**Mitchell Feigenbaum**'s father is Abraham Joseph Feigenbaum, an analytic chemist whose parents had emigrated from a town near Warsaw in Poland to the United States. Mitchell's (or Mitch's as he is known) mother is Mildred Sugar whose parents emigrated to the United States from Kiev. Mitchell was the middle child of his parents three children, having an older brother Edward and a younger sister Glenda.

Mitchell entered a public school for gifted children when he was five years old. Unlike Edward who displayed all the characteristics of a child prodigy, reading from a very young age, Mitchell could not read when he entered school and he needed tutoring from his mother to bring him up to the level of the other children. Moved to a different school, he became somewhat bored and had no friends among the other children. In fact up until the time he went to university Mitchell would not enjoy the company of his fellow pupils.

Feigenbaum's mother taught him algebra when he was in the fifth form but reading continued to be something that he did not like much. Perhaps the reason was that he tried reading articles in *Encyclopaedia Britannica* which, given that he was so young, proved too difficult for him to understand. When he was twelve years old he started his high school education in Brooklyn. About the same time he began to develop certain obsessive tendencies such as excessive cleanliness which meant that he was continually washing his hands. He suffered these difficulties for quite a few years but overcame them when a university student.

The school system seemed unable to provide Feigenbaum with the right stimulus for he tried as hard as he could to avoid classes despite making remarkable academic progress and scoring full marks in mathematics and science in the examinations covering the State. Even when he went to Tilden High School in Brooklyn, a school with a fine reputation, Feigenbaum found the education there no more enjoyable, despite once again excelling in examinations.

In [1] Feigenbaum described how his love of calculating started at school:-

... starting in junior high school, I decided that I could calculate the logarithm table myself, and later the trigonometric tables. I loved Newton's method for solving transcendentals, and in high school I already knew that starting values can make a big difference and lead to non-convergent jumps up to the limit of patience of manual arithmetic. My father showed me his beautiful ivory-on-mahogany slide rule in junior high school, and I quickly realised its idea. I was allowed to use the new Friden calculating machine which, shortly before its transformation into a relic, could also extract square roots. I love numbers and always as an amusement, and more seriously than that, invented new algorithms to calculate them.

In fact while at school Feigenbaum had usually learnt more in studying by himself than in the formal lessons. He had already taught himself to play the piano when he was about 12 years old, but at high school he taught himself calculus. Also at high school a friend of his father gave him a mechanical device with switching circuits that could play nim and other games. The machine came with a paper by Shannon on Boolean logic which fascinated Feigenbaum with his self-learning attitude.

In February 1960, at the age of sixteen, Feigenbaum entered the City College of New York. There he studied electrical engineering but attended all the mathematics courses and the physics courses in addition to those in electrical engineering. Completing the five year course in less than four years he graduated with a Bachelor's degree in 1964. In the summer of that year he began his graduate studies at Massachusetts Institute of Technology. He entered MIT with the intention of researching in electrical engineering for his doctorate but after only one term he changed to physics and began to study general relativity.

Now again general relativity was a topic which he studied on his own, reading the book *Course of Theoretical Physics* by Lev Landau and Evgenii Lifshitz. His official courses were on quantum mechanics, classical mechanics, and complex function theory. It was while he was at MIT that Feigenbaum first used a computer but not as part of his studies there. It was when he was visiting Brooklyn Polytechnic that he found they had a programmable digital computer. He writes [1]:-

This was the first computer I ever used, and within an hour had programmed it to take square roots by Newton's method.

At MIT Feigenbaum's doctoral studies were supervised by Francis Low and he was awarded a doctorate in 1970 for a dissertation on dispersion relations. Following this he went to Cornell as an instructor/research associate, a post which was half funded by an NSF postdoctoral grant, and half funded as a teaching post. During his two years at Cornell he taught courses on variational techniques and on quantum mechanics. He used a HP computer at Cornell which perhaps could be better described as a programmable calculator. The machine had only one other user, Ken Wilson, so he was able to spend time mastering its use.

After the two years at Cornell, Feigenbaum went to Virginia Polytechnic Institute as a postdoctoral worker, again with a two year position. He again taught, giving courses on Banach spaces and *C**-algebras. Certainly these short term posts were not ideal. As Feigenbaum said (see [7]):-

These two year positions made serious work almost impossible. After one year you had to start worrying about where you could go next.

After the two years at Virginia Polytechnic Institute, Feigenbaum was offered a long term position on the staff of the theory division at Los Alamos. He writes [1]:-

When I arrived at Los Alamos, the theory division head, P Carruthers, felt that the time was right, and I was the appropriate person, to see if Wilson's renormalisation group ideas could solve the century and a half old problem of turbulence. In a nutshell, it couldn't - or so far hasn't - but led me off in wonderful directions.

The 'wonderful directions' that Feigenbaum refers to here involve the study of chaos where he was to make a remarkable discovery. It was made since data was available from computing and, as Feigenbaum himself has noted, only became obvious because the computers he used calculated so slowly that he could see the intermediate steps of the calculation. Feigenbaum's involvement with computers moved forward in December 1974 when he got his own programmable calculator for the first time, the HP65. With this machine [1]:-

In swift order, I invented new ODE solvers, minimisation routines, interpolation methods, etc. For someone who cares for numbers, much of the tedium was eliminated.

In 1976 Sir Robert May, then a professor of biology at Princeton, pointed out that the logistic map led to chaotic dynamics. The logistic mapping *g* is defined by

x_{n+1}=g(x_{n}) =λx_{n}(1 -x_{n}).

It models the relative population *x*_{n} which is the ratio of the actual population to the maximum population. Each iteration gives the new relative population in terms of the old one. The parameter *λ* is the effective growth rate. We must have 0 < *x*_{n} ≤ 1 and 0 ≤ *λ* ≤ 4.

For *λ* < 1, *x*_{n} tends to 0. For 1 ≤ *λ* ≤ 3, *x*_{n} tends to 1 - ^{1}/_{λ}. Beyond 3 a bifurcation occurs (corresponding to high and low populations in alternate years). Further bifurcations occur until at about *λ* = 3.57... chaotic dynamics sets in.

In 1973 it had been conjectured that the behaviour of the logistic equation was the same in a qualitative sense for all *g*(*x*) which have a maximum value and decrease monotonically on either side of this maximum. The remarkable result obtained by Feigenbaum was to show that not only was the behaviour qualitatively similar but there was a very precise mathematical result which held for all such logistic equations.

Feigenbaum did not actually work with the precise logistic equation which May studied and in fact his work was independent of that by May. What Feigenbaum pointed out, if we state it in terms of the notation set up above, was that if *λ*_{n} is the parameter value at which the *n*th bifurcation occurs then

(

λ_{n}-λ_{n-1})/(λ_{n+1}-λ_{n})→ 4.669201609102... asn→ ∞.

When Feigenbaum first found 4.669 in August 1975, which he only found to three places due to the limit of the accuracy of his HP65, he spend some time trying to see if it was a simple combination of 'well-known' numbers. He did not find anything. Of course, now the number is 'well-known' and called the *Feigenbaum number*.

This in itself was surprising but in October 1975 Feigenbaum found that this number is the same for a large class of period doubling mappings. This was indeed remarkable and Feigenbaum realised the significance of it immediately [1]:-

I called my parents that evening and told them that I had discovered something truly remarkable, that, when I had understood it, would make me a famous man.

By April 1976 Feigenbaum had completed his first paper on the topic. He submitted it to a journal but after taking six months to referee the paper they rejected it. By 1977 he had been asked by over a 1000 scientists for a copy of it. He eventually managed to get it published in 1978. His second, more technical, paper finished in November 1976, suffered a similar fate and was rejected when first submitted. It eventually appeared in print in 1979. Feigenbaum presents an elementary review on period-doubling bifurcations in nonlinear dynamical systems in [4].

Feigenbaum has made other contributions to the theory of chaos and he has also written two papers on the mathematics of making maps. In one of these (the paper [2]) Feigenbaum writes:-

Constructing maps from a digital database requires the development of a number of special tools. These, amongst others, include methods for generalising linework and for the automated placement of type. Additionally, granted the numerical power of a computer with its attendant indifference to whether it plots lines and circles or analytically much more complicated curves, an opportunity exists to craft projections of much higher fidelity than have previously been possible. Thus, one should develop tools to capitalise on this power and modernise cartography. ... The modernisation of cartography done to archival standards poses many problems, the solutions for which are strongly illuminated by the ideas and methods of nonlinear systems. The maps constructed with these methods all appeared for the first time in The Hammond Atlas of the World, published exactly one year ago.

The Introduction to the *Hammond Atlas* notes [6]:-

Using fractal geometry to describe natural forms such as coastlines, mathematical physicist Mitchell Feigenbaum developed software capable reconfiguring coastlines, borders, and mountain ranges to fit a multitide of map scales and projections. Dr Feigenbaum also created a new computerised type placement program which places thousands of map labels in minutes, a task which previously required days of tedious labour.

It might at this point be reasonable to wonder whether Feigenbaum considers himself a mathematician or a physicist. His view is that there is no hard distinction between physics and mathematics. We agree with him and certainly in constructing this archive we have taken the view that mathematics includes theoretical physics.

In 1982 Feigenbaum left Los Alamos when he was appointed to a professorship at Cornell. Four years later he became the first Toyota professor at Rockefeller University. In the same year that he was appointed to Rockefeller University he was awarded the Wolf Prize in physics. The citation for the prize said that it was awarded to Feigenbaum:-

... for his pioneering theoretical studies demonstrating the universal character of non-linear systems, which has made possible the systematic study of chaos.

The press release made at the time that he was awarded the prize, sums up nicely his contribution:-

The impact of Feigenbaum's discoveries has been phenomenal. It has spanned new fields of theoretical and experimental mathematics ... It is hard to think of any other development in recent theoretical science that has had so broad an impact over so wide a range of fields, spanning both the very pure and the very applied.

Among other awards that Feigenbaum has received we mention the Los Alamos National Laboratory's Distinguished Performance Award in 1980, the Ernest O Lawrence Award by the U.S. Department of Energy in 1982, a MacArthur Foundation award in 1984, and in 2005 the New York City Mayor's Award for Excellence in Science and Technology:-

... for his pioneering studies in chaos theory.

**Article by:** *J J O'Connor* and *E F Robertson*