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## Abordagem histórico-epistemológica do ensino da geometria fazendo uso da geometria dinâmica; Historical-epistemological approach geometry teaching making use of dynamic geometry.

Waldomiro, Tatiana de Camargo
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
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A presente pesquisa, de cunho quantitativo, tem como propósito responder a seguinte questão: De que modo e em que alcance o trabalho pedagógico articulado com a história, geometria e meio computacional tem refletido sobre posturas e caminhos que levassem os alunos a se envolver com o conhecimento matemático? Desse modo, fizemos uma investigação e análise sobre os efeitos de uma articulação entre o ensino da história da matemática e o uso de ferramentas computacionais como solução para as dificuldades apresentadas no Ensino de Geometria, principalmente no Ensino Médio. Utilizamos a obra de Lakatos e a primeira proposição (do livro 1) de Euclides para realizar a verificação de sua demonstração através de um software de Geometria dinâmica. Os resultados serão utilizados para a construção de um novo software que envolva o ensino e aprendizagem de história da matemática e geometria. Outros objetivos podem ser assim colocados: Refletir sobre as condições e viabilidade da integração de recursos computacionais para o ensino da Matemática no âmbito Ensino Médio em especial a partir do produtos/softwares propostos para a educação matemática; Compreender o potencial de softwares de geometria dinâmica para a educação matemática escolar; Analisar as necessidades matemáticas de uma instrumentação eficaz...

## A importância do ensino de geometria nos anos iniciais do ensino fundamental : razões apresentadas em pesquisas brasileiras; The importance of teaching geometry in the early years of elementary school : reasons presented by Brazilian researches

Wagner Aguilera Manoel
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Dissertação de Mestrado Formato: application/pdf
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O ensino e a aprendizagem de Geometria é tema presente em diversas pesquisas em Educação Matemática. Muitas pesquisas apontam que nos Anos Iniciais do Ensino Fundamental (AIEF), nota-se ainda uma maior ênfase no ensino de outras áreas da Matemática, em comparação aos conteúdos relacionados aos conhecimentos geométricos, mas apontam também que é importante ensinar Geometria nos AIEF. Muitos autores consideram fundamental a presença da Geometria no ambiente escolar, seja pela importância dessa disciplina na cultura e na história da humanidade, seja pelas habilidades cognitivas que ela desenvolve, ou mesmo pelo fato de ela estar presente no cotidiano do aluno. Diante dessa problemática, a questão que emergiu e que norteou esta pesquisa foi: quais as razões para ensinar Geometria nos AIEF apresentadas pelos autores de pesquisas brasileiras no período de 2006 a 2011? O objetivo dessa investigação foi realizar uma compilação e um estudo analítico da importância de se ensinar Geometria nos AIEF e produzir novas interpretações e resultados. A metodologia escolhida foi a pesquisa bibliográfica do tipo meta-análise qualitativa e o material a ser analisado foram teses e dissertações com o tema Geometria nos Anos/Séries Iniciais do Ensino Fundamental. As razões encontradas na literatura foram classificadas em onze eixos de análises (currículo...

## Geometria hiperbólica : uma proposta para o desenvolvimento de atividades utilizando o software livre NonEuclid; Hyperbolic geometry : a proposal for the development of activities using the software NonEuclid

Armando Staib
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Dissertação de Mestrado Formato: application/pdf
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## Geometria do táxi : pelas ruas de uma cidade aprende-se uma geometria diferente; Taxicab geometry : learning a different geometry through the streets of a city

Vivianne Tasso Perugini de Oliveira
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Dissertação de Mestrado Formato: application/pdf
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Neste trabalho apresentamos o estudo sobre a Geometria do Táxi, uma Geometria não-Euclidiana de fácil compreensão e muito próxima do cotidiano das pessoas, uma vez que tem uma ampla gama de aplicações em situações relacionadas à geografia urbana. A Geometria do Táxi é uma geometria muito semelhante à Geometria Euclidiana, diferindo desta apenas pela definição de distância. Enquanto que, na Geometria Euclidiana, a distância entre dois pontos é o comprimento do segmento de reta que os une, podendo ser obtida com o auxílio do Teorema de Pitágoras, na Geometria do Táxi, a distância entre dois pontos é o comprimento do menor caminho percorrido por linhas horizontais e verticais de um ponto a outro. Esse pequeno detalhe sob o ponto de vista matemático, apresenta grandes diferenças, principalmente nas figuras geométricas que estão relacionadas à distância. Abordamos esse aspecto sob a forma de exemplos e apresentamos no final do trabalho uma sugestão de atividades pedagógicas para serem trabalhadas em sala de aula.; In this paper we present the study of the Taxicab Geometry, a non-Euclidean Geometry of easy understanding and very close to people's daily lives, as it has a wide range of applications in situations related to urban geography. The Taxicab Geometry is a geometry very similar to Euclidian Geometry...

## Geometry of Parallelizable Manifolds in the Context of Generalized Lagrange Spaces

Wanas, M. I.; Youssef, N. L.; Sid-Ahmed, A. M.
Tipo: Artigo de Revista Científica
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In this paper, we deal with a generalization of the geometry of parallelizable manifolds, or the absolute parallelism (AP-) geometry, in the context of generalized Lagrange spaces. All geometric objects defined in this geometry are not only functions of the positional argument $x$, but also depend on the directional argument $y$. In other words, instead of dealing with geometric objects defined on the manifold $M$, as in the case of classical AP-geometry, we are dealing with geometric objects in the pullback bundle $\pi^{-1}(TM)$ (the pullback of the tangent bundle $TM$ by $\pi: T M\longrightarrow M$). Many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this more general context. We refer to such a geometry as generalized AP-geometry (GAP-geometry). In analogy to AP-geometry, we define a $d$-connection in $\pi^{-1}(TM)$ having remarkable properties, which we call the canonical $d$-connection, in terms of the unique torsion-free Riemannian $d$-connection. In addition to these two $d$-connections, two more $d$-connections are defined, the dual and the symmetric $d$-connections. Our space, therefore, admits twelve curvature tensors (corresponding to the four defined $d$-connections), three of which vanish identically. Simple formulae for the nine non-vanishing curvatures tensors are obtained...

## K3 en route From Geometry to Conformal Field Theory

Wendland, Katrin
Tipo: Artigo de Revista Científica
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To pave the way for the journey from geometry to conformal field theory (CFT), these notes present the background for some basic CFT constructions from Calabi-Yau geometry. Topics include the complex and Kaehler geometry of Calabi-Yau manifolds and their classification in low dimensions. I furthermore discuss CFT constructions for the simplest known examples that are based in Calabi-Yau geometry, namely for the toroidal superconformal field theories and their Z2-orbifolds. En route from geometry to CFT, I offer a discussion of K3 surfaces as the simplest class of Calabi-Yau manifolds where non-linear sigma model constructions bear mysteries to the very day. The elliptic genus in CFT and in geometry is recalled as an instructional piece of evidence in favor of a deep connection between geometry and conformal field theory.; Comment: 39 pages, no figures; lecture notes for the author's contribution to the 2013 Summer School "Geometric, Algebraic and Topological Methods for Quantum Field Theory" in Villa de Leyva, Colombia

## Discrete differential geometry. Consistency as integrability

Bobenko, Alexander I.; Suris, Yuri B.
Tipo: Artigo de Revista Científica
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A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra), the discrete differential geometry aims at the development of discrete equivalents of notions and methods of smooth surface theory. Current interest in this field derives not only from its importance in pure mathematics but also from its relevance for other fields like computer graphics. Recent progress in discrete differential geometry has lead, somewhat unexpectedly, to a better understanding of some fundamental structures lying in the basis of the classical differential geometry and of the theory of integrable systems. The goal of this book is to give a systematic presentation of current achievements in this field.; Comment: A preliminary version of a book. 157 pp; See http://www.ams.org/bookstore-getitem/item=GSM-%5C98 for the final version appeared as: A.I. Bobenko, Yu.B. Suris. Discrete Differential Geometry. Integrable Structure. Graduate Studies in Mathematics, Vol. 98. AMS, 2008

## Asymmetric Nondegenerate Geometry

Rylov, Yuri
Tipo: Artigo de Revista Científica
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Nondegenerate geometry (T-geometry) with nonsymmetric world function is considered. In application to the space-time geometry the asymmetry of world function means that the past and the future are not equivalent geometrically. T-geometry is described in terms of finite point subspaces and world function between pairs of points of these subsets, i.e. in the language which is immanent to geometry and free of external means of description (coordinates, curves). Such a description appears to be simple and effective even in the case of complicated T-geometry. Antisymmetric component of the world function generates appearance of additional metric fields. This leads to appearance of three sorts of Christoffel symbols and three sorts of geodesics. Three sorts of the first order tubes (future, past and neutral) appear. If the fields connected with the antisymmetric component are strong enough, the timelike first order tube has a finite length in the timelike direction. It was shown earler that the symmetric T-geometry explains non-relativistic quantum effects without a reference to principles of quantum mechanics. One should expect that nonsymmetric space-time T-geometry is also characteristic for microcosm, and it will be useful in the elementary particle theory...

## Riemannian geometry over different normed division algebra

Leung, Naichung Conan
Tipo: Artigo de Revista Científica
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We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, Hyperkahler and G_2-manifolds respectively. For vector bundles over such manifolds, we introduce (special) A-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Similarly we introduce (special) A/2-Lagrangian submanifolds as maximally real submanifolds. They include (special) Lagrangian, complex Lagrangian, Cayley and (co-)associative submanifolds. We also discuss geometric dualities from this viewpoint: Fourier transformations on A-geometry for flat tori and a conjectural SYZ mirror transformation from (special) A-geometry to (special) A/2-Lagrangian geometry on mirror special A-manifolds.; Comment: 45 pages. To appear in Journal of Differential Geometry

## D-manifolds, d-orbifolds and derived differential geometry: a detailed summary

Joyce, Dominic
Tipo: Artigo de Revista Científica
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This is a long summary of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html . A shorter survey paper on the book, focussing on d-manifolds without boundary, is arXiv:1206.4207, and readers just wanting a general overview are advised to start there. We introduce a 2-category dMan of "d-manifolds", new geometric objects which are 'derived' smooth manifolds, in the sense of the 'derived algebraic geometry' of Toen and Lurie. They are a 2-category truncation of Spivak's 'derived manifolds' (see arXiv:0810.5174, arXiv:1212.1153). The category of manifolds Man embeds in dMan as a full (2-)subcategory. We also define 2-categories dMan^b,dMan^c of "d-manifolds with boundary" and "d-manifolds with corners", and orbifold versions of these dOrb,dOrb^b,dOrb^c, "d-orbifolds". Much of differential geometry extends very nicely to d-manifolds and d-orbifolds -- immersions, submersions, submanifolds, transverse fibre products, orientations, orbifold strata, bordism, etc. Compact oriented d-manifolds and d-orbifolds have virtual classes. There are truncation functors to d-manifolds and d-orbifolds from essentially every geometric structure on moduli spaces used in enumerative invariant problems in differential geometry or complex algebraic geometry...

## Extended Absolute Parallelism Geometry

Youssef, Nabil. L.; Sid-Ahmed, A. M.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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In this paper, we study Absolute Parallelism (AP-) geometry on the tangent bundle $TM$ of a manifold $M$. Accordingly, all geometric objects defined in this geometry are not only functions of the positional argument $x$, but also depend on the directional argument $y$. Moreover, many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this different framework. We refer to such a geometry as an Extended Absolute Parallelism (EAP-) geometry. The building blocks of the EAP-geometry are a nonlinear connection assumed given a priori and $2n$ linearly independent vector fields (of special form) defined globally on $TM$ defining the parallelization. Four different $d$-connections are used to explore the properties of this geometry. Simple and compact formulae for the curvature tensors and the W-tensors of the four defined $d$-connections are obtained, expressed in terms of the torsion and the contortion tensors of the EAP-space. Further conditions are imposed on the canonical $d$-connection assuming that it is of Cartan type (resp. Berwald type). Important consequences of these assumptions are investigated. Finally, a special form of the canonical $d$-connection is studied under which the classical AP-geometry is recovered naturally from the EAP-geometry. Physical aspects of some of the geometric objects investigated are pointed out and possible physical implications of the EAP-space are discussed...

## An introduction to C-infinity schemes and C-infinity algebraic geometry

Joyce, Dominic
Tipo: Artigo de Revista Científica
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This is a survey of the author's paper arXiv:1001.0023 on "Algebraic Geometry over C-infinity rings". If X is a smooth manifold then the R-algebra C^\infty(X) of smooth functions c : X --> R is a "C-infinity ring". That is, for each smooth function f : R^n --> R there is an n-fold operation \Phi_f : C^\infty(X)^n --> C^\infty(X) acting by \Phi_f: (c_1,...,c_n) |--> f(c_1,...,c_n), and these operations \Phi_f satisfy many natural identities. Thus, C^\infty(X) actually has a far richer structure than the obvious R-algebra structure. We explain a version of algebraic geometry in which rings or algebras are replaced by C-infinity rings. As schemes are the basic objects in algebraic geometry, the new basic objects are "C-infinity schemes", a category of geometric objects generalizing manifolds, and whose morphisms generalize smooth maps. We also discuss "C-infinity stacks", including Deligne-Mumford C-infinity stacks, a 2-category of geometric objects generalizing orbifolds. We study quasicoherent and coherent sheaves on C-infinity schemes and C-infinity stacks, and orbifold strata of Deligne-Mumford C-infinity stacks. This enables us to use the tools of algebraic geometry in differential geometry, and to describe singular spaces such as moduli spaces occurring in differential geometric problems. Many of these ideas are not new: C-infinity rings and C-infinity schemes have long been part of synthetic differential geometry. But we develop them in new directions. In a new book...

## Axiomatization of geometry employing group actions

Dydak, Jerzy
Tipo: Artigo de Revista Científica
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The aim of this paper is to develop a new axiomatization of planar geometry by reinterpreting the original axioms of Euclid. The basic concept is still that of a line segment but its equivalent notion of betweenness is viewed as a topological, not a metric concept. That leads quickly to the notion of connectedness without any need to dwell on the definition of topology. In our approach line segments must be connected. Lines and planes are unified via the concept of separation: lines are separated into two components by each point, planes contain lines that separate them into two components as well. We add a subgroup of bijections preserving line segments and establishing unique isomorphism of basic geometrical sets, and the axiomatic structure is complete. Of fundamental importance is the Fixed Point Theorem that allows for creation of the concepts of length and congruency of line segments. The resulting structure is much more in sync with modern science than other axiomatic approaches to planar geometry. For instance, it leads naturally to the Erlangen Program in geometry. Our Conditions of Homogeneity and Rigidity have two interpretations. In physics, they correspond to the basic tenet that independent observers should arrive at the same measurement and are related to boosts in special relativity. In geometry...

## Algebraic Geometry over $C^\infty$-rings

Joyce, Dominic
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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If $X$ is a smooth manifold then the $\mathbb R$-algebra $C^\infty(X)$ of smooth functions $c:X\to\mathbb R$ is a $C^\infty$-$ring$. That is, for each smooth function $f:{\mathbb R}^n\to\mathbb R$ there is an $n$-fold operation $\Phi_f:C^\infty(X)^n\to C^\infty(X)$ acting by $\Phi_f:(c_1,\ldots,c_n)\mapsto f(c_1,...,c_n)$, and these operations $\Phi_f$ satisfy many natural identities. Thus, $C^\infty(X)$ actually has a far richer structure than the obvious $\mathbb R$-algebra structure. We develop a version of algebraic geometry in which rings or algebras are replaced by $C^\infty$-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are $C^\infty$-$schemes$, a category of geometric objects which generalize smooth manifolds, and whose morphisms generalize smooth maps. We also study quasicoherent and coherent sheaves on $C^\infty$-schemes, and $C^\infty$-$stacks$, in particular $Deligne$-$Mumford$ $C^\infty$-$stacks$, a 2-category of geometric objects generalizing orbifolds. This enables us to use the tools of algebraic geometry in differential geometry, and to describe singular spaces such as moduli spaces occurring in differential geometric problems. This paper forms the foundations of the author's new theory of "derived differential geometry"...

## Geometry without topology as a new conception of geometry

Rylov, Yuri A.
Tipo: Artigo de Revista Científica
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A geometric conception is a method of a geometry construction. The Riemannian geometric conception and a new T-geometric one are considered. T-geometry is built only on the basis of information included in the metric (distance between two points). Such geometric concepts as dimension, manifold, metric tensor, curve are fundamental in the Riemannian conception of geometry, and they are derivative in the T-geometric one. T-geometry is the simplest geometric conception (essentially only finite point sets are investigated) and simultaneously it is the most general one. It is insensitive to the space continuity and has a new property -- nondegeneracy. Fitting the T-geometry metric with the metric tensor of Riemannian geometry, one can compare geometries, constructed on the basis of different conceptions. The comparison shows that along with similarity (the same system of geodesics, the same metric) there is a difference. There is an absolute parallelism in T-geometry, but it is absent in the Riemannian geometry. In T-geometry any space region is isometrically embeddable in the space, whereas in Riemannian geometry only convex region is isometrically embeddable. T-geometric conception appears to be more consistent logically, than the Riemannian one.; Comment: 29 pages

## Geometry without Topology

Rylov, Yuri A.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (sigma-immanent description). Constructing the geometry, one does not use topology and topological properties. For instance, the straight, passing through points A and B, is defined as a set of such points R that the area S(A,B,R) of the triangle ABR vanishes. The triangle area is expressed via metric by means of the Hero's formula, and the straight appears to be defined only via metric, i.e. without a reference to (topological) concept of curve. (Usually the straight is defined as the shortest curve, connecting two points A and B). Such a construction of geometry is free from constraints (continuity, dimensionality of space), generated by a use of topology, but not by geometry in itself. At such a description all information on the geometry properties (such as uniformity, isotropy, continuity and degeneracy) is contained in metric. Modifying the metric, one changes the geometry automatically. The Riemannian geometry is constructed by two different ways: (1) by conventional way on the basis of metric tensor, (2) as a result of modification of the metric in the sigma-immanent description of the proper Euclidean geometry. The two obtained geometries are compared. The convexity problem in geometry and the problem of collinerity of vectors at distant points are considered. The nonmetric definition of curve is shown to be a concept of only proper Euclidean geometry. It is inadequate to any non-Euclidean geometry.; Comment: 25 pages...

## Generalized Kahler geometry

Gualtieri, Marco
Tipo: Artigo de Revista Científica
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Generalized Kahler geometry is the natural analogue of Kahler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We explore the fundamental aspects of this geometry, including its equivalence with the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2,2) supersymmetry, as well as the relation to holomorphic Dirac geometry and the resulting derived deformation theory. We also explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kahler geometry.; Comment: 45 pages

## Non-Archimedean analytic geometry as relative algebraic geometry

Ben-Bassat, Oren; Kremnizer, Kobi
Tipo: Artigo de Revista Científica
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We show that Berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of To\"{e}n--Vaqui\'{e}--Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we can define a topology on certain subcategories of the of the category of affine schemes with respect to this category. By examining this topology for the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry and in this way we also get definitions of (higher) non-Archimedean analytic stacks. We demonstrate that the category of Berkovich analytic spaces embeds fully faithfully into the category of varieties in our version of relative algebraic geometry. We also include a treatment of quasi-coherent sheaf theory in analytic geometry. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.; Comment: added material on quasi-coherent modules, connection to derived analytic geometry, corrected mistakes

## General Geometry and Geometry of Electromagnetism

Shahverdiyev, Shervgi S.