The outstanding Fibonacci mathematicians of the 20th century

In the 20th century the interest in Fibonacci numbers in modern mathematics again increases. In the first half of the 20th century the considerable results in this area were obtained by the Dutch mathematician Willem Abraham Wythoff and the Belgian mathematician Edouard Zeckendorf.

Willem Abraham Wythoff (1865-1939). Many combinatorialists and number theorists know about the Wythoff game but few know anything about the man for whom this subject is named.

Willem Abraham Wythoff (1865-1939)
Willem Abraham Wythoff (1865-1939)

W. A. Wythoff was born in Amsterdam in 1865, the son of an operator of a sugar refinery. He received Ph.D. in mathematics from the University of Amsterdam in 1898. From 1899 to 1929, Dr. Wythoff was the collaborator of Revue Semestrielle des Publications Mathematiques, a forerunner of Mathematical Reviews. The Wythoff game originates in W. A. Wythoff, "A modification of the game of nim," Nieuw Archief voor wiskunde 2 (1905-07) 199-202.

Dr. Wythoff's own words:

"The game is played by two persons. Two piles of counters are placed on the table, the number of each pile being arbitrary. The players play alternately and either take from one of the piles an arbitrary number of counters or from both piles an equal number. The player who takes up the last counter or counters, wins."

The solutions of the Wythoff game, involving the Fibonacci numbers, can be found in numerous articles in The Fibonacci Quarterly, and also in H. S. M. Coxeter, "The golden section, phyllotaxis, and Wythoff's game," Scripta Mathematica, 19 (1953) 135-143.

Edouard Zeckendorf (1901-1983)
Edouard Zeckendorf (1901-1983)

Edouard Zeckendorf (1901-1983) M.D., Colonel Belgian Medical Corps, number-theorist. Many number theorists know about Zeckendorf sums, but few know anything about man for whom these sums are named. At the end of the nineteenth century, his father established himself as a dentist in Liege, Belgium. He was a Dutch citizen. Edouard was born in Liege, and in 1925 he qualified as a medical doctor at the University of Liege and became a Belgian Army officer. Before 1930, he also obtained a License for Dental Surgery. Both Dr. Zeckendorf and his wife were artists.

In 1940, Dr. Zeckendorf was taken prisoner by the Germans, and in that capacity, he provided medical care to allied POW's until 1945. During 1949-50, he was in charge of a 500-mile long Zeckendorf sums originate with an article published in 1939. Each positive integer has a unique representation as a sum of two nonconsecutive Fibonacci numbers. In many ways, these representations can be compared with binary representations. Numerous articles published in The Fibonacci Quarterly discuss Zeckendorf sums and their many generalizations.

The small brochure of the Russian mathematician Vorobyev "Fibonacci Numbers" published in 1961 rendered a huge influence on increase of mathematical interest in Fibonacci numbers. The brochure issued repeatedly and was translated to many World languages.

The Vorobyev's brochure 'Fibonacci Numbers'

However the establishment of the mathematical Fibonacci Association by the group of the American mathematicians in 1963 became by the most outstanding event in the history of the Fibonacci numbers theory. Since 1963 Fibonacci Association began to issue "The Fibonacci Quarterly". The American mathematicians Verner Emil Hoggatt (1921-1981) and Alfred Brousseau (1907-1988) are founders of the Fibonacci Association. Verner Hoggatt, along with Brother Alfred Brousseau, published the first volume of The Fibonacci Quarterly in 1963, thereby founding the Fibonacci Association. The Quarterly has grown into the well-recognized number theory journal, and the Association will sponsor its Ninth International Conference on Fibonacci Numbers and Their Applications during the year 2000.

Verner Emil Hoggatt (1921-1981)

The April 4, 1969 issue of TIME Magazine reported on the phenomenal growth of the Fibonacci Association. That same year, Houghton Mifflin published Vern's book, Fibonacci and Lucas Numbers, perhaps the world's best introduction to the Fibonacci numbers theory. Howard Eves has written that "during his long and outstanding tenure at San Jose State University, Vern directed an enormous number of master's theses, and put out an amazing number of attractive papers... He became the authority on Fibonacci and related numbers."

Alfred Brousseau (1907-1988)

Alfred Brousseau (1907-1988) was other outstanding person involved to establishment of the Fibonacci Association. The order, to which Brother Alfred belonged, is Fratres Scholarum Christianarum, meaning Brothers of the Christian Schools, or simply the Christian Brothers. Brother Alfred was received into the Order on July 31, 1923. He was assigned to St. Mary's College in Moraga, California, in 1930.

While teaching at St. Mary's, Brother Alfred also continued his own studies, in physics, and in 1937 he received the Ph.D. degree from the University of California. In 1941, he was appointed Principal of Sacred Heart High School, and in 1959 in returned to St. Mary's College. He served as Chair of the School of Science there for many years, and was active in the classroom and activities of the California Mathematics Council until 1978.

Brother Alfred was an avid photographer. He made a collection of some 20,000 slides of California wildflowers. Images of more than half of these, and another of Brother Alfred's collections as well, are preserved in the form of widely used websites at the University of California, Berkeley.

In 1969 the Time Magazine featured the two founders of the Fibonacci Association in the article "The Fibonacci Numbers". Brother Alfred is quoted-"We got a group of people together in 1963, and just like a bunch of nuts, we started a mathematics magazine," The Fibonacci Quarterly. "We have a backlog of articles, and we've been accepted by the mathematical fraternity." Brother Alfred is pictured in the Time article holding a pineapple, which has well-known connections with Fibonacci numbers. The column tells of other natural occurrences of these numbers-male bees reproduce "Fibonaccially," and Fibonacci numbers occur in the spiral floret formations visible in many sunflowers, scales on pine cones, branching patterns on trees, and leaf-positions on branches. People who learn about the Fibonacci sequence, Brother Alfred said, "tend to find an esthetic satisfaction in it. They think that there's some kind of mystical connection between these numbers and the universe."

After 20 years after the book of the Verner Hoggat the Publisher "Ellis Horwood Limited" published in 1989 the book by Prof. S. Vaida "Fibonacci & Lucas Numbers and the Golden Section", which attracted a broad attention of Fibonacci mathematicians and is at present the best mathematical book in this area. Who the writer of this book? From his brief scientific biography we learn that the mathematician Steven Vaida at the moment of the book publication was the Professor of mathematics of University Sussex (England). He obtained a degree of the Doctor of Philosophy at the Vienna University. In 1965 he began to work as the Professor of operational researches of the Birmingem University. Also it is noticed that he was the honorary member of many mathematical organizations, in particular, of the Mathematics and Statistics Institute and статистки and of the Society of operational researches (England).

Among the modern Fibonacci mathematicians it is necessary to select Professor Herta Taussing Freitag, the member of the Fibonacci Association. She was born 6 Dec 1908 in Vienna, Austria and died 25 Jan 2000 in Roanoke, Virginia, USA.

Herta Taussing Freitag (1908-2000)
Herta Taussing Freitag (1908-2000)

Herta Freitag received the degree Magister Rerum Naturalium, in Mathematics and Physics, from the University of Vienna in 1934, and graduated M.A. (1948), Ph.D. (1953) from Columbia University, USA.

Her lectures, always meticulously crafted and beautifully illustrated in her inimitably artistic calligraphy, are delivered so enthusiastically and yet so modestly, as if she fears that her personality might take any of the glory or attention away from Mathematics. One of her most inspired remarks concerns mathematicians' fondness for generalizing results:

A mathematician is like a lover - give him a little finger and he wants the whole hand!

A colleague, with echoes of Gauss's description of Mathematics as the Queen of the Sciences, and Number Theory as the Queen of Mathematics, named Herta Freitag as the Queen of the Fibonacci Association. For she has attended and given a paper at every International Conference of the Association since the first one in 1984. She has also contributed prodigiously to the Elementary Problems and Solutions Section of the Fibonacci Quarterly and published many papers in that journal. Most appropriately, the Fibonacci Quarterly chose to honor her not on her 90th birthday, but on the threshold of her 89th year, since 89 is a Fibonacci number.

It is curiously to note that Prof. Gerta Freitag delivered at the 7th International Conference on Fibonacci Numbers and Their Applications the lecture "Elements of Zeckendorf Arithmetic" (co-author G.M. Phillips). The fact of the appearance of the lecture with such title is rather distinctive. He testifies that the Fibonacci mathematicians closely came to the creation of the new computer arithmetic, based on Fibonacci numbers and Zeckendorf's sums. As was shown at the previous pages of our Museum, just this direction became by the subject of the scientific and engineering developments in the 70th and 80th years of the 20th century in the Soviet science.