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"Geometrical constructions."
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Discrete groups and geometric structures : Workshop on Discrete Groups and Geometric Structures, with Applications III, May 26-30, 2008, Kortrijk, Belgium
This volume reports on research related to Discrete Groups and Geometric Structures, as presented during the International Workshop held May 26-30, 2008, in Kortrijk, Belgium. Readers will benefit from impressive survey papers by John R. Parker on methods to construct and study lattices in complex hyperbolic space and by Ursula Hamenstadt on properties of group actions with a rank-one element on proper CAT (0)-spaces. This volume also contains research papers in the area of group actions and geometric structures, including work on loops on a twice punctured torus, the simplicial volume of products and fiber bundles, the homology of Hantzsche - Wendt groups, rigidity of real Bott towers, circles in groups of smooth circle homeomorphisms, and groups generated by spine reflections admitting crooked fundamental domains.
eBook
Interactions between geometrical forms and microstructural features in culm of square bamboo
Bamboo culms can alter the bamboo’s geometric shape by adjusting the hierarchical organization of anatomical components as a means of adapting to different living conditions. Therefore, a square-like culm has been found commonly in the Chimonobambusa bamboo species. However, the underling mechanism for how these anatomical components assemble into a square culm in the species remains to be considered. Furthermore, the relationship between the geometrical construction of culm and its corresponding organization of anatomical components within also needs clarification. Therefore, the geometrical construction of cross-sections was examined in this work. A super-ellipse based on the Lamé curve was confirmed. Additionally, the transitional zone, at 3/4 in the radial direction, was detected as an inflection point where the geometric parameters clearly changed. Meanwhile, anatomical observation also suggested that the transitional zone can be identified as an inflection point because the fibre morphology difference in circumferential regions becomes more apparent in this area. It is worth mentioning that there is a coherence between the geometrical and microstructural features in circumferential and radial variation. These findings are meaningful to manifest the controlling mechanism of hierarchical structures on the geometrical shape of bamboo culm.
Journal Article
Subject-Specific Genres and Genre Awareness in Integrated Mathematics and Language Teaching
The increasing attention devoted to the role of language in the different school subjects calls for approaches of integrated subject matter and language teaching and learning. In this article we argue for the importance of subject-specific genres for integrated mathematics and language teaching. Based on an exemplary analysis of geometric construction texts we show that subject-specific genres in the context of schooling might be influenced by different academic and institutional contexts. In a case study of a classroom discourse in 7th grade about geometric construction texts we show how these different contexts pose a challenge for teaching this genre. As a result, genres in school mathematics might appear as blended genres. Based on our findings we refine the notion of genre awareness as an important aspect of teacher knowledge in order to better prepare teachers for the challenges of integrated subject matter and language teaching.
Journal Article
Geometric Constructions
Written in an informal style that intersperses history and philosophy with mathematics, this class-tested, self-contained book demonstrates how some simple construction tools can be associated with various fields of real numbers through coordinate geometry.
eBook
Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry
I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid (implicit diagrammatic reasoning, superposition) can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised.
Journal Article
Visualizing mathematics with 3D printing
The first book to explain mathematics using 3D printed models.
Winner of the Technical Text of the Washington Publishers
Wouldn't it be great to experience three-dimensional ideas in three dimensions? In this book—the first of its kind—mathematician and mathematical artist Henry Segerman takes readers on a fascinating tour of two-, three-, and four-dimensional mathematics, exploring Euclidean and non-Euclidean geometries, symmetry, knots, tilings, and soap films. Visualizing Mathematics with 3D Printing includes more than 100 color photographs of 3D printed models. Readers can take the book's insights to a new level by visiting its sister website, 3dprintmath.com, which features virtual three-dimensional versions of the models for readers to explore. These models can also be ordered online or downloaded to print on a 3D printer.
Combining the strengths of book and website, this volume pulls higher geometry and topology out of the realm of the abstract and puts it into the hands of anyone fascinated by mathematical relationships of shape. With the book in one hand and a 3D printed model in the other, readers can find deeper meaning while holding a hyperbolic honeycomb, touching the twists of a torus knot, or caressing the curves of a Klein quartic.
eBook