This report examines the mathematics of paper folding. One can solve cubic polynomials by folding a common tangent to two distinct parabolas. This then leads to constructions that cannot be done with a straightedge and compass such as angle trisection and doubling a cube. / text

The design of a pop-up book or card has hitherto been labour intensive with tasks of trials and errors. The constructions of collapsible pop-up structures can be demanding and inefficient without adequate knowledge of their geometric properties. This paper examines the properties of creases in 90° pop-up structures. A 90° pop-up structure is one that erects fully when two adjacent base pages, on which it sits, are opened to a right angle. In particular, we define a boundary region for creating 90° pop-ups. Similarly, paper folds are able to achieve pop-up effects and can be integrated with 90° pop-up constructions. The development of these pop-up structures can be represented graphically. Through this study, a fundamental foundation for pop-up topology and geometry is built. This foundation would be vital for understanding the applications of pop-up making techniques. The mathematical relationships devised would be useful for developing computer-enhanced pop-up design. / Singapore-MIT Alliance (SMA)

A crease pattern is an embedded planar graph on a piece of paper. An m × n map is a rectangular piece of paper with a crease pattern that partitions the paper into an m × n regular grid of unit squares. If a map has a configuration such that all the faces of the map are stacked on a unit square and the paper does not self-intersect, then it is flat foldable, and the linear ordering of the faces is called a valid linear ordering. Otherwise, the map is unfoldable. In this thesis, we show that, given a linear ordering of the faces of an m × n map, we can decide in linear time whether it is a valid linear ordering or not. We also define a class of unfoldable 2 × n crease patterns for every n ≥ 5. / Graduate / 0984

The diploma thesis Geometric constructions by paper folding is focused on the use of paper folding in teaching mathematics. The main part is formed by worksheets, where the paper folding replaces drawing geometric tasks. For verification and practice of geometrical knowledge additional questions to each construction are added. To simplify and unify approach for developing worksheets constructions of basic axioms which enable to construct all the tasks are described.

This thesis considers all of Ernest McClain's published writings, from March, 1970, to September, 1976, from the standpoint of their present-day acoustical significance. Although much of the material comes from McClain's writings, some is drawn from other related musical, mathematical, and philosophical works. The four chapters begin with a biographical sketch of McClain, presenting his background which aided him in becoming a theoretical musicologist. The second chapter contains a chronological itemization of his writings and provides a synopsis of them in layman's terms. The following chapter offers an examination of some salient points of McClain's work. The final chapter briefly summarizes the findings and contains conclusions as to their germaneness to current music theory, thereby giving needed exposure to McClain's ideas.

Este trabalho tem como foco a dobradura em sala de aula, auxiliando o professor em sua prática docente. Com dobras simples de serem realizadas a dobradura pode auxiliar o aluno a desenvolver a concentração, estimular a criatividade, concretizar uma ideia ou pensamento no momento em que surge a foma no papel e, consequentemente, o aluno interioriza o aprendizado desejado. Os tópicos estudados versam sobre a construção dos principais polígonos regulares e de um sólido espacial, o tetraedro. São também estudadas algumas aplicações aritméticas, como divisão de segmentos e raízes quadradas e cúbicas. / This work aims to study the activity of paper folding in the classroom as an auxiliary resource for the teacher. The folders are quite simple and will improve the students skills on concentration, creativity, and the ability to realize on paper his/her thoughts and ideas. The covered topics range from the construction of the main regular poligons, a spatial solid (tetrahedron), through some arithmetic applications, like division of a segment and square and cubic roots.

Protein folding is considered to be one of the grand challenge problems in biology. Protein folding refers to how a protein's amino acid sequence, under certain physiological conditions, folds into a stable close-packed three-dimensional structure known as the native state. There are two major problems in protein folding. One, usually called protein structure prediction, is to predict the structure of the protein's native state given only the amino acid sequence. Another important and strongly related problem, often called protein folding, is to study how the amino acid sequence dynamically transitions from an unstructured state to the native state. In this dissertation, we concentrate on the second problem. There are several approaches that have been applied to the protein folding problem, including molecular dynamics, Monte Carlo methods, statistical mechanical models, and lattice models. However, most of these approaches suffer from either overly-detailed simulations, requiring impractical computation times, or overly-simplified models, resulting in unrealistic solutions. In this work, we present a novel motion planning based framework for studying protein folding. We describe how it can be used to approximately map a protein's energy landscape, and then discuss how to find approximate folding pathways and kinetics on this approximate energy landscape. In particular, our technique can produce potential energy landscapes, free energy landscapes, and many folding pathways all from a single roadmap. The roadmap can be computed in a few hours on a desktop PC using a coarse potential energy function. In addition, our motion planning based approach is the first simulation method that enables the study of protein folding kinetics at a level of detail that is appropriate (i.e., not too detailed or too coarse) for capturing possible 2-state and 3-state folding kinetics that may coexist in one protein. Indeed, the unique ability of our method to produce large sets of unrelated folding pathways may potentially provide crucial insight into some aspects of folding kinetics that are not available to other theoretical techniques.

Este trabalho tem como foco a dobradura em sala de aula, auxiliando o professor em sua prática docente. Com dobras simples de serem realizadas a dobradura pode auxiliar o aluno a desenvolver a concentração, estimular a criatividade, concretizar uma ideia ou pensamento no momento em que surge a foma no papel e, consequentemente, o aluno interioriza o aprendizado desejado. Os tópicos estudados versam sobre a construção dos principais polígonos regulares e de um sólido espacial, o tetraedro. São também estudadas algumas aplicações aritméticas, como divisão de segmentos e raízes quadradas e cúbicas. / This work aims to study the activity of paper folding in the classroom as an auxiliary resource for the teacher. The folders are quite simple and will improve the students skills on concentration, creativity, and the ability to realize on paper his/her thoughts and ideas. The covered topics range from the construction of the main regular poligons, a spatial solid (tetrahedron), through some arithmetic applications, like division of a segment and square and cubic roots.

Les études sont mitigées sur les séquelles cognitives des commotions cérébrales, certaines suggèrent qu’elles se résorbent rapidement tandis que d’autres indiquent qu’elles persistent dans le temps. Par contre, aucunes données n’existent pour indiquer si une tâche cognitive comme l’imagerie mentale visuelle fait ressortir des séquelles à la suite d’une commotion cérébrale. Ainsi, la présente étude a pour objet d’évaluer l’effet des commotions cérébrales d’origine sportive sur la capacité d’imagerie mentale visuelle d’objets et d’imagerie spatiale des athlètes. Afin de répondre à cet objectif, nous comparons les capacités d’imagerie mentale chez des joueurs de football masculins de calibre universitaire sans historique répertorié de commotions cérébrales (n=15) et chez un second groupe d’athlète ayant été victime d’au moins une commotion cérébrale (n=15). Notre hypothèse est que les athlètes non-commotionnés ont une meilleure imagerie mentale que les athlètes commotionnés. Les résultats infirment notre hypothèse. Les athlètes commotionnés performent aussi bien que les athlètes non-commotionnés aux trois tests suivants : Paper Folding Test (PFT), Visual Object Identification Task (VOIT) et Vividness of Visual Imagery Questionnaire (VVIQ). De plus, ni le nombre de commotions cérébrales ni le temps écoulé depuis la dernière commotion cérébrale n’influent sur la performance des athlètes commotionnés. / The research is mitigated on the cognitive after-effects of a concussion. Some studies suggest the effects disappear rapidly whereas others observe a continuation in their manifestation. However, no research has been done to indicate whether a cognitive task like mental imagery brings out these effects following a concussion. This study will evaluate the effects of sport-related concussions on object and spatial visual mental imagery of athletes. To achieve this goal, we compare the mental imagery capacity between two groups of male football athletes of University level. The first group (n=15) with no history of concussions and the second group (n=15) with one or more concussions. We hypothesize that the non-concussed athletes visualize better than the concussed athletes. Our results invalidate our hypothesis. Both groups have similar results on the three following measures: Paper Folding Test (PFT), Visual Object Identification Task (VOIT) and Vividness of Visual Imagery Questionnaire (VVIQ). Furthermore, the quantity of concussions and the time past since the last concussion seems to have no impact on the visual mental imagery performance of concussed athletes.