The master's thesis Teaching geometry at primary school using origami is focused to increasing symmetry knowlidges, especially to reflection symmetry and central symmetry. The thesis contains analysing maths textbooks. It deals with creating worksheets and metodology of them and evaluation of tested worksheets.

Made available in DSpace on 2014-06-11T19:24:52Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-11-16Bitstream added on 2014-06-13T20:13:17Z : No. of bitstreams: 1 neves_prv_me_rcla.pdf: 1296931 bytes, checksum: b45ea16895bf9e20db063325be68a349 (MD5) / Este trabalho teve o objetivo de produzir um conjunto de atividades para analisar como o uso do caleidoscópio associado ao estudo dos ornamentos planos pode contribuir no ensino de grupos de simetria e transformações geométricas em um curso de graduação em Matemática. Esta pesquisa tem caráter qualitativo e foi desenvolvida segundo a proposta metodológica de Romberg. Elaborou-se uma proposta de ensino baseada na metodologia de Resolução de Problemas que foi aplicada a um grupo de professores (alguns em fase de formação) de matemática. As atividades tiveram a finalidade de fazer com que os alunos usassem o caleidoscópio para reproduzir ornamentos planos e, a partir de então, discutissem, com base em argumentos geométricos e algébricos, quais as possibilidades (e impossibilidades) que esse instrumento oferece para obtenção desses ornamentos e suas respectivas justificativas. A coleta de dados ocorreu, essencialmente, por observação participante em sala de aula por meio do uso de questionários, anotações e registros fotográficos. Após a coleta de dados, foi feita uma análise das possibilidades e limitações do material desenvolvido para o ensino de grupos de simetria e transformações geométricas, bem como o uso do caleidoscópio enquanto recurso didático / The purpose of this work was to develop a set of activities to analyze how the use of kaleidoscope associated to the study of ornaments can contribute to the teaching of symmetry groups and geometric transformations on a undergraduate course in Mathematics. This is a qualitative research and it was developed according to the methodological proposal of Romberg. A teaching proposal was drafted and was applied to a group of mathematics teachers. Activities were designed following the methodology of problem-solving and intended to make students to use the kaleidoscope to reproduce some ornaments and thereafter, discuss, based on geometric and algebraic arguments, the possibilities and impossibilities that this tool provides to obtain ornaments and their respective justifications. Data collection occurred primarily by participant observation in the classroom through the use of questionnaires, notes and photographic records. After the end of the course a viability analysis of the activities was done (possibilities and limitations) for teaching symmetry groups and geometric transformations as well as the use of Kaleidoscope as a didactic tool

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The relationship between symmetry and apparent reversals of enantiomorphic (mirror-reflected) objects was investigated. Subjects were presented with a series of standard and enantiomorphic books with various structural symmetries. The object directions (top-front-right) assigned to standard books were compared with the directions assigned to their enantiomorphs and the axes of apparent reversal determined. The primary finding was that apparent reversals were not limited to the left-right dimension. Reversals of top-bottom and front-back were also obtained. In most cases, apparent reversals occurred along the axis of structural (geometric) symmetry. However, symmetry defined in structural terms did not always predict apparent reversals. In certain cases, subjects perceived reversals most often along the left-right axis, even though (depending on the book) reversals of top-bottom or front-back were equally possible. The concept of perceived symmetry, which includes but is not limited to structural symmetry, is developed. Also, the influence of perceived symmetry on frames of reference is discussed.

Symmetry – measured as the size asymmetry of a group of symmetrical body traits such as ear height or elbow circumference – has often been used as an index of the capacity to develop normally despite stress and correlates with a wide range of outcomes including intelligence, health and aspects of behaviour. However, theoretical debate continues over the underlying causes of these associations and outstanding methodological issues – such as the reliance on small sample sizes of college age students – makes the robustness of the findings uncertain. The present work advances the existing empirical literature in six separate domains. It also improves upon past methodology by using novel methods of digital measurement of asymmetry as well as for the first time digitally measuring endogenous asymmetry as indexed by the bones and linking bone asymmetry to intelligence. The research was conducted on four samples. Numbers given are for participants who provided asymmetry measures. Firstly, a sample of elderly participants from the Lothian Birth Cohort 1921 (LBC1921, n = 216) tested around ages 11, 79, 83, and 87. Secondly, the Science Festival Sample (SFS), a group of children recruited at a public science event aged between 4 and 15 (n = 856). Thirdly, a group of Orkney residents aged 18 to 86 (the ORCADES, n = 1200). Fourthly the Berlin Sample (BS), a group of Berlin residents (n = 207) between 20 and 30 years old. In the LBC 1921, men with poorer socioeconomic status in childhood had higher facial asymmetry in old age (β = -.25, p = .03). While investigating issues related to asymmetry in the same sample it was found that relatively more severe digit curvature – a minor physical anomaly – was associated with relatively greater cognitive decline (β = -.19, p = .02). Within the SFS asymmetry decreased across human childhood (β = -.16, p = .01), and more asymmetrical children exhibited slower choice reaction times (β = .0.17, p = .002). In the ORCADES sample, the more asymmetrical participants (as indexed by bone asymmetry) were less intelligent (β = -.24, p = .01). In the Berlin Sample and the LBC 1921 no consistent associations were found between personality traits and asymmetry. Collectively, these findings suggest symmetry functions as a measure of overall well-being as the trend is for higher asymmetry to be associated with a relatively poorer score on a variety of outcome measures. The findings considerably expand the number of existing studies in these empirical areas and in several cases – particularly asymmetry’s association with socioeconomic status in the elderly and reaction times among children – represent the first work on those areas. The present work confirms the finding that asymmetry is linked to adverse outcomes. However, the underlying mechanisms by which symmetry is linked to such outcomes remain underexplored and require clarification.

Pietro Perugino's oeuvre is characterized by the reappearance of figures and motifs replicated through the reuse of cartoons. Perugino's deliberate self-plagiarism, despite being rooted in quattrocento compositional methods, exhibits an exploitation of the reproductive nature of the cartoon. While this practice allowed him to develop an efficient design process, the results of this imitation endowed Perugino's work with a formulaic quality, as was first noted by Giorgio Vasari in his Lives of the Most Eminent Painters, Sculptors and Architects (1568). Significantly, in the sixteenth century, theorists revised the concept of imitation to incorporate not only the notion of replication, but emulation as well. An examination of Perugino's reproductive practices alongside this revised view of imitation elucidates the nature of Vasari's criticism, ultimately revealing why the critic placed him among artists of the quattrocento, rather than that of the cinquecento.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in fulfilment of the requirements for the degree of Master of Science. 21 March 2016. / The problem of turbulent hydraulic fracturing is considered. Despite it being a known phenomenon, limited mathematical literature exists in this field. Prandtl’s mixing length model is utilised to describe the eddy viscosity and a mathematical model is developed for two distinct cases: turbulence where the kinematic viscosity is sufficiently small to be neglected and the case where it is not. These models allow for the examination of the fluid’s behaviour and its effect on the fracture’s evolution through time. The Lie point symmetries of both cases are obtained, and a wide range of analytical and numerical solutions are explored. Solutions of physical significance are calculated and discussed, and approximate solutions are constructed for ease of fracture estimation. The non-classical symmetries of these equations are also investigated. It was found that the incorporation of the kinematic viscosity within the modelling process was important and necessary. / MT2016

An MSc dissertation by Shivani Singh. University of Witwatersrand Faculty of Science, School of Mathematics. August 2016. / These symmetries have interesting applications to enumerative combinatorics and to Ramsey theory. The aim of this thesis will be to present some important results in these fields. In particular, we shall enumerate the r-ary symmetric bracelets of length n. / LG2017

In this work we will consider the steady and two-dimensional potential flow of an incompressible fluid past a body without friction. Contrary to common experience, we will show that it is possible to calculate the Lie point symmetries that will leave the boundary value problem invariant. We are able to do this by solving the determining equation for the Lie point symmetries subject to a side condition. The side condition is a consequence of the boundary condition that occurs in the boundary value problem. We will show that solutions of the boundary value problem that were obtained previously using the method of conformal transformations are also group invariant solutions of the boundary value problem. We will also show that every group invariant solution of the boundary value problem can be used to generate new group invariant solutions of the same boundary value problem.

A research report submitted to the Faculty of Science, University of the Witwatersrand, in ful lment of the requirements for the degree of Master of Science. School of Mathematics Johannesburg February 15, 2016 / We obtain a solution of the Black-Scholes equation with a non-smooth bound- ary condition using symmetry methods. The Black-Scholes equation along with its boundary condition are rst transformed into the one dimensional heat equation and an initial condition respectively. We then nd an appro- priate general symmetry generator of the heat equation using symmetries of the heat equation and the fundamental solution of the heat equation. The method we use to nd the symmetry generator is such that the boundary condition is left invariant and yet the symmetry can still be used to solve the heat equation. We then use the help of Mathematica to nd the solution to the heat equation. Then the solution is then transformed backwards to a solution of the Black-Scholes equation using the same change of variables that were used for the forward transformations. The solution is then nally checked if it satis es the boundary condition of the Black-Scholes equation.