Margherita Beloch Piazzolla (12 July 1879, in

Frascati Frascati () is a city and ''comune'' in the Metropolitan City of Rome Capital in the Lazio region of central Italy. It is located south-east of Rome, on the Alban Hills close to the ancient city of Tusculum. Frascati is closely associated with ...

– 28 September 1976, in

Rome , established_title = Founded , established_date = 753 BC , founder = King Romulus (legendary) , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, Italy).svg , map_caption ...

) was an Italian mathematician who worked in algebraic geometry, algebraic topology and photogrammetry.

Biography

Beloch was the daughter of the German historian

Karl Julius Beloch Karl Julius Beloch (21 January 1854 in Nieder-Petschkendorf – 1 February 1929 in Rome) was a German classical and economic historian. Biography From 1872 to 1875, he studied classical philology and ancient history in Freiburg, Heidelbe ...

, who taught ancient history for 50 years at

Sapienza University of Rome The Sapienza University of Rome ( it, Sapienza – Università di Roma), also called simply Sapienza or the University of Rome, and formally the Università degli Studi di Roma "La Sapienza", is a Public university, public research university l ...

, and American Bella Bailey. Beloch studied mathematics at the

and wrote her undergraduate thesis under the supervision of

Guido Castelnuovo Guido Castelnuovo (14 August 1865 – 27 April 1952) was an Italian mathematician. He is best known for his contributions to the field of algebraic geometry, though his contributions to the study of statistics and probability theory are also signi ...

. She received her degree in 1908 with Laude and "dignità di stampa" which means that her work was worthy of publication, and in fact her thesis "Sulle trasformazioni birazionali nello spazio" (On Birational Transformations in Space) was published in the Annali di Matematica Pura ed Applicata.

was very impressed with her talent and offered her the position of assistant which Margherita took and held until 1919, when she moved to Pavia. In 1920 she moved to Palermo to work under Michele De Franchis, an important figure of the Italian school of algebraic geometry at the time. In 1924, Beloch completed her "libera docenza" (a degree that at that time had to be obtained before one could become a professor), and three years later she became a full professor at the

University of Ferrara The University of Ferrara ( it, Università degli Studi di Ferrara) is the main university of the city of Ferrara in the Emilia-Romagna region of northern Italy. In the years prior to the First World War the University of Ferrara, with more than 5 ...

where she taught until her retirement (1955). Piazzolla is Beloch's married surname.

Scientific work

Her main scientific interests were in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

and

photogrammetry Photogrammetry is the science and technology of obtaining reliable information about physical objects and the environment through the process of recording, measuring and interpreting photographic images and patterns of electromagnetic radiant ima ...

. After her thesis, she worked on classification of algebraic surfaces studying the configurations of lines that could lie on surfaces. The next step was to study rational curves lying on surfaces and in this framework Beloch obtained the following important result: "Hyperelliptic surfaces of rank 2 are characterised by having 16 rational curves." Beloch also made some contributions to the theory of skew algebraic curves. She continued working on topological properties of algebraic curves either planar or lying on ruled or cubic surfaces for most of her life, writing about a dozen papers on these subjects. Around 1940 Beloch become more and more interested in

and the application of mathematics, and in particular algebraic geometry, to it. She is also known for her contribution to the mathematics of paper folding: In particular she seems to have been the first to formalise an origami move which allows, when possible, to construct by paper folding the common tangents to two parabolas. As a consequence she showed how to extract cubic roots by paper folding, something that is impossible to do by

ruler and compass In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

. The move she used has been called the

Beloch fold Beloch is a European surname. Notable people with the surname include: * Karl Julius Beloch (1854–1929), German classical and economic historian * Margherita Piazzola Beloch Margherita Beloch Piazzolla (12 July 1879, in Frascati – 28 Septe ...

.Ken Liu
"The Magic and Mathematics of Paper-Folding"
''Tor.com'', June 29, 2017.

References

1879 births 1976 deaths People from Frascati Italian mathematicians Women mathematicians {{Mathematics of paper folding