Dissertations / Theses on the topic 'Geometry in art'

When I design an object or a piece of furniture I do it with geometry in my mind. I use simple shapes like squares, rectangles and lines to create forms that then become an object. I organize the shapes in space thinking about the negative and positive space that they create. I also think of how each shape interacts with each other. The overall shape of an object contains other shapes inside, and these shapes create a relationship between each other in space. I also consider how geometry relates to the function of the object. In my designs, geometry and function shape the object. In this essay I describe the design and production process for a cheese tray, a night lamp and a candle holder. I talk about how I used geometry to design each one of these objects and the production process involved.

With my research in Concrescence planters I was able to design set of planters using specific methods that I generally use in designing. The methods that I utilized the most are the Golden Rectangle and the Fibonacci sequence as Golden Rectangle talks about the symmetry and design in nature and how I apply the same systems to my own design to make it as natural for human interaction. The research also utilized some of the same design techniques and element that then to translate from one design to another which make the design process simpler and fluid.

My thesis is that geometry is an abstract and universal language that can reflect the inner being of the world in concrete form. I propose that the ideal forms of geometry, like those of harmony in music, have an aesthetic and metaphysical dimension that is capable of touching the most essential part of our being in the world. In this context, I suggest that painting geometry may be understood as an art practice that is closely aligned with the ideals of philosophical reflection, and how, as a consequence of this understanding, my approach to painting geometry is directed towards the realisation of the ideals of beauty, truth and freedom in particular; an approach that I claim shares much in common with the origins of both abstract and concrete art in post-Kantian German Idealist thought and Romantic art. On this basis, I argue that my painting practice is engaged with the possibility of the realisation of an ideal form of expression. This goal may be summarised as the achievement in painted form of a visual or spatial equivalent to the formal language of harmony of music. The paintings that I have submitted for examination may be understood as a direct consequence of my research findings, in view of which my intention is to make a contribution to the current and evolving language of abstract and concrete art. To this end, my thesis serves as an exegesis for the paintings submitted for examination in fulfilment of the requirements of my doctoral candidature.

Placing security cameras in buildings, finding good locations for cameras to enforce speed limits or placing guards to defend a border are some of the problems we face everyday. A nation that wishes to defend its border with armed guards wants to be sure the entire border is secure. However, hiring more guards than necessary can be costly. A start-up company moving into a new building wants to be sure every room in the building is seen by some security camera. Cameras are expensive and the company wants to install the smallest number of cameras; at the same time the company wants to be sure the building is secure. These problems, and many other visibility type problems, are not easy to solve in general. In some specific cases, optimal solutions can be obtained quickly. In general, finding an optimal solution may take a very long time. The original results of this thesis address some of these problems. We show some positive results for solving some of these visibility problems. We also give some negative results for some of these problems. These negative results are useful because they tell us that we are unlikely to find a fast algorithm to solve a particular problem optimally.

Painters of Christian subjects in the late Middle Ages and early Renaissance developed a complex system of geometry which they used to order the various elements in the image. They did this because they were convinced that the aesthetic dimension of their work resided in the structure of the work. More specifically, the artists of the late Middle Ages and the Renaissance believed that the particular aesthetic experience which geometric compositional structure provides corresponded to Christian mystical experience. Thus a work of art that combined geometric structure, naturalistic style, and Christian imagery could provide an experience analogous to that of Christian revelation. This paper traces the development of this idea from its origin in the Old Testament tradition, its formalization in Greek thought and its full flowering in early Christian painting.

The dissertation explores the centrality of the Platonic Solids, and polyhedral geometry generally, to the artistic and mixed-mathematical cultures of Renaissance Germany. Beginning with Albrecht Dürer’s groundbreaking treatise on geometry, the Underweyung der Messung (1525), the dissertation redefines sites of early modern experimentation to include the graphical spaces in which new geometrical knowledge was practiced, invented, contested, manipulated, discarded, and presented. The research describes the historical contexts and development of the practice of polyhedral geometry over the course of the 16th century, expanding from Dürer to the lesser-known textbooks for practical geometry that his work inspired in Germany, and continuing with epitomes of the polyhedral genre, namely Wenzel Jamnitzer’s Perspectiva corporum regularium (1568) and the drawings of the Augsburg artisan Lorentz Stöer. The dissertation then follows the migration of polyhedra into intarsia and turned-ivory artifacts used for teaching applied geometry to European aristocracy, and concludes by addressing the polyhedral cosmology of the astronomer Johannes Kepler. By tracing the lifespan of polyhedra from their use as perspectival tools and pedagogical devices in Renaissance workshops into courtly Kunstkammern and onto the precious surfaces of domestic objects, the dissertation uncovers the influence that the decorative arts had on the conceptualization of geometrical knowledge and its new engagement with materials and concepts of materiality.
History of Science

Nesta dissertação apresentamos a concepção, implementação e validação de uma proposta para o ensino de transformações geométricas no plano usando o ambiente de geometria dinâmica GeoGebra. A proposta integra Geometria e Arte através da construção de pavimentações do plano e de mosaicos de Escher e foi dirigida para professores do ensino fundamental, tendo como objetivo apresentar uma nova alternativa de trabalho na Geometria escolar e também capacitá-los para o uso de mídias digitais nas suas salas de aula. O trabalho foi desenvolvido dentro dos princípios da Engenharia Didática. Na análise e validação da implementação da proposta tomamos como base a teoria Sócio-Histórica, cuja referência principal é a obra de Vygotsky; também utilizamos o trabalho de Duval sobre registros de representação semiótica no processo de aprendizagem da Matemática. A partir das análises a priori e a posteriori observamos que os professores participantes da oficina, através do uso do GeoGebra, se apropriaram dos princípios da geometria dinâmica e dos conceitos da geometria das transformações.
This work presents the conception, implementation and validation of an experiment to teach geometric transformations in the plane using the dynamic geometry environment GeoGebra. The proposal integrates geometry and art through the construction of tessellations of the plane, including Escher's mosaics, and it was directed to elementary school teachers, aiming to present a new alternative to work with geometry using digital media. The work used the principles of Didactic Engineering and the analysis of the experiment was based on the Socio-Historical theory, whose main reference is the work of Vygotsky and on the work of Duval about registers of semiotic representation in the process of mathematics learning. The analysis a priori and a posteriori showed that the teachers, through the use of GeoGebra, learned the principles of dynamic geometry and the concepts of geometry transformations.

This work is comprised of two parts: The Inspiration and The Institution. The Inspiration concerns what originated the work—the conception of the idea. It lies within the realm of those things which are timeless. Therefore, it is what gives character to the building of the place or the institution. The inspiration is the beginning. The Institution is the formulation of the work--the "building" of the idea. It is a place crafted with the methods of its time. ln this sense, the institution is circumstantial, and therefore representing the end. However, in its completion there is the reflection of its beginning, its inspiration. What we call the beginning is often the end And to make an end is to make a beginning. The end is where we start from.¹ What inspires this work is the architecture of the ancient communities of the Anasazi. More specifically and fundamentally, the inspiration for this work lies in the phenomenon of geometry and material in these ruins. Further, it is seated in such ideas as concentricity or nestedness and the opening of a wall. These are the ideas which are timeless. This is the beginning and the end. What formulates the work is a school. As an institution of learning, it already constitutes fertile ground for teaching. Therefore, with architecture as the medium, the building can teach about the play of geometry and the use of material. The function of the school is purely circumstantial, and it has little to do with the inspiration. Still, the geometry and material of the place made are founded in the inspiration. Hence, the architecture will continue to be a place for learning regardless of the functions of its past or future. The aspiration of the work is the development of a work of architecture as a place which nurtures the position of learning and as an institution which becomes a revelation of its inspiration.
Master of Architecture

Reims Cathedral holds a great deal of significance for the history of Gothic architecture, as well as the larger history of France as the coronation church. Given the historic significance of Reims, it is not surprising that much scholarship has been dedicated to the building’s sculpture, glass, and architecture. Most studies dealing with the cathedral’s architecture are based on stylistic and archaeological analysis, augmented by the use of surviving documents related to the construction. Although much fruitful work has been done in this vein, important questions about the building’s chronology and design still remain unresolved. The extent to which the design of the cathedral was established at the start of its construction, for example, continues to be disputed. The most recent monograph on the cathedral, published by Alain Villes in 2009, suggests that dramatic revisions to the overall plan and elevation were introduced during the course of its construction, going beyond the alterations to the façade designs that many previous authors have noted, but his theses remain controversial. Subsequently, Robert Bork has produced geometric models of the cathedral, which suggest that its plan was more coherent and unified. Additionally, French archaeologist Walter Berry has conducted new excavations, which further reveal additional archaeological evidence not yet taken into account by other Reims scholars. My dissertation, “Measuring the Past: The Geometry of Reims Cathedral,” examines the architectural design from a geometric perspective, augmented by archaeological, stylistic, and historic evidence. The primary contribution that my dissertation makes to art history is the development of a new, modern plan of the cathedral. I developed this plan by taking thousands of measurements using handheld devices and laser mapping, which I then incorporated into a single data set. This work allowed Bork and me to further refine the underlying geometry that created the cathedral’s layout and proportions. This new plan indicates that a master plan devised by the first architect governed the whole church, with subsequent modifications affecting its articulation rather than its overall layout. In addition to explaining how this plan was originally conceived, my dissertation also examines the anomalies and mistakes made during construction, which at times forced minor deviations from the plan. Some of these building errors and the obvious attempts to correct them give clues to the order of construction, in addition to supporting the notion that the masons repeatedly returned to the uniform scheme. This allows me to reassess the scholarship written about the cathedral and the complex history of the building project, while resolving some of the disputes over the cathedral’s construction and design.

This thesis examines the intersection of math and art by focusing on three specific branches of math: the fourth dimension, non-Euclidean geometry, and chaos and fractals. Different genres of art interact with each of these branches of math. The influence of the fourth dimension can easily be seen in Cubism and Russian Constructivism. Non-Euclidean geometry guided some of M.C. Escher’s work, and it inspired the Crochet Coral Reef project. Chaos and fractals can be found in art and architecture throughout history, but Vincent van Gogh and Jackson Pollock are notable examples of artists who used chaos in their work. Some artists incorporate math into their work in a rigorous, exacting manner, while others take inspiration from a general concept and provide a more abstract interpretation. Regardless of mathematical accuracy, mathematically inspired art can provide a greater understanding of mathematical concepts.

The various surviving disc and composite brooches provide proof of the skill and craftsmanship of Anglo-Saxon metalsmiths. Surprisingly, no one has conducted a full geometrical analysis of these brooches to discover the design process preceding the casting and decoration. This thesis endeavors to rectify this through a geometrical investigation of the sophisticated geometrical planning principles used by Anglo-Saxon craftsmen in the creation of these elaborate brooches. Through the use of simple geometrical constructions, smiths were able to create works of great beauty and sophistication. Closer inspection reveals that Anglo-Saxon smiths produced all the composite disc brooches in this study using similar processes of planning. In order to plan out the compositions of each brooch, master smiths would only need a compass, a straightedge, and some material on which to write. Each brooch reveals the same kind of coherent geometry, sharing traits and patterns; with proportions tend to be governed by a series of modular association. Although the master smiths or designers of the composite brooches used simple tools to create the composition, the figures in this thesis were created using the Vectorworks CAD program. This significantly expedited the analytical process and allowed for exact measurements. Despite using the computer program to replicate the planning process, all the figures can be recreated with just a compass and straightedge. While a complete geometric study of all the composite disc brooches needs to be done, this study examines five of the best preserved and well-crafted of that type, ranging from some of the simplest to the most elaborate, as an introduction to the subject.

Pretendemos entender a filosofia de Espinosa, em especial, a sua Ética ordine geometrico demonstrata, a partir de uma operação conflituosa bem específica entre, por um lado, a perspectiva do transcendente (ou a teologia racional) e, por outro, um desejo de salvação mundana; entre o projeto da filosofia imanentista de Espinosa e um mundo submetido ao poder teológico-político; e entre o texto teológico e o método da escrita da filosofia de Espinosa. Tais operações estruturam o cerne de nosso trabalho, no qual visamos entender o nexo causal na passagem de um Deus sive natura absolutamente infinito para nós, os modos finitos desta mesma natureza, de maneira a chegarmos a um entendimento que possa nos garantir não apenas ser, mas tomar parte ativamente neste absolutamente infinito. Não só procuraremos caminhar neste solo conflituoso, mas ainda proporemos tratá-lo com um procedimento que em si enfatiza conflitos, pois visamos responder às nossas questões acerca da filosofia da imanência, de Deus, da passagem do infinito ao finito a partir de uma aproximação entre a obra de Espinosa e o complexo universo artístico da literatura, das artes plásticas e da música do século XVII barroco. Além disto, procuramos demonstrar a hipótese de que a singularidade da Ética enquanto texto, expressa por uma forma textual filosófica sem precedentes, produz uma questão conceitual extremamente complexa que se funde à própria idéia do absolutamente infinito. Pois se a síntese da geometria dos indivisíveis, do século XVII, fornece-nos uma nova idéia de infinito (como amplamente discutiremos) e se a ordem geométrica da demonstração da Ética é fruto desta mesma síntese, então o livro deve necessariamente trazer, já, em sua fartura textual esta idéia de infinito. (Continua)
We intended to understand Espinosa\'s philosophy, especially, his Ethics ordine geometrico demonstrata, starting from a very specific conflicting operation against, on one side, the perspective of the transcendent (or the rational theology) and, on other, a desire for a mundane salvation; between the project of Espinosa\'s immanentist philosophy and a world submitted to the theological-political power; and between the theological text and the method of writing of Espinosa\'s philosophy. Such operations structure the core of our work, in which we seek to understand the causal connection in the passage from a God sive natura, absolutely infinite, to us, the finite manners of his same nature, in way that we can arrive to an understanding that can guarantee to us not to be a part, but to take part actively in this absolutely infinite. Not only we will try to walk in this conflicting path, but we intend to treat it with a procedure that emphasizes conflicts in itself, for we aim to answer our subjects - concerning the philosophy of the immanence, God, and the passage from the infinite to the finite - dealing with an approach between Espinosa\'s work and the complex artistic universe of literature, visual arts and music from the Baroque XVII century. Farther, we intend to demonstrate the hypothesis that the singularity of the Ethics while a text, expressed by an unprecedented philosophical textual form, produces an extremely complex conceptual subject that merges to the same idea of the absolutely infinite present in the Ethics. For if the synthesis from the geometry of the indivisibles, of the XVII century, provide us a new idea of the infinite (as we will extensively discuss) and if the geometric order on the demonstration of the Ethics is a fruit of this same synthesis, then the book should necessarily bring, already, in its textual profusion 7 this idea of the infinite. In other words, the idea of the geometric-synthetic order, key to the formulation of the absolutely infinite, already takes place in the textual structure ordine geometric demonstrata of the Ethics. Thus, we look forward to demonstrate that the order of exposition of the text in the Ethics operates with the same idea expressed by its ontology (the idea that is also expressed in mathematics by the geometrical synthesis). Farther on, we will insist that the formal articulation of the Ethics renders to us patent the fruition of the infinite, because we believe that such work while a text and as text, already expresses to its reader the experience of this new synthesis of an indivisible absolutely infinite.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
Esta dissertação busca investigar o processo de produção de Performances Matemáticas Digitais (PMD) com o auxílio do software GeoGebra em ambientes educacionais voltados à formação de estudantes e professores de Matemática. Especificamente, o objetivo é investigar o papel das Artes e das tecnologias digitais na comunicação de ideias matemáticas. Dessa maneira, a pergunta que norteou essa pesquisa foi: Como estudantes de Matemática produzem PMD sobre Geometria considerando o uso do software GeoGebra? PMD concerne no uso das Artes e das tecnologias digitais em Educação Matemática, com o objetivo de compartilhar ideias matemáticas de uma maneira alternativa. O GeoGebra é um software livre e possui muito prestígio no ensino e aprendizagem de Matemática, pois ele possibilita a exploração de aspectos como a visualização e experimentação com tecnologias. Com o intuito de investigar interpretativamente tais aspectos envolvendo a produção de PMD, esta pesquisa é de natureza qualitativa. Os dados foram produzidos por meio da realização de um curso de extensão universitária de 18 horas sobre Geometria que envolveu a produção de PMD e o uso do software GeoGebra. Este foi ministrado na Universidade Estadual Paulista “Júlio de Mesquita Filho” (UNESP), Campus de Rio Claro. Durante os encontros os estudantes e professores de Matemática criaram três tipos de PMD: Graphics Interchange Format (GIFs), música e clipe. As ideias matemáticas comunicadas tiveram origem em uma atividade sobre um triângulo equilátero e outra sobre um tetraedro regular. Os registos dos dados foram feitos por meio do diário de campo e consideraram-se como dados as PMD, as filmagens das seções do curso, gravações das telas e dos áudios dos computadores dos participantes, interações em um grupo no Facebook e as anotações dos cursistas nas atividades impressas. A análise dos dados explora aspectos artísticos, tecnológicos e matemáticos desenvolvidos durante o curso e no processo de produção PMD. Os resultados mostram que a criação de PMD pode ser uma possibilidade para o ensino de Matemática destacando-se, principalmente, fatores artísticos e os coletivos pensantes a respeito do processo de pensar-com-PMD. Podendo transmitir emoções para os espectadores dessas PMD.
This work aims to investig ate the production process of Digital Mathematical Performances ( DMP ) with the use of GeoGebra software in educational environments aimed at the training of students and teachers of Mathematics. Specifically, the objective is to investigate the role of the Arts and Digital Technologies in the communication of mathematical ideas. Thus, th e question that guided this research was: h ow do math students develop DMP about Geometry considering the use of GeoGebra software? DMP refers to the use of the Digital Arts and Technologies in Mathematics Education, aiming to share mathematical ideas in a n innovative way. GeoGebra is a free software and has educational prestige in the teaching and learning of mathematics, as it enables the exploration of aspects such as visualization and experimentation with technologies. In order to investigate such aspec ts involving the production of DMP , this research is quali tative in nature. The data were produced through the accomplishment of a 18 - hours knowledge mobilization course about Geometry that involved t he production of DMP and the use of GeoGebra software. I t was taught at Sao Paulo State Univ ersity “Júlio de Mesquita Filho” (UNESP), Campus of Rio Claro. During the sessions , the students and teachers of Mathematics created three types of DMP : Graphics Interchange Format ( GIF ), music and video clip. The mathem atical ideas communicated had originated in an activity on an equilateral triangle and another on a regular tet rahedron. The data records were made by means of field diaries, the produced DMP , video recordings of the sessions , videotape of the participants' computer screens, interactions in a Facebook group , and notes of the participants in the acti vities were considered as data printed. The data analysis explores artistic, technological and mathematical aspects present during the course develop ment and the DMP . The results show that creation of DMP can be a possibility for the teaching of mathematics emphasizing, mainly, artistic factors and the collective thinkin g about the process of thinking - with - DMP . It can convey emotions to the audience of these DMP .
CAPES: 1607542.

La tesi titulada Geometria, ordre i domini. Del post minimalisme a la crítica geopolítica, té com apunt de partida i objecte establir les noves significacions adquirides per la geometria com a representació a l’actualitat, així com analitzar i establir el procés artístic i social a què responen aquests canvis. Des dels inicis de la modernitat, la representació geomètrica ha jugat un paper important en la vida social i artística, tant de forma formal com física o ètica. Així doncs, si històricament aquesta havia estat contenidora d’una sèrie d’ideals positius i ordenadors de la vida humana, des de l’aparició del Minimalisme aquests vincles ètics i conceptuals han patit una enorme deriva i mutació, tant com a sistema de representació, com a estructura del pensament al servei de certs ordenaments espacials. A partir d’aquí, la recerca ens duu a un nou canvi de direcció que conforma una de les principals hipòtesis de la tesi i que configura tota la segona part. El motiu d’aquest nou objectiu d’atenció per part dels artistes a dia d’avui és la resposta a noves estratègies adoptades pel poder com eines de control i especulació, les quals centren el seu objectiu en l’espai habitat i el territori. Així doncs, si les crítiques artístiques a aquella geometria de domini van partir en un inici d’un àmbit bàsicament formal amb les respostes al Minimalisme, sent la geometria la seva forma de representació del poder més visible, posteriorment els artistes han anat dirigint-se a un àmbit més sociològic, concretament situat en la geografia i el territori. D’aquesta forma els artistes realitzen les seves aportacions com un acte d’alliberació front l’actual acció geopolítica, la qual se’ns presenta com rígida, inhumana i excloent. Aquests artistes cerquen generar intersticis de llibertat, de convivència col·lectiva i percepció de la realitat tal com són definits per Nicolás Bourriaud. Així doncs, les crítiques artístiques a la geometria com element formal, de domini i control, han derivat de forma paral·lela a noves consideracions i significacions adoptades per aquesta geometria, i que a dia d’avui estan vinculades a la geografia i l’ordenació de l’espai, en el que s’ha anomenat: gir etnogràfic, tal i com ha estat definit per Hal Foster. Aquest gir etnogràfic emmarcat en els territoris, que segueix al post minimalisme més formal, marca la necessitat de diferenciar la present trajectòria de crítiques a l’ordre en dos parts diferenciades la primera part dedicada a les crítiques més formalistes, i la segona que estudia tota aquesta crítica la qual ha convertit als artistes en una espècie d’etnògrafs, geòlegs, antropòlegs, etc. Per aquest motiu es culmina la primera part amb les aportacions de Robert Smithson, ja que aquest artista, amb la seva crítica, ha estat un dels principals referents per als treballs geològics posteriors i el seu anàlisi dóna llum i facilita la comprensió d’aquest gir etnogràfic, descrivint amb eficàcia aquesta nova deriva adoptada pels artistes i la continuïtat que suposa en respecte les crítiques post minimalistes anteriors. Per tant, aquest camp d’estudi analitza com s’ha dut a terme aquesta evolució i com aquesta ha estat motiu per a diferents accions artístiques i teories crítiques, a la llum de les quals, s’han anat redefinint les nocions de geometria i ordre fins l’actualitat. En segon lloc, es mostra a mode de síntesi, aquesta darrera derivació front l’acció geopolítica en el nostre context actual globalitzat. Així doncs, el plantejament d’aquest estudi és establir aquesta trajectòria a partir de les principals obres dels artistes i filòsofs implicats i les seves intervencions, organitzats segons la seva cronologia i afinitats conceptuals des dels anys setanta fins a dia d’avui.
La tesis titulada Geometría, orden y dominio. Del post minimalismo a la crítica geopolítica, tiene como punto de partida y objeto dilucidar las nuevas significaciones adquiridas por la geometría como representación en la actualidad, así como analizar y establecer el proceso artístico y social a que responden estos cambios. Des de los inicios de la modernidad, la representación geométrica ha jugado un papel importante en la vida social y artística, tanto de manera formal, como física o ética. Así pues, si históricamente ésta había sido supuestamente contenedora de una serie de ideales positivos y ordenadores de la vida humana, desde la aparición del Minimalismo estos vínculos éticos y conceptuales han sufrido una enorme deriva y mutación, tanto como sistema de representación, como estructura del pensamiento al servicio de ciertos ordenamientos espaciales. A partir de este punto la investigación nos lleva a ver un nuevo cambio de dirección que conforma una de las principales hipótesis de esta tesis y que configura toda la segunda parte. El motivo de este nuevo objetivo de atención por parte de los artistas a día de hoy es la respuesta a nuevas estrategias adoptadas por el poder como herramientas de control y especulación, las cuales centran su objetivo en el espacio habitado y el territorio. Así pues si las críticas artísticas a aquella geometría de dominio partieron en un inicio de un ámbito básicamente formal con las respuestas al Minimalismo, siendo la geometría su forma de representación de poder más visible, posteriormente los artistas han ido dirigiéndose a un ámbito más sociológico, concretamente situado en la geografía y el territorio. De este modo los artistas realizan sus aportaciones como un acto de liberación frente la actual acción geopolítica, la cual se nos presenta como rígida, inhumana y excluyente. Estos artistas buscan generar intersticios de libertad, de convivencia colectiva y percepción de la realidad tal como son definidos por Nicolás Bourriaud. Así pues, las críticas artísticas a la geometría como elemento formal, de dominio y control, han derivado de forma paralela a nuevas consideraciones y significaciones adoptadas por esta geometría, y que a día de hoy están vinculadas a la geografía y la ordenación del espacio, en lo que se ha llamado: giro etnográfico, tal como ha sido definido por Hal Foster. Este giro etnográfico enmarcado en los territorios, que sigue al post minimalismo más formal, marca la necesidad de diferenciar la presente trayectoria de críticas al orden en dos partes diferenciadas: la primera parte dedicada a las críticas más formalistas, y la segunda que estudia toda esta crítica la cual ha convertido a los artistas en una especie de etnógrafos, geólogos, antropólogos, etc. Por este motivo se culmina la primera parte con las aportaciones de Robert Smithson ya que este artista, con su crítica, ha sido uno de los principales referentes para los trabajos geológicos posteriores y su análisis arroja luz y facilita la comprensión de este giro etnográfico, describiendo con eficacia esta nueva deriva adoptada por los artistas y la continuidad que supone frente a las críticas post minimalistas anteriores. Por tanto, este campo de estudio analiza cómo se ha llevado a cabo esta evolución y cómo esta ha sido motivo para diferentes acciones artísticas y teorías críticas, a la luz de las cuales, se han ido re definiendo las nociones de geometría y orden hasta la actualidad. En segundo lugar, se muestra a modo de síntesis, esta última derivación frente la acción geopolítica en nuestro contexto actual globalizado. Así pues, el planteamiento de este estudio es establecer esta trayectoria a partir de las principales obras de los artistas y filósofos implicados y sus intervenciones, organizados según su cronología y afinidades conceptuales desde los años setenta hasta día de hoy.
The thesis titled Geometry, order and rule. Criticism of post minimalism geopolitics, is about to start and establish new meanings acquired by the geometry representation as to the present, as well as analyze and establish the artistic and social process responsible for these changes. Since the dawn of modernity, geometric representation has played an important role in the social and artistic, both for physical or formal ethics. So if this had been historically container of a series of computers and positive ideals of human life, from the appearance of minimalism, these ethical and conceptual links mutated tremendously, both as a system of representation as a structure of thought in the service of certain legislations space. From there the investigation leads to a new change of direction, which forms one of the major hypothesis of the thesis, and configure the entire second half. The reason for this new object of attention by artists today is the answer to new strategies adopted by power as tools of control and speculation, which focused its target in space habitat and territory. So, artistic criticism to the geometry, started at the beginning in a formal field of responses to Minimalism, with its geometry like the more visible form of representation of power. Later, artists have been addressing a sociological level, specifically located in the geography and territory. Thus artists made their contributions as an act of liberation front the current geopolitical action, which is presented as rigid, inhuman and exclusive. These artists seek to generate interstices of freedom, coexistence and collective bargaining perception of reality as they are defined by Nicolas Bourriaud. So, artistic criticism to geometry as formal element, of domain and control, have resulted in parallel with new meanings and considerations adopted by this geometry, which today are linked to the geography and the spatial ordination, which is called: ethnographic turn, as has been defined by Hal Foster. This ethnographic turn framed in the territories, which continues to more formal post minimalism, marks the need to differentiate the present trajectory of criticism order in two distinct parts: the first part dedicated to more formalist criticism, and second that studies all this criticism which has become artists in a kind of ethnographers, geologists, anthropologists etc. For this reason the first part ends with the contributions of Robert Smithson, as the artist, with his critics has been one of the main references for geological work and its subsequent analysis sheds light and facilitates the understanding of this ethnographic turn, describing effectively this new drift adopted by artists and continuity with previous post minimalism criticism. Therefore, this study examines how this evolution has taken place, examines how this trend has been the subject of several critical theories and artistic actions, examines how this trend has been the subject of several critical theories and artistic actions, and as from which have redefined notions of geometry and order until today. Secondly, it is shown by way of summary, the latter derivation front geopolitical action in our current globalized context. So the approach of this study is to establish the path from the main works of the artists and philosophers involved and their interventions are organized according to chronology and conceptual affinities since the seventies until today.

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Este trabalho de investigação é um estudo que visa a melhoria do ensino de Matemática. Tem como objetivo a revitalização do ensino de Geometria numa perspectiva interdisciplinar entre Matemática e Arte. O estudo envolveu alunos de uma turma do 6.º ano do Ensino Fundamental - Anos Finais de uma Escola Pública Estadual do Interior do Estado de São Paulo. A pesquisa aplicada em três etapas se deu por meio de Sequências didáticas A, B e C compostas por conteúdos de Geometria. Foram analisadas as transformações geométricas, em especial a simetria de reflexão, rotação e translação, bem como, a aprendizagem de conceitos matemáticos. Conceitos como: simetria, proporção, polígonos, poliedros, pontos, retas, curvas, ângulos, cores, figuras e formas geométricas, dentre outros, foram verificados em recursos como vídeos, softwares educativos Simetrizadores, obras de arte e banco de questões, envolvendo habilidades de leitura visual e geométrica. O desenvolvimento metodológico se deu por pesquisa de abordagem qualitativa, do tipo descritiva, intencionando a retomada da “Geometria Básica”, no sentido de valorização de tais conceitos geométricos, como elementos relevantes para estabelecer a conexão entre a Arte e a Matemática. Os dados foram recolhidos a partir da aplicação das Sequências didáticas A – Transformações Geométricas, B – Obras de Arte e C – Banco de Questões envolvendo materiais manipuláveis, recursos tecnológicos, análise de imagem, contexto histórico e fazer artístico. Os resultados mostraram uma oportunidade de minimizar as defasagens de aprendizagens em conceitos geométricos; houve evolução na construção e ampliação do conhecimento matemático com a interação aluno/aluno, professor/aluno e trabalho em equipe. Apresenta-se, anexo a dissertação, o produto educacional, elaborado com os dados do trabalho, cuja finalidade é fornecer aos professores de Matemática e de Arte, Sequências didáticas envolvendo a Geometria Básica de forma interdisciplinar e contribuir para o ensino interdisciplinar.
This research is a study aimed at improving Mathematics teaching. Its purpose is to revitalize the teaching of Geometry in an interdisciplinarity perspective between Mathematics and Art. The study involved students from a sixth grade Elementary School group of a State Public School in the interior of the State of São Paulo The research applied in three stages was done through didactic Sequences A, B and C and composed og Geometry contents. The geometrical transformations, especially the symmetry of reflection, rotation and translation, as well as the learning of mathematical concepts were analyzed. Concepts such as: symmetry, proportion, polygons, polyhedra, points, lines, curves, angles, colors, figures and geometrical shapes, geometrical thoughts, among others, were verified in resources such as videos, symmetrizing educational software, works of art and question bank, involving visual and geometrical reading skils. The methodological development happened through a qualitative research, descriptive type, intending the “Basic Geometry” resumption, considering the importance of such geometrical concepts, as relevant elements to establish a connection between Art and Mathematics. The data were collected from the application of the didactic Sequences A - Geometric Transformations, B - Works of Art and C - Question Bank involving manipulable materials, technological resources, image analysis, historical context and artistical making. The results showed an opportunity to minimize the lags of learning in geometrical concepts; there was an evolution in the construction and expansion of mathematical knowledge with the student/student, teacher/student interactions and teamwork. An educational product and the dissertation, prepared with the data of all the work is presented in an annex, whose purpose is to provide teachers of Mathematics and Art, didactic Sequences involving the "Basic Geometry" in an interdisciplinary way and to contribute to interdisciplinary teaching.

My diploma work is an installation, made from multiple instances of a device iniciating string resonance via electromagnetic field. These devices along with strings are placed on the wall in geometrical shape. Installation creates loop on multiple levels. Except the fact that installation have a circular shape, position of each string starts at the end of another string. Second, more inconspicious loops takes place in the electromagnetic device resonating the strings, that do that with feedback loop. Strings consist every step of chromatic scale, that repeats itself, just an octave higher. Amplification of the final sound of strings is done purely acoustically, with help of the wall on wich the piece is installed. This piece is in its nature concerned with spirituality in music, not necessarily in sense of evoking a spiritual experience, but rather demonstrating metaphors and parallels, that exists between physical aspects of tonal music and different religious ideas. The symmetrical shape of installation refer to religious and occult visuality, built f.e. in cabal on Fibonacci numbers, that is present not only in nature ( for example, the veins of the leaves grow by these numbers), but also in tonal music system (ancient philosophers were working with this concept, see Plato's Music of the Spheres). Strings in this piece produce drone sound, that is naturally evoking spirituality (most visible in buddhist monk meditation). This sound in the piece demonstrates immutability and constancy, the fact that all the chromatic tones are playing demonstrates wholeness (this fact may produce interesting resonances emerging between chromatic steps), so to speak, the unchangeable laws of physics, or to put it in religious lingo, the god law. The symbol of loop also refers to religion, like the eternal return of the same, the periodicity of history. Strings can be viewed as astrophysical symbol. Everything stated is nothing but my recourse, that should not ultimately determine the perception of the piece by viewer. The goal of the work is to offer experience without need to be put into context

Le mouvement Nouvelle tendance (NT) est un groupe international d’artistes formé pendantles années soixante (1961 - 1973) autour d’un programme d’expositions de la Galerie d’ArtContemporain (Galerija suvremene umjetnosti) de Zagreb. Au cours de son existence, lemouvement NT a rassemblé presque deux cents artistes et plusieurs groupes tels que GRAV,T, N, Zero, Equipo 57, Dvizhenije, MID etc. La première étape de ce mouvement jusqu’en 1968est caractérisée par l’abstraction géometrique et l’art lumino-cinétique tandis que dans laseconde partie (1968 -1973), Nouvelle tendance ouvre le chapitre de l’art numérique
New Tendancy movement (NT) is an international group of artists united in the sixties(1961 - 1973) around the exhibition programme at the Gallery of Contemporary Art (Galerijasuvremene umjetnosti) in Zagreb. During its existence, the movement gathered around twohundred artists and differents groups such as GRAV, T, N, Zero, Equipo 57, Dvizhenije, MID etc.The first phase of the movement that lasted until 1968 was characterized by the geometricabstraction and lumino-kinetic art. During the second phase, New Tendancy opened thechapter of numerical arts

I base my spatial installations upon working with waste materials. I use characteristic and aesthetics of cardboard and wooden clippings to express dialogue with geometrical drawings, which I have been producing since the last year. Protracted and unexpectedly-carved shapes resemble timber beams or ruins and can lead to ecological or eschatological questions. The title "Monument" is an ironical disputation with human trace and materials we use without giving them any further attention.

Orientadores: Cid Carvalho de Souza, Pedro Jussieu de Rezende
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação
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Resumo: Nesta dissertação, apresentamos nossa pesquisa sobre o Problema da Galeria de Arte (AGP), um dos problemas mais estudados em Geometria Computacional. O AGP, que é um problema NP-difícil, consiste em encontrar o número mínimo de guardas suficiente para garantir a cobertura visual de uma galeria de arte representada por um polígono. Na versão do problema tratada neste trabalho, usualmente chamada de Problema da Galeria de Arte com Guardas-Ponto, os guardas podem ser posicionados em qualquer lugar do polígono e o objetivo é cobrir toda a região, que pode ou não conter buracos. Nós estudamos como aplicar conceitos e algoritmos de Geometria Computacional, bem como Técnicas de Programação Inteira, com a finalidade de resolver o AGP de forma exata. Este trabalho culminou na criação de um novo algoritmo para o AGP, cuja ideia é gerar, de forma iterativa, limitantes superiores e inferiores para o problema através da resolução de versões discretizadas do AGP, que são reduzidas a instâncias do Problema de Cobertura de Conjuntos. O algoritmo foi implementado e testado em mais de 2800 instâncias, de diferentes tamanhos e classes. A técnica foi capaz de resolver, em minutos, mais de 90% de todas as instâncias consideradas, incluindo polígonos com milhares de vértices, e ampliou em muito o conjunto de casos para os quais são conhecidas soluções exatas. Até onde sabemos, apesar do extensivo estudo do AGP nas últimas quatro décadas, nenhum outro algoritmo demonstrou a capacidade de resolver o AGP de forma tão eficaz como a técnica aqui descrita
Abstract: In this dissertation, we present our research on the Art Gallery Problem (AGP), one of the most investigated problems in Computational Geometry. The AGP, which is a known NP-hard problem, consists in finding the minimum number of guards sufficient to ensure the visibility coverage of an art gallery represented as a polygon. In the version of the problem treated in this work, usually called Art Gallery Problem with Point Guards, the guards can be placed anywhere in the polygon and the objective is to cover the whole region, which may or not have holes. We studied how to apply Computational Geometry concepts and algorithms as well as Integer Programming techniques in order to solve the AGP to optimality. This work culminated in the creation of a new algorithm for the AGP, whose idea is to iteratively generate upper and lower bounds for the problem through the resolution of discretized versions of the AGP, which are reduced to instances of the Set Cover Problem. The algorithm was implemented and tested on more than 2800 instances of different sizes and classes of polygons. The technique was able to solve in minutes more than 90% of all instances considered, including polygons with thousands of vertices, greatly increasing the set of instances for which exact solutions are known. To the best of our knowledge, in spite of the extensive study of the AGP in the last four decades, no other algorithm has shown the ability to solve the AGP as effectively as the one described here
Mestrado
Ciência da Computação
Mestre em Ciência da Computação

Esta tesis pretende reconsiderar la geometría, dentro del campo del arte contemporáneo, intentando analizar y definir un cambio en la naturaleza misma de su significado y valor, que en nuestra hipótesis de trabajo se inició en los años 60 del siglo pasado. El presente estudio aspira a poner en cuestión, de manera paralela, la concepción clásica de la geometría, dentro del arte más reciente, como orden ideal, representación de la forma “pura” y “neutra”, atribuciones ambas que pertenecen más a una lógica intemporal que al tiempo real de la experiencia. A través del desarrollo de esta tesis hemos estudiado cómo ha variado el significado proyectual del arte geométrico en las últimas décadas, dentro de la evolución de la pintura y la escultura contemporáneas, y cómo la puesta en obra de estas variaciones conceptuales tiene, consecuentemente, un sentido diferente de la recepción, los modos de presentación y la percepción de la obra también para el espectador. Para el desarrollo de la presente investigación he utilizado elementos de análisis formal en un conjunto representativo de obras pictóricas y escultóricas, y de sus intereses expresivos. Estas obras han renunciado a una concepción autonomista o ideal del arte y por el contrario, privilegian el efecto de la geometría en conexión física con el espacio real de la experiencia, social o cultural, diferenciándose así del marco representacional y mental propio del modelo idealista. A partir de esta perspectiva, examino inicialmente las obras de ciertos artistas norteamericanos de la década de los 80, para los que la abstracción geométrica, por su actitud crítica, es un nuevo dispositivo codificado, con base material, que organiza referencias a la sociedad y cultura contemporáneas, alejado de un lenguaje formalista «puro» que tratara de separarse de la cultura material. Si bien fue en los años ochenta cuando se planteó la discusión pública sobre la ‘crisis’ de la geometría, ya en las décadas de los 60 y 70 habían surgido los primeros indicios de esa ‘crisis’. En los Estados Unidos algunos representantes del minimalismo de ese periodo empezaron, de una manera más o menos consciente, a vincular la geometría a la producción material de la sociedad industrial de su tiempo, paralelamente, otros artistas desarrollaron un análisis crítico y social del poder cultural, implícito en sus significantes: formas, materiales, escala, etc. A partir de 1980, en la década de los 90 y hasta la actualidad, aquella imagen dominante de la geometría, a menudo agresiva o brutal, cambiará radicalmente con la aparición de un nuevo tipo de obras geométricas, producto en cierta medida de esta sociedad postindustrial, ligada al consumo, la tecnología digital y al espectáculo. En manos de esta generación de artistas, no tan vinculados como las anteriores a la experiencia industrial de los materiales, sino a la del ordenador y sus posibilidades como tecnología de la imagen- que permite proyectar y trabajar en campos característicos de las nuevas geometrías no euclidianas: fractales, topológicas o en la integración de modelos naturales- la geometría emerge, de esta manera, como una poderosa herramienta de experimentación artística generando espacios y formas de creciente complejidad en variaciones ilimitadas. Estas nuevas obras, ya se trate de instalaciones, proyectos algorítmicos o escultóricos, se constituyen ellas mismas en entornos dinámicos, interactivos y multisensoriales. La geometría se presenta así, como un proyecto y proceso creativo estimulante, capaz de movilizar un compromiso frágil, múltiple y abierto entre la obra como sistema material y experimental y un espectador que piensa, siente y actúa. Esta reconsideración global de la geometría operaría tanto sobre la base de códigos culturales, que son los registros representacionales que constituyen nuestra imagen de la realidad y definen nuestra experiencia, como sobre la base de una compleja y nueva instrumentalidad. En otras palabras: la geometría subyacente se plantearía como una experiencia artística diferente, con unas marcadas características y singularidades. El siguiente interrogante será, naturalmente, el de cómo debemos mirar y reconsiderar estas nuevas posibilidades de creación artística.
This thesis reconsiders geometry within the field of contemporary art, trying to analyze and define a change in the nature of its meaning and value that in our hypothesis of work began in the 60s of the last century. This study aspires to question, paralelly, the classic conception of geometry within the most recent art, as the ideal order, representation of the “pure” and “neutral” form, both attributes belonging to a timeless logic rather than real-time experience. Throughout this thesis we have studied how the projective meaning of geometric art has changed over the last decades, within the evolution of contemporary painting and sculpture, and how these variations in concept have, consequently, a different meaning in reception, the manner of presentation and the spectator’s perception of the work. To develop the present research I have used elements of formal analysis in a representative group of paintings and sculptures, and their expressive interests. These works have renounced an autonomist or ideal conception of art and, on the contrary, favour the effect of geometry in a physical interface with the real space of the experience, social or cultural, thus differing from the representational and mental framework of the idealist model. From this perspective, I initially examine the works of various North American artists from the 80s, for whom the geometric abstraction, for their critical attitude, is a new coded device, with a material basis, which organizes references to contemporary society and culture, far from a formalist “pure” language that tries to separate itself from material culture. Although it was in the eighties when the public discussion on the “crisis” of geometry was laid out, in the decades of the 60s and 70s the first indication of the “crisis” had already come up. In the United States some representatives of the minimalism of this period started, in a more or less conscious way, to link geometry to the material production of the industrial society of that time, in parallel, others developed a critical and social analysis of cultural power, implicit in their signifiers: shapes, materials, scale etc. From 1980, during the decade of the 90s and until the present day, that dominant image of geometry, often aggressive or brutal, would radically change with the appearance of a new type of geometrical works, to a certain degree the product of this post-industrial society, tied to consumption, to digital technology and to entertainment. In the hands of this generation of artists, not so tied as the previous to the industrial experience of the materials as to that of the computer and its possibilities as image technology –allowing the projection and work in the characteristic fields of new non-Euclidian geometries: fractals, topological or in the integration of natural models- geometry emerges, in this way, as powerful tool for artistic experimentation generating increasingly complex spaces and shapes in unlimited variations. These new works, be they installations or algorithmic or sculptural projects, are constituted within dynamic, interactive and multisensorial surroundings. Geometry is thus shown as a stimulating, creative project and process, capable of triggering a fragile, multiple and open engagement between the work as a material and experimental system and the spectator who thinks, feels and acts. This comprehensive reconsideration of geometry would work as much on a cultural codes basis, these being the representational registers which make up our image of reality and define our experience, as on the basis of a complex and new instrumentality. In other words, the underlying geometry would be considered as a different artistic experience, with marked characteristics and singularity. The next question will be then, of course, how we should view and reconsider these new possibilities of artistic creation.

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This dissertation portrays a research carried out with students of the Integrated Technical Course on Agropecu?ria PRO-EJA Indigenous of the Federal Institute of Amazonas - IFAM, in the Municipality of Tabatinga, located in the western part of the state of Amazonas. It has, among others, the purpose of proposing didactic strategies for the teaching and learning processes of Geometry, based on the relation of the contents of geometry with geometric patterns observed in the confection processes and handicrafts of the Ticuna indigenous peoples of the Umaria?? Indigenous Community, as well as suggest some pedagogical activities to be worked using these elements. The methodology of this work consists in the application of a questionnaire to evaluate the level of understanding and the importance of geometry for the students and for the course, in detailed observations of the student?s presentation during the seminars where the students presented results of their researches. The realization of craft workshops aiming to establish a relationship between the geometric patterns studied and those found in this process, and their possible application in problems of their daily life. During the seminars and in the confection activities it was noticed that the handicrafts facilitated the understanding of the basic contents of geometry because they are part of the socio-cultural context of the student. The satisfaction and motivation for the recognition of their culture were evidenced in the evaluation. In this way, it can be said that indigenous handicrafts can facilitate the teaching and learning processes of geometry for these students. This work also intends to make a modest contribution to the mathematics teachers of the indigenous schools with some suggestions of activities that can be developed by the students of the community with the intention of making the learning more meaningful and pleasant for the students and also to strengthen the traditional culture of the Ticunas
Esta disserta??o retrata uma pesquisa realizada com alunos do Curso T?cnico Integrado em Agropecu?ria PRO-EJA Ind?gena do Instituto Federal do Amazonas ? IFAM, situado no Munic?pio de Tabatinga, localizado no oeste do estado do Amazonas. Tem, entre outras, a finalidade de propor estrat?gias did?ticas para os processos de ensino e aprendizagem da Geometria, baseada na rela??o dos conte?dos de geometria com padr?es geom?tricos observados nos processos de confec??o e nos artesanatos dos povos ind?genas da etnia Ticuna da Comunidade ind?gena Umaria??, bem como sugerir algumas atividades pedag?gicas para serem trabalhadas utilizando esses elementos. A metodologia deste trabalho consiste na aplica??o de um question?rio para avaliar o n?vel de entendimento e a import?ncia da geometria para os alunos e para o curso, em observa??es detalhadas da apresenta??o dos alunos durante os semin?rios onde os discentes apresentaram resultados de suas pesquisas. A realiza??o de oficinas de confec??o de artesanatos visando estabelecer rela??o entre os padr?es geom?tricos estudados com os encontrados nesse processo, e sua poss?vel aplica??o em problemas do seu cotidiano. Durante os semin?rios e nas atividades de confec??o percebeu-se que os artesanatos facilitaram o entendimento dos conte?dos b?sicos de geometria por fazerem parte do contexto sociocultural do discente. A satisfa??o e motiva??o pelo reconhecimento de sua cultura foram evidenciados na avalia??o. Desta forma, pode-se afirmar que os artesanatos ind?genas, podem facilitar os processos de ensino e aprendizagem da geometria para estes discentes. Este trabalho pretende ainda dar uma modesta contribui??o aos docentes de matem?tica das escolas ind?genas com algumas sugest?es de atividades que podem ser desenvolvidas pelos alunos da comunidade com o intuito de tornar a aprendizagem mais significativa e prazerosa para os discentes e tamb?m fortalecer a cultura tradicional dos Ticunas.

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Fundo Mackenzie de Pesquisa
This is a paper about the mathematics and art teacher´s training in which are approached the theoretical basis of this two subjects, such as, with its interdisciplinarity among the others areas of the knowlegment, language, codes and its technologies, the nature of the Science and its technologies. This basic research points to the interdisciplinarity of these two areas of the knowlegment, by the mathematics and art subjects existent together with the mathematical and visuals elements, dots, line and shape. Analyzing the life of the mathematician Euclides de Alexandria and his work The Elements, composed by 23 books, pointing out his first book, in which such concepts are cited. The same way it s approached Paul Klee s life and his abstracted work, such as his teachings in Bauhaus schools, where the very same concepts are developed by the visual language. Establishing a parallel between the mathematicians Euclides, Lintz and Machado, such as the artists Klee, Kandinsky and Mondrian, stands out the concepts: dots, line and shape in same of their art work. It s fixed, so that the interdisciplinarity in both area of the knowlegment is providing an important contribution to the education.
Este é um trabalho sobre formação de professores de matemática e arte no qual são abordados os fundamentos teóricos dessas duas disciplinas, bem como a sua interdisciplinaridade com as demais áreas do conhecimento: linguagens, códigos e suas tecnologias, as ciências da natureza e suas tecnologias. Essa pesquisa bibliográfica aponta para a interdisciplinaridade dessas duas áreas do conhecimento, por meio das disciplinas Matemática e Arte, existente justamente nos conceitos dos elementos matemáticos e visuais: ponto, linha e forma. Abordando a vida do matemático Euclides de Alexandria e sua obra Os Elementos, composta de 23 livros, destacando-se o livro primeiro, onde tais conceitos são citados. Da mesma forma aborda-se a vida de Paul Klee e sua obra abstrata, bem como o seu ensino na escola de Bauhaus, onde os mesmos conceitos são trabalhados por meio da linguagem visual. Fazendo um paralelo entre os matemáticos Euclides, Lintz e Machado, bem como os artistas Klee, Kandinsky e Mondrian, destacam-se os conceitos: ponto, linha e forma em algumas de suas obras de arte. Estabelece-se, então, a interdisciplinaridade nessas duas áreas de conhecimento, proporcionando uma relevante contribuição para a Educação.

For the purpose of cultural heritage preservation, the task of recording and reconstructing visually complicated architectural geometrical patterns is facing many practical challenges. Existing traditional technologies rely heavily on the subjective nature of our perceptual power in understanding its complexity and depicting its color differences. This study explores one possible solution, through utilizing digital techniques for reconstructing detailed historical Islamic geometric patterns. Its main hypothesis is that digital techniques offer many advantages over the human eye in terms of recognizing subtle differences in light and color. The objective of the study is to design, test and evaluate an automatic visual tool for identifying deteriorated or incomplete archaeological Islamic geometrical patterns captured in digital images, and then restoring them digitally, for the purpose of producing accurate 2D reconstructed metric models. An experimental approach is used to develop, test and evaluate the specialized software. The goal of the experiment is to analyze the output reconstructed patterns for the purpose of evaluating the digital tool in respect to reliability and structural accuracy, from the point of view of the researcher in the context of historic preservation. The research encapsulates two approaches within its methodology; Qualitative approach is evident in the process of program design, algorithm selection, and evaluation. Quantitative approach is manifested through using mathematical knowledge of pattern generation to interpret available data and to simulate the rest based on it. The reconstruction process involves induction, deduction and analogy. The proposed method was proven to be successful in capturing the accurate structural geometry of the deteriorated straight-lines patterns generated based on the octagon-square basic grid. This research also concluded that it is possible to apply the same conceptual method to reconstruct all two-dimensional Islamic geometric patterns. Moreover, the same methodology can be applied to reconstruct many other pattern systems. The conceptual framework proposed by this study can serve as a platform for developing professional softwares related to historic documentation. Future research should be directed more towards developing artificial intelligence and pattern recognition techniques that have the ability to suplement human power in accomplishing difficult tasks.

Magistrinio darbo temą pasirinkau – „Atspindžio interpretacijos abstrakčioje tapyboje“. Kolekcijoje tapau efemeriškus vaizdus, kuriuos pastebėjau ant stiklinių daugiaaukščių pastatų, pamatytus atspindžio vaizdus norėjau įamžinti tapyboje. Kolekciją atlikau dviejų stilių deriniu – geometrine abstrakcija ir veiksmo tapyba (abstraktusis ekspresionizmas), nes šie deriniai tokie parodoksalūs. Sąmoningai atsisakiau optinio meno. Svarbiausia darbe įsikišimas realybės – dangaus vaizdai atspindyje. Tapybos kūrinuose – atspindys užpildo architektūros tuštumą. Kolekcijoje „Atspindžio interpretacijos abstrakčioje tapyboje“ ieškojau harmonijos ir pusiausvyros, taip pat jiems giminingų aspektų, tokių, kaip pasikartojimai, supriešinimai statiškos geometrijos ir judesio, variacijos ir simetrijos, stengiausi sukurti išbaigtą darnų, juntamą dviem lygmenimis atvaizdo ir vaizdinio junginį. Aštuoni kolekcijos paveikslai labai skirtingi, vienuose išryškinau atspindžio siluetą, kituose labai svarbus konstrukcijos erdviškumas, dar kitus perteikiau labai dekoratyviai.
I choose theme of master‘s degree work – „Reflection interpretation in abstrakt paintings“. In paintings collection I paint ephemeral reflection view, this reflection I saw on the multistory buildings. I want to perpetuate in my paintings view of reflection. I was painting my collection with two different styles combination – of abstract geometry and abstract expressionism, because this combination is very paradoxical. Conscious i do not use optical art. Above all in paintings are intervention of reality – view of sky in reflection. In collection "Reflection interpretation in abstract paintings" I look for harmony and equilibrium, also similar aspects, like repetition, contrary static geometry and movements, variation and symmetry. I try to create completed painting, palatable by two sides like picture and image combination. Seven collection paintings are very different, one of them I try to bring to light reflection silhouette, in next painting are very important spatial character of construction, in others paintings are important decorative.

Orientador: Ernesto Giovanni Boccara
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Artes
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Resumo: Esta dissertação é o resultado da reflexão sobre o significado da mandala, através da abordagem associativa de temas correlatos. A palavra mandala vem do sânscrito, e significa círculo. Qualquer que seja o juízo que lhe façamos, fato é que está associada 'a ação artística de nossa espécie, desde o tempo em que habitávamos cavernas. Pela natureza de sua essência, diz-se que são arquétipos, e ilustram tudo o que se refere 'as noções de ordem, centro e totalidade no Universo. Desta possível relação entre manifestações materiais advindas de movimentos de causas não materiais, surge a noção de que determinadas formas ou arranjos, podem possuir significação independentemente de seu contexto espaço-temporal, estabelecendo nexos e coerências entre as diversas áreas do conhecimento e planos da existência. A geometria é abordada em seus sentidos matemático e simbólico como o princípio que afere critérios, modelos, leis e coesão 'as qualidades dinâmicas do princípio da transformação, também relacionado ao conceito de mandala. Há aqui um interesse em reconhecer que vivemos, nestes tempos, a falência de um modo de ver o mundo e a vida. A dicotomização da realidade em áreas de conhecimento, representa um paradigma a ser transposto por uma visão sistêmica que ofereça síntese no lugar da análise, associações e irmanamentos de princípios e idéias, em lugar de cisão e fragmentação. Encontramos mandalas entalhadas nas cavernas de nossos ancestrais, na arte religiosa, na arquitetura, no traçado das cidades, na arte moderna e contemporânea, nos desenhos das crianças e dos esquizofrênicos, no design, na estruturação de diversos sistemas de conhecimento filosófico, na nossa relação com o sagrado e nas formas criadas pela razão e pela natureza. São evidências da presença da dimensão do mistério. A dissertação é ilustrada de modo a se reforçar estas idéias
Abstract: This essay resulted from the contemplation of the meaning of Mandala, by associative considerations about other reciprocally related subject-matters. The word ¿Mandala¿ originates from the Sanskrit and means circle. In spite of the thoughts we might have about Mandala, it is certain that it is linked to the artistic events of our kind ever since the times we lived in caves. Due to the nature of its essence, it is said that Mandala are archetypes that symbolize the totality of existence in the universe, inner or outer. From this eventual relationship between material manifestations deriving from non-material causes, arises the conception that some determined forms and dispositions may have a meaning, independently from its temporal-spatial context, that established connections and coherences between the several knowledge areas and life plans. The mathematic and symbolic senses of Geometry are approached as the theory that brings to balance criteria, models, laws and harmony within the qualities of the maxims of transformation dynamics, which is also related to the Mandala concept. In here there is the interest of recognizing that nowadays we experience the collapse of the way of perceiving the world and life. Dichotomizing reality in knowledge areas represents a paradigm to be trespassed by a systemic vision which offers synthesis instead of analysis, associations and union of principles and ideas in the place of scission and fragmentation. Mandalas are found engraved in our ancestral¿s caves, in the religious art, in architecture, in cities¿ delineations, in modern and contemporaneous arts, in children¿s and schizophrenic persons¿ drawings, in design, in the structure of many philosophic knowledge systems, in our relationship with sacred things and in the forms created by reasoning power and by nature. Mandalas are the evidences of the existence of mystery magnitude. This essay is illustrated so as to reinforce these ideas
Mestrado
Mestre em Artes

Acompanha: O ensino de arte e matemática: abordagens geométricas (material didático)
A presente dissertação teve como objetivo da introduzir conceitos básicos de geometrias não-euclidianas em aulas de Matemática do Ensino Médio, usando Arte e Matemática. Para tanto, utilizou-se de abordagem triangular, fundamentada por Barbosa e de registros de representações semióticas, baseados nos estudos de Duval. O estudo envolveu estudantes de 2as séries do Ensino Médio de um colégio público estadual do município de Rio Negro (PR). A pesquisa aplicada constou de duas etapas. Num primeiro momento, analisou-se pêssankas à procura de conceitos matemáticos empregados em sua composição e verificou-se a utilização, instintivamente, pelos artesãos, de conceitos como simetria, proporção, polígonos, elipses, biláteros, retas e pontos. Na segunda etapa abordou-se o ensino de geometrias não-euclidianas no Ensino Médio, usando Arte e Matemática. Do ponto de vista metodológico a abordagem foi qualitativa, de natureza interpretativa, com observação participante. Os dados foram recolhidos a partir da aplicação de sequências de atividades envolvendo anamorfose, geometria espacial e projetiva e, da aplicação de oficina investigativa, envolvendo geometrias plana, espacial, elíptica e projetiva. As atividades desenvolvidas com os alunos envolveram materiais manipuláveis, recursos tecnológicos, análise de imagem, contexto histórico e fazer artístico. Os resultados mostram a validade do trabalho docente com metodologia interdisciplinar, tornando as aulas de Matemática motivadoras e desafiantes. Como produto final, apresenta-se um manual pedagógico que tem por finalidade fornecer aos professores de Matemática e de Arte, interessados no assunto, informações sobre conexões entre Arte e Matemática que se fazem presentes no ensino de noções de geometrias não-euclidianas.
The present dissertation had as objective to introduce basic concepts of non-euclidian geometries in Mathematics classes of the Medium Teaching using Art and Mathematics. Therefore, it was used the triangular approach, supported by Barbosa and of registrations of semiotic representations, based in the Duval studies. The study involved students of second grades of High School in a public school from Rio Negro (PR). The applied research consisted of two stages. In a first moment, pysanky was analyzed mathematical concepts used in its composition and the use was verified instinctively, by the artisans, of concepts as symmetry, proportion, polygons, ellipses, biláteros, straight line and points. In the second stage the teaching of geometries was approached non-euclidian in the Medium Teaching, using Art and Mathematics. The methodological point of view the approach was qualitative, of interpretative nature, with participant observation. The data were picked up starting from the application of sequences of activities involving anamorphosis, space geometry and projective and, of the application of investigative shop involving plane, space, elliptic and projective geometries. The activities developed with the students involved materials that there manipulated, technological resources, image analysis, historical context and how to produce artistic activities. The results showed the validity of the educational work with interdisciplinary methodology, making the Math lessons motivating and challenging. As the final product, It presents a pedagogical manual that has for purpose to provide teachers of Mathematics and Art, interested in the subject, information connections between Art and Mathematics that are present in the teaching notions of non-euclidian geometries.

TumbleTime is a children's customizable play area that is both fun and educational. Planned to be produced in foam and vinyl fabric, TumbleTime resembles a gymnastic mat. The design is involved geometry that allows children to discover geometric relationships while playing with their furniture. TumbleTime has thousands of configurations demonstrated by 3D computer renderings and animations produced using Autodesk 3ds Max.

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This study aims to present the "State of the Art" of Brazilian researches in the period 1991-2014 on the teaching and learning of Analytic Geometry in Brazil. We have adopted the assumptions of Content Analysis to carry out the methodological procedures and we have used the theoretical ideas of Dreyfus (1991) on which the processes of Advanced Mathematical Thinking (AMT) have emerged implicitly from pedagogical approaches addressed in academic productions for data analysis. Data were collected through bibliographic survey in theses and dissertations in the thesis database of CAPES and through sites from Strictu Sensu Graduate Programs in Mathematics Education in Brazil. We have identified forty-one academic productions about teaching and learning Analytical Geometry. We created two axes of analysis: the academic productions that used ICTs as a research subject and the academic productions that have not used ICTs as a primary focus. In researches using ICT as a research focus, the authors have used dynamic geometry software, spreadsheets and Moodle platform as a tool for teaching Analytic Geometry. In academic productions that did not target ICTs as focus, we have used tools such as compass and ruler for the construction of geometric entities, manipulative materials and techniques of isometric perspective, construction of geometric figures in 3D for the resolution of problem situations and the diversification among the semiotic representation registers. We detected that AMTs processes such as visualization, change of representation and abstraction were implicit in the activities proposed by these academic productions. We have concluded that the themes of analytic geometry covered in the research have not changed over the period considered. What have changed were the teaching and learning strategies that are centered on the student now, thus allowing him to create a more active role in the learning process without relying strictly on the teacher
A presente pesquisa tem como objetivo apresentar o “Estado da Arte” das pesquisas brasileiras no período de 1991 a 2014 sobre o ensino e a aprendizagem da Geometria Analítica no Brasil. Utilizamos os pressupostos da Análise de Conteúdo para realizar os procedimentos metodológicos e para a análise dos dados fizemos uso das ideias teóricas de Dreyfus (1991) sobre quais processos do Pensamento Matemático Avançado (PMA) emergiram implicitamente das estratégias pedagógicas abordadas nas produções acadêmicas. A coleta de dados foi feita por meio do levantamento bibliográfico das teses e dissertações no banco de teses da CAPES e dos sites de programas de Pós-Graduação Strictu Sensu em Ensino de Matemática no Brasil. Identificamos quarenta e uma produções acadêmicas sobre o ensino e a aprendizagem da Geometria Analítica. Criamos dois eixos de análise: as produções acadêmicas que utilizaram as TICs como objeto de pesquisa e as produções acadêmicas que não utilizaram as TICs como foco principal. Nas pesquisas que utilizaram as TICs como foco de pesquisa, os autores utilizaram softwares de geometria dinâmica, planilhas eletrônicas e a plataforma Moodle como ferramenta para o ensino da Geometria Analítica. Nas produções acadêmicas que não visaram as TICs como foco, foram utilizados instrumentos como compasso e régua para a construção dos entes geométricos, materiais manipulativos e técnicas da perspectiva isométrica, construção das figuras geométricas em 3D para a resolução de situações-problema e a diversificação entre os registros de representação semiótica. Detectamos que os processos do PMA tais como visualização, mudança de representação e abstração estavam implícitas nas atividades propostas por estas produções acadêmicas. Concluímos que os temas da Geometria Analítica abordados nas pesquisas não mudaram ao longo do período estudado. O que mudou foram as estratégias de ensino e aprendizagem, agora centradas no estudante, possibilitando que o mesmo criasse uma postura mais ativa no processo de aprendizagem sem depender estritamente do professor

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This study aims to present an overview work on the Mathematics Education thesis and dissertations at Pontifícia Universidade Católica de São Paulo from 1994 to 2007. These dissertations and theses were related to computational environment as tools in the Geometry context therefore, they were able to help this research to identify the most used tendencies on the subject matter so that the current studies can have a funded base of information where they can keep on going the new researches. The study analysis source is composed by thirty-two (32) selected pieces of work searched by titles, abstracts and key-words. The State of Art was the methodology process base. An outline of each work was elaborated based on the 10 activities by Romberg (1992). The study results show that six (06) Math objects were preferred, nine (09) where the computational environment were used and twenty-seven (27) where the author have chosen Cabri. Among the categorized Math objects, we could notice that fifteen (15) of them were related to the Geometric Transformations
Essa pesquisa tem como objetivo apresentar um estudo do panorama das teses e dissertações em Educação Matemática da Pontifícia Universidade Católica de São Paulo no período de 1994 a 2007. Estas dissertações e teses fizeram uso de ambientes computacionais como ferramenta no contexto da Geometria e possibilitaram que esta pesquisa identificasse as tendências e o que tem sido privilegiado sobre o tema de modo a permitir que estudos posteriores tenham uma base consolidada de informações da qual possam prosseguir suas pesquisas. A fonte de análise desse estudo constitui-se de trinta e dois (32) trabalhos selecionados por meio de títulos, resumos, linha de pesquisa e palavraschave. A metodologia adotada para essa pesquisa foi o Estado da Arte. Baseando-se nas dez (10) atividades propostas por Romberg (1992), foi elaborado um fichamento de cada pesquisa. Os resultados da pesquisa mostraram seis (06) objetos matemáticos foram privilegiados e nove (09) ambientes computacionais utilizados, sendo o Cabri optado por vinte e sete (27) autores. Dentre os objetos matemáticos categorizados, observou-se quinze (15) trabalhos referentes a Transformações Geométricas

In der Erforschung der Städtebaukunst stehen die bildenden Künste im Vordergrund. Die Frage nach den technischen Künsten wird kaum gestellt, obwohl der Entwurf des Architekten nicht nur von individueller Intuition, sondern im gleichen Maße auch vom technisch Machbaren geprägt ist. Um aber das der Planung frühneuzeitlicher Städte zugrunde liegende Konstruktionsschema und die Planungsintention erkennen zu können, ist es notwendig die Städtebau- mit der Technikgeschichte zu verbinden. Die in den Tratakten zur geometria practica und zur architectura militaris beschriebenen Konstruktions- und Vermessungsmethoden werden städtebaulichen Planungen der Frühen Neuzeit gegenübergestellt. An einzelnen Fallstudien, die vom Ende der mittelalterlichen Stadtplanung bis hin zu barocken Stadterweiterungen reichen, wird untersucht, wie das Planungswerkzeug die Entwurfssprache des Architekten beeinflusste und die Formensprache der Stadt- und Landschaftsplanung nachhaltig veränderte. Der Paradigmenwechsel im Städtebau vollzog sich in Mitteleuropa um die Wende vom 15. zum 16. Jh. mit dem Bau der Erzgebirgsstädte Annaberg und Marienberg. Die in Annaberg noch praktizierte rhythmische und räumliche Grundrissgestaltung wurde in Marienberg zugunsten eines egalisierten Stadtgrundrisses aufgegeben. Überlegungen zur Stadtstruktur und Hygiene führten zu diesem Wandel, die Aufteilung des Grundrisses selber aber wurde durch das verwendete Instrumentarium bestimmt. Im Barock stand den Planern Geometrie als allgemeine Kulturtechnik zur Verfügung, das Denken in geometrischen Formen und Proportionen bestimmte den Entwurfsprozess. Geometrie war nicht mehr nur Planungsmittel, sondern wurde - wie bei der Anlage der Berliner Torplätze zu Beginn des 18. Jh. zu sehen ist - Planungsziel. Die Rekonstruktion der Planungsmaße beweist, dass nicht nur die Namensgebung - Rondell, Oktogon und Quarré – auf die Quadratur des Kreises hinweist, sondern die Proportion der Plätze aus ihr heraus entwickelt wurde.
In the research of the art of urban development, the fine arts are mainly taken into account. The question of the technical arts is seldom raised, even though the architect’s design is a work of personal intuition as well as of the technically possible. In order to recognize the construction scheme and the planning intention in the urban planning of Early Modern Times it is necessary to merge the history of urban development with the history of technology. The construction and surveying methods described in the essays of the geometria practica and the architectura militaris are compared to the urban planning of the Early Modern Times. In case studies reaching from urban planning at the close of the Middle Ages to Baroque city expansion it is shown how the planning tools influenced the design language of the architect and the form language of urban development. The paradigm shift in Middle European urban planning took place at the end of the 15th century with the construction of Annaberg and Marienberg in the Ore Mountains. The rhythmic and spatial floor plan design still used in Annaberg was abandoned for a leveled out town plan in Marienberg. Deliberations of city structure and hygiene lead to this change. The design of the layout itself, however, was dictated by the implemented tool. In the Baroque period, geometry was available to the planers as general cultural knowledge and technology: the use of geometrical forms and proportions determined the design process. Geometry was not only planning method, but became planning intention – as can be seen by the Berlin Gate Plazas built in the beginning of the 18th Century. Not only do the names given - Rondell, Oktogon and Quarré – refer to the squaring of the circle, but the reconstruction of the design measurements proves the connection. Only the development of practical geometry enabled the variable form language of Baroque city construction.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Recent academic works comes approaching fear from the perspective and as the three-dimensional space can be represent two-dimensionality. This work propose contribute in the pole of the view and in the pole of the know about the three-dimensional space across of arising from techniques of the Italian Renaissance. The inquiry gives-itself about the discovery from the perspective in the Quattrocento, where Italian cities they provided, graces to an assembly of factors, the development of a technique from the painting that aimed at better represent the space. Inserted inside this historical-social context, went elaborated a sequence of activities inspired in works of the painters and architects of that epoch, in order to prepare the look at the comprehension of the techniques from the perspective and from the projective geometry and spatial. It intends, like this, cause to an analysis about the domain of the perspective techniques use in Renaissance works, that, with the progress of the look, will permit the acquisition of the space painted, reconstructed in a representation by means of models
Recentes trabalhos acadêmicos vêm abordando o tema da perspectiva e como o espaço tridimensional pode ser representado bidimensionalmente. Este trabalho propõe contribuir no pólo do visto e no pólo do sabido do espaço tridimensional através de técnicas oriundas do Renascimento Italiano. A investigação dá-se sobre a descoberta da perspectiva no Quattrocento, onde cidades italianas propiciaram, graças a um conjunto de fatores, o desenvolvimento de uma técnica da pintura que visava melhor representar o espaço. Inserido dentro deste contexto histórico-social, foi elaborada uma seqüência de atividades inspiradas em trabalhos dos pintores e arquitetos daquela época, a fim de preparar o olhar para a compreensão das técnicas da perspectiva e da geometria projetiva e espacial. Pretende-se, assim, levar a uma análise sobre o domínio do uso de técnicas de perspectiva em obras renascentistas, que, com o aprimoramento do olhar, permitirá a aquisição do espaço pictórico, reconstruído em uma representação por meio de maquetes

I do not know what inspires me to use geometric shapes in my artwork. It may have started when I took my first jewelry and metalworking class in college. For one assignment, I had to find an artist and make an artwork that was inspired by his artworks. I choose is Sol Lewitt because I simply liked his opened cubes. Since I started studying in Jewelry and Metalwork and Sculpture at the University of Iowa, I my artworks are still based on geometric shapes. I enjoy them because they are shapes that I see around me everyday; such as textbooks, chairs, doors etc. In my thesis, I will explain my artworks and the techniques that I used, why I made them, what inspired me. My first semester, I just made containers which were based on geometric shapes. After that my ideas and concepts evolved to a simple object based on geometric shapes with no function. A simple geometric object without a function is better for exploring beautiful geometric. Firstly, I find a subject that I can use my ideas and concepts on. It can be something that I see everyday such as myself, faucet, jewelry etc. Sometimes, I find an interesting thing in textbooks, magazines and catalogs. Other times, I find them when I buy materials for my artworks; just shopping at the hardware. There are many things around me that I could use to create my artworks. I just choose some of them to make art and finish assignments.

Orientador: Prof. Dr. Marcus Antônio Mendonça Marrocos
Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional - PROFMAT, Santo André, 2018.
Neste trabalho, apresentamos a geometria diferencial das curvas planas de um modo mais acessível para um leitor não especialista no assunto, mas de forma a despertar seu interesse. A Teoria Local das Curvas Planas é desenvolvida por meio de exemplos e, em particular, exibimos a família das curvas pedais. Ilustramos a Teoria Global por meio do Teorema dos Quatro Vértices e apresentamos, também, formas de explorar os conceitos de geometria diferencial na Educação Básica, com resultados geométricos interessantes e visualmente atraentes. Para isso, contamos com o auxílio do GeoGebra, um software de matemática dinâmica, e da string art, um estilo de arte caracterizado por um arranjo de cordas que formam padrões geométricos. Com isso, buscamos proporcionar ao leitor uma forma diferente de experimentar a geometria diferencial das curvas planas, bem como proporcionar aos alunos do Ensino Médio um aprendizado interessante de geometria analítica.
In this work, we present the differential geometry of the plane curves in an accessible way for not specialized readers in the subject, but in order to arouse their interest. The Local Theory of Plane Curves is developed by means of several examples and, in particular, we bring out the class of pedal curves. In order to ilustrate the Global Theory we present the Four-Vertex Theorem and we also present a way to introduce differential geometry concepts to secondary school students with interesting and visually attractive geometric results. To do this, we use the software GeoGebra, a interactive geometry and algebra application, and string art, a sort of art characterized by an arrangement of strings that form geometric patterns. We hope to provide to the readers a pratical experience of differential geometry of plane curves, as well as providing them the students of High School with an interesting learning of analytical geometry.

My project, Contemporary Non-objective Art, is a practice-led investigation into the dominant narratives surrounding non-objective or abstract art, and how they can be revisited and reimagined. This project aims to challenge typical and chronological ideologies that have defined non-objective art and to demonstrate a new outlook. Particularly by demonstrating how theatricality in art can be embraced, filling the work with both irony and symbolic weight. Throughout this project I use the term “non-objective art” as a way of describing and focusing on the aesthetic and technical elements of my work and practice. I want to stress that this is a visual language of modernist painting and sculpture as a form of practice— even when it is carrying the historical values of its time. I argue that the modernist narrative has changed, and that this is an important element of contemporary reductive practices—in particular, geometric abstraction. The social and cultural narrative of modernity is now self-consciously inflected with the self-critique that led to postmodernity. This has created new contexts and interests in art practice that are no longer aligned with the reductive idealism of early 20th century non-objective or “abstract” art.1 At a time when artists are struggling to maintain sense of non-objective art, I hope to use this paper and my artwork to add strangeness and irony as a way to find new interest and vitality in non-objective art. My research illustrates a resistance to the principals and ideas of the early 20th century through the 21st century’s visual language of unconventionality.

The world is slowly moving into increased human-robot interaction where both humans and robots can co-exist in the same domain. For the robot to be able to operate effectively in a man’s designed environment, it becomes necessary to model the robot with human capabilities as humans are seen as more capable. Replicating human becomes a huge challenge due to numerous degrees-of-freedom (DOFs) that human possess resulting into too many variables and nonlinear equations. Other challenges do occur like singularities.   In this thesis, the singularity challenge of a redundant humanoid arm is explored while maintaining a simple 7 DOF serial chain structure. As opposed to the 30 DOF human arm, a simpler 7 DOF humanoid arm is adopted and studied to eliminate the singularity challenges. The singularity problem mainly comes from the elbow and the spherical joints at the shoulder and wrist. A step-by-step review of available inverse kinematics techniques is made with more focus on the iterative Jacobian-based methods. A step-by-step approach is adopted so as to identify the source of singularities while using the iterative Jacobian-based techniques that are able to handle the nonlinearities of the equations.   The Singular Value Filtering (SVF) technique coupled with Selectively Damped Least Squares (SDLS) is employed. Without any restrictions to the stretch of the arm or end-effector pose, the method demonstrates, in conjunction with Euler angle singularity avoidance method, the elimination of singularity problems. This is achieved with no adjustment to kinematic model of the manipulator.

Acompanha: Desenho com as formas geométricas + Cadê meu instrumento?
Essa pesquisa reporta-se à duas áreas de conhecimento, Artes Visuais e Matemática. A questão que apresentamos é quanto à possibilidade de relacionarmos essas duas áreas a partir das formas. Teve como objetivo desenvolver uma proposta interdisciplinar a partir das áreas de Artes Visuais e Matemática que utilize o desenho e as formas geométricas para criação de um livro ilustrado. A pesquisa caracterizada como qualitativa, de natureza interpretativa, pelo problema apresentado e pela abordagem da pesquisa, em especial pela relação de sujeito com o mundo real, que envolve uma aproximação entre o objeto e a subjetividade, de forma a coletar, descrever e analisar de maneira contextualizada. Os produtos da pesquisa foram dois: o livro ilustrado Cadê meu instrumento? E o Guia de Aprendizagem O Desenho com as formas Geométricas. O primeiro caracterizado como produção do autor, como ilustrador, com aplicação da proposta de construção da imagem utilizando as formas geométricas na ilustração e, o segundo, apresenta a relação entre a Matemática e as Artes Visuais a partir das formas geométricas, com a proposta de experimentação do desenho. A aplicação prática da pesquisa foi desenvolvida a partir da construção de um curso de extensão Aproximações entre Artes Visuais e Matemática a partir das Formas Geométricas e do Desenho. No curso, foram apresentados os conteúdos teóricos e as atividades com desenho a partir das formas geométricas contidas no Guia de Aprendizagem. A aplicação envolveu sete estudantes do último ano do curso de Artes Visuais, que tinham experiências como professores e produtores de arte. Entre os resultados da pesquisa, destacamos a aproximação entre as Artes Visuais e a Matemática com a elaboração dos dois produtos, com potencialidade ao ensino, esses validados pelas aplicações teóricas e práticas. Espera-se com essa proposta auxiliar, além da aproximação entre as duas áreas de conhecimento, Matemática e Artes Visuais, potencializar o desenvolvimento prático, intelectual, criativo e sensível dos alunos.
This research relates to two areas of knowledge, Visual Arts and Mathematics. The question that presents the question as to the possibility of relating the two areas from the forms. It aimed to develop an interdisciplinary proposal from the areas of Visual Arts and Mathematics that use drawing and geometric forms for the creation of an illustrated book. A research characterized as qualitative, of an interpretive nature, by problem and research approach, especially by the relation of subject to the real world, which involves an approximation between the object and subjectivity, in order to collect, describe and analyze in a Contextualized way. The research products were two: the illustrated book Where's my instrument? And the Learning Guide The Drawing with Geometric Forms. The first character as production of the author, as illustrator, with application of the proposal of construction of the image using as geometric forms in the illustration and, second, present a relation between Mathematics and Visual Arts from the geometric forms, with a proposal of Experimentation of the drawing. The practical application of the research was developed from the construction of an extension course between ApproachesVisual Arts and Mathematics from Geometric Forms and Drawing. There is no course, the theoretical contents and the activities with drawing from the geometric forms contained in the Learning Guide. The application involved seven seniors from the Visual Arts course, who are the teachers and art producers. Among the results of the research, we highlight the approximation between Visual Arts and Mathematics with an elaboration of the two products, with potential for teaching, these are valid for theoretical and practical applications. It is hoped that this auxiliary proposal, besides the approximation between two areas of knowledge, Mathematics and Visual Arts, will enhance the students' practical, intellectual, creative and sensitive development.

This thesis is concerned with the design of Škoda 130 RS rear Semi-Trailing Arm Suspension and fixtures on the body. The main requirement of this thesis is to reduce weight of suspension. It is carried out the kinematic analysis and optimization of the kinematic. The construction part of this thesis provides a design of suspension based on the analysis of forces acting on the car wheel. Last part of the thesis includes of new rear suspension stress analysis.