# Vera Martha Winitzky de Spinadel

### Quick Info

Born
22 August 1929
Buenos Aires, Argentina
Died
26 January 2017
Buenos Aires, Argentina

Summary
Vera W de Spinadel was the first woman to be awarded a mathematics Ph.D. by the University of Buenos Aires. She was an Argentine mathematician whose main contributions were to mathematics in architecture, art, and design. She introduced the "metallic means family" which generalises the Golden Ratio.

### Biography

Vera Spinadel was given the name Vera Martha Winitzky and only became known as Vera Spinadel, Vera de Spinadel or Vera W de Spinadel after her marriage to Erico Spinadel. She was the daughter of Alejandro Winitzky and Rosa Schajnovidze. Vera attended both primary and secondary school in Buenos Aires before entering the Faculty of Exact and Natural Sciences of the University of Buenos Aires in 1947.

When Vera began her studies of mathematics at the University of Buenos Aires, Julio Rey Pastor was head of mathematics and Alberto González Domínguez (1904-1982) was a full professor. Vera was taught by González Domínguez who had strong links to a broad range of leading mathematicians throughout the world. He worked on mathematical analysis, in particular on the theory of probability, the theory of distributions and applications to quantum physics. Vera's mathematical interests were broad and she studied applications of mathematics to architecture, economics and engineering. While an undergraduate, in 1949 she attended lectures on meteorology at the Servicio Meteorológico Nacional. From 1949 to 1952 she undertook research on seismology in the Department of Geophysics, Servicio Meteorológico Nacional, under the direction of Otto Schneider. In July 1952 Vera graduated with the degree of Bachelor of Physical-Mathematical Sciences.

After the award of her Bachelor's degree, Vera was appointed as a Special Mathematics Teaching Assistant in the Faculty of Economic Sciences at the University of Buenos Aires. There, in 1952, she took part in two mathematics seminars directed by Alberto González Domínguez on Divisors of Zero in the Algebra D and Fundamentals of Mikusinski's Operational Calculus [3]:-
Vera reported that she felt insecure thinking about how much she still needed to learn to work as a teacher, but she ended up accepting [the position] and started teaching classes in mathematics and architecture. Passionate about mathematics, she decided to deepen her studies by studying for a doctorate in Mathematical Sciences at the same university where she had studied and was working.
In July 1954 she was a delegate from the University of Buenos Aires to the Colloquium "Some Mathematical Problems that are being studied in Latin America", which was held in Villavicencio, Mendoza.

Two people now played a large part in her life, Erico Spinadel (1929-2020) and Emilio Oscar Roxin. We should look briefly at this.

Erico Spinadel (1929-2020) was an Austrian, born in Vienna on 6 May 1929. He moved to Argentina in 1938 and studied at the Escuela Industrial de la Nación Otto Krause in Buenos Aires from 1943 to 1948. He then entered the University of Buenos Aires and was in the same classes as Vera for some of the mathematics courses he took as part of his engineering degree. Both Vera and Erico graduated in 1952 and Erico then worked from 1952 to 1955 for a firm that manufactured centrifugal and deep well pumps. Vera and Enrico married on 30 June 1955. They had four children: Laura Patricia Spinadel (born 7 July 1958), who became a leading architect and university professor; Pablo Spinadel, who became an electronic engineer with a Ph.D. in flexible automation from the University of Vienna; Irene Spinadel, who studied psychology, worked as a psychotherapist but from 2002 has had a literary career in Spain; and Andrea Gisela Spinadel, who became an artistic director.

Emilio Oscar Roxin undertook research at the University of Buenos Aires with Alberto González Domínguez as his thesis advisor. González Domínguez was appointed as a member of the board of directors of the National Atomic Energy Commission of Argentina in October 1955. Vera Spinadel attended a course on Nuclear Reactors at the Comisión Nacional de Energía Atómica in San Carlos de Bariloche in 1955 and for the following few years undertook research on nuclear reactors advised by Fidel Alsina Fuertes. For this research she collaborated with Emilio Roxin and they published two joint works in the Internal Publications Series of the Comisión Nacional de Energía Atómica, namely Cálculo del Factor de Utilización Térmica (1956) and Sobre un Problema de Cinética de Reactores (1957). The Argentine Atomic Energy Commission was planning for the start of construction of the RA-1 reactor, the first reactor to start up in all of Latin America and the southern hemisphere, and, in 1956, Spinadel worked on the calculation of the critical mass of the RA-1 Reactor. Construction of the reactor began in April 1957. Beginning in 1956 Spinadel participated in a seminar on Non-Linear Differential Equations, directed by Emilio Roxin.

In May 1958 Spinadel submitted her thesis Teoría de las Zonas Alcanzables en Sistemas Bidimensionales and was awarded the degree of Ph.D. in September of that year. With this she became the first woman to obtain a doctorate in mathematics from the University of Buenos Aires. You can read a translation of the Introduction to her thesis at THIS LINK.

Several references give Spinadel's thesis advisor as Emilio Roxin but this seems to be an error. In her thesis Spinadel does not acknowledge help from any thesis advisor but she does reference her joint paper with Emilio Roxin, namely Sobre un Problema de Sistemas de Ecuaciones Diferenciales Lineales , as "to appear". It was published later in 1958. Emilio Roxin submitted his thesis Puntos y zonas alcanzables en sistemas autónomos perturbados en forma arbitraria to the University of Buenos Aires in April 1958, one month before Spinadel submitted her thesis. In it he expresses his thanks to Vera Spinadel and to Alberto González Domínguez and also references his joint paper with Spinadel as "to appear".

In 1957 Spinadel was appointed as Acting Adjunct Professor, Chair of Mathematics II, Faculty of Architecture and Urbanism. She was promoted to Regular Associate Professor in the same chair in 1962. In addition, she later held professorships in the Faculty of Exact Sciences and in the Faculty of Economic Sciences.

In 1985 the Faculty of Architecture and Urbanism set up five new courses concerning design. Spinadel discussed the content of these new courses with her students and realised that since there was only a very basic mathematical content, students were exposed to no in-depth mathematical studies. She decided that the design courses required content which explored the mathematical-scientific side so that students could develop more creative processes. It was important, she believed, to emphasise that nature was non-linear and could not be modelled by a linear approach. She had also read Benoit Mandelbrot's book The fractal geometry of nature published in 1982 and wanted to introduce graphic representation and the use of fractals into teaching design. She published papers on teaching graph theory, for example Las redes y sus aplicaciones (1989), and the paper Divisibility and Cellular Automata in the journal Chaos, Solitons and Fractals in 1995. In 1995 she was appointed Director of the Centre for Mathematics and Design in the Faculty of Architecture, Design and Urbanism.

Spinadel began publishing work on "Metallic Numbers" in 1997. The paper [6] introduces these concepts and has the following Abstract:-
The objective of this work is to introduce a new family of quadratic irrational numbers. The family is called Metallic Numbers and its most conspicuous member is the Golden Number. Other members of the family are the Silver Number, the Bronze Number, the Copper Number, the Nickel Number, etc. All of them have interesting common mathematical properties, which are analysed in detail.

The main results obtained in this research work are:

1) the members of the family are closely related to the quasi-periodic behaviour in non-linear dynamics, thus being of great help in the search for universal paths that lead from "order" to "chaos";

2) the sequences based on the members of this family have many additive properties and are simultaneously geometric sequences, which is why they have been the basis of various systems of proportions in Design.

These two facts indicate the existence of a promising bridge that unites the most recent discoveries in technology with art, through the analysis of fundamental relationships between Mathematics and Design.
The reader has almost certainly heard of the Golden Number but may wonder about "the Silver Number, the Bronze Number, the Copper Number, the Nickel Number, etc." Spinadel's Metallic Numbers belong to her "metallic means family" (MMF) and are positive solutions of the equation
$x^{2} - px - q = 0$
for different values of $p$ and $q$, namely $\sigma_{p,q} = \large\frac{1}{2}\normalsize (p + √(p^{2}+ 4q))$.

For $p = 1, q = 1$ we get the Golden Number $\sigma_{1,1} = \large\frac{1}{2}\normalsize (1+ √5)$ which is the limit of the ratio of two consecutive Fibonacci numbers.

For $p = 2, q = 1$ we get the Silver Number $\sigma_{2,1} = 1+ √2$ which is the limit of the ratio of two consecutive Pell numbers.

For $p = 3, q = 1$ we get the Bronze Number $\sigma_{3,1} = \large\frac{1}{2}\normalsize (3 + √13)$ which plays an important role in the study of dynamical systems and quasicrystals.

For $p = 1, q = 2$ we get the Copper Number $\sigma_{1,2} = 2$.

For $p = 1, q = 3$ we get the Nickel Number $\sigma_{1,3} = \large\frac{1}{2}\normalsize (1 + √13)$.

Here are Spinadel's Conclusions in the paper [10]:-
In analysing, from a mathematical point of view, the similarities as well as the differences among the members of the MMF, it is obvious that these characteristics are strongly linked with the transition from periodic to quasi-periodic dynamics. But simultaneously, from the beginning of humanity, there have been philosophical, natural and aesthetic considerations that have given them primacy in the establishment of geometrical proportions based on some members of this family. Such a broad range of applications opens the road to new multi-disciplinary investigations that undoubtedly will contribute to clarifying the relations between art and technology, building a bridge that should join rational scientific thinking with aesthetical emotion. Hopefully, this new perspective could help us to confer on technology, from which we depend every day more and more for our survival, a more human character.

In an interview Spinadel gave (see [2]) she explained about the Golden Number:-
I know it's a bit difficult to explain these mathematical things, but what matters is that the Greeks, who knew everything, wrote that number in public places in Athens with golden characters, hence the name golden number. The figure represented the order of each of the 19 years in which the new moons occur again on the same days. What does it have to do with architecture? Everything! Because the Greeks themselves said that the harmony of forms depended on the golden number and, let me explain to you, the Parthenon is completely designed with that divine proportion. Wherever you look at it, the number 1.61803398 appears. Be careful, the Greeks did not invent it, there is data that civilisations as old as Babylon and Assyria used that number four thousand years ago.
In the same interview she explained that other numbers had been used by architects:-
This is not a condition exclusive to the golden number: there are other systems of proportions - such as the Cordovés - resulting from the research of a Spanish architect, Rafael de La Hoz, who was commissioned [in 1973] to study the domes of the churches of the Córdova region of Spain. He assumed that there he would find the Golden Ratio, which arises from the pentagon, but what he found was a different ratio, which arises from the octagon. Thus he discovered a number that was used for the design of these domes and is called the Cordovan number. Here, in the province of Córdoba, it was also applied, of course, in the churches.
Let us note that the Cordovan number is $\Large\frac 1 {\sqrt{}(2-\sqrt{}2)}$. It is the ratio between the radius of a regular octagon and the length of its side.

Spinadel was an organiser of the First International Conference on Mathematics and Design held in 1995 and of the Second International Conference on Mathematics and Design held in San Sebastian, Spain in June 1998. She was one of the organisers of the Third International Conference on Mathematics and Design held in Melbourne, Australia in 2001. She edited the Proceedings of the Fourth International Conference on Mathematics and Design which was held in Buenos Aires. The 9th International Congress on Mathematical Education was held in the convention centre in Makuhari outside Tokyo, Japan, from 31 July to 6 August 2000. Spinadel attended the Congress having been invited to organise the Study Group on Art and Mathematics Education. In 2001 she served on the Board of the 'International Society for Mathematical and Computational Aesthetics'.

In April 2005, Spinadel founded the Mathematics and Design Laboratory of the University of Buenos Aires. Three years later she was elected president of the International Mathematics and Design Association, which organises international congresses and publishes the Journal of Mathematics and Design. She served on the editorial board of the Smarandache Notions Journal and of the journal Nonlinear Dynamics and Systems Theory.

In 2010 Spinadel retired and, in September of that year, was made Professor Emeritus in the Faculty of Architecture, Design and Urbanism.

Let us end by quoting Spinadel's own description of her work [1]:-
The main objective of my work is to bring together mathematicians, architects, engineers and designers interested in the interaction between mathematics and design. I use the word design in its broadest sense, that is, a design is a planning resource that constitutes a basic element in interdisciplinary communication between human beings, whether architectural, graphic, visual or sound, as well as any other simple or combined interaction.
In 2018 the Vera W de Spinadel Award was set up. It invites students from any part of the world to discover and interpret new applications to the design of the proportions of Spinadel's Family of Metallic Numbers.

### References (show)

1. El Mundo Matemático de Vera, Zibilia (18 November 2018).
https://www.zibilia.com/=ElMundoMatematicoDeVera
2. M Jurado, Vera Spinadel: mujer de arquitectura y matemáticas, Clarín.com (4 May 2018).
https://www.clarin.com/arq/mujer-arquitectura-matematicas_0_B1fFzwZsf.html
3. J A Schifler, A Família dos Números Metálicos no Ensino e Aprendizagem de Conteúdos de Matemática na Educação Básica (Curitiba, 2020).
https://repositorio.utfpr.edu.br/jspui/bitstream/1/24146/1/numerosmetalicosensinoaprendizagem.pdf
4. Vera Spinadel: mujer de arquitectura y matemáticas, Biblioteca de Matemáticas (6 April 2018).
5. Vera W de Spinadel, Nexus Network Journal (2006).
6. V M Winitzky de Spinadel, La familia de números metálicos en Diseño, Seminario Nacional de Gráfica Digital, Sesión de Morfología y Matemática, Ediciones Facultad de Arquitectura, Diseño y Urbanismo, Universidad de Buenos Aires 2 (1997), 173-179.
7. V M Winitzky de Spinadel, The Metallic Means and Design, in Kim Williams (ed.), Nexus II: Architecture and Mathematics (Edizioni dell'Erba, 1998), 143-157.
8. V M Winitzky de Spinadel, The family of Metallic Means, Visual Mathematics 1 (3) (1999), 317-338.
9. V M Winitzky de Spinadel, "Triangulature" in Andrea Palladio, Nexus Network Journal, Architecture and Mathematics 1 (1-2) (1999), 117-120.
10. V M Winitzky de Spinadel, A new family of irrational numbers with curious properties, Humanistic Mathematics Network Journal 19 (1999), 33-37.
11. V M Winitzky de Spinadel, Conference Report: 9th International Congress on Mathematical Education (ICME-9), Nexus Network Journal 2 (4) (2000), 214-215.
12. V M Winitzky de Spinadel and H S Nottoli, Herramientas matemáticas para la arquitectura y el diseño (Ediciones Facultad de Arquitectura, Diseño y Urbanismo, Universidad de Buenos Aires. Cuadernos de Cátedra, 2005).
13. V M Winitzky de Spinadel, In Memoriam: Slavik Jablan 1952-2015, Symmetry 7 (3) (2015), 1261-1274.
14. Winitzky de Spinadel, Vera Martha, Curriculum Vitae, International Mathematical Union.