Parametric equations are commonly used to describe surfaces. Looking at parametric equations does not provide tangible information about an object. Thus through the use of physical materials, an understanding of the limitations of the materials allows someone to gain a broader understanding of the surface. A M$\ddot{o}$bius strip and Figure 8 Klein bottle were created through knitting due to the precision and steady increase in curvature allowed through knitting. A more standard Klein bottle was created through crochet due to the ease in creating quick increases in curvature. Both methods demonstrate the change in curvature for both surfaces where the M$\ddot{o}$bius strip and Figure 8 Klein bottle have slower changes in curvature, but the classic Klein bottle has a quicker change in curvature.

The main results of this dissertation show evidence supporting the Successive Surface Scaffolding Conjecture. This is a new conjecture that, if true, guarantees the existence of all the wye-delta-order minimal obstructions of a surface S as subgraphs of the irreducible triangulations of the surface S with a crosscap added. A new data structure, i.e. an augmented rotation system, is presented and used to create an exponential-time algorithm for embedding graphs in any surface with a constant-time check of the change in genus when inserting an edge. A depiction is a new formal definition for representing an embedding graphically, and it is shown that more than one depiction can be given for nonplanar embeddings, and that sometimes two depictions for the same embedding can be drastically different from each other. An algorithm for finding the essential cycles of an embedding is given, and is used to confirm for the projective-plane obstructions, a theorem that shows any embedding of an obstruction must have every edge in an essential cycle. Obstructions of a general surface S that are minor-minimal and not double-wye-delta-minimal are shown to each have an embedding on the surface S with a crosscap added. Finally, open questions for further research are presented. / Graduate

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Neste trabalho apresentamos uma versão do Teorema de Poincaré-Bendixson para campos vetoriais contínuos na garrafa de Klein. Como conseqüência, mostramos que a garrafa de Klein não possui campo vetorial contínuo com trajetória injetiva recorrente / We present a version of the Poincaré-Bendixson Theorem on the Klein bottle for continuous vector fields. As a consequence, we obtain the fact that the Klein bottle does not admit continuous vector fields having a recurrent injective trajectory

Made available in DSpace on 2016-06-02T20:13:15Z (GMT). No. of bitstreams: 1 5889.pdf: 1158628 bytes, checksum: eb3e8512fbb67f17e03cf8a0d7697ebe (MD5) Previous issue date: 2014-04-10 / Universidade Federal de Sao Carlos / This research had as a goal a better definition of the notion of unconscious in the work of Claude Lévi-Strauss. In the early texts, such as The Symbolic Efficacy (1949), this concept is introduced as a set of structural laws, named Symbolic function. Nevertheless, throughout the path of the levi-straussian work, this terminology fades bit-by-bit, making room for concepts like the symbolic thought. In its turn, this concept imprints the determination of symbolism, characteristic that defines mankind, and thus, the unconscious defines humanity itself. Placed between nature and culture, the unconscious therefore is paradoxically plural and universal, allowing the apprehension of an integrated reality. The pair unconscious/ conscious characterizes the different levels of the phenomenon that Lévi-Strauss analyses in his anthropology, granting two types of investigation: inductive and deductive. The proposal of an unconscious facet of the social phenomenon made possible the prescription of two discontinued methods. The exam of the empirical and transcendental realities is orchestrated in a way of establishing, at the same time, an irreducibly statement between nature and culture and a refusal to old-fashioned dualisms. In this work, throughout the developments of the notion unconscious in the work of Lévi-Strauss, we hope to demonstrate that this concept is symbolic, and that such an analysis implies the investigation of the passage between nature and culture. Therefore, we hope to set the implications of this notion to the definition of mankind in the work of Lévi- Strauss. / Esse trabalho teve como objetivo melhor definir a noção de inconsciente na obra de Lévi-Strauss. Em textos iniciais, como a A eficácia simbólica (1949), esse conceito é apresentado como um conjunto de leis estruturais, nomeado função simbólica. Não obstante, ao longo do percurso da obra levistraussiana, pouco a pouco, o termo função simbólica se desvanece, dando lugar a conceitos como: pensamento simbólico. Este, por sua vez, marca a determinação do simbolismo na obra do antropólogo, noção, esta, que define o estatuto de humanidade. Desse modo, o inconsciente passa a caracterizar não só o pensamento simbólico, mas a própria definição da humanidade (LÉVI-STRAUSS, 2012). Situado entre natureza e cultura tal qual o homem e, portanto, paradoxalmente plural e universal, o inconsciente levistraussiano permite a apreensão de uma realidade integrada. O par consciente/inconsciente, ao qualificar os diferentes níveis dos fenômenos aos quais se dedica Lévi-Strauss, operacionaliza o uso tanto da indução quanto da dedução na investigação a que se propôs. A postulação de uma faceta inconsciente dos fenômenos sociais foi o que permitiu uma abordagem completa da realidade, prescrevendo métodos aparentemente descontínuos.. Desse modo, a abordagem do empírico e do transcendental é orquestrada de maneira a prescrever uma irredutibilidade entre as ordens natureza e cultura, ao mesmo tempo, uma recusa à antigos dualismos (sensível/inteligível, corpo/espírito, natureza/cultura). Neste texto, esperamos, através da marcha demonstrativa da função simbólica ao longo dos desenvolvimentos da obra do antropólogo, ter contribuído para uma compreensão do inconsciente, na antropologia estrutural, como simbólico. Ordem que subjaz a pluralidade cultural, assim como confere o universal almejado pela cientificidade esquadrinhada por Lévi-Strauss. A análise do conceito de função simbólica implica, inelutavelmente, o exame da passagem entre natureza e cultura, bem como o lugar que cabe a humanidade mediante estas duas esferas.

Neste trabalho apresentamos uma versão do Teorema de Poincaré-Bendixson para campos vetoriais contínuos na garrafa de Klein. Como conseqüência, mostramos que a garrafa de Klein não possui campo vetorial contínuo com trajetória injetiva recorrente / We present a version of the Poincaré-Bendixson Theorem on the Klein bottle for continuous vector fields. As a consequence, we obtain the fact that the Klein bottle does not admit continuous vector fields having a recurrent injective trajectory

En 1949, C. Loewner a demontré dans un travail non publié l'inégalité systolique optimale du tore T reliant l'aire au carré de la systole. Par la systole on désigne la longueur du plus court lacet non contractile de T. De plus, l' égalité est atteinte si et seulement si le tore est plat hexagonal. Ce résultat a donné naissance à la géométrie systolique. Dans cette thèse, nous étudions des inégalités de type systolique portant sur les longueurs minimales de différentes courbes et pas seulement la systole.Dans un premier temps, nous démontrons trois inégalités géométriques optimales conformes sur la bouteille de Klein reliant l'aire au produit des longueurs des plus courts lacets noncontractiles dans des classes d'homotopie libres différentes. Pour chaque classe conforme, nous décrivons la métrique extrémale réalisant le cas d'égalité.Nous établissons ensuite des inégalités géométriques optimales sur le ruban deMobius muni d'une métrique de Finsler. Ces inégalités géométriques relient la systole et la hauteur du ruban de Mobius à son volume de Holmes-Thompson. Nous en déduisons une inégalité systolique optimale sur la bouteille de Klein munie d'une métrique de Finsler avec des symétries. Nous décrivons également une famille de métriques extrémales dans les deux cas.Dans le troisième travail, nous démontrons une inégalité systolique critique sur la surface de genre deux. Plus précisément, il est connu que la surface de genre deux admet une métrique Riemannienne plate à singularités coniques qui est extrémale parmi les métriques à courbure nonpositive pour l' inégalité systolique. Nous montrons que cette métrique est en fait critique pour des variations lentes de métriques, cette fois-ci sans hypothèse de courbure, pour un autre problème systolique portant sur les longueurs des plus courts lacets non contractiles dans certaines classes d'homotopie libres données. Ces classes d'homotopie correspondent aux lacets systoliques et deux-systoliques de la surface extrémale / In 1949, C. Loewner proved in an unpublished work that the two-torus T satisfies an optimal systolic inequality relating the area of the torus to the square of its systole. By a systole here we mean the smallest length of a noncontractible loop in T. Furthermore, the equality is attained if and only if the torus is flat hexagonal. This result led to whatwas called later systolic geometry. In this thesis, we study several systolic-like inequalities. These inequalities involve the minimal length of various curves and not merely the systole.First we obtain three optimal conformal geometric inequalities on Riemannian Klein bottles relating the area to the product of the lengths of the shortest noncontractible loops in different free homotopy classes. We describe the extremal metrics in each conformal class.Then we prove optimal systolic inequalities on Finsler Mobius bands relating the systoleand the height of the Mobius band to its Holmes-Thompson volume. We also establish an optimalsystolic inequality for Finsler Klein bottles with symmetries. We describe extremal metric families in both cases.Finally, we prove a critical systolic inequality on genus two surface. More precisely, it is known that the genus two surface admits a piecewise flat metric with conical singularities which is extremal for the systolic inequality among all nonpositively curved Riemannian metrics. We show that this piecewise flat metric is also critical for slow metric variations, this time without curvature restrictions, for another type of systolic inequality involving the lengths of the shortest noncontractible loops in different free homotopy classes. The free homotopy classes considered correspond to those of the systolic loops and the second-systolic loops of the extremal surface