# Vera M Winitzky de Spinadel's publications

1. Vera M Winitzky de Spinadel, Teoría de las zonas alcanzables en sistemas bidimensionales, Doctoral Thesis (Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, May 1958).

Introduction
In the year 1676, Gottfried Wilhelm Leibniz first introduced the concept of a differential equation, thus initiating a new era in the field of mathematics. Over the next 50 years, a school of mathematics dedicated to obtaining analytic expressions for the solutions of linear and nonlinear differential equations (by Bernoulli, Euler, Riccati, Clairaut, etc.) developed.
Only in 1739, Euler began a systematic study of linear equations and in 1760, Lagrange formulated the general superposition theorem for linear systems, establishing with it a real split between linear and nonlinear mathematics. This concept had such a strong impact on research that the next 125 years were devoted almost exclusively to the development of the theory of linear differential equations, culminating in the concepts of Fourier and Laplace transforms and the study of systems of orthogonal functions. This is why it was only in 1881 that H Poincaré, inspired by the problems of celestial mechanics, published the first fundamental work on the properties of the solution curves of a system of nonlinear differential equations. Shortly after, A Liapounov, in a classic memoir, expounded his theory of motion stability, laying the foundations on which the Russian research school in this domain was later built.
At the same time, I Bendixson applied the theory to the real field and G D Birkhoff, with his classic studies on problems of dynamics, introduced functional analysis in the theory of dynamical systems. Among the classic works, it is also necessary to mention those of O Perron dedicated to the study of the asymptotic behaviour of solutions and analysis of singular points.
In the field of electronics, the oscillating circuit is the most important example of a nonlinear device. And as such, its study has been the subject of numerous and interesting works due to its application in different branches of technology. The nonlinear differential equation of the circuit
$\large\frac{d^2 y}{dt^2}\normalsize + f(y)\large\frac{dy}{dt}\normalsize + ay = 0$
was first studied by E V Appleton and B van der Pol. Later, van der Pol used the method of isoclines to solve it for the case where
$f(y) = -\epsilon (1 - y^{2})$ with $\epsilon = 0.1; 1.0; 10$,
and he found that, under certain conditions, an oscillating circuit performs relaxation oscillations, which are the limiting case when the parameter ε is large. On the other hand, he also showed that relaxation phenomena appear in many branches of science, for example, physiology, the heartbeat being an oscillation of relaxation.
The problem of oscillations in systems with nonlinear elements has been studied by A Andronow and C E Chaikin, J J Stoker, and many others. Various methods for solving nonlinear equations of this type were developed by N Krylov and N Bogoliubov. The Poincaré perturbation method, originally developed for astronomical problems, was also applied (with limitations).
The search for periodic solutions and Poincaré limit cycles, of such great interest in applications, has been studied in recent imes using functional analysis methods, by S Lefschetz, M L Cartwright, N Levinson, J L Massera, etc.
Total differential equations of the type
$\large\frac{dx}{dt}\normalsize = f(x) + \omega(t)$
where $f(x)$ is a fixed function and $\omega(t)$ an arbitrary function, with certain restrictions, are of great importance in applications, due to the existence of numerous physical problems whose behaviour is described by an equation like the preceding one. That is why there is great interest in the study of the solution curves of such an equation, when the function $\omega(t)$ is varied, which generally represents an external variable action applied to the physical system. In particular, it is important to investigate what are the possible states of the physical system that can be reached, starting from a given initial state, by properly choosing the control term $\omega(t)$. This is precisely the problem to which the theory of reachable zones, formulated by E O Roxin in $n$-dimensional spaces, gives an answer.
This work applies this theory to the planar case, where reasoning is more intuitive due to its corresponding geometric representation, and studies the attainable areas in numerous cases of immediate application in branches of technology and physics.

2. Vera M Winitzky de Spinadel, Sobre Teoremas de Comparación de Juegos Diferenciales, Revista de la Unión Matematica Argentina 26 (2) (1972), 107-114.

Abstract
Recently, A N V Rao has obtained remarkable results referring to the comparison of differential games. To prove his theorems, Rao uses the concept of strategy given by A Friedman. The proof does not subsist if one tries to use classic strategies of R Isaacs, of the "feedback" type. However, by adding additional conditions similar to those of L Berkovitz, the validity of the proofs is maintained.

3. Vera M Winitzky de Spinadel, An Application of Pontrjagin's Principle to the Study of the Optimal Growth of Population, International Atomic Energy Agency, Vienna, Austria 2 (1976), 189-199.

Abstract
This paper examines the consequences of an optimal control of population growth and allows to derive criteria referring to the economic basis of expenditure on population control and to obtain optimal paths for a model in which such a control is possible. By means of very simple assumptions one can reduce the problem to a two-state variable control problem and, in consequence, apply Pontrjagin's maximum principle to solve it.

4. Vera M Winitzky de Spinadel, Las redes y sus aplicaciones, Revista de Educación Matemática 4 (1989), 55-82.

Abstract
Network theory is a branch of Operational Research that is applied in the treatment of various problems from the economic, sociological and technological fields. Historically, it is proven that man, when faced with a problem, tends to make a diagram in which the points represent individuals, locations, activities, stages of a project, etc., joining them by means of lines that indicate a certain existing relationship among them. D Konig was the first to propose that such diagrams receive the name of networks, making a systematic study of their properties.
Strictly speaking, the exposition of these properties would include a number of concepts and theorems, among which some are relatively complicated. Since our goal is to present this topic in a way that is accessible to a large number of readers of different scientific backgrounds, we will present the basic concepts in the simplest way possible, show how to use them, and give some methods that can be of fruitful use in applications.

5. Vera M Winitzky de Spinadel, Acotación Uniforme Local de las Soluciones en un Sistema de Control con Ruido, Revista de la Universidad de Buenos Aires 39 (1-2) (1994), 18-26.

Abstract
The numerical treatment of a control problem, using a step-by-step procedure, usually involves a certain informational noise of one type or another. Very often, it happens that small informational errors produce instability in the solutions and the problem of regularisation arises. In this paper, a local uniform boundedness of the solutions is proved, which turns to be useful in the design of "control with guidance".

6. Vera M Winitzky de Spinadel (with Cecilia Crespo Crespo and Christiane de Ponteville), Divisibility and Cellular Automata, Chaos, Solitons and Fractals 6 (1995), 105-112.

Abstract
Cellular automata (CA) are perfect feedback machines which change the state of their cells step by step. In a certain sense, Pascal's triangle was the first CA and there is a strong connection between Pascal's triangle and the fractal pattern formation known as Sierpinski gasket.
Generalising divisibility properties of the coefficients of Pascal's triangle, binomial arrays as well as gaussian arrays are evaluated mod p. In these arrays, two fractal geometric characteristics are evident: a) self-similarity and b) non integer dimension.
The conclusions at which we arrive, as well as the conjectures we propose, are important facts to take into account when modelling real experiments like catalytic oxidation reactions in Chemistry, where the remarkable resemblance of the graph:
number of entries in the $k$th row of the Pascal's triangle which are not divisible by 2 versus $k$
and the measurement of the chemical reaction rate as a function of time, provides the reason to model a catalytic converter by a one-dimensional CA.

7. Vera 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.

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.

8. Vera M Winitzky de Spinadel, On Characterization of the Onset to Chaos, Chaos, Solitons and Fractals 8 (10) (1997), 1631-1643.

Abstract
The purpose of this paper is to introduce the family of Metallic Means, whose members are the well known Golden Mean and its relatives, the Silver Mean, Bronze Mean, Copper Mean, Nickel Mean, etc. Bringing out, from the mathematical point of view, their similarities as well as their differences, it is possible to find a universal behaviour on the roads to chaos, a major problem which is still open for further research.

9. Vera M Winitzky de Spinadel, The Metallic Means and Design, in Kim Williams (ed.), Nexus II: Architecture and Mathematics (Edizioni dell'Erba, 1998), 143-157.

Abstract
In this paper a new family of positive quadratic irrational numbers is introduced: the family of "Metallic Means". Its most well-known member is the Golden Mean. Other members of the family are the Silver Mean, the Bronze Mean, the Copper Mean, the Nickel Mean, etc. These Metallic Means share important mathematical properties that transpose them into a basic key and constitute a bridge between mathematics and design.
Modern research in mathematics, usually conceived of as a highly structured system, has uncovered unsuspected channels that one may follow and try to interpret fractal geometries, so common in natural systems and human, animal and plant morphology, or to look for universal roads that indicate the onset to chaos, present in phenomena that go from DNA microscopic structure to the cosmic macroscopic galaxies. In this richness lie numerous contact points between mathematical tools and their application to creative design. The Metallic Means Family is one such tool and their many interesting properties will help us in the future to travel the difficult roads that connect one field of human knowledge with another, overcoming the isolation of specialties and resuming the global, Renaissance approach to problem resolution, more affine to the thinking of the twenty-first century.

10. Vera M Winitzky de Spinadel, The family of Metallic Means, Visual Mathematics 1 (3) (1999), 317-338.

Introduction
Let me introduce you to the Metallic Means Family (MMF). Their members have, among other common characteristics, the property of carrying the name of a metal. Like the very well known Golden Mean and its relatives, the Silver Mean, the Bronze Mean, the Copper Mean, the Nickel Mean and many others. The Golden Mean has been used in a very big number of ancient cultures as a proportion basis to compose music, devise sculptures and paintings or construct temples and palaces. Some of the relatives of the Golden Mean have been used by physicists in their latest researches trying to analyse the behaviour of non-linear dynamical systems in going from periodicity to quasi-periodicity. But in quite a different context, Jay Kappraff uses the Silver Mean to describe and explain the roman system of proportions, referring to a mathematical property that, as we shall prove, it is common to all the members of this remarkable family.

Conclusions
The members of the MMF are intrinsically related with the onset from a periodic dynamics to a quasi-periodic dynamics, with the transition from order to chaos and with time irreversibility, as proved by Ilya Prigogine and M S El Naschie.
But, simultaneously, there are philosophical, natural and aesthetically considerations that have impelled the utilisation of proportions based on some members of the MMF, from the beginning of human history. They appeared in the Egyptian sacred art as well as in India, China, Islam and many other ancient civilisations. They have dominated the Greek art and architecture, they extended to the magnificent monuments of the Gothic Middle Age and they reappeared with all its splendour in the Renaissance period.
In many instances, the harmony and beauty of a pattern is the result of the influence of the Golden and Silver Means at a fundamental level.
Such a wide range of applications where the members of the MMF appear, opens many roads to new inter-disciplinary investigations that will undoubtedly clear up the existent relations between Art and Technology, establishing a bridge among the rational scientific approach and the esthetical emotion. And perhaps, this new perspective could help us to give to Technology, from which we depend increasingly for our survivorship, a more human aspect.

11. Vera M Winitzky de Spinadel, A new family of irrational numbers with curious properties, Humanistic Mathematics Network Journal 19 (1999), 33-37.

Abstract
The "metallic means family" (MMF) includes all the quadratic irrational numbers that are positive solutions of algebraic equations of the type
$x^{2} - nx - 1 = 0$
$x^{2} - x - n = 0$
where $n$ is a natural number. The most outstanding member of the MMF is the well-known "golden number." Then we have the silver number, the bronze number, the copper number, the nickel number and many others. The golden number has been widely used by a great number of very old cultures, as a base of proportions to compose music, to create sculptures and paintings or to build temples and palaces. With respect to the many relatives of the golden number, a great part of them have been used in different researches that analyse the behaviour of non linear dynamical systems when they proceed from a periodic regime to a chaotic one. Notwithstanding, there exist many instances of application of these numbers in quite different knowledge fields, like the one described by the mathematician Jay Kappraff in his study of the old Roman proportion system of construction. This system was based on the silver number, on account of a mathematical property, which is not unique but is common to all members of the MMF, as we shall prove. Being irrational numbers, all the members of the MMF have to be approximated by ratios of integer numbers in applications to different scientific fields. The analysis of the relation between the members of the MMF and their approximate ratios is one of the goals of this paper.

12. Vera M Winitzky de Spinadel, First Interdisciplinary Conference of The International Society of the Arts, Mathematics and Architecture, ISAMA 99, Nexus Esecutivo (1999), 186-188.

Overview of the conference by Vera Spinadel
The first interdisciplinary conference of ISAMA 99 was held in San Sebastian, Spain, June 7-11, 1999. The conference directors were Nathaniel A. Friedman, SUNY (State University of New York), Albany, USA, and Javier Barrallo, Universidad del País Vasco, San Sebastián, Spain. The main purpose of this conference was to bring together persons interested in relating mathematics with the arts and architecture. This set included teachers, architects, artists, mathematicians, scientists and engineers. ISAMA focussed on the following fields related to mathematics: Architecture, Computer Design and Fabrication in the Arts and Architecture, Geometric Art and Origami, Music, Sculpture and Tesselations and Tilings. These fields included graphics interaction, CAD systems, algorithms, fractals and mathematical software like Maple, Derive, Mathematica, etc.
The International Scientific Committee was formed by twelve scientists, including the author of this report. Sixty-four papers were presented and published in a beautiful volume, edited by the Department of Applied Mathematics, School of Architecture, University of the Basque Country. There was a one day excursion to Gernika, where we could admire the monumental sculptures by Henry Moore and Eduardo Chillida, the world's foremost sculptor born in San Sebastián, whose work is inspired by architecture. In the afternoon, we made a tour of the Guggenheim Museum in Bilbao, designed by Frank Gehry, which is a crowning achievement of contemporary architecture. Fortunately, at this time there was also at the Guggenheim a magnificent exhibit of Chillida's work as well as the widely known architectural sculpture by Richard Serra, inspired by elliptical forms. Three other highlights at ISAMA 99 should be mentioned. They are the excursion on Thursday afternoon to the wonderful Chillida's private sculpture park Zabalaga, the world premier of "A Flame In Flight" for solo violinist by Robert Cogan, performed by Michael Appleman, and the granite sculpture "Oushi-Zoukei" carved by Keizo Ushio during the conference.
Without doubt, the goal of sharing information and discussing common interests to enrich interdisciplinary education, was achieved. Finally, we missed the presence of our friends from Yugoslavia, in particular, Slavik Jablan, who was present at the successful conference Mathematics & Design (M&D-98) held also in San Sebastian, Spain, June 1-4, 1998.

13. Vera M Winitzky de Spinadel, "Triangulature" in Andrea Palladio, Nexus Network Journal, Architecture and Mathematics 1 (1-2) (1999), 117-120.

Abstract
At the June 1998 workshop on the architecture of Andrea Palladio, the dimensions of the rooms were much remarked. Vera Spinadel convincingly argues that Palladio used precise mathematical relationships as a basis for selecting the numerical dimensions for the rooms in his villas. The integer dimensions are demonstrated to be approximants linked to continued fractions, and a particular way of deriving these integers through the use of a continued fraction expansion that approximates by excess is introduced.

14. Vera M Winitzky de Spinadel, Conference Report: 9th International Congress on Mathematical Education (ICME-9), Nexus Network Journal 2 (4) (2000), 214-215.

The Report
Flying from Japan back to my home in Buenos Aires, Argentina, I imagined the following dialogue with an unknown reporter:
Question: What did you do in Japan?
Answer: I attended the 9th International Congress on Mathematical Education (ICME-9), held in Tokyo/Makuhari from July 31 to August 6, 2000.
Question: What was your role as a participant?
Answer: I have been invited to be the Chief Organiser of a Topic Study Group TSG 20: Art and Mathematics Education.
Question: What was the main purpose of this group?
Answer: To gather contributions from different countries and cultures, so as to have a great display of how Art interacts with Mathematics Education.
Question: How was it organised?
Answer: There were two 90 minutes Sessions and a very nice exhibition that ran parallel to them.
Question: Tell me about the Sessions.
Answer: The first one was devoted to Visual Arts and Cultural History. The speakers were Javier Barrallo and Paquita Blanco (Spain), who presented an interesting mathematical simulation of Gothic cathedrals, then Liu Keming (China) talked about mathematical issues in Chinese ancient painting and drawing and finally, Muneki Shimono (Japan) showed his program of teaching the cultural history of Mathematics. The second session was devoted to Mathematical education and its relation with Art. Julianna Szendrei (Hungary) talked about Art and Mathematics in primary teachers' training, then María V Ponza (Argentina) showed a beautiful video about how to link Mathematics with dance, and finally I invented a fable related by a strange old man: the famous Golden Mean!
Question: And which were the conclusions of this group?
Answer: As the approach was quite multidisciplinary, we agreed that Art, in any of its many forms, has to be used as a main tool in teaching Mathematics to ANY student, not only to students engaged in artistic studies.
Question: What are your next plans?
Answer: Extending this globalizing idea to the research field, we are organizing the Third World Conference Mathematics & Design 2001 Mind/Ear/Eye/Hand/Digital at Deakin University, Geelong, Australia, 3-5 July, 2001. You are kindly invited to attend!

The reporter
Vera W de Spinadel is a Full Consultant Professor at the Faculty of Architecture, Design and Urbanism at the University of Buenos Aires, Argentina. She is the Director of the research centre Mathematics and Design, which comprises a team of interdisciplinary professionals working on the relations among Mathematics and Informatics with Design, where the word "design" is understood in a very broad sense (architectonic, graphic, industrial, textile, image and sound design, etc.). She organised the First and Second International Conferences on Mathematics and Design. She is the author of several books and has published many research papers in international journals. She has received several research and development grants as well as several research and technological production prizes.

15. Vera M Winitzky de Spinadel, Nonlinearity and the metallic means family, Nonlinear Analysis: Theory, Methods & Applications 47 (7) (2001), 4345-4353.

Abstract
In the analysis of many complex situations of the real world, we face some routes to chaotic behaviour. Considering globally these chaotic phenomena, we notice they have some properties in common:
1) they correspond to nonlinear dynamical systems;
2) they depend strongly on initial conditions;
3) they pass from a periodic motion to a quasi-periodic one and finally, in the state of "global chaos" to a frankly aperiodic motion.
The transition to chaos is produced going from commensurable ratios (Like periods or winding numbers) to incommensurable ones. In consequence, the more irrational this ratio is, the nearest are we from chaos. The Golden Mean, being the most irrational of all irrational numbers, plays together with other means of the Metallic Means Family (recently introduced by the author), a key role in the roads to chaos.

16. Vera M Winitzky de Spinadel, Geometría fractal y geometría euclidiana, Revista Educación y Pedagogía 15 (35) (2002), 84-91.

Abstract
The elements of Euclidean geometry are points, lines, curves, etc., that is, ideal entities conceived by man to model natural phenomena and quantify them by measuring lengths, areas or volumes. But these entities can be so complex and irregular that measurement using the Euclidean metric becomes meaningless. However, there is a way to measure the degree of complexity and irregularity, evaluating how fast the length, surface or volume increases, if we measure it on smaller and smaller scales. This approach was adopted by Mandelbrot, a Polish mathematician, who in 1980 coined the term fractal to designate highly irregular but self-similar entities.

17. Vera M Winitzky de Spinadel, Symmetry Groups in Mathematics, Architecture and Art, Symmetry: Art and Science 2 (1-4) (2002), 385-403.

Abstract
The word "symmetry" has two meanings. A symmetric object is well proportioned but the concept is not restricted to concrete objects; the synonym "harmony" refers to its use in Acoustics and Music. The second meaning is that of the geometric bilateral symmetry, the symmetry so evident in superior animals, especially in men. From the mathematical point of view, a whole symmetry theory can be considered for applications. From this theory, we have chosen the "symmetry group of the square" to present interesting uses in Architecture and art.

Symmetry in Culture
The word symmetry comes from the Greek symmetria, meaning "the right proportion". From the historical point of view, the term symmetry has denoted many meanings, depending on the field of human knowledge where it was used. Notwithstanding, "symmetry is a unifying concept", as Hargittai's, Magdolna and István, have proved in their beautiful and unique book "Symmetry". Indeed, the concept of symmetry can provide a connecting link among many different fields of endeavour, perhaps the best and more appropriate link to protect human studies from the increasing and separating compartmentalization within our scientific world.
Going back to the year 27 B.C., we found a monumental work: the 10 books written by the roman architecture Vitruvio (probably Marco Vitruvio Pollione) and dedicated to the Emperor August. Architecture, says Vitruvio, depends from order, disposition, eurhythmy, property, symmetry and economy. These terms have today, completely different meanings. E.g., order, says Vitruvio, confers the appropriate measure to the elements of a certain building, when considered separately and symmetry, gives concordance to the proportions of the different parts of the construction. This approach to the meaning of symmetry is quite similar to the mutually corresponding arrangement of the various parts of a human body around a central axis, producing a proportioned balanced form.
Vitruvio dedicated much time to the study of the proportions in the human body in his considerations on symmetry. Symmetry, says Vitruvio, comes from proportion, that is, from a correspondence between the dimensions of the parts of a whole and of the whole with respect to a certain part selected as a model, the module. Such a selection of parts of the human body as a module, initiated probably by Vitruvio, was the very beginning of a historical ergonomic chain linking Vitruvio, Albrecht Dürer, Leonardo da Vinci and many, many other artists, including the modern contemporary architect Le Corbusier.

18. Vera M Winitzky de Spinadel, Number theory and Art, in Javier Barrallo, Nathaniel Friedman, Reza Sarhangi, Carlo Séquin, José Martínez and Juan A Maldonado (eds.), ISAMA-Bridges 2003. Proceedings of Meeting Alhambra, University of Granada, Granada, Spain (2003), 415-422.

Abstract
The Metallic Means Family (MMF), was introduced by the author, as a family of positive irrational quadratic numbers, with many mathematical properties that justify the appearance of its members in many different fields of knowledge, including Art. Its more conspicuous member is the Golden Mean. Other members of the MMF are the Silver Mean, the Bronze Mean, the Copper Mean, the Nickel Mean, etc.

19. Vera M Winitzky de Spinadel, La familia de Números Metálicos, Cuadernos del Cimbage, Instituto de Investigaciones en Estadística y Matemática Actuarial, Facultad de Ciencias Económicas, Universidad de Buenos Aires 6 (2004), 17-45.

Abstract
In this paper we introduce a new family of positive quadratic irrational numbers. It is called the Metallic Means family and its most renowned member is the Golden Mean. Among its relatives we may mention the Silver Mean, the Bronze Mean, the Copper Mean, the Nickel Mean, etc. The members of such a family enjoy common mathematical properties that are fundamental in the present research about the stability of macro- and micro- physical systems, going from the internal structure of DNA up to the astronomical galaxies. The most important results of this new investigation are the following:
1) The members of this family intervene in the determination of the quasi-periodic behaviour of non linear dynamical systems, being essential tools in the search of universal routes to chaos.
2) The numerical sequences based on the members of this family, satisfy many additive properties and simultaneously, are geometric sequences. This unique property has had as a consequence the use of some Metallic Means as a base for proportion systems.

20. Vera M Winitzky de Spinadel (with Hernán 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).

Preface
While mathematics works with abstract spaces and concepts, design operates on concrete spaces, those inhabited by man with his everyday objects.
This book deals with the points of contact between these two fields: it develops contents of the mathematical discipline by applying them to topics directly linked to the work of architects and designers. The eclecticism of the set responds to the chosen cut, and to the intention of emphasizing the links with professional practice.
The book begins with a chapter devoted to the geometry of forms. The following two ("Graphs" and "Theory of symmetry") allow us to know what basic guidelines have historically regulated the canons of beauty or the proportions of designed objects. Chapter 4, "Applications of Derivatives and Integrals," provides basic computational tools for analysing the behaviour of structures. Numbers 5 and 6, "Probability Theory" and "Statistics", provide notions of indisputable utility in the development process of a work or product. Chapter 7, finally, deals with the study of topography and includes information on measurement devices, planimetry and altimetry.
We trust that this book will be of interest to its specific audience, students of architecture and design, and we hope that it can provide them with valuable tools for their future professional development.

21. Vera M Winitzky de Spinadel, Introducción de los números irracionales por descomposición en fracciones continuas, Premisa 35 (2007), 37-45.

Summary
It was Pythagoras of Samos who discovered the incommensurability of the hypotenuse of a right triangle, thus introducing the first "irrational" number, that is, one that cannot be written as a ratio of two integers. Mathematically, the set of rational numbers together with the set of irrational numbers forms the set of real numbers, which has the property of being dense (that is, it does not have any holes). And the irrational numbers are defined by Dedekind cuts, stating that a number on the real axis divides it into two disjoint sets: the set of real numbers greater than it and the set of real numbers less than it. Therefore, if the chosen number is not an integer or a rational, then the irrational is defined. But this definition does not allow quantifying the degree of irrationality, that is, the degree of approximation of the rational approximants to the irrational number. This degree of irrationality turns out to be of importance in the experiences that are designed looking for the borders between a physical system that behaves periodically and its transformation into a chaotic system, where it is impossible to predict the behaviour since very similar initial conditions originate totally disparate results. To detect this degree of irrationality, we will use the decomposition into continued fractions.

22. Vera M Winitzky de Spinadel, Más resultados sobre el Número de Plata, Forma y Simetría: Arte y Ciencia Congreso de Buenos Aires, 2007 (Universidad de Buenos Aires, 2007), 440-444.

Abstract
The Metal Number Family (FNM) is an infinite set of positive quadratic irrational numbers discovered by Dr Spinadel in 1994, whose common mathematical properties make them highly suitable for application in interdisciplinary design problems. The most preponderant member of the FNM is the well-known Golden Number = 1.618... The one that follows is the Silver Number $\sigma_{Ag} = 1 + √2$. Both numbers have a decomposition into pure periodic continued fractions and the first objective of this work was to relate the decomposition corresponding to the Silver Number with a succession of gnomonic rectangles. Based on it, we were able to build different forms corresponding to the silver spiral, similar to the traditional golden spiral. We not only built the flat versions of the silver spiral but also made a spherical rhumb line representation, animated with ad hoc software. Finally, we relate the Silver Number and its successive powers with architectural designs, ranging from the Roman ruins found in the city of Ostia to contemporary projects. As an interesting detail, we find a relationship between the Silver Number and the famous "Cordovan proportion".

23. Vera M Winitzky de Spinadel, Characterization of the onset to chaos in Economy, Proceedings of the Seventh All-Russian Conference on Financial and Actuarial Mathematics and Related Fields FAM'2008 2 (2008), 250-265.

Abstract
Basically, any process evolving with time is a dynamical system. Dynamical systems appear at every branch of Science and virtually at every aspect of life. Economy is an example of a dynamical system: the prices variations at the Stock Exchange is a simple illustration of the temporal evolution of this system. The main objective of the study and analysis of a dynamical system is the possibility of predicting the final result of a process.
Some dynamical systems are predictable and some are not. There are very simple dynamical systems depending only on one variable that show a highly non predictable behaviour, due to the presence of "chaos", that means they possess a sensitive dependence on the initial values.
The main aim of this paper is to investigate which are the factors that produce alternative roads to pass from order to chaos in economic problems.

24. Vera M Winitzky de Spinadel, Visualización y tecnología, Cuadernos del Cimbage, Instituto de Investigaciones en Estadística y Matemática Actuarial, Facultad de Ciencias Económicas, Universidad de Buenos Aires 10 (2008), 1-16.

Abstract
The objective of this work is to show how the visualization obtained by means of current mathematical/computer tools such as computerised graphics, constitutes an indispensable element in the application in the investigation of mathematical concepts that go from fractal structures, knots and the transition to chaos to the most general topological transformations.

25. Vera M Winitzky de Spinadel (with Antonia Redondo Buitrago), Towards van der Laan's Plastic Number in the Plane, Journal for Geometry and Graphics 13 (2) (2009), 163-175.

Abstract
In 1960 D H. van der Laan, architect and member of the Benedictine order, introduced what he calls the "Plastic Number" $\psi$, as an ideal ratio for a geometric scale of spatial objects. It is the real solution of the cubic equation $x^{3} - x - 1 = 0$. This equation may be seen as example of a family of trinomials $x^{n} - x - 1 = 0, n = 2, 3, ...$. Considering the real positive roots of these equations we define these roots as members of a "Plastic Numbers Family" (PNF) comprising the well known Golden Mean $\phi = 1.618...$, the most prominent member of the Metallic Means Family and van der Laan's Number $\psi = 1.324...$ Similar to the occurrence of $\phi$ in art and nature one can use $\psi$ for defining special 2D- and 3D-objects (rectangles, trapezoids, ellipses, ovals, ovoids, spirals and even 3D-boxes) and look for natural representations of this special number.
Laan's Number $\psi$ and the Golden Number $\phi$ are the only "Morphic Numbers" in the sense of Aarts et al., who define such a number as the common solution of two somehow dual trinomials. We can show that these two numbers are also distinguished by a property of log-spirals. Laan's Number $\psi$ cannot be constructed by using ruler and compass only. We present a planar graphic construction of a segment of length $\psi$ using a dynamical graphics software as well as a computer- independent solution by intersecting a circle with an equilateral hyperbola. This allows to deduce and analyse "Laan-Number figures" like $\psi$-rectangles with side length ratio $1: \psi$ and a $\psi$-pentagons with sides of ratio $1:\psi:\psi ^{2}:\psi ^{3}:\psi ^{4}$. To this $\psi$-pentagon we also find a "$\psi$-Pythagoras Theorem".

26. Vera M Winitzky de Spinadel, Characterization of the onset to chaos in Economy, Cuadernos del Cimbage, Instituto de Investigaciones en Estadística y Matemática Actuarial, Facultad de Ciencias Económicas, Universidad de Buenos Aires 11 (2009), 25-38.

Abstract
Basically, any process evolving with time is a dynamical system. Dynamical systems appear at every branch of Science and virtually at every aspect of life. Economy is an example of a dynamical system: the prices variations at the Stock Exchange is a simple illustration of the temporal evolution of this system. The main objective of the study and analysis of a dynamical system is the possibility of predicting the final result of a process.
Some dynamical systems are predictable and some are not. There are very simple dynamical systems depending only on one variable that show a highly non predictable behaviour, due to the presence of "chaos", that means they possess a sensitive dependence on the initial values.
The main aim of this paper is to investigate which are the factors that produce alternative roads to pass from order to chaos in economic problems.

27. Vera M Winitzky de Spinadel, Use of the powers of the members of the Metallic Means Family in artistic Design, Journal of Applied Mathematics 4 (4) (2011), 333-340.

Abstract
The Metallic Means Family (MMF) was introduced by the author more than ten years ago. In the meantime, there have been published many applications of the members of this family to every type of design, particularly artistic design. The most preponderant of the MMF are the Golden Mean, the Silver Mean, the Bronze Mean, the Copper Mean, the Nickel Mean, etc. As is well known, the Golden Mean is linked to pentagonal geometry and the Silver Mean, to octagonal geometry. There has not been found yet direct relations of the rest of the members to any type of specified geometrical construction. But as all of them are irrational numbers, one should look for optimal rational approximations. In the case of the regular pentagon and the regular inscribed star in it, there appears not only the Golden Mean but also integer powers of it. Something similar happens with the regular octagon. All these positive irrational numbers have a continued fraction expansion which rational approximants are successively in excess and in defect. We are going to prove that the powers of the members of the MMF can be approximated by an "excess continued fraction expansion", which rational approximants converge always in excess and therefore, much more quickly than the normal one. This circumstance opens the new possibility of using in artistic design any of the powers of the members of the MMF.

28. Vera M Winitzky de Spinadel, Fractals and multifractals, in Oleg Vorobyev (ed.), Proceedings of the XV International EM'2011 Conference (Krasnoyarsk, 2011), 35-40.

Abstract.
The word fractal comes from the Latin adjective "fractus" that means interrupted or irregular. As is well known, this name was introduced in the seventies by the polish mathematician Benoit B Mandelbrot.
Fractals are sets with two characterizing properties:
1. They have a non-integer Hausdorff dimension which, in some cases, may be an integer.
2. They are self-similar, in the sense that they are invariant in the presence of "scale changes".
Fractals are often encountered in many growth processes like clouds formation, seashores, trees development, mountains, length of a coastline, etc. But there are certain nontrivial physical phenomena that possess a spectrum of scaling indices. Sets of this kind are called "multifractals" and are characterized by an entire spectrum of exponents, of which the Hausdorff dimension is only one.
The consideration of how a multifractal is decomposed in its self-similar components is very important in the search of universal scenarios of the roads to chaos.

29. Vera M Winitzky de Spinadel, In Memoriam: Slavik Jablan 1952-2015,
Symmetry 7 (3) (2015), 1261-1274.