Ion Barbu (Romanian pronunciation: [iˈon ˈbarbu], pen name of Dan Barbilian; 18 March 1895 –11 August 1961) was a Romanian mathematician and poet. His name is associated with the Mathematics Subject Classification number 51C05, which is a major posthumous recognition reserved only to pioneers of investigations in an area of mathematical inquiry. As a poet, he is known for his volume Joc secund ("Mirrored Play"), in which he sought to fulfill his vision of a poetry which adhered to the same virtues that he found in mathematics.
Early life Born in Câmpulung-Muscel, Argeș County, he was the son of Constantin Barbilian and Smaranda, born Șoiculescu. He attended elementary school in Câmpulung, Dămienești, and Stâlpeni, and for secondary studies he went to the Ion Brătianu High School in Pitești, the Dinicu Golescu High School in Câmpulung, and finally the Gheorghe Lazăr High School and the Mihai Viteazul High School in Bucharest. During that time, he discovered that he had a talent for mathematics, and started publishing in Gazeta Matematică; it was also then that he discovered his passion for poetry.
He was a student at the University of Bucharest when World War I caused his studies to be interrupted by military service. After being sent to Botoșani in December 1916, he attended the Reserve Officers' School in Bârlad and was promoted to the rank of corporal in April 1917. Serving under the command of major Barbu Alinescu, he advanced to platoon leader by April 1918, and went into reserve as a sub-lieutenant in 1919. Barbilian completed his undergraduate degree in 1921. The next year he won a doctoral grant to go to the University of Göttingen, where he studied number theory with Edmund Landau for two years. However, he attended few classes, suffered from cocaine and ether addiction, and eventually abandoned his studies at Göttingen. Returning to Bucharest, chronically ill as a result of drug intoxication, he was hospitalized for rehabilitation from August 1924 to January 1925. In 1925 he began to teach mathematics at Spiru Haret High School, along with his German wife, Gerda, who taught German literature. He then studied with Gheorghe Țițeica, completing in 1929 his Ph.D. thesis, Reprezentarea canonică a adunării funcțiilor ipereliptice (Canonical representation of the addition of hyperelliptic functions). The thesis defense committee was presided by David Emmanuel and included Țițeica and Dimitrie Pompeiu. In the spring of 1929 he bought a house at 8, Carol Davila Street, Bucharest, where he would live for the rest of his life. For a while, he taught at the Cantemir Vodă High School. In the summer of 1937, he served as president of the commission administering the Baccalaureate at the Gheorghe Lazăr High School in Sibiu, after which he issued a scathing report to the Ministry of Education.
Achievements in mathematics
Apollonian metric In 1935, Barbilian published his article describing metrization of a region K, the interior of a simple closed curve J. Let xy denote the Euclidean distance from x to y. Barbilian's function for the distance from a to b in K is
d ( a , b ) = log max p ∈ J ( p a / p b ) + log max q ∈ J ( q b / q a ) . {\displaystyle d(a,b)=\log {\underset {p\in J}{\max }}(pa/pb)+\log {\underset {q\in J}{\max }}(qb/qa).}
As Barbilian noted, this construction generates various geometries that are generalizations of the Klein projective model; he highlighted four special cases, including the Poincaré disk model in hyperbolic geometry. At the University of Missouri in 1938 Leonard Blumenthal wrote Distance Geometry. A Study of the Development of Abstract Metrics, where he used the term "Barbilian spaces" for metric spaces based on Barbilian's function to obtain their metric. And in 1954 the American Mathematical Monthly published an article by Paul J. Kelly on Barbilian's method of metrizing a region bounded by a curve. Barbilian claimed he did not have access to Kelly's publication, but he did read Blumenthal's review of it in Mathematical Reviews and he understood Kelly's construction. This motivated him to write in final form a series of four papers, which appeared after 1958, where the metric geometry of the spaces that today bears his name is investigated thoroughly. He answered in 1959 with an article which described "a very general procedure of metrization through which the positive functions of two points, on certain sets, can be refined to a distance." Besides Blumenthal and Kelly, articles on "Barbilian spaces" have appeared in the 1990s from Patricia Souza, while Wladimir G. Boskoff, Marian G. Ciucă and Bogdan Suceavă wrote in the 2000s about "Barbilian's metrization procedure". Barbilian indicated in his paper Asupra unui principiu de metrizare that he preferred the term "Apollonian metric space", and articles from Alan F. Beardon, Frederick Gehring and Kari Hag, Peter A. Häströ, Zair Ibragimov and others use that term. According to Suceavă, "Barbilian's metrization procedure is important for at least three reasons: (1) It yields a natural generalization of Poincaré and Beltrami–Klein's hyperbolic geometries; (2) It has been studied in the context of the study of Apollonian metric; (3) Provides a large class of examples of Lagrange generalized metrics irreducible to Riemann, Finsler, or Lagrange metrics."
Ring geometry Barbilian made a contribution to the foundations of geometry with his articles in 1940 and 1941 in Jahresbericht der Deutschen Mathematiker-Vereinigung on projective planes with coordinates from a ring. According to Boskoff and Suceavă, this work "inspired research in ring geometries, nowadays associated with his, Hjelmslev's and Klingenberg's names." A more critical stance was taken in 1995 by Ferdinand D. Velkamp:
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