Tessellations have a rich mathematical structure and are especially appropriate as a context for teaching geometry in the middle grades. Few studies have researched how children conceptualize and learn tessellations in spite of their international use in educational contexts. This exploratory study looks at how fourth grade students conceptualize tessellations before instruction. The analysis is done from a Piagetian, cognitive viewpoint and from an aesthetic viewpoint. It is argued that the aesthetic viewpoint is crucial and foundational to children's mathematical understanding, just as it is for mathematicians. A series of clinical interviews was conducted with six fourth grade children. The results identified common themes of children's understanding, strategies, reasoning, and aesthetic criteria for tessellations. Children's ontology varied between object and process conceptions of tessellations. Children struggled especially with the infinite space of mathematical tessellations. Children's aesthetics, including symmetry, influenced their choices in creating tessellations and are shown to have played a cognitive role in children's mathematical exploration of tessellation structures. Mathematics influences students' aesthetic appreciation of tessellations and, more importantly, aesthetics drives the study of the mathematical structure of tessellations. Children's aesthetic criteria were the same as mathematicians', but with much different emphases. Other results are discussed, including the mathematical content elicited by the tasks, the influence of the tools used to create tessellations, the children's epistemology of their tessellations, and the role symmetry played in giving children confidence. Recommendations for future research and possible implications for curriculum and instruction are noted. / text

The methods presented in this study are aimed at the identification of subpopulations (clones) of genetically similar cells within tissue samples through measurement of loci-specific Fluorescence in-situ hybridization (FISH) spot signals for each nucleus and analyzing cell spatial distributions by way of Voronoi tessellation and Delaunay triangulation to robustly define cell neighbourhoods. The motivation for the system is to examine lung cancer patient for subpopulations of Non-Small Cell Lung Cancer (NSCLC) cells with biologically meaningful gene copy-number profiles: patterns of genetic alterations statistically associated with resistance to cis-platinum/vinorelbine doublet chemotherapy treatment. Current technologies for gene-copy number profiling rely on large amount of cellular material, which is not always available and suffers from limited sensitivity to only the most dominant clone in often heterogeneous samples. Thus, through the use of FISH, the detection of gene copy-numbers is possible in unprocessed tissues, allowing identification of specific tumour clones with biologically relevant patterns of genetic aberrations. The tissue-wide characterization of multiplexed loci-specific FISH signals, described herein, is achieved through a fully automated, multicolour fluorescence imaging microscope and object segmentation algorithms to identify cell nuclei and FISH spots within. Related tumour clones are identified through analysis of robustly defined cell neighbourhoods and cell-to-cell connections for regions of cells with homogenous and highly interconnected FISH spot signal characteristics. This study presents experiments which demonstrate the system’s ability to accurately quantify FISH spot signals in various tumour tissues and in up to 5 colours simultaneously or more through multiple rounds of FISH staining. Furthermore, the system’s FISH-based cell classification performance is evaluated at a sensitivity of 84% and specificity 81% and clonal identification algorithm results are determined to be comparable to clone delineation by a human-observer. Additionally, guidelines and procedures to perform anticipated, routine analysis experiments are established.

The methods presented in this study are aimed at the identification of subpopulations (clones) of genetically similar cells within tissue samples through measurement of loci-specific Fluorescence in-situ hybridization (FISH) spot signals for each nucleus and analyzing cell spatial distributions by way of Voronoi tessellation and Delaunay triangulation to robustly define cell neighbourhoods. The motivation for the system is to examine lung cancer patient for subpopulations of Non-Small Cell Lung Cancer (NSCLC) cells with biologically meaningful gene copy-number profiles: patterns of genetic alterations statistically associated with resistance to cis-platinum/vinorelbine doublet chemotherapy treatment. Current technologies for gene-copy number profiling rely on large amount of cellular material, which is not always available and suffers from limited sensitivity to only the most dominant clone in often heterogeneous samples. Thus, through the use of FISH, the detection of gene copy-numbers is possible in unprocessed tissues, allowing identification of specific tumour clones with biologically relevant patterns of genetic aberrations. The tissue-wide characterization of multiplexed loci-specific FISH signals, described herein, is achieved through a fully automated, multicolour fluorescence imaging microscope and object segmentation algorithms to identify cell nuclei and FISH spots within. Related tumour clones are identified through analysis of robustly defined cell neighbourhoods and cell-to-cell connections for regions of cells with homogenous and highly interconnected FISH spot signal characteristics. This study presents experiments which demonstrate the system’s ability to accurately quantify FISH spot signals in various tumour tissues and in up to 5 colours simultaneously or more through multiple rounds of FISH staining. Furthermore, the system’s FISH-based cell classification performance is evaluated at a sensitivity of 84% and specificity 81% and clonal identification algorithm results are determined to be comparable to clone delineation by a human-observer. Additionally, guidelines and procedures to perform anticipated, routine analysis experiments are established.

Various industries have embraced 3D printing for manufacturing on-demand, custom printed parts. However, 3D printing requires intelligent data processing and algorithms to go from CAD model to machine instructions. One of the most crucial steps in the process is the slicing of the object. Most 3D printers build parts by accumulating material layers by layer. 3D printing software needs to calculate these layers for manufacturing by slicing a model and calculating the intersections. Finding exact solutions of intersections on the original model is mathematically complicated and computationally demanding. A preprocessing stage of tessellation has become the standard practice for slicing models. Calculating intersections with tessellations of the original model is computationally simple but can introduce inaccuracies and errors that can ruin the final print. This dissertation shows that a point cloud approach to preprocessing and slicing models is robust and accurate. The point cloud approach to object slicing avoids the complexities of directly slicing models while evading the error-prone tessellation stage. An algorithm developed for this dissertation generates point clouds and slices models within a tolerance. The algorithm uses the original NURBS model and converts the model into a point cloud, based on layer thickness and accuracy requirements. The algorithm then uses a gridding structure to calculate where intersections happen and fit B-spline curves to those intersections. This algorithm finds accurate intersections and can ignore certain anomalies and error from the modeling process. The primary point evaluation is stable and computationally inexpensive. This algorithm provides an alternative to challenges of both the direct and tessellated slicing methods that have been the focus of the 3D printing industry.

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.

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.

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.

While the games industry grows, the need for visually appealing graphics increase. This drives the demand for new hardware and the development of new techniques forward. With the latest generation of 3D graphic APIs the new rendering pipeline technique tessellation was announced. With each generation of games getting more advanced, the demand for highly detailed models increase. This leads to a higher workload for the GPU. The purpose of this thesis is to research the fi eld of tessellation and develop a prototype to optimize the tessellation pipeline. The prototype is designed to help reduce the number of generated triangles by dividing the mesh into regions, each region with its own tessellation factor. The prototype implementation supports both manual and automatic generation of these regions. A new fi le format was developed to support the region data. The implementation was split into three parts; one Python based Maya plug-in used for preparing the model for exporting, one exporter for Maya that generated the output fi le, and one viewer application capable of rendering the resulting mesh. The geometry output of the proposed technique was evaluated by comparing it to uniformly tessellated reference mesh geometry. Measurements show that the proposed technique is capable of reducing the face count with minor or maintained correlation to the reference geometry. / Medans spelindustrin växer ökar behovet av visuellt tilltalande grafik. Detta driver utvecklingen av hårdvara och nya tekniker framåt. Med den senaste generationens 3D grafik API kom renderingssteget tesselering. Eftersom varje generation av spel blir mer avancerat ökar efterfrågan på detaljerade modeller. Detta leder till en högre arbetsbörda för GPUn. Syftet med denna avhandling är att undersöka tesselering och utveckla en prototyp för att optimera tesseleringspipelinen. Prototypen är utformad för att hjälpa till att minska antalet genererade trianglar genom att dela upp figuren i regioner, varje region med egen tesseleringsfaktor. Prototypen stödjer både manuell och automatisk generering av dessa regioner, ett nytt filformat utvecklades för att stödja dessa. Prototypen delades upp i tre delar, ett Python baserat Maya plug-in som används för att förbereda modellen för export, en Maya exportör som genererar filerna, och ett visningsprogram som kan rendera den färdiga figuren. Figuren från den föreslagna tekniken utvärderades genom att jämföra den med motsvarande figur med universal tesseleringsfaktor. Mätningar visar att den föreslagna tekniken klarar att reducera antalet trianglar med låg felmarginal.

Kontext: Displacement Mapping är en teknik som används inom 3D-spel för att skapa detaljrikedom i geometri utan att behöva triangelobjekt bestående av oönskad geometrikomplexitet. Tekniken har även andra användningsområden i 3D-spel, till exempel terränggeometri. Tekniken skänker detaljrikedom genom att i samband med tesselering förskjuta geometri i en normalriktning eller längs annan specificerad riktning. Vector Displacement Mapping är en teknik liknande Displacement Mapping där skillnaden är att Vector Displacement Mapping förskjuter geometri i tre dimensioner. Mål: Syftet med arbetet är utforska Vector Displacement Mapping i sammanhanget 3D-Spel och att antyda att tekniken kan användas i 3D-spel likt Displacement Mapping. Arbetet jämför Vector Displacement Mapping med Displacement Mapping för att urskilja skillnader i exekveringstid mellan teknikernas centrala skillnader. Skillnaderna i exekveringstid ställs i kontrast mot diskussion av teknikernas grafikminnesanvändning. Metoder: Jämförelsen baseras på en implementation av de båda teknikerna tillsammans med tesselering. Prestandamätningar genereras med implementationen som grund. Implementationen använder sig av Direct3D 11. Resultat: Resultatet som erhålls genom jämförelsen visar att exekveringstiderna mellan teknikernas centrala skillnader varierar svagt. Grafikminnesanvändningen mellan teknikerna skiljer sig med en faktor 3 eller en faktor 4 där Vector Displacement Mapping använder mer grafikminne. Slutsatser: Slutsatser som dras baserat på resultatet är att Vector Displacement Mapping i situationer där överhängande geometri är ett önskat resultat kan ersätta Displacement Mapping. Vidare diskussion förs kring slutsatser, avgränsningar och framtida forskning som arbetet berör.

This dissertation presents new concepts and methodology in designing shape-morphing structures using bistable elements. Developed using the Pseudo-Rigid-Body Model (PRBM), linear bistable compliant mechanism elements produce predictable and controllable length changes. Step-by-step design procedures are developed to guide the design process of these bistable elements. Two different examples of Shape-Morphing Space Frames (SMSFs) were designed and prototyped utilizing the bistable linear elements in a single-layer grid, in addition to flexures and rigid links, to morph a cylindrical space frame into both a hyperbolic and a spherical space frame. Moreover, bistable unit-cell compliant-mechanism elements were also developed to morph a compact structure from a specific initial shape to a final specific shape. The detailed design of those unit cells were done using Computer-aided design (CAD) software following a novel design procedure to transform a one-degree-of-freedom mechanism into a structure with sufficient compliance within its links to toggle between two chosen stable positions. Two different design examples were investigated in this research and prototyped to demonstrate the ability to morph disks into a hemisphere or a sphere with the structure being stable in both states (disk and sphere).