profoundly. Klaus Schmitt made the connection with Richard Riesenfeld, the head of the Computer Graphics Lab in the Department of Computer Science. Richard gave us permission to use his equipment, when we were not in the way of his group, in the evenings on weekends. We, Dietmar Saupe and I,  became acquainted with the early pixel based computer graphics, which is entirely different from the vector based computer graphics in a PS/2, and of course became the standard later on. And then came December 6, 1982. Leo Kadanoff, a famous physicist from the University of Chicago came to lecture. We were expecting that he would talk about phase transitions, since his student Ken Wilson had just received the Nobel Prize in Physics. But he didn't. Instead he lectured about chaos in a rather popular form. We made an appointment for next morning to exchange ideas at the blackboard. I told him about my stuff around Poincare-Birkhoff and he talked about something, which I found extremely puzzling. He talked about the dynamics of

in the complex plane, c is a small constant, and he drew weird chalk pictures on the blackboard and explained quasi-circles. I must confess that I did not understand a thing. But I was interested and so I twisted Dietmar Saupe's arm that we should run experiments in Richard's lab in the afternoon. And in the evening we saw a stunning picture of a Julia set in color, probably the very first one of this kind. The beauty of the picture was breathtaking at the time. So we could not stop. Soon we had a collection of images. When we showed them to Frank Hoppenstedt, a mathematician in the department, he said, go and look at that book by a guy named Mandelbrot in the library, which I saw recently. In fact, Benoit Mandelbrot's "The Fractal Geometry of Nature" had just come out and here we found everything about Julia Sets and of course the Mandelbrot Set. But compared to his graphics, we were at an advantage. We had our hands on the prime computer graphics equipment of the time. Soon we were lost in running experiments most evenings and weekends, over Christmas and New Year until we returned to Germany in the spring of 1983. Dietmar did not get the expected use out of his newly purchased skiing equipment. When we returned to Bremen we had almost a whole suitcase filled with 35mm slides. In those days that was the way to capture images from a computer screen. 


Back in Bremen we had only one goal: How could we get similar equipment? Peter Richter, with whom I was heading a research group was immediately fascinated too and eventually we obtained with funding from the Stiftung Volkswagenwerk and the Deutsche Forschungsgemeinschaft very competitive equipment, an AED 767 raster graphics terminal and the Rolls Royce of vector graphis, the legendary Evans & Sutherland PS 300. In fact the Deutsche Forschungsgemeinschaft was not immediately convinced that matematicians and physicists would need such top of the line equipment. But we prevailed, and that is how our own computer graphics lab came into existence and some golden years of experimentation followed and we all got deeper and deeper into fractal geometry and computer graphics. We produced scientific movies with the the Göttingen Institut für den Wissenschaftlichen Film (IWF), which received several international awards.


Towards the Goethe Exhibits

What also followed quite unexpectly was a public exhibit of framed pictures selected from the Salt Lake City slides together with a catalogue in the Bremer Sparkasse, a bank in Bremen with a beautiful art noveau main hall.


This exhibit early in 1984 attracted quite an unexpected interest in the public and media, and in particular from Manfred Eigen, recipient of the Nobel Prize in Chemistry in 1967 in Göttingen and Friedrich Hirzebruch in Bonn, both headed a Max-Planck-Institute that they had founded. Soon we had follow-up exhibits in Bonn and Göttingen. And then Arnold Mandell, a professor of psychiatry from UCSD gave a crucial impulse. I knew him from conferences in interdisciplinary dynamical systems and when he saw our images he insisted that we should aim for an exhibit in the US. He knew the director of the Museum of Modern Art in San Francisco and made the connection with him. I visited and he made the connection with the late Frank Oppenheimer, then the director of the San Francisco Exploratorium. When Frank saw the pictures he was immediately enthusiastic: "We must show them here. We must have them, but I don't have funding." He made the connection with the director of the San Francisco Goethe-Intsitute, whom I visited with Frank that same evening. Of course he had no money either, but connections. He made the connection with the head of the Goethe-Institute organisation in Munich who was interested, but not sure that he could have the idea approved by his committee, a committee of art historians etc., which was in charge of selecting exhibits for world-wide shows in the more than 150 Goethe-Institutes.

Manfred Eigen and HOP in Göttingen

Naturally the question came up: "Is this art?" And the answer was: "No, not at all. It is pure mathematics." They had never seen something like this. We needed high level help. Manfred Eigen wrote a letter of recommendation, which could not be neglected by the committee and we were on. The opening of the first exhibit was scheduled for 1985 and  we began to spend late evenings and weekends to redo many of our Salt Lake City experiments in better image quality and did many new experiments, in particular around the Mandelbrot Set. The title of the exhibit was chosen to be "Schönheit der Fraktale - Bilder aus der Theorie Komplexer Dynamischer Systeme". 



                          Catalogue for Goethe-Institute Exhibits                  

                                                   GEO Magazin June 1984         

While we were frantically working on our experiments, the press officer, Klaus Sondergeld,  of the University of Bremen had the idea to present our images to GEO, a very popular monthly magazin in Germany, devoted to top photography (at that time ) of geological themes. We were allowed to present to the editor in chief, Rolf Winter, and he did not even consult with anybody, when he decided on the spot to have our mathematics pictures augmented by a wonderful cover story. On top of that he offered 15 000 Deutsche Mark, which gave our small company MAPART GbR, which we had founded to market our pictures and copyrights, quite a boost. NFrom there on we used the proceeds from MAPART to improve our lab equipment.


The Goethe-Exhibit started its world tour in the middle of 1985. In fact we had prepared two copies of some 100 beautifully framed cibachrome images. The first showings were at the Oxford Museum of Modern Art, sponsored by Oxford University and in particular Sir Michael Atiyah, a close friend of Hirzebruch, and the second almost simultaneous showing was at the San Franciso Exploratorium, sponsored by Stanford University and in particular by the late Robert Ossermann. Sadly Frank Oppenheimer, who was key to winning Goethe-Institute,  had just passed away a few months before the opening of the exhibit in San Francisco. Later the two copies toured all continents and far more than 100 cities.


Before the exhbit left our lab we wanted to have a showing in Bremen. The Goethe-Institute is mostly active outside of Germany and therefore the only chance to show our work in Germany was to organize a show by ourselves. So we did and rented art exbition space in the Bremen Böttcher Strasse, which is devoted to the German painter Paula Modersohn-Becker. The exhibit ran from May 11 to June 18, 1985.

The Books and Mandelbrot

Two major paths evolved in connection with the Goethe-Institute exhibits and the fare-well-exhibit in Bremen. One day during the preparation Peter Richter said:"Wouldn't it be great if  Mandelbrot came for the vernissage?" We digged out his phone number and I dared to just call him and propose our idea. To our surprise he accepted and that is how we first met him in person and soon a life-long friendship evolved which led to many new adventures. Mandelbrot opened my eyes to worlds that previously were unknown to me, and my soon revitalized love for the arts and music were an indirect consequence of the interactions with Mandelbrot himself and his work and what his work initiated in me.  Mandelbrot himself had started his journey when he studied the rather obscure Zipf Law, which describes the frequency distribution of words in a language. This galvanized his later work on fractals, and in a similar way, when I crossed his path that galvanized my later work in medicine, or to put it differently prepared me for my work in medicine. Incidentally, there is a fantastic video about Zipf's Law, which I believe is a must see, by Michael Stevens called The Zipf Mystery


While we were preparing the images for the exhibit the irresistabel idea came up to have an  exhibition catalogue. Hence we started to work on that simultaneously and eventually produced the catalogue including some xx color images. Having color printing at that time was still quite expensive, but we fortunately had the financial resources of MAPART.  One of our outside guests at the vernissage in Bremen was Joachim Heinze, at that time the mathematics editor at Springer-Verlag in Heidelberg. He saw the catalogue and immediately had the idea to use the material for a mathematics book. During a several hourlong walk along the river Wümme we hammered out the outline of the mathematical content that should be written around the images taken from the catalogue. Joachim Heinze had to present the project to his advisory board, made up of leading mathematicians. They were rather skeptical because in the 1980s pictures were not yet acceptable amoung Bourbak style mathematicians, who were still dominating the mathematical society. But with the backing of his boss, the late Heinz Götze, he was able to outvote his board and thus the book "The Beauty of Fractals" was conceived and was published one year later in August of 1986, a record time for writing and producing a book! Incidentally, when Springers' planning team was confronted with the large volume of color images they were helpless regarding the high cost of production. As a result it would have been necessary to overprice the book and quite likely it would have become a non-seller.  The solution was that Peter Richter and I subsidized the color printing so that the bookstore price came out to be around 70$ (of course with a contract, which would pay back to us, if the book would be a success). And it became a success. It became the all-time bestseller of a mathematics book of Springer-Verlag. It was later translated into Russian, Italian, Chinese and Japanese.




In 1992 we published the book "Chaos and Fractals: New Frontiers of Science" with Springer-Verlag New York, co-authors Hartmut Jürgens and Dietmar Saupe. Paul Manning at Springer was responsible for pushing it to a large best selling success in US.  In 2004 a second edition was published with one significant change: The first edition contained two invited appendices, one about Fractal Image Compression by Yuval Fisher, and one on Multifractals by Carl Evertsz and Benoit Mandelbrot.  A Chinese translation appeared in 2008. The idea of the book was to present the topics of chaos and fractals as comprehensively as possible and as correctly as possible with using the usual high level mathematics. One measure to achive this was to use illustrations, experiments, and tables. In fact the second edtion still had more than 600 (!) illustrations,  and another measure was to write two books in one binding: A running text in a more popular science style of presentation in a serif font, and a more technical style of persentation inserted in the running text in a sans serif font. Similar to the publication of "The Beauty of Fractals" and its excessive number of color plates, this time the publisher was concerned about the excessive cost for typesetting and layout of the book with this excessive number of tables and illustrations. The solution was that we, in fact it was Dietmar Saupe, took on the typesetting and complete layout of the book ourselves. This was quite an undertaking, which Dietmar mastered with excellence sqeezing LaTeX to its limits.

There is an amusing annecdote connected with "Chaos and Fractals".  Barnes and Nobles had the idea for a special sales campaign for Christmas 1993. They wanted to have 1500 copies hand signed by me. Paul Hrdy, then heading marketing and sales at Springer flew me in from Germany and I sat for an entire day in Springer's warehouse in New Jersey signing books. Every so often I had to change pens from ball point to soft and take a rest because I hardly could write my own signature anymore. In fact, during the day my signature degraded from well readable to much less readable and shorter and that is what it is still today. The following spring term I gave a course on chaos and fractals at FAU based on my book and students didn't have to but were better off having the book. One student came and said: "I bought your book at Barnes and Nobles in their University Commons store and I got a great price on it. Guess why? They had several copies. I found one with your personal signature in it and went to the front desk asking for a huge reduction, because someone had scribbled something in it."

Twice "Chaos and Fractals" made an appearance in popular sitcoms: Murphy Brown and Seinfeld. In Season 8, Episode 9 of Seinfeld,  "smart George" is seen carrying around the textbook "Chaos and Fractals: New Frontiers of Science" throughout his scenes at the diner. While Murphy Brown (Candice Bergen) is testing a date on her just aquired knowledge about fractals.



A few years later Paul Hrdy made me a wonderful gift. Springer had decided to give up their own warehouse and have outsourced the fullfilling of book orders, following the advice of one of the multinational advising companies for process optimization (not much later they regretted that step and had there own warehouse again). As a result they had to reduce the books in stock dramatically. Those which were sorted out would go to the trash. I was allowed to spend a day in the warehouse and pick books. In fact I picked two huge piles of some several hundred books, one for the math department at FAU and one for my institute CeVis in Bremen. The latter one later became a gift to Jacobs University.


Mathematics in the Later Years

Mathematically, I and my students had moved from partial and functional differential equations and their numerical anlysis to chaos and fractals, studied Newton's Method in the complex plane and in two real dimensions from a fractal perspective, found and discussed connections between the theory of phase transitions in a renormalization setting and Julia sets, and so on.


The second half of the 1980s were the time of accepting mathematical experiments as a way of life. The establishment in mathematics was not sure about this. A small anecdote may emphasize this. At the end of the 1980s, around the time when I received an offer for a chair at the University of Bonn (which I declined), Hirzebruch invited me to run a Semester on Dynamical Systems in his Max-Planck-Institute for Mathematics. In the middle of that he asked me to prepare a proposal for modernizing the computer equipement for the Institute. At the time the Institute had hardly anything (except a few PCs), and the most powerful computer was some HP desk computer, barely more powerful than the PCs at the time. I proposed a lab with up-to-date SUN or SGI workstations. Hirzebruch asked me to provide a detailed plan with financials suitable for submission to the Max Planck Society. I did. The proposal was never submitted. One of the members, Don Zagier, at the time the young star of the Institute, who was a heavy user of the infintely slow HP for problems in number theory opposed. His argument was: "A mathematician should use a rather powerless computer. Otherwise there is the temptation to compute too much and too fast without thinking." I am sure he meant a good mathematician. In other words, he meant powerful computers were for the less gifted. Of course today computers and computer experiments have become completely natural and integrated into mathematics almost everywhere, and pictures are allowed, although some mathematicians still speak about Bildchen, as if they feel a need to apologize.


Later, in the beginning of the 1990s my mathematical interest included some number theoretical problems around automatics sequences, finite and cellular automata, some of which have applications in theoretical computer science, and IFS (Iterated Function Systems). Here is a nice popular site for IFS.


Guentcho Skordev and my student Fritz von Haeseler at CeVis were instrumental in these investigations, which started with a rather innocent question from fractal geometry and elementary number theory: If one looks at binomial coefficients mod 2, in other words an even coefficient is replaced by 0 and and odd coefficient by 1, then a pattern emerges which is reminiscent of the Sierpinski Triangle. What happens for mod p, where p is a prime, or a prime power, or a composite number were increasingly challenging problems. It turned out that a combination of renormalization techniques with finite automata and hierarchical iterated function system gave us answers.


A frequent guest from France, Jean-Paul Allouche,  galvanized our interest. I met Guentcho when he and I were still working on fixed point theory in algebraic topology and when he came to Bremen we continued these investigations further, after I had not looked at such problems for more than a decade. 


When Carl Evertsz arrived at CeVis around 1991, after his post doc years with Mandelbrot,  our interest expanded into the mathematics of finance along Mandelbrot's critique of the traditional theory of efficient markets. Peter Singer, an expert in martingales joined us from the University of Erlangen and after I had aquired some funding from the Deutsche Investment Trust (at that time a daughter company of Dresdner Bank) we engaged heavily in freeing the Black-Scholes formalism from the hypothesis of efficient markets using intensive computer experiments. Key was to build a new theory in which Einstein's Square-Root-Law would be replaced by Hurst-Exponents. It looked like this would be the new direction of our mathematics for quite a while. But it didn't. An unexpected dinner table conversation with Klaus J. Klose, then Head of Radiology at the University Hospital in Marburg, shot me into a new world, like a ball hitting a bumper in a pinball machine, the world of radiology and surgery. I would never have guessed that.

Computer Science: 


Enough of Computers

I had my full dose in using computers in mathematics around 1970, when I used computers in algebraic topology. I remember this time as both wonderful and stressful. I had nobody to talk to. The algebraic topology community was not interested in computers (yet). There were horrendous problems and computer limitations that I had to overcome. It was not at all clear whether or not it could be done what I attempted and eventually succeded to do: Compute the cohomology ring of simplicial pairs. At that time on the IBM 7090 the language to use was FORTRAN and some ASSEMBLER and very little system language. Everything lived on punched cards. When it was done and published I felt that I wanted to fully concentrate on pure mathematics. I had enough of computers. It is hard to describe what using computers meant at the time, standing in line for a punch card machine (some sort of special typewriter punching holes in cards), assorting the cards in a special box and delivering the box to a shelf in the computer center, where operators would pick up the cards and feed  them into the computer, picking up the paper print-out from a personalized shelf box in the compuer center, looking at the typically unexpected results (something wrong, but what), and start all over. We would never know when our job would be processed. That depended on the workload of the computer and the resources required by our own program. The computer would run 24/7. As a consequence we would often hang out near the computer in crammed space, smoke or eat cheap food and wait and check, wait and check long hours for our paper print out, often late evenings, often during the night. There was no information systems by email or texting (not serious). What we did in a week at that time, probably could be done in an hour today. And there were no parsers as we know them today and finding mistakes was a pain. But we didn't know better and thought that we were a very exclusive club. Only two years later the IBM 7090 was replaced by an IBM 360 with terminals for input and communication and that was a major revolutionary step which came too late for me. Had I continued my work in making algebraic topology computational I would have had a much easier life now. But as I said, I had it with computers for a while and missed out on UNIX and C and all that great stuff. 


Salt Lake City - Salt Lake City

Before the two sabbaticals in Salt Lake city my group  had worked intensely in simplicial continuation methods but my contact with computers was very loose at best. This changed during the first sabbatical from 1979-80, when Robert Barnhill let me and my students use his legendary Evans & Sutherland PS/2. THis raised our interest for computer graphics hard- and software for the first time.


During the second sabbatical from 1982-83 Richard Riesenfeld let us use all the equipment in his absolutely fantastic and world class lab in the Computer Science Department. As a by-product we learned much about his groundbraking work in computer graphics and were inflamed. Richard made the connection with SIGGRAPH (Special Interest Group Computer Graphics) and he showed our work on fractals (Julia Sets) on the annual meeting of SIGGRAPH in 1983 in Detroit. His lab and his way of running his lab was an inspiration for me to build a similar lab in Bremen. 


Back in Bremen in the summer of 1981 our lab development, headed by mz student Hartmut Jürgens, went through several stages, all inspired by the computer graphics role models in Salt Lake City:


1981:  NORD 100 Computer

         Tektronix 4014 (loan from the University Computing Center)

1983:  Evans & Sutherland PS300

         Matrix Camerasystem for slides and film

         AED 767 Rastergraphics

1986: Bull SPS9/60 Computer

         Raster Technology One/360

1988: Management Graphics Solitaire

         Sony U-Matic editing recorder

1989:  Silicon Graphics Iris

1993:  Silicon Graphics Onyx

1996:  Silicon Graphics O2 


During the years from 1981 to 1988 our lab was probably the only computer graphics lab in the hand of mathematicians and physicists. Peter Richter and his group as well as I and my group produced several movies with the Goettingen Institut fuer den Wissenschaftlichen Film (IWF). And we produced two movies for Spectrum-Verlag, Heidelberg, which was responsible for the the German edition of Scientific American. The second movie titeled "Fractals: An Animated Discussion". It was distributed 1990 by W.H. Freeman as a VHS video. It featured two interviews that I did with Edlorenz and Benoit Mandelbrot and computer graphical experiments around the Mandelbrot Set, Julias Sets and the Lorenz. Attractor. In one of the scenes we zoom continuously into a particular spot of the Mandelbrot Set which we named "Seahorse Valley". This was inspired by the legendary movie by the late Charles Eames titeled "Powers of Ten". I had met Charles Eames through my friend the late Raymond Redheffer from UCLA, who was the mathematical advisor for the movie and had this movie in the back of my mind all the time.  



Computer Science
Juergens und Evertsz

There is another scene in this video, which originated in our goal to show how chaos and fractals are intertwined. The experiment with which we came up turns out to be one of the nicest ways to explain what chaos means and how it connects with fractals. We choose a simple pendulum (iron ball swinging) in interaction with three magnets. 


In 1985 there were two major events which impacted our visibilty beyond expectation. In August of 1985 Scientific American chose to publish a simplified version of our computer code to generate images of the Mandelbrot Set and they chose one of our images as the front cover. This led to a world-wide fractal mani. Probably the Mandelbrot Set is the one object of mathematics on which the cumulated computer time that has been used world wide exceeds by far verything. The net ist still saturated with images and videos. But few have really done a good job of discovery or beauty. For me two more recent ones stand out:


- the mantric video and music  (Hunting for Fossils with music from PrinsJan)

- the iPad App by Frax HD by Kai Krause and friends.


The second was the SIGGRAPH annual meeting in July of 1985 held in San Francisco. Benoit Mandelbrot was invited to run a course on fractals and Dietmar Saupe and I were invited by Mandelbrot to teach our stuff. This was the first meeting with Richard Voss and SIGGRAPH was an experience that words cannot describe. Some 40,000 attendees (not in our course though) celebrated the merging power of computer graphics in scientific visualization and Disney animation. We all felt like witnessing history. The year after we run a course on the annual meeting in Dallas and in 1987 I was invited to run the course and I invited among others D. Saupe, Robert Devaney, Richard Voss and Michael Barnsley. From the beginning I intended to publish the course notes and so we did. The book "The Science of Fractal Images", co-edited with Dietmar Saupe, came out in 1988, published by Springer-Verlag, New York, and became an astounding success due in part to the efforts of Ruediger Gebauer, the math editor at Springer. In 1989 it won an award from the Society for Technichal Communication. This book brought fractals to the computer science and in particular to the computer graphics community. In summary, the core team of Dietmar Saupe, Richard Voss and me, and several with Benoit Mandelbrot,  did a total of 10 successive invited courses on SIGGRAPH annual meetings from 1985 - 1994 (from San Francisco, to Dallas, Anaheim, Atlanta, Boston, Dallas, Las Vegas, Chicago, Anaheim and Orlando). 

The cover shows a picture combining a rendering of the true electrostatic potential of the Mandelbrot Set near its boundary in fantasy colors (here like a snow landscape) generated by Hartmut Juergens together with fractal clouds, based on fractional Brownian motion, generated by Dietmar Saupe. One of my all time favorite images, which brings together the worlds of mathematical fractals and random fractals. Incidentally, the part of the Mandelbrot Set shown on the cover of the book is the same as on the cover of Scientific American: Here we see the electrostatic potential rendered as a mountain ridge with cliffs falling off (in fact the potential falls of exponentially), and there we a rendering of the electrostatic potential with some highlighted equi-potential curves. Note however that for comparing the two renderings a flip like mirror imaging should be performed. The image on top of my website is another beautiful rendering of the Mandelbrot Set by Hartmut Jürgens. Again we see the 3D-rendering of its electrostatic potential.  A Japanese translation of the book appeared in 1989.

That is how we got involved into computer graphics and from 1991 onward our involvement took another turn and became very serious, when we began our journey into medical image computing which is filled with innovative visualization and rendering techniques. But our emphasis soon changed from purely scientific within computer graphics and computer science to a much broader ore even different set of goals.  

Medical Image Computing: 


For more reading go to MeVis - The First Ten Years.


Founding MeVis 

The not for profit research center MeVis Research gGmbH was founded in August 1995.  During the years from 1991 to 1995 our work in medical image computing was out of CeVis, the institute within the Department of Mathematics of the University of Bremen. It became more and more obvious to me that we were at a bifurcation point. Should the primary focus of our research be on merit in medicine, or should it be on merit in mathematics and computer science? I knew what I wanted, I wanted medicine with no compromise. Consequently the imbedding of our research within a disciplinary department of mathematics or computer science would be problematic. With the help of the City State of Bremen I founded MeVis and became its CEO. Initially the research staff were a handful of visionary people who dared with me to chart unknown territory, because there was nothing like MeVis in Germany at the time. There were somewhat comparable institutions like Max Viergever's  ISI at Utrecht University, Hans Reiber's LKEB  at Leiden University Medical Center, or Kunio Doi's Kurt Rossmann Lab at the University of Chicago. I visited all, prior to founding MeVis, to learn and pick up ideas for my own center.  In two essential regards MeVis was designed differently: Firstly, it was and remained independent from a university. I very much wanted independence and freedom. I did not want to work under a chairperson of a department, a dean or a president, because I knew that they have to balance many competing groups in their realm. I wanted to follow our own and nothing but our own trajectory with the obvious consequence that we had to take care of ourselves without the umbrella  and safety net functions of a university.  The City State of Bremen promised a base funding and some 20-30% and we had to cover the remaining budget by research contracts from the DFG (Deutsche Forschungsgemeinschaft), BMBF (Federal Ministry of Research), European agencies and industrie. Secondly, Unlike ISI, LKEB or the Rossamann Lab we were not imbedded within a strong research environement of a universiy hospital, which appears to be a must-have according to the prevailing opinion, if you want to have necessary medical input and impact.


The University of Bremen did not and still has no university hospital under its wings nor is it associated with a hospital. Thus, when designing und building MeVis this strategic disadvantage had to be turned into a strength. The idea was to grow a regional, national, European and finally world-wide network of clincal research partners. This is nice to say but it was more than  difficult to achieve, because this was not the style of research at university hospitals and we were largely unknown in our first years. Funding was also a tremendous problem in the first years for two reasons: Firstly, we had no track record yet. Secondly, the digital transformation in radiology, which was our main theme at the beginning, was still in its infancy and the majority of radiologists, certainly in Germany, were still doubtful that research would be key for their future workplace. Thus funding in national programs virtually did not exist yet.



Medical Image C

Mammography and Breast MRI

In 1995 I met the late Jan Hendriks, from Radboud University in Nijmegen. Jan, a radiologist, was responsible at the time and had built the Dutch national breast cancer screening program with his friend Roland Holland, a Hungarian born pathologist. Jan and Roland were European pioneers in breast cancer screening and were very open minded about technological and methodological improvements. They soon became life-long friends and mentors. Through their vision and enormeous experience I was put on a steep learning curve regarding the intricacies, pitfalls and tremendeous challenges and responsibilities of screening. Mammography at the time was film-based and digital detectors were far from being sufficient in terms of the necessary image resolution that is required in mammography.


However, Jan was looking ahead and he and I knew that it would only be a matter of time, 3 years, 5 years, 10 years, when digital detectors would be available with sufficient detail. We started to talk about the features of software, which could support the mammography reading by a breast radiologist. And at MeVis we started a first prototype development, which had to overcome huge challenges. One example is worth mentioning: Of course at that time we could only work with digitized film mammograms. But to maintain their information integrity they had to be digitized at very high resolution, resulting in huge amounts of data per film. Straight forward loading times from memory into a display monitor were exhibitive and peculiar software innovations were required. This lead to a very fruitful collaboration which upruptly ended when Jan died in 2004. But in between we expanded our joined activities from year to year, entered two Europe-wide research programs, which combined the expertise of all European breast cancer experts, went through numerous cycles of development,  field testing, and redesign, and by 2001 we had solutions which were ready to go into routine use in European screening centers. That is when MeVis BreastCare GmbH was founded with Carl Evertsz as CEO, a joined venture between Siemens Healthcare and MeVis Technology GmbH, our initial commercial spin-off. Incidentally, Siemens a few years prior had dismanteled their digital  mammography efforts, because they simply didn't believe it would be viable. Another company, Hologic, in the US, however introduced detectors, which promised to have the proper high image resolution, but they had no software solutions for reading digital mammograms. Siemens chose to license Hologic's detectors and being partner in our company offered our software in return. Thereafter, MeVis BreastCare grew with the success of Hologic and Siemens and became the world-wide market leader in digital mammography-reading software solutions. If MeVis would have been under the umbrella of a university medical center it is very likely that this success would have never happened. Unlike in my original scientific home, mathematics, where international collaboration is always promoted, medical research is very peculiar. Cross institutional collaboration is not cultivated and often even prohibited.


Jan Hendriks

Henny Rijken, Roland Holland, HOP, Jan Hendriks, Carl Evertsz

There was another early impuls in the development of MeVis. Around 1996 we learned from Burckhard Terwey, who was running a very successful private MR practice (ZEMODI) in Bremen, about the benefits and challenges of Breast MRI, which had been introduced by the German radiologists Werner Kaiser and Sylvia Heywang-Köbrunner. In fact there is a great anecdote about Sylvia, which I will not forget to tell. Breast MRI is based on a very peculiar property common to many cancers: As they grow they stimulate the growth of nearby vascular systems to feed them (angiogenesis). These new vessels, however, have some sort of defect, when compared with regular vessels: They are more leaky. Now, if one administers a contrast agent and follows the flow of it in the breast, one is able to detect the leakyness by measureing and comparing the dynamics of contrast agent enhancement in the tissue, and thus finding a sign for a possible cancer.  There were many problems with breast MRI: it was not adopted by the insurances, the exam was very slow, there were no clear criteria how and when to use it, and most radiologists were overwhelmed by the amount of data and  information. At the time a radiologist could easily spend an hour on a single case. The need of software support was eminent.  Thus it looked very promising to start working on breast MRI, also because it promised to fill some of the gaps that mammography left.  Our assessment with Burckhard Terwey was that a dedicated software for analyzing and reading the massive amounts of data should be extremely beneficial for speeding up the diagnostic process and for improving its fundamentals. And hopefully this would lead to a broader adoption of breast MRI in the community of breast cancer diagnosticians. His advice was the lift-off for our breast MRI project, which started in 1996 and is still ongoing. Another radiologist in Germany who played a key role in guiding our research was Joachim Teubner from Heidelberg. When the first prototype was ready in 1997, running on an SGI O2 computer (PCs or MACs were far from enough powerful at the time) it was chosen to become the first product for the commercial spin-off company which we (Carl Evertsz, Hartmut Jürgens and I) founded in 1997 with our own funding (we had to take a sizable bank loan) and venture capital. The company was named MeVis Technology GmbH. We hired a CEO from outside, who did not work out however, and as a consequence Hartmut took the position of CEO. 

Multimodal breast view fusing breast MRI with digital mammograpgy: the way to go in the future

Our product had a moderate success in European breast cancer centers, in particular in the Netherlands, Sweden and the UK. But the time was not ripe for this innovation yet. As a result of my professorship at  Florida Atlantic University I met Kathy Schilling and Jon Wiener. Jon is an MR radiology specialist of the first rank and very open to innovation and Kathy was heading the Boca Women's Center, one of the largest breast cancer centers in the US.  Like Jon, Kathy was very open to innovation and collaboration. As a result, one of our SGI O2s arrived in Boca Raton and an extremly productive and wonderful collaboration started which is still ongoing.  In fact we did many joined projects, for example mammotome breast biopsy and computer supported teaching projects for Johnson & Johnson. But most importantly our software for breast MRI matured in the collaboration and Kathy and Jon conducted a significant patient study, which they published in the American Journal of Roentgenology in 2005:  Without going into too many medical details, their findings were that in the group of women, where mammography had found a cancer which would recommend itself for local surgery, breast MRI found additional cancer to the extend that a local surgery would have been the wrong treatment. This finding helped greatly  towards adopting breast MRI in the US. Today breast MRI is standard practice in the US and is established for a number of indications and is of proven benefit for certain subgroups of women. 

Left to right: Richard Voss, Kathy Schilling, Karin Peitgen, HOP, President Frank Brogan, Nanette Peitgen, Benoit Mandelbrot, Carl Evertsz in the FAU's President residential mansion 2006, after Benoit and I had been inducted into the Hall of Fame of the Charles E. Schmidt College of Science 

Eventually our product became the leading software for breast MRI analysis and reading in the US through a partnership with the company MRI Devices, which later was merged into Invivo, which is now a part of Philips Healthcare. This match was perfect for us and let to several innovations, the most important from a clinical point of view was the first ever software for computer guided breast biopsy under MR. The matchmakers were not surprisingly Jon Wiener and Kathy Schilling. The head of the original company MRI Devices, Tom Schubert wanted to enhance his business, which at the time was strictly hardware, for example the leading head and breast MRI coils in the market, to a business of complete application solutions, in which software was smoothly integrated into and fine tuned for their product. A whole palette of innovative software solutions was developed, first for breast cancer, then for neurological applications, and for prostate cancer, each under the brand name DYNACAD, and finally for lung cancer. The name was combining two essential features which were common to each type of application: CAD = Computer Aided Diagnosis, and DYNA, meaning that the diagnosis was based on assessing the dynamics of the flow of a MRI specific contrast agent in the respective organ.


But I promised an annecdote about Sylvia Heywang-Köbrunner. Around 2000 we wanted to intensify our research at MeVis in breast MRI and looked for advice from her. She was always extremely busy, had a position at the University of Halle in East Germany at the time, and I convinced her to meet with us in Bremen between two other obligations, which she had. The deal was that I would pick her up at the airport Leipzig-Halle with my small single engine Piper Arrow and take her to Bremen. Remarkably she was not afraid although our flight was in instrument conditions all the way. Descending with 500 fpm on the  ILS 27 instrument approach into Bremen we still were in the clouds. Suddenly the clouds broke and we found ourselves between two layers, and within seconds we saw a Cessna plane coming direct towards us. I deviated, the other pilot deviated, we barely passed each other, and already we were back in the clouds, since we were descending at 500 fpm. I informed the tower of the incident and they said that they did not have the other plane on their radar. After we landed safely Sylvia was still in a good mood and surprisingly calm. We went up to the air traffic controller's office and learned what they believed had happened: The Bremen Approach Radar was and is computer supported. The computer has many functions and one is to sort out signals, which are likely to be erroneous radar signals. How did the computer manage the sorting out? Flying in the approach zone of controlled airports requires permission by the controllers. IFR traffic, like we were, has that permission automatically and is squawking a designated Squawk Code for identification. VFR traffic has to squawk 0021 at all times. Now the solution for the almost tragedy: The pilot in the other plane did switch off his transponder, so no VFR squawk code was transmitted, thus the computer came to the conclusion there is a target but no squawk, hence it is erroneus and filtered it out from being displayed to the air traffic controllers, and consequently, they could not warn me and direct me away from the conflicting traffic, as they have to for instrument traffic, and even worse, they could not talk to the other pilot, because they were not aware of him, because he didn't squawk and did not ask for permission to fly into the controlled airspace. Worse, he should have never been flying in between layers as a VFR pilot, because that is another serious violation. This pilot, who almost killed us was never identfied and caught.


Sylvia and I learned quite a lesson about the mistreatment of "false" signals by computer support, which is one of the great, and will always remain so, challenges in computer aided decision support in medicine at large, and breast cancer detection in particular.  

Liver Surgery Planning

Our liver surgery planning efforts started in 1992, (long before MeVis) in the institutional framework of CeVis, which I had founded in the same year as an institute with the mathematics department. The instigator for our efforts was Klaus J. Klose, then head of radiology at the University hospital in Marburg. He and his group had struggled with a rather profound problem in liver surgery for some time and when he became aware of fractals and my research he managed to get to talk to me in a very peculiar way. In November of 1991 I gave a keynote lecture for a company, Schering Pharma, at Schloss Reinhartshausen, which is a famous German winery and hotel. The audience was a few hundred private practice radiologists from the region. As a university chief, Klaus did not belong there,, but his wife did, and so he slipped in. He managed to be seated rigth next to me at the festive dinner and though I was exhausted from my 2 hour long lecture, he talked to me with a contagious enthusiasm about a particular challenge in liver surgery, for which he wanted to win my collaboration. Shortly after, a small team from Bremen visited Marburg, and his team gave us comprehensive introduction.

When back home, I liked the problem(s), I guess in particular, because they made two connections with our expertise: fractals and numerical continuation methods. In approximation the vascular trees in a liver, the portal vein, the liver vein, the liver artery, and the bile ducts, form a perfect fractal each, and one of the complications is given by the fact that they vary enormously from patient to patient and that they are wildly intertwined. 



The picture is a photograph of the portal venous tree obtained by a delicate casting method. Jean H.D. Fasel, a liver anatomist from the University of Geneve provided the image. He is one of a few anatomists, who still cultivates the skill of casting anatomical objects. Jean contacted me, I believe in 1994,  after our first publication on computer supported liver anatomy and offered his help.  His expertise and in particular his extremely rare and therefore valuable liver casts became of key importance in our own work.  So what was the problem which Klaus J. Klose gave us?


In oncological liver surgery a surgeon needs to take out a tumor, which is surrounded by these wildly intertwined vascular trees. In fact, the surgeon needs to take out the tumor including a certain safety margin, making sure that all tumor cells are removed. This means that the surgery will take place in healthy liver tissue, which is densely permeated by vascular structures. Thus the problem is to remove the tumor safely with minimal damage to the liver and its remaining function. 


In 1957 the great liver surgeon and anatomist Claude Couinaud devised a schematic solution after nearly 100 years of anatomical research. He had studied numerous livers from deceased patients and proposed to bring unifying order to the formation of the vascular trees. This can be seen in the following picture: 


In essence he claimed that the anatomy of the liver is guided by the portal venous tree and can be paritioned into 8 segments, the famous Couinaud-Segment. His schematic partition became the guiding principal of liver surgery! Every surgeon used it and probably most thought that Couinaud discovered a secret of nature and that nature of the vascular trees really was like that. When we started to work on the problem the Couinaud's partition was so deeply ingrained in surgeons minds and beliefs that in retrospect it appears naive to fight it or to get rid of it. In fact, one of his colleagues and greatest liver surgeons of the century, Henry Bismut,  defended Couinaud up until very recently. In one of our last anatomical papers with Jean Fasel 2014 


Majno, P., Mentha, G., Toso, C., Morel, P., Peitgen, H.O., and Fasel, J.H.D. Anatomy of the liver: an outline with three levels of complexity. A further step towards tailored territorial liver resections. J Hepatol. 2014; 60: 654–662


we did show how far from nature the schematic partition actually is and Bismut wrote a remarkable editorial about our paper, in which he shows his entire greatness. We quote:


"Why is this leap in complexity needed? Because specialized hepatobiliary surgeons now operate on more advanced disease, with more frequent reoperations on more fragile patients, and are under the constant pressure of minimizing complications and of sparing as much liver tissue as possible. Surgery based on the real vasculo-biliary anatomy is the only way to achieve these goals, and I am not surprised that it is becoming more customised. The call to free the operation from Couinaud’s 8 segment scheme is in fact not subversive, it is the natural evolution of the concept we introduced in 1982 to the new imaging and surgical techniques, and to the identification of patients who can benefit from it. Of note, the articles I wrote in 1982 were before CT imaging, and the surgeon had to rely on preoperative and intraoperative ultrasound that by definition was done on the individual patient. There was no discrepancy to worry about."


Two of Bismut's insights are important: Firstly, that more sophisticated and daring liver surgery needs to take the real anatomy into account and will benefit from it. Secondly, at the time when (our) computer methods did not yet exist, the schematic view was better than nothing and therefore helpful.


But to get there wasn't a smooth ride for us. I still remember when famous liver surgeons seeing our results and methods would say: "A good surgeon doesn't need this. We have learned to work with Couinaud that is good enough."


In the images below we see a study that we worked out with Jean Fasel, where the competing schematic assertions before and at Couinaud's time are compared with the real vascular anatomy. By just observing their vast differences one can see how problematic the schematic approach was and is: None of them is right, none of them is wrong. They are all forcing the variety of natures freedom into a cage. We see partitions relating to the schemes of Couinaud with susegments, Hjortsjo, Couinaud, the original tree, Goldsmith, Platzer.

So how did the liver surgical community come to accept that there was promise beyond Couinaud? Our first software solutions to assist in risk determination, risk minimization and surgery planning were ready around 1995/96. The software was collosal. It needed more than one expert in our lab to actually run it. But the response in the German surgical community was not encouring. This changed when I met Karl Oldhafer in 1998 at a conference in Hannover.  Karl immediately recognized the potential of our work and became a loyal research partner and friend. He brought me into contact with Christoph Broelsch, Jürgen Fuchs and Peter Neuhaus. All four surgeons were students of the legendary liver surgeon Rudolf Pichlmayr from Hannover. Broelsch was famous for introducing living-donor-liver-transplantation in 1990 at the University of Chicago and later used our methods in transplantation surgery, Fuchs is a world renowned abdominal pediatric surgeon with  special expertise in the surgery of neuroblastoma, and Neuhaus is a world renowned liver surgeon. Eventually we would work with all, but Karl caught fire immediately.  Broelsch later became the prefered doctor of German President Johannes Rau. 

Koichi Tanaka

In 2002 our efforts to convince surgeons that our software support could help them obtained an unexpected boost. In June of 2002 Japanese and German surgeons held a scientific meeting in Berlin, celebrating their long history of friendly relations and professional exchange. Peter Neuhaus was in charge of the conference and had invited me to give a keynote lecture.  There was my window of opportunity to demonstrate our work to high ranking surgeons from Japan. That was not exactly the topic for which Peter Neuhaus had invited me, but I did not want to miss this opportunity, which appeared like handed to me on a silver plate. After the lecture I was surrounded by several of them. They were eager to get more information and : "How can we use this? Can you make this available to us?".  


Incidentally, the evening before the conference there was a festive dinner with many speeches as it is typical for such events. But then something happend which I couldn't believe and will never forget. The Japanese surgeons stood up, gathered in front of us at the head of the room and began to sing German folk songs in German (!), and invited us to join them!


One of the surgeons who cornered me after my lecture was Hiroshi Shimada, head of abdominal surgery at Yokohama City University. Soon we entered into an extremly fruitful collaboration, which was continued after Hiroshi retired by Itaru Endo, Shimada's successor. Our specialty with them was the fine tuning of our methods for the surgery of cholangiocarcinoma (bile duct cancer), which are a huge challenge to operate for liver surgeons. Later Itaru demonstrated in a study that using our software support the one-year-survival-rate increased by 20%.


In the Fall of 2004 Hiroshi invited me to give a keynote at the 7th Meeting of the Japanese Society of Clinical Anatomy in Yokohama. My lecture was the only one in English. In the middle of it I noticed that the people in the audience disappeared under tables in front of them and I was wondering. Seconds later I understood why. We were experiencing an earthquake.


At a lecture in Yokohama minutes before an earthquake hit

The digital transformation of medicine is benefitting from the still growing power of computers at large, and microprocessors and storage in particular. Between 1990 and 2015 the power of computers has increased by a factor of 10 (Moore's Law) and the newest IPads from Apple are delivered with 128 GByte of storage and about 50 GigaFLOPS processing power at a size of 12 by 12 mm. Increase of power and decrease of size (dramatic miniaturization) and decrease of price per GigaFLOPS is still happening in the world of computing. As a result an iPdad of today has roughly the same computing power and storage at a cost of less than one hundredth compared to our graphics supercomputer Onyx from Silicon Graphics Inc., which measured the size of two to three dish washers. This has a huge impact on surgery planning. The video shows our work with Itaru Endo, from Yokohama City University, where the results of a surgical plan prepared by MeVis Distant Services are used during the operation with the aid of an iPad. 

The other surgeon who cornered me after my lecture in Berlin was Yasuhiro Fujimoto. He was the right hand of the legendary living-donor-liver-transplantation surgeon Koichi Tanaka form Kyoto. I proposed to Yasuhiro that he should come to Bremen before his flight back to Japan and with permission of his boss he did. There we learned about the achievements of Tanaka and Yasuhiro learned as much as he could within a day about our liver surgery planning software. When Yasuhiro left we were hopeful that next we would be able to connect with Tanaka. But there were months of silence. We asked Yasuhiro whether Tanaka would accept that we would visit and demonstrate our work. Eventually he did and we visited late October of 2002. With me were Guido Prause and Holger Bourquain from MeVis. Holger, a neuroradiologist by training,  had  substantially improved the handling of our software and was a master in life demonstrations.


Arriving in Kyoto we were told that Professor Tanaka would be very busy during the next days but would have time on Monday evening at 8 pm in his office, maybe for an hour. There something happened which should give our efforts an unimaginable boost for years to come. I introduced what our research was about and soon Holger started a life demonstration of our software along a case study. At first Tanaka sat at his table while he kept a notable distance to us. Then he moved closer and soon he notably shifted from "let's see what they got" to "extremely interested and engaged".  

First meeting with Koichi Tanaka in his office October 28, 2002

(left to right: Holger Bourquain, HOP, Koichi Tanaka, Yasuhiro Fujimoto)

The meeting which was supposed to be for an hour lasted until after midnight and eventually Tanaka asked: " I will perform a living-donor-liver-transplantation on Wednesday. Do you think that you can use your software to analyze the risk for the donor and generate proposals for minimizing the risk?" I looked at Holger (begging he would not say no) and said: "Yes, I think we can give it a try". Then Tanaka said: "Please prepare your results, so that we can show them in the Wednesday morning meeting with the international surgeons, prior to the surgery. And one more thing. I am just finishing a book on living-donor-liver-transplantion surgery. I invite you to write a chapter for that book. Will you accept?" "With greatest pleasure" I replied.


These were two unbelievable instantly extended recognitions and endorsements of our work from the greatest surgeon and pioneer of living-donor-liver-transplanation. When we returned to our hotel we walked on water and used several drinks to calm down again. Incidentally, at that time Tanaka had already performed more than 1300 operations. He had introduced the very particular art of this surgery in Japan and had a reputation which words cannot describe in his community. Actually, Japan at the time was leading world wide in the number of cases because in Japan liver transplants from cadaveric livers, as they are very frequent in the US and Europe, are not an option, because of religious reasons. 

On Tuesday Holger managed a couple of miracles: Firstly, he managed to get the CT-scans of the donor into his laptop. Not a small achievement, when you  remember we were in a hospital in Japan with NO knowledge of Japanese, but we had Yasuhiro on our side, who had spent professional time in the US and was himself fluent on his beloved Mac (a devotion, which I happen to share since my first MacIntosh in 1985 at the University of California in Sanat Cruz). Secondly, Holger was able to work with the data and, with Yasuhiro looking over his shoulder, to analyze the individual surgical risks of the donor and generate planning proposals. On Wednesday morning we gathered in a lecture theater with dozens of doctors, many from all over the world coming to learn from the master. Holger and Yasuhiro were asked by Tanaka to present our results.  Tanaka asked questions, exchanged arguments and then stopped and said this: "Because of the MeVis software results I have changed my mind. I will now operate this way." And he explained his new insight obtained via our risk analysis. I was allowed to observe him performing this historical operation during the entire day. In fact there was one more thing: We used an ad-hoc perfusion experiment on the part of the liver that had been obtained as a transplant after its removal from the donor along our planning proposal and were thus able to verify that the risk assessment that we had provided was indeed correct. This was a monumental breakthrough for us. And a collaboration and friendship began for which I will be forever grateful. 

Koichi Tanaka, left, with HOP on October  30,  2002

after the first living-donor-liver-transplantation based of the MeVis planning results

Back home we developed rapidly a piece of software that would allow secure data exchange over the internet with Tanaka's group. Not that easy at the time, because of bandwidth limitations, data protection issues and the huge amount of  CT data that had to be moved. Tanaka since then has used our methods for all of his operations, even when he was operating in other countries and he has used our methods to extend his surgical techniques and push the envelope. Moreover, he has propagated and disseminated our method all over the world in his lectures and communications and has inspired other surgeons to follow him. The most remarkabe thing about this attitude of Tanaka in my mind is this: Here we have Mr. living-donor-liver-transplantation with all the experience you can ask for and he doesn't say: "A good surgeon doesn't need this." These are the men who move things forward and take down barriers. These are the men who are very confident in themselves (otherwise they wouldn't dare to remove about half of a liver from a healthy donor) but somehow deep down they know there is room for improvement beyond their own skills. These are the men who take patient care seriuos in an exemplary fashion. I have been very lucky to meet Koichi Tanaka and many other great doctors around the world with this attitude and make them partners of our research at MeVis. They are the prerequisite of medical progess, but they are relatively rare, and it is not easy to find them. Actually, this is not a men thing. I have been blessed to be able to work with outsatnding women with this attitude as well.

Although the world of transplantation surgery is dominated by males. In the US, however, Liz Pomfret, chair of the Lahey Clinic Department of Transplantation in Boston is a notable exception. Liz has picked up our methods and used them now for more than a decade in her own work and has been a great stimulation for our work, together with Christoph Wald, who initiated and built our collaboration in many other clinical domains, like lung cancer screening.


Our methods for surgical planning are not restricted or limited to liver surgery, although we started with that. Susequently we extended the methods to mostly oncological surgery of the lung, brain, kidney and pancreas.

MeVis Distant Services

What we did with Tanaka, receive CT data, work out the risk analysis and generate a surgical proposal in the form of  images and movies, was the template for a world-wide service which we first established in a research setting, funded by the German BMBF and then commercialized into an internet-based service, which in fact was imbedded into a   significant amounts of quality control, to the extend that we were able to secure FDA clearance. The work in Bremen was initially supervised by Holger Bourqain and executed by a group of specially trained radiology technicians.

The company which we founded in 2004 was named MeVis Distant Services AG. It was merged with MeVis Medical Solutions (MMS) AG in 2006. MMS has continued to extend and improve the service over the years. The liver surgery service is available for transplantation and oncological surgery. Far more than 7000  cases have been supported for surgeons on all continents. 

Breast Cancer Screening in Germany: The First Model Project in Bremen

Towards the end of the 1990s Germany engaged into a hefty political discussion about mammography screening. Countries like Sweden (the first in Europe) already had a population based breast cancer screening program, which they based on extremely tedious and elaborate epidemiological studies. The Netherlands had followed Sweden with a national program and then the UK set up their national screening program within their very particular NHS. The Dutch program distinguished itself by very rigorous quality measures and a quality control system with great finess.


The situation in Germany was abysmal. Studies had indicated that despite the fact that many women received mammography (grey screening: screening without quality control) the outcome was totally unacceptable: The vast majority of cancers which were detected had grown to a large size (difficult to overlook in a good mammogram) and very likely the desease was already metastatic. The programs in Sweden, Holland and the UK had proven that with very dedicated quality measures screening was able to catch baby cancers. The paradigm is that baby cancers most likely have not yet metastasized and often can be treated without much negative aesthetic imapct of the breast appearance. And the Swedes even had provided proof that screening reduces the breast cancer mortality, which is the ultimate goal and criterion of success. However, screening come with huge challenges. Some of the challenges are based in the fact that although mammography has been proven to be the most appropriate method for screening in many studies it is not without problems: For example, the breast tissue of a woman may be dense and appear more or less opaque in a mammogram and thus hiding a possible breast cancer.  Other challenges originate in the fact that screening programs deal with populations which are overwhelmingly not having the desease which a screening program is targeting.

Quite typical for medicine, everything good comes with side effects and those can be so severe that screening is doing more damage than good. This is the case when the quality of screening is low (like in Canada in the 1990s and in the US at that time). Screening is for non-symptomatic women between 50 and 70, when the number of breast cancer cases is at its peak in a general population. Non-symptomatic basically means most likely healthy. There is an excellent website which I would like to refer to by the American Cancer Society.  This website contains numerous statistical details that demonstrate how difficult it is to speak about screening in a few words and not oversimplify (unfortunately that is what the antagonists too often do).  So let's say 5 out of 1000 non-symptomatic women should be detected with cancer. That means that more than 99% should come out of screening with a negativ result. Now, if the mammogram is technically poor or badly positioned, the radiologist may not have a chnace to find the cancer. Ot, if the radiologist has not enough experience cancers likely will be missed. At the same time the unexperienced radiologist may select many cases with suspicious spots in mammograms, which would lead to unnecessary biopsies and surgical interventions, although they are not cancer. Therefore quality and quality control is mandatory in screening or it can turn into a nightmare.  In order to be experienced a radiologist needs very dedicated training for screening and has to see a lot of cases per year (some 5000). In the best of all worlds the radiologist's performance is continuously checked and there is retraining if necessary.


This is basically the route that the Dutch program had chosen and the late Jan Hendriks together with his colleague Roland Holland were the pioneers in setting up a very rigorous system of quality control, where every element in the screening chain is under observation, assessment, control and continuous improvement. 

So here we are at the end of the 1990s in Germany and I knew all of that and I get involved. Now Germany has a very peculiar health care system. With the legislature, executive power and jurisdiction, the German health care system acts almost like the 4th power. It is largely self-organized between the interest groups of insurances and providers. They decide what to do and how to do it. Healthcare politics in the executive and legislature branches can set boundaries and frameworks, but the healthcare players share the real power in closed non-transparent circles. 


Here is a simple example which shows what I mean: Pilots have to take recurrent training and have to pass profiency tests periodically and need a medical to fly. The simplest thing is that they get their eyes examined periodically (an air traffic pilot every 6 months). And that is all very desirable in view of the safety of passengers. Imagine for a minute pilots could determine themselves what their proficiency is and whether their eyesight is good enough, to keep it simple. That is basically the situation in general radiology. Eye exams periodically? No. Bad eyes, stop to work? No. 


Now you tell them that they need to be trained, that they need recurrent training, that they need to be monitored and so on. And you understand what was going on in Germany when the discussion was about which kind of screening do we want, and which kind of screening should we have. Moreover, in view of the 5000 cases or more per year requirement it became immediatley clear to the community of radiologists and gynaecologists that many of them who were  providing screening exams so far would not be needed anymore. And it becomes worse when someone without a medical degree, like me, makes such demands.

Eventually the situation in Germany led to a nation wide competitive solicitation for proposals of model projects which would be evaluated by a group of European experts. I tried to organize an effort in Bremen out of MeVis. Initially there was a broad support in the Bremen community of interested doctors. But when they learned about my Dutch, so to speak, underpinnings and convictions in terms of quality here , and quality there, the broad support turned into opposition from many.  But with the help of some brave people at MeVis and in town we pushed for a proposal which was honest in regard to the Dutch insights. The fights to get to the submission were unbelievable. But it appeared to me that giving in on quality requirements would be a missed opportunity. Or to put it differently, it is relatively easy to relax quality in a program if appropriate, but very difficult to increase quality in a given set program.


But we prevailed and eventually secured the support of the Bremer Cancer Society of Bremen under the presidency of Heinrich Schmidt and some other institutions, including the Senator for Healthcare of the City State. We submitted a proposal without a single compromise on quality.  Christian Beck, Antje Boedicker, and Markus Lang from MeVis were my co-authors on the proposal. The result was that only the Bremen proposal was chosen by the evaluation committee among 17 applicants for 3 slots.  Now, this was also the most unusual proposal I have ever done, because no money or funding whatsoever came with it. Being chosen meant that we were now on our own to secure funding. I managed to orchestrate the funding support from local insurance companies. And the next step was to find and recruit an appropriate medical head for the program who would be thruthful to the essence of the proposal. This was mission critical, because I figured a lip server but potential reactionist could destroy everything. I was convinced that the very experienced and nationally recognized Hans Junkermann from Heidelberg would be the perfect choice.  But my Bremen opposition argued, why bring someone from outside, when we have good people in town? And they fought my proposal with teeth and claws and some did not refrain from playing dirty. I knew I had to win this or all was lost, because I was certain that a doctor from the Bremen hospitals would have found it very difficult to resist the enormous pressure, and there was nobody with national recognition in breast cancer diagnosis in Bremen. Jan Hendriks was my advisor during these weeks. He made me strong. He reminded me of the fights that he had to go through and always said: "If you want honest screening, now is the time and you must not cave in." So I persevered and Hans actually left his work in Heidelberg at the University Hospital and took on his heavy responsibility. To this day I am full of admiration for Hans. What a great man. What a great attitude.

The official start of the screening model project program in Bremen was Juli 2001. Hans implemented the program, made it practical and kept it going despite all the hostilities and opposition in Bremen and Germany. I assisted as well as I could, for example shielding him from the most ridiculous opposition and by serving in the steering committee. 


So why all of that struggle and excitement, when there were at least three exemplary models in Sweden, Holland and the UK? The answer is easy to give but not easy to understand. The healthcare systems in each of these countries is significantly different from the German one. Hence, the model project(s) were supposed to find a way and structure, and show that the very demanding requirements laid down in the European Standards would be achievable in the German system. There were ongoing monitoring assessments by a European group of experts and a final evaluation which came to the conclusion that Hans'  Bremen project had exceeded expectations by far and would be a suitable template for expanding breast cancer screening throughout Germany. And that is how it was done. 


Two anecdotes resemble the friendly and enemy fire. My older daughter, who is now a radiologist, as is my younger daughter,  went to medical school. Somewhere in the curriculum she had to get practical experience in a hospital. She chose orthopedic surgery. The chief surgeon treated her with resepct and she loved the work. Then from one day to the next she apparently had fallen from grace. During the morning meeting with all surgeons present he accused her of "being one of Peitgen's brood, the man who was destroying the business of doctors in Bremen". This occured while the project was running already.


Also during that time the Senator for Healthcare of the City State of Bremen (the one that had supported us in the submission phase of the project had been replaced)  one day commanded Hans and myself to appear in her office. She was furious and told us: " If something like this happens again I am closing down the project immediately! I cannot afford this to happen."  Here is what caused her to threaten us: Part of the screening chain is an invitation system, in our case under the supervision of the Einwohnermeldeamt (a goverment office for registration of citizens). Invitation system means that in some sequencing along town quarters all women between 50 and 70 are invited by letter accompanied with information leaflets to participate every two years. Now it turned out that some women, to my best recollection three, had replied by letter that they would not particapte and would request not to receive any further invitation. 


The assistants working in the invitation office started to make a note on a sheet of paper, because the computer supported invitation system did not have such provisions, and should not have them, in order not to violate the German data protection laws. On occasion of a routine audit, the auditor had asked: "What do you do with women who don't want to be invited?" They answered, : "We have started a paper list." The auditor regarded this incident as a violation of the data privacy laws. Obviouly they had been trapped. They were not supposed to track anybody, although they just wanted to do things right for the women. 


These were the kinds of attacks we had to endure, and of course Hans much more so than I. Almost every month there was another one of these attacks and sometimes even from some women activists who felt screening was just another invention by men to exert dominance over women. The difference between my life in mathematics and medicine could not have been uglier and at the same time more satisfying. And I am eternally grateful to Hans Junkermann for letting me share this experience.

Mathematics Education

Mathematics Education: 


How I Was Lured into Education

In my Bonn years and the years in Bremen until 1991 I never had any contact  with mathematics education. Since the schools which I experienced were great and prepared me more than well for my univerity studies, I probably thought that everything was in good order regarding the university preparation of future math teachers. How wrong and naive that was, I only began to understand in 1988.


Our book "The Beauty of Fractals" had been out for two years and had made quite a splash in the mathematical community by raising some eyebrows and mostly generating tremendeous approval. Ronald Reagan, during his first presidency put a committee at work to understand the weakness and challenges of the educational system. The 1983 report "A Nation at Risk: The Imperative for Educational Reform" had a huge impact within the mathematical community in the US. One of the great achievements were that the various mathematical societies, like the AMS, MAA, SIAM, and NCTM which previously were more aware of their differences came together with the Joint Policy Board for Mathematics trying to unify along the grand goals of the report in order to manage joined efforts for mathematics education reform. One of great leaders in this was the late Kenneth Hoffman, an algebraist and former head of the mathematics department at MIT. I knew him through Springer-Verlag in New York and one day he called me in Germany and his words were: "I need your help. I want you to do three things for us. I want you to give a special evening lecture on the 100th Anniversary Annual Meeting of the AMS (American Mathematical Society) in Atlanta, I want you to lecture to the teachers at their NCTM (National Council of Teachers of Mathematics) Annual Meeting in Chicago, and thirdly I want you to come and give a keynote at the Presidential Awards for Excellence in Mathematics and Science Teaching (PAEMST) in Washington." Each of these events took place in 1988, the AMS event in January, the NCTM event in April, and the PAEMST in October.

Museum of Modern Art, New York, June 1989; Ken Hoffman (right), next to Benoit Mandelbrot, next to the President of  Springer-Verlag after my keynote lecture (still exhausted) on occasion of the 25th Anniversary of Springer-Verlag New York


100th Anniversary of the AMS in Atlanta

I was quite proud about the AMS assignment, which took place in a ballroom of the Atlanta Hyatt Regency Hotel after dinner. About the time when Peter Hilton was supposed to introduce me,  we could not start, because there were still hundreds of mathematicians trying to get into the room. The lecture had to be postponed by an hour so that the hotel could open up an adjacent ballroom and set up furniture. People had to walk out and come back later. I will never forget how it felt to speak to many hundred colleagues in a totally packed double ballroom. 


The National Council of  Teachers of Mathematics

The NCTM lecture in Chicago did not mean much to me before, but a lot after the event. Some two thousand math teachers from across the USA were present and heard for the first time about fractals and the Mandelbrot set. My lecture was recieved with many outbursts of enthusiam, something I had never experienced in this way. I was touched and when the president of NCTM told me right after the lecture that now I had an obligation to bring this to more teachers, I sad immediately : "I would be glad to." So I gave keynotes on several more national NCTM meeting (1989 Orlando, 1990 Salt Lake City, 1991 New Orelans, 1992 Nashville (where I brought B. Mandelbrot along), and 1993 Seattle (where I brought M. Feigenbaum along)  and regional NCTM meetings throughout the US. In 1989 I committeed to write a small booklet for teachers that the NCTM could distribute. Meanwhile I had met the late Lee E. Yunker, a high school teacher from Chicago, and I had the idea that I should get some advice from teachers for my NCTM piece. When I met Lee again in Orlando, he was an Officer and Director of the NCTM at the time, he suggested to bring Terry Perciante and Evan Maletsky on board. This beacme the team, together with Dietmar Saupe and Hartmut Jürgens back in Bremen.  However the small booklet turned into the massive two volume "Fractals for the Classroom" books (appeared in 1991 and 1992) and the three volumes "Fractals for the Classroom: Strategic Activities" books (appeared in 1990, 1992, and 1999), all published jointly by Springer-Verlag New York and NCTM, and pushed to success by Springer-Verlag's Rüdiger Gebauer, who served as their math editor at the time and became their president a few years later. The books and workbooks were later translated into many languages such as German, Polish, Korean. 


Lee was connected with Fermi Lab in Chicago. It should be mentioned that the national labs of the US at the time were required to use a fraction of their funds for supporting education. As a result we ran  a number of Summer Institutes at Fermi Lab and later at the Princeton Plasma Physics Lab for an audience of regional and national math and science teachers.


The Presidential Award of Excellence in Mathematics and Science Teaching

The Washington keynote for the PAEMST audience was something I was looking foward to with great expectations. To meet President Reagan, maybe even lecture in his presence is something very special. But as it turned out he wasn't there. As we later learned he had preferred to be at a Dodgers game in Los Angeles and we had to be content with the Vice President George H.W. Bush. All teachers received gift packages and Springer-Verlag had managed to slide in "The Beauty of Fractals".  One of the awarded teachers, Glenn Govertsen from Montana, became a colleague andfriend in my teacher enhancement activities in South Florida for two decades. My lectures in Atlanta, Chicago and Washington mostly used 35mm slides to present the images. But I also did some magic. I did a life experiment, which I had learned from Ralph Abraham and Jim Crutchfield during my years in Satan Cruz and perfected to some extend, which became well known as "Video-Feedback": Take a video camera and direct into onto a TV and feed what the camera sees into the TV. If brightness and contrast on the TV and the zoom factor of  the camera are just right, and the camera is turned  around its own long axis something magical will happen (see page 19 in "Chaos and Fractals"). 

Videofeedback Composition by Japhy Riddle


From California to Florida

In 1990 Arnold Mandell lured me away from my position at UCSC to take a professorship at Florida Atlantic University (FAU). In fact, President Anthony Catanese hired me on the spot, hoping I would attract massive NSF funding for the department. I took the position in 1991 and started to write grant applications for teacher enhancement projects in South Florida, which were supposed to develop and foster a close collaboartion between FAU and neighboring school districts. The first one was granted in 1994 and we were able to maintain high level funding for regional and nation-wide projects until I retired in 2012. During the years 2004 - 2012 our grants were part of the national Math and Science Partnership (MSP) Initiative (here is the MSP.NET Abstract and Award Abstract #0412342). 


South Florida and National Science Foundation (NSF)

To be eligible for funding we had to build a strong partnership with a school district. Back in 1992/93 I tried both the School District of Palm Beach County and School District of Browad County. I had little luck. Their attitude was, how can a mathematician help us with our problems, when we know so well how notoriously inappropriate the mathematics teacher preparation is on the college and university level. All my previous credentials didn't mean much to them. But then unexpected rescue came from Marybeth Johnson, Vicky Quinn and Nancy Barba.  Marybeth and Vicky, teachers in Broward,  had been in the audience of my Nashville NCTM lecture and were enthused. They were friends with Nancy, and Nancy had clout in the administration of the district. So they tested us. Before they made their committment they let us a run a small summer program. I brought in all the weapons that I had: Evan Maletsky, Lee Yunker and Terry Perciante, plus Dietmar Saupe, all co-authors on the workbooks "Fractals for the Classroom: Strategic Activies". The program was a success and the district bought into our NSF plans and a partnership was founded which lasted two decades. Nancy, Vicky, Marybeth and later Ana Escuder became my great partners and friends, who carried these highly complex and ambitious grants with me. When I was able to attract Richard Voss to FAU, he joined my efforts and became co-principal investigator. 


In the beginning our work with teachers of math and science circled around discovery learning using chaos and fractals as an example. Later we built an entire 8 courses mathematics curriculum for a masters degree for middle school teachers. More than 60 teachers graduated from the program with a masters degree. Some of the grants were just summer institutes with a summer school program for kids attached, others provided summer institutes and year around classes. From the inception of our work in South Florida computers in the classroom played a major role, which initially was a struggle, because teachers and schools were not ready for it.


In that regard there are two developments that supported our efforts greatly. One was that we hired a few math graduate students and developed a series of some 10 Java-Applets , which should be available on this website, but they are not. Java has recently updated their machine to the extend that our apps don't run anymore because of network security reasons. So much for web continuity! This is very annoying to a person who is a great promoter of the internet age.


Computers into the Classroom

The other developement fortunately is still alive and as vivid as it gets: Around 2003 - 2005 we started to use dynamic geometry software in our NSF programs. There were several  products, all commercial. We preferred Cabri from France at the time. However, there was a huge problem. Our teachers were enthused and wanted to use the software in their own classroom. District licenses were prohibitivly expensive. Thus we looked for non-commercial alternatives. Most were not a fraction as capable as Cabri. But then we discovered GeoGebra and the young Austrian, Markus Hohenwarter, who had produced his software as part of his PhD thesis at the University of Salzburg.  One day I called him and learned that he was looking for a university job and had basically accepted a position in Munich. I tried to lure him and his wife Judith, who was still working on her PhD, to Florida. Markus accepted, came to Florida, took a visiting position in the Math Department of FAU and became a key figure in our NSF grants. In fact, we were able to ammend one on his behalf and in the following one we made him a part of the faculty of the grant. This had an enormous impact on our efforts to integrate computers into math teaching. Our teachers were able to work with the creator of GeoGebra, which inspired them beyond words. Simultaneously, GeoGebra for some years had found a new home in South Florida for its continued development with funding from the NSF (for example the great spreadsheet feature was conceived), and being part of the national MSP community, we could help to introduce and spread GeoGebra in the US.  Markus is a great math educator and entrepreneur. He had, and still has, the philosophy that dynamic geometry and algebra software should be freely available for teachers and students. Meanwhile his system is available in all languages (for example, Kazak, Nepalese and Arabic (various)). And Markus has cultivated an open source community of world-wide GeoGebra developers. Markus left South Florida for a prossorship at the University of Linz in Austria. He is more than likely one of the most influential people in math education, when it comes to computer use, and he is at the same time as modest as it gets. 


From South Florida to Bremen

Around 1991 it was time to bring what I had learned and developed about teacher education and teacher enhancement to Bremen. The attention that future math teachers receive at German universities is not more than in US universities. By and large mathematics professors hold that a teacher should get a good dose of real mathematics. In Germany this is even the attitude towards instructing elemenatry school teachers, when they were supposed to master a course in calculus.


Some day I was invited by German physicists to their annual Spring Meeting to lecture about my experiences with bringing fractals and chaos to the high school classroom. There I met a young physics high school teacher, Gisela Gründl. She was offering help to organize efforts in Germany provided I would get her a teaching position in Bremen. This was managed and there was Gisela and started to organize teacher enhancement efforts in Bremen wih amazing efficiency. Eventually we hosted annual four day teacher enhancement workshops in Bremen which drew teachers from Germany, Switzerland and Austria and a small very powerful team of teachers in Bremen became expert disseminators: Reimund Albers, Heidi Christiansen, Claudia Homburg and Klaus Lies. Between 1992 and 2012 we organized 25 workshops (between 1992 and 1995 two per year) and starting with 1996 we included activites for music teachers and teachers of German by involving the shakespeare company bremen and The Deutsche Kammerphilharmonie Bremen.  Soon it became desireable to have some legal framework for our activities and we founded Lehreakademie Bremen e.V.  


Around 2000, for several years to come, we organized and facilitated weekend workshops for math teachers in Bremen to help them how to enhance their own classroom using computers effectively. This initiave became known under the name Bremer Netzwerk Mathematik und Computer


The final effort was within the university. Reimund and I developed and wrote a grant for the Deutsche Telekom Stiftung, which was granted from 2007 - 2010, and which funded our work on developing and implementing an entirely new mathematics curriculum for future elementary school teachers at the University of Bremen. The project was called Mathematik Neu Beginnen, and the idea was to present mathematics in a discovery style and assemble a body of mathematics from a higher mathematical view point, which is the backbone of elementary school mathematics. 



Three Presidents

The late Johannes Rau, former President of Germany, a good friend of the surgeon Christoph Broelsch once spend time in Bremen after his visit to the Bremer Schaffermahlzeit, playing cards. Christoph suggested it would be great if I would personally meet him. I felt awkward but eventually accepted and met them in a hotel room where they played, smoked big cigars and apparently had enjoyed some great Bourdeaux wines. The emphasis is on enjoyed. It was around midnight  and I was introduced as a mathematician.  Immediately the President lived up and told a joke. He was known for good jokes and an artful way of telling and imbedding them. A wonderful mathematical joke emerged, which I have used in many of my lectures to teachers, reminding them that it is insufficient to tell a student that he or she is wrong. It is more important to analyze what kind of misconception or misunderstanding led to the wrong answer. The joke uses addition and multiplication in an odd number system, which sounds completly logical in itself, but produces very strange results. I always wondered how the President could have remembered the details, because in this joke every detail matters, or it is broke. Later I discovered that the President's joke goes at least back to the 1950s. Here is a wonderful rendering of it:

In 1995 I had the pleasure of meeting German President Roman Herzog. When he made his first official visit to the City State of Bremen it was decided that he should make a brief visit to the University of Bremen and I was supposed to demonstrate our work in liver surgery in our lab. The idea was to use our computers and the power of live demonstrations. A last minute change was made because of security concerns. Students of the University decided to organize a demonstration against the President, and it was decided that I should demonstrate in a large and well protected lecture hall, without windows and secured doors. There I was at the blackboard and he was sitting with his entourage in the first row in an otherwise empty and dark lecture hall. A rather ghostly and unreal atmosphere. We had a deep and long interaction about our work and the time schedule ran completely out of order, something I am notorious for. One year later I met him again in his residence in Berlin, when he awarded me with the Federal Cross of Merit First Class. I have very fond memories of President Richard von Weizäcker in 2003, after the Iraq War had just begun. I was at a lunch table right next to him, and when he learned about my relations with the USA he was eager to learn about my view of the acceptance of the war in the US society, and then he gave me deep insight into his assessment of President Bush's war politics. He was deeply concerned and when we look at the state of chaos in  the Middle East in 2015 I have to admit that President von Weizäcker must have had some access to predicting the future. Meetings with President Christian Wulf and President Joachim Gauck were interesting because they gave me insight into their personalities but are not related to my story. 

Early Years and High School:


I was born and grew up in a tiny village east of Cologne. My early childhood was village life in the purest form. My horizon was limited to what happend in the village or the nearby villages. School was a typical village school with the first 4 grade levels in one single classroom. Students were encouraged to solve problems in higher grade levels. Pranks, some of them wild and dangerous, played a central role in the after school hours.


At that time Germany had a segregated school system: one track (Volkschule) from 1st to 8th grade for the poor and/or less gifted, and another track (Gymnasium) from 5th grade to 13th grade for rich and/or gifted students. Volksschule was tuition free, while Gymnasium required significant tuition payments, which were a stretch even for many middle class families. Around, when I was in 7th grade, things changed and Gymnasium went tution free. Here was my chance. The Wuellenweber Gymnasium in Bergneustadt, some 2.5 bus hours away from my village, took students from Volkschule into special accelerated tracks after passing a stiff entrance exam. I was on and that was the biggest adventure so far and a huge window into the future. 


Controversial political discussions at home were frequent, very much to the dismay of my mother Herta Peitgen. My father, Walter Peitgen,  belonged to the the FDP, while I eventually joined the SPD. I was a great fan of Willy Brandt. My father was quite popular in our borough (Gemeinde) and served as mayor for 12 years, although his party was always in the minority position. In fact he was awarded with the Federal Cross of Merit and several years later with the Federal Officers Cross of Merit. Unfortunately he did not live to see that I followed his footsteps in that regard in 1996.



When I was 15 I had an experience which caused  nightmares for decades. I lost my best friend Bodo Jürges in a fatal car accident. Bodo’s brother had a motor assisted bicycle (Moped), which Bodo borrowed occasionally without his brother’s knowledge. I had a street bike, which Bodo liked very much. Both of us were not allowed to drive the Moped, because we were still to young to have a license. One day I drove the Moped and Bodo rode my bike on a narrow winding road. In fact I pulled Bodo who held onto my left shoulder. Suddenly there was a car coming ahead and Bodo let go of my shoulder, swerved to the middle of the road and the car hit him and pushed him into the curb of the road. Bodo died within minutes at the site. The driver of the car was under the influence of alcohol.  He got convicted for manslaughter through culpable negligence. I also had to stand a trial but was cleared of all charges. Nevertheless the death of my friend and my role in the accident burdened and troubled me deeply for many years to come. 


Germany at the time was post war Germany, still deeply entangled in its horrific past, unwilling to talk about the past and by and large unwilling to cope with its monumental guilt. I still remember the solitary fight of Fritz Bauer to have the Auschwitz Trial, and I remember how important it was for me at that time that there were poets and playwrights like Bertolt Brecht, who had resisted the Nazis. This was the time when we had many questions and too many of the adults were unwilling to speak. Some of our teachers were hugely different. They understood that we needed to talk and they talked, and I know that this molded me at least as much as the academic programs at Wuellenweber Gymnasium. One of them, Herbert Heidtmann, our teacher of religious education, read excerpts of the Hoess diaries with us. Here was a man, Rudolf Hoess, who listened to Beethoven and Schubert in the evening after his day "work" as commander of Auschwitz. When I was sixteen this remarkable teacher took some of us students to a visit of Weimar. This was quite an accomplishment, because Weimar is in East Germany, the former German Democratic Republic, and visits were rarely granted at that time. In Weimar we were confronted with hard to swallow dichotomy of the town of Goethe and Schiller and the nearby concentration camp Buchenwald (just a few kilometers outside of Weimar). This left marks on us students, which have remained important for my entire life. In Buchenwald we laid flowers in the cell, in which Dietrich Bonhoeffer was held shortly before he was executed on April 9, 1945.  We studied Bonhoeffer's writing and discussed the fact that in 1956 the Bundesgerichtshof (German Federal Court of Justice) had ruled that Bonhoeffer's trial (SS-Standgericht), which was held on the evening before his execution on special order of Adolf Hitler, without a defense attorney for Bonhoeffer and any prosecution or defense witnesses, was due procedure within the framework of the former law and would be still valid.  This one sad matter explains better than anything the dark clouds which were still hanging over post war Germany and why it was so difficult for some of us to identify postively with Germany for several decades. 


Academically I was going through a huge transformation from a village boy to a young adult with many aspirations, many windows opened, and with rock solid convictions formed by my teachers. In particular Gerhard Klawitter, my math teacher deserves credit for my aspiration to become a professional mathematician, which is not a small goal. But passion carries a long way and passion for mathematics it was which he infused into me with durable success.


For some personal reasons my entrance into university life was not without obstacles. The enrollment deadline for the Summer Semester at the University of Bonn had long elapsed when I arrived in the Spring of 1965. The semester was on its way for a month already and the univerity's  enrollment office told me: "No way we can enroll you, unless the professors who run the courses ask for granting an exception in writing." With this seemingly hopeless expectation I met Friedrich Hirzebruch for the first time in his office in Beringstrasse 1. He listened, gave me this wonderful feeling of caring for me,  asked a few questions and then said: "Ja, Herr Peitgen, was machen wir denn da? Geben Sie mir einen Tag und ich suche eine Lösung." (Well, Mr. Peitgen, what can we do here? Give me a day and I will try to find a solution.) Next day he had already conversed with Professor Jürgen Schmidt, who was in charge on Linear Algebra (Hirzebruch was in charge of Real Analysis) and offered that two of his assistants Dr. Klaus Lamotke (later professor at the University of Cologne) and Dr. Klaus Jänich (later professor at the University of Regensburg) would coach me for one week to catch up with the lost month, assess me, and if positive, he would write a letter to the enrollment office of the University of Bonn asking them to grant an exception. This is how I got in and due to the excellent preparation that I received from my teacher Gerhard Klawitter and my personal coaches, I was able to finish both courses with top grades. 


When I first met Hirzebruch I had no clue who he was and what kind of giant mathematician he was. Soon I learned that and  I was even more impressed and grateful for his help to get me enrolled and up to speed with the semester's progression. But my attachment to Hirzebruch had deeper reasons. He was the absolute best teacher I have ever seen and enjoyed. He became my role model for how to teach and he pulled me firmly into mathematics and demonstrated from the first days on that becoming a research mathematician would be an attractive and very exciting profession. Above all, however, be became a life long fatherly friend. 

PhD Students and Habilitations:


Advising students to earn a doctoral degree is one of the great privileges of a university professor, because seeing and helping a student make the transition towards becoming an independent researcher is a great pleasure and satisfaction. The topics of the theses mark the development and changes of my own research interests. Furthermore, after their doctorate many of my students became co-authors of papers and books, co-founders of companies, partners in building the institutions, partners in outreach projects, such as breast cancer screening or teacher enhancement, and most importantly life-time friends. The topics of the 31 theses are listed in the Mathematics Genealogy Project. Among mathematicians it is a sport to note your mathematical origin: According to my genealogy I am a great-great-great-grandson of Davild Hilbert and a great-great-great-grandson of Sophus Lie.  This says nothing about my mathematical abilities, but it is fun to draw these lines. Once in 1990s, when I bragged about the fact that I can document a direct line of decendants originating with Charles the Great, Dietmar Saupe announced a lecture about the statistics of decendants. He proved that a huge fraction of all Germans share this origin with me, even when they cannot document it. 




Dissertations in Mathematics, Computer Science and Math Education  (1978 – 2015):

17 mathematics, 12 computer science, 2 mathematics education, *8 are serving as professors 


Dr. rer. nat. Michael Prüfer, 1978, mathematics

Dr. rer. nat. Norbert Angelstorf, 1981, mathematics

Dr. rer. nat. Hubert Peters 1981, mathematics

Dr. rer. nat. Hans-Willi Siegberg, 1982, mathematics

Dr. rer. nat. Dietmar Saupe, 1982, mathematics, *

Dr. rer. nat. Hartmut Jürgens, 1983, mathematics

Dr. rer. nat. Fritz von Haeseler, 1985, mathematics, 

Dr.-Ing. Cornelia Zahlten, 1995, computer science

Dr. rer. nat. Jürgen Gerling, 1995, mathematics

Dr. rer. nat. Ehler Lange, 1996, mathematics

Dr. rer. nat. Kathrin Berkner, 1996, mathematics

Dr. rer. nat. Antje Ohlhoff, 1996, mathematics, *

Dr. rer. nat. Anna Rodenhausen, 1996, mathematics, *

Dr. rer. nat. Thomas Netsch, 1998, computer science

Dr.-Ing. Mark Haidekker, 1998, computer science, *

Dr. rer. nat. Wilhelm Berghorn, 1999, mathematics

Dr. rer. nat. Dirk Selle, 1999, computer science

Dr.-Ing. Christian Beck, 2002, computer science

Dr. rer. nat. Björn Engelke, 2002, mathematics

Dr. rer. nat. Ralf Hendrych, 2002, mathematics, *

Dr. rer. nat. Tobias Boskamp, 2003, mathematics

Dr. rer. nat. Martina Döhrmann, 2004, mathematics education, *

Dr.-Ing. Jens Breitenborn, 2004, computer science

Dr.-Ing. Horst Hahn, 2005, computer science, *

Dr. rer. nat. Reimund Albers, 2006, mathematics education

Dr.-Ing. Jan-Martin Kuhnigk, 2008, computer science

Dr. rer. nat. Inga Altrogge, 2009, mathematics

Dr.-Ing. Tobias Böhler, 2011, computer science

Dr.-Ing. Anja Hennemuth, 2012, computer science, *

Dr.-Ing. Andrea Schenk, 2012, computer science

Dr.-Ing. Markus Harz, 2015, computer science



Habilitations: 5 mathematics, 1 computer science


Dr. Dietmar Saupe


Dr. Fritz von Haeseler


Dr. Juri Suris


Dr. Carl Evertsz


Dr. Bernhard Preim


Dr. Tobias Preußer



Benoit B. Mandelbrot:

Benoit Mandelbrot belongs into the class of scientists, who made monumental contributions to many branches of the tree of science (mathematics, physics, chemistry, biology, medicine, engineering sciences, geological and geophysical sciences, financial mathematics and economics), was extremely popular and well known in the general society, and at the same time was not exactly most welcome by many of his colleagues, although he was awarded significant prizes and honors.


The recurring thread in Mandelbrot’s creation, Fractal Geometry, is that much of  the world is ruled by power laws. It is noteworthy that he was inspired to take this world view in his dissertation from 1952.  The first half was on Zipf’s Law and makes claims about word frequencies and the second about statistical thermodynamics.


There is a powerful video explaining Zipf’s Law on the Vsauce YouTube Channel created by Michael Stevens, which is worth viewing.


I remember the days when Benoit called me and we would talk for hours on the phone and discuss misrepresentations about his work that he had become aware of and which deeply concerned him.


For much of his life he didn’t belong to the scientific establishment, which is usually seeking their home in reputed universities. Instead he joined and stayed with IBM Research for most of his professional life, before he joined Harvard and Yale as a professor. At IBM Research he was a Fellow, which is short for boundless scientific freedom, financial security and almost total independence without the typical academic politicking.  It is quite likely that his maverick style and his crisscrossing between many sciences was only possible in an institution like IBM Research, which did not cultivate disciplines and the separation between disciplines, as many universities do.  And if that is true, then probably fractal geometry, his creation, would not exist at all or be a subspecialty of  mathematics, or physics, or …, if he were confined by the walls of a particular university department.


This role model example of Benoit has probably rubbed off on me more than anything, when I decided to leave the department of mathematics at the University of Bremen with my medical research ideas in 1995, and subsequently founded MeVis Research GmbH.


So what is it about Mandelbrot’s work that is so important, and was so important for myself?


Loosely speaking, his work opened my eyes in the literal sense and in a more philosophical sense. I was brought up as a mathematician in Bonn under the constraints and inspiration of the Bourbaki School.


In short, Bourbaki style stood for absolute purity of mathematical presentation and a total ban of pictures (or the use of the eye) in mathematical expositions. We would say often: “As you can see”, but we didn’t mean the eye. Mandelbrot, however, an outspoken Anti-Bourbakist, reported again and again that the eye played a key role in his discoveries and in his understanding and solving mathematics problems.


The Mandelbrot Set is a great example. He did the first mathematical experiments in 1980 using graphical representations of the results, and was soon lead to believe that he was on top of a gold mine. When we compare his first images with what we have today this also is a striking testimonial for the evolution of computer graphics over the last three decades. Below we see one of Mandelbrot’s original images (left) next to one of our renderings of the electrostatic potential from 1990. There were only 10 years in between these two images and that demonstrates some of the computer graphics revolutions in that decade.

Impacted by our own experiments in 1982/83 in Salt Lake City (make link) I moved from “eyes shut” to “eyes wide open” in my work. Incidentally,  the well known movie Eyes Wide Shut, has no connection to what I am trying to say, except, that one of my most favorite compositions by my friend G. Ligeti, Musica Ricercata, is used in the movie "Eyes Wide Shut".


The Mandelbrot Set has many properties which are very difficult to accept. One of them is that if you zoom in onto details there is no limit, even if you zoom infinitely close. This movie Hunting for Fossils zooms in by a magnification factor of an unbelievable  10 to the power of almost 200! Look at it.


The other eye-opener was that there was something interesting in nature that I, and most of the science community, had missed, and which inspired my initial interest in medicine. The Fractal Geometry of Nature!


So what is it that Mandelbrot gave us? Foremost and most importantly he gave us a theory of roughness. What does that mean? From Newton and Leibniz to today we witness a huge and monumental strand in mathematics, the theory of smoothness. From calculus to differential equations and modern day differential manifolds the theme is the same:  approximating  reality and real world phenomena by models which are smooth, or are differentiable, or allow the application of differentiable calculus in some way or another. This has become so dominant that we were mislead to believe that reality indeed is smooth and can be perfectly modeled by mathematical objects, such as curves, differential equations, manifolds, etc. , which are smooth at heart. In this extremely successful approach which is at the heart of much of physics and engineering we have completely neglected objects and features of nature (and society) which are far from being smooth. Mandelbrot liked to say in his interviews: “Clouds are not spheres, coast lines are not straight and mountains are not cones.”


Let’s look at a chart of stock. The moving prizes over time generate a geometric shape (curve) which is as far from smooth as a rugged mountain range, and in order to be able to treat it mathematically, economists apply a simple trick, the method of moving averages, which results in a smooth shape (curve). And their belief is: “By filtering out noise and displaying key trends, moving averages can show you points at which a stock's price is likely to reverse course. That simple fact alone can help make you a better investor and improve your investing results!” I would say, however, if you force a chart into the smooth world, as you do with moving averages, you may have just lost what you were looking for. Incidentally, economics uses words like trend without caution. In fact, talking about trends means absolutely nothing unless you specify a time scale (meaning, at which detail are you obsevring the movement of prices) like a week, a month, or a year, and you specify a method of grabbing a trend. For example, you could pick a point on the chart, say every 10 days, and then connect these points with a straight line segment.  This would be a piece-wise linear approximation of the chart. And now you could choose those segments, which pick up the movement of prizes neatly (just with some small variation). Maybe, none does: thus no trend; maybe a few do: thus a few trends with this crude method would be found. Or you could apply moving averages and whenever, say the 50 day moving average has a min or max (or vice versa) you got your  start points and endpoints of trends.


It seems pretty obvious that this is completely arbitrary and cannot be consistently useful, because a trend, its existence and detection, depends on your choices of detail and measurement. And moreover, there would be as many differing trends as you want. There are two things, one demand and one question:


The demand: Whenever someone talks about a trend in moving prices of whatever, without specifying the time scale or detail of observation and the method of detection, this is totally worthless information.


The question: Why is that? Well the reason leads us to the heart of the matter, and that is: price movements are of a fractal nature. And it was Mandelbrot who told the world to pay attention to this fact, because it is not a negligible fact. Rather it matters a great deal if you want to understand stock and do anything about them, practically as an investor, or theoretically as a researcher. There is a whole array of fractal properties of stocks. The most fundamental one is that a chart is statistically self-similar, very much like coastlines and many patterns in nature.


In very popular science terms that means that the whole can be partitioned into parts, and each of the parts look like the whole (would be difficult to distinguish from the whole after proper scaling). (Note that I said “look like” and did not say “similar”. If the parts are similar to the whole, we have the more strict form of self-similarity”.) In other words, a chart over a short time span like a few days looks, or a few hours, very much like a chart over a month or a year. Obviously, if that is the case, then presenting a chart without reference to the time span is very useless information and the same then is true for trends in them. Or to put it differently, we can expect trends, short ones and long ones, for any time span. Another fractal property is hidden somewhat deeper and connects with the efficient market hypothesis. If the markets were indeed efficient, then stocks would be best represented by random walks and the mathematical treatment of stocks would be relatively simple. For example, we would know how to scale the 1-day volatility to a n-day volatilty by simply scaling with the square root of n. 


There is and has been a lot of discussion and controversy about this issue. As a mathematician I would say, markets are efficient in the same way patterns of nature are smooth. Meaning, when it is convenient and useful we can make that assumption as an approximation, in particular when it yields useful insights or results. For example, the Black-Scholes formalism for option pricing uses Einstein’s Square Root Law, which is a random walk feature. In my view the controversy is pointless. Markets are clearly not efficient and patterns of nature are clearly not smooth. However, that is not a profound remark, unless one can distinguish the difference between smooth and rough, or efficient and not efficient. In that regard fractal geometry offers a whole bag of tools for measurements and models to simulate. Price movements are more appropriately approximated by fractional Brownian motion processes than random walks.


In fact, fractional Brownian motions are characterized by the Hurst Exponent H, which derives from a power law consideration and is related to the long-term memory in such processes.


Now, one observes that 0 < H < 1, where H=1/2 singles out Brownian motion or random walks. A general measure for roughness that Mandelbrot gave us is the Fractal Dimension D of a pattern, and D, unlike more familiar notions of dimension, typically is not an integer, but can be. The fractal dimension is related to the Hausdorff  Dimension but has one important differing feature: it can be algorithmically computed. So for example, the geometric pattern of a chart has a distinguished D, where 1 < D < 2.


And for fractional Brownian motion there is a beautiful mathematical relationship:


                                                                                    H + D = 2.


In other words, if a fractional Brownian motion process is a reasonable approximation for a particular stock, then either H or D carry information about how close or how far this stock is from a random walk. We used this approach for our work on generalizing the Black-Scholes formalism, in order to obtain more realistic option pricing models in the 1990s.



Mathematics and Music: György Ligeti, Iannis Xenakis and Jean-Claude Risset:

I met György Ligeti in 1985. Manfred Eigen had shown him our catalogues and György wanted to meet. He was totally taken by the fractal images. He kept asking questions without limits. He wanted to understand, which was not easy. He wanted to go into the depth of mathematics. He wanted to understand what chaos meant in rigorous mathematical terms.  From 1985 to our last meeting before his death it was always the same. There was very little small talk about life, family or job. He wanted to talk about mathematics. He never had any mathematical education beyond high school, but he always had a deep interest. It took a while for me to understand that without knowing about chaos and fractals somehow in his compositions Ligeti had dealt with phenomena which were not chaos- or fractal-like in a rigorous scientific manner, but came very close. He somehow had almost intuitively smelled that when he first saw our work in 1985 and thus was fascinated about the parallels in his work and in the sciences. For more details I refer to my chapter Continuum, Chaos, and Metronomes – A Fractal Friendship and the interview, which Volker Banfield and I did.

In the late 1980s and 1990s Ligeti’s recognition as a composer of contemporary music rose to Olympic heights. At the same time chaos and fractals had their climax in the public perception. Im not implying that the latter impacted Ligeti’s recognition but it was an interesting coincidence which Ligeti liked to some extend but also was somewhat nervous  about. However, he and I, did several events together with Volker Banfield, where Volker would usually play from the grandiose Etudes  and György and I would lecture and discuss relationships on stage.


I remember one of those events very well. The City of Bremen has an annual Festival of Music. I was able to convince the artistic director to have an evening around Ligeti. He agreed under the condition that I would persuade Ligeti to be present and participate in a stage discussion that I was supposed to moderate. The assumption was that this would draw a huge audience and it did. Quite on purpose the concert took place in the Satellite Assembly Hall of what is now AIRBUS in Bremen. The whole concert was arranged around Ligeti's Violin Concerto.The marvelous Christian Tetzlaff was chosen to play the violin together with the Deutsche Kammerphilharmonie Bremen.​


There was a final rehearsal late in the afternoon before the concert. I picked up Ligeti at the train station and drove him over to the site. When he entered he heard Tetzlaff play and said without hesitation: ” This will not work. The acoustics is terrible. We cannot have my concert here. Absolutely not.” Every attempt to sway him was unsuccessful. At about 7 pm we arrived at the compromise that if he would not attend the concert we were allowed to have his violin concerto being performed. Now the center piece of the evening, the live experience of the great master on stage, had evaporated but at least the concert was rescued. I had maybe 30 minutes to prepare moderating the evening just by myself, whereas otherwise I would have asked a few key questions and Ligeti would have talked in his marvelously associative style. The next day we met in his hotel room and talked about the various concepts of infinity in mathematics, one of his favorite subjects, and he didn’t even ask how the evening went. The concert was a phenomenal success, although it was rather demanding contemporary music, with compositions by Ligeti, Iannis Xenakis and Jean-Claude Risset. Nobody complained about the acustics.


On one of my birthdays he called, apologized for not coming and told me about the gift that he had decided to present: the dedication of his 17th Etude for Piano. It wasn’t composed yet but when it eventually came out it felt like a jewel for me. During his very last years it was increasingly difficult to communicate. He spoke less and less and then stopped speaking long before he died. I was asked to speak at his funeral in Vienna and, although this was difficult for me, it helped me to accept that we would never have our great music versus mathematics conversations again. 

The other great composer, whom I met were Iannis Xenakis, Jean-Claude Risset and Steve Reich. I was able to lure Iannis to Bremen and be present at a concert covering some of his great work. The concert was financially subsidized by the German TV-Station 3sat under the condition that I would engage Iannis in an interview that 3sat would broadcast together with the concert. One year earlier we had done a similar project with Ligeti (see video) and one year later we did another project like this with Detlev Mueller-Siemens. When I say we, I am referring to my friend Volker Banfield, the pianist, who would join me in the interview (and play in the Ligeti event), and my friend Hans-Helmut Euler, who had a professional relationship with 3sat and was responsible for the deals with 3sat.  My role was programming the concerts, luring the composers and running the interviews.​


I want to communicate a few first hand impressions about each of the composers. Reich’s work, like Piano Phase, has many connections to brain research and mathematics and acoustical physics. I had studied that quite deeply, because I like his work so much. One of my favorite pieces is DrummingWhen I briefly met him in Miami I was looking forward to learn more details. However, it turned out that he was not aware of many of these connections and didn’t seem to be interested. What a difference when compared with Ligeti!


Xenakis had a very profound education in mathematics and much of his work is deeply motivated and structured by stochastic processed and probability theory. What I also liked about him was his anti-facist background during the second world war. When I listen to his music, I don’t hear that but it certainly helps enormously to appreciate what is going on.


Jean-Claude and I probably were able to communicate like colleagues. His mind and thinking is scientifically structured through and through and I have used his ingenious synthetic work around Shepard-tone glissandi in numerous public lectures. His work makes one point very clear: Our perceptual system is a signification part of music beyond what is on the score. I also was perplexed when I found traces of Risset’s glissandi in some of Bach’s organ music. Here is a great example: BWV 542 - Fantasia & Fugue in G Minor. Listen to the glissandos after some 3:35 minutes. On the background of Risset’s endless falling and rising glissandi, I perceive Bach very differently and quite enriched by a new dimension.




Peter H. Richter:

Peter Richter was a colleague and dear friend at the University of Bremen. Peter was a theoretical physicist. We did many things together, which probably we wouldn’t have done just on our own. Peter was always ready for scientific adventures but at the same time extremely modest and sincere. He kept my feet on the ground and yet, most likely, without Peter I would not have entered chaos theory. We run seminars to educate ourselves about continuous (Hamiltonian) dynamical systems, Hamiltonian chaos, discrete dynamical systems, area preserving diffeomorphisms and eventually dynamical systems in the complex plane. Peter always had an inclination to maintain a connection with real physics in these journeys: classical mechanics and the theory of phase transitions, in particular.


I want to communicate one anecdote from 1993, which is connected with our bestselling book “The Beauty of Fractals” and which culminated in a triumphant victory over the German news magazine Der SPIEGEL. Peter Brügge, one of their long-time journalists, published three successive articles under the headline “Der Kult um das Chaos”.

In the months before he had visited Peter and myself a couple of times in Bremen, had spent time with us in our lab and we must have spent several days total with discussions about what chaos means in rigorous mathematical terms and how it is used and misused or even abused in overly euphoric public discussions in some media and popular science books and articles. In fact, we struggled many times when we tried  to make him understand the real mathematics behind chaos and to understand its limitations.


Overall, we found him quite interesting, because he appeared to us as someone who really wanted to understand what was the meat in chaos and what was  a bluff. Then the articles appeared and we were stunned. Apparently he had wanted our confidence and had concealed his true intentions very deceivingly. On one hand he spoke about my appreciation for him on the other hand he claimed that we had published results about chaos in celestial systems (in “The Beauty of Fractals”), which he had found not to be chaotic at all, based on the testimony of a mathematician from Karlsruhe. His remarks were derogatory and tried to put us in the same corner with some chaos evangelists who made unfounded claims. Specifically his “insight” was that chaos did not really exist but rather was a result of sloppy computer simulations.  We were very disappointed, discussed matters with the president of Springer-Verlag in Heidelberg, our publisher, and came to the conclusion that we should go after Der SPIEGEL legally. We hired the top legal firm in Hamburg for matters of misrepresentations in the press and, guided by legal council, demanded that Der SPIEGEL would publish a counter statement (Gegendarstellung). At the court hearing, we were offered the opportunity to settle outside of court. Der SPIEGEL offered that Peter and I could publish 3 articles in Der SPIEGEL, without any restriction. Our legal council advised that this would indicate their fear for a counter statement and consequently we declined the offer. The court ruled that a counter statement, setting things straight on our behalf, had to be published by der SPIEGEL.  Compared with the damage that the article had done this was a small effect but a triumphant victory over Der SPIEGEL, because there is nothing which news magazines hate more than being forced to publish counter statements. 


The Karlsruhe mathematician played a dubious role in this affair. However, some time later he published a paper confirming our resuts, which he had denounced to be wrong, at least that is what Brügge had claimed in the court proceedings. 


Later we summarized our experiences and provided many mathematical details in an article which was published under the title: "Der SPIEGEL, das Chaos - und die Wahrheit” in Physikalische Blätter, Nr. 4, 1994 (The Physikalische Blätter was an official journal of the Deutsche Physikalische Gesellschaft. It was renamed to Physik Journal in 2002).


In his later years Peter became Konrektor (Vice-President) of the University of Bremen and was responsible for a smooth transition in the transformation processes towards bachelor and master study programs at the University of Bremen. He had a big heart for physics education. After his retirement he worked very successfully for an improvement of the training for future physics teachers, which was a nice parallel with my efforts for math teachers.

bottom of page