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This is a book on Euclidean geometry that covers the standard material in a completely new way, while also introducing a number of new topics that would be suitable as a junior-senior level undergraduate textbook. The author does not begin in the traditional manner with abstract geometric axioms. Instead, he assumes the real numbers, and begins his treatment by introducing such modern concepts as a metric space, vector space notation, and groups, and thus lays a rigorous basis for geometry while at the same time giving the student tools that will be useful in other courses. Jan Aarts is Professor Emeritus of Mathematics at Delft University of Technology. He is the Managing Director of the Dutch Masters Program of Mathematics.
Mathematics. --- Geometry. --- Mathématiques --- Géométrie --- Electronic books. -- local. --- Geometry, Plane. --- Geometry, Solid. --- Geometry, Plane --- Geometry, Solid --- Geometry --- Mathematics --- Physical Sciences & Mathematics --- Solid geometry --- Plane geometry --- Euclid's Elements --- Math --- Science
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Did you know that any straight-line drawing on paper can be folded so that the complete drawing can be cut out with one straight scissors cut? That there is a planar linkage that can trace out any algebraic curve, or even 'sign your name'? Or that a 'Latin cross' unfolding of a cube can be refolded to 23 different convex polyhedra? Over the past decade, there has been a surge of interest in such problems, with applications ranging from robotics to protein folding. With an emphasis on algorithmic or computational aspects, this treatment gives hundreds of results and over 60 unsolved 'open problems' to inspire further research. The authors cover one-dimensional (1D) objects (linkages), 2D objects (paper), and 3D objects (polyhedra). Aimed at advanced undergraduate and graduate students in mathematics or computer science, this lavishly illustrated book will fascinate a broad audience, from school students to researchers.
Polyhedra --- Polyèdres --- Models --- Data processing --- Modèles --- Geometrical models --- Polyhedral figures --- Polyhedrons --- Geometry, Solid --- Shapes --- Data processing. --- Models.
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These lecture notes are based on a series of lectures given at the Nankai Institute of Mathematics in the fall of 1998. They provide an overview of the work of the author and the late Chih-Han Sah on various aspects of Hilbert's Third Problem: Are two Euclidean polyhedra with the same volume "scissors-congruent", i.e. can they be subdivided into finitely many pairwise congruent pieces? The book starts from the classical solution of this problem by M Dehn. But generalization to higher dimensions and other geometries quickly leads to a great variety of mathematical topics, such as homology of gr
Tetrahedra. --- Volume (Cubic content) --- Characteristic classes. --- Classes, Characteristic --- Differential topology --- Cubic measurement --- Volumetry --- Units of measurement --- Geometry, Solid --- Polyhedra --- Tetrahedra --- Characteristic classes
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The publication of the first edition of Lagerungen in der Ebene, auf der Kugel und im Raum in 1953 marked the birth of discrete geometry. Since then, the book has had a profound and lasting influence on the development of the field. It included many open problems and conjectures, often accompanied by suggestions for their resolution. A good number of new results were surveyed by László Fejes Tóth in his Notes to the 2nd edition. The present version of Lagerungen makes this classic monograph available in English for the first time, with updated Notes, completed by extensive surveys of the state of the art. More precisely, this book consists of: a corrected English translation of the original Lagerungen, the revised and updated Notes on the original text, eight self-contained chapters surveying additional topics in detail. The English edition provides a comprehensive update to an enduring classic. Combining the lucid exposition of the original text with extensive new material, it will be a valuable resource for researchers in discrete geometry for decades to come.
Mathematics. --- Math --- Science --- Convex surfaces. --- Polyhedra. --- Sphere. --- Geometry, Solid --- Shapes --- Orbs --- Polyhedral figures --- Polyhedrons --- Convex areas --- Convex domains --- Surfaces --- Superfícies convexes --- Poliedres --- Esfera
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Convex Cones
Functional analysis --- Cone. --- Cône. --- Analyse fonctionnelle --- Functions of real variables. --- Convex bodies. --- Functions of real variables --- Convex bodies --- Cone --- Geometry, Descriptive --- Geometry, Solid --- Convex domains --- Real variables --- Functions of complex variables --- Géometrie convexe
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This introductory text in graph theory focuses on partial cubes, which are graphs that are isometrically embeddable into hypercubes of an arbitrary dimension, as well as bipartite graphs, and cubical graphs. This branch of graph theory has developed rapidly during the past three decades, producing exciting results and establishing links to other branches of mathematics. Currently, Graphs and Cubes is the only book available on the market that presents a comprehensive coverage of cubical graph and partial cube theories. Many exercises, along with historical notes, are included at the end of every chapter, and readers are encouraged to explore the exercises fully, and use them as a basis for research projects. The prerequisites for this text include familiarity with basic mathematical concepts and methods on the level of undergraduate courses in discrete mathematics, linear algebra, group theory, and topology of Euclidean spaces. While the book is intended for lower-division graduate students in mathematics, it will be of interest to a much wider audience; because of their rich structural properties, partial cubes appear in theoretical computer science, coding theory, genetics, and even the political and social sciences.
Mathematics. --- Graph Theory. --- Graph theory --- Geometry --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Graphic methods --- Graph theory. --- Bipartite graphs. --- Cube. --- Graphs, Theory of --- Theory of graphs --- Extremal problems --- Combinatorial analysis --- Topology --- Math --- Science --- Geometry, Solid --- Graphic methods.
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Molecules, galaxies, art galleries, sculptures, viruses, crystals, architecture, and more: Shaping Space—Exploring Polyhedra in Nature, Art, and the Geometrical Imagination is an exuberant survey of polyhedra and at the same time a hands-on, mind-boggling introduction to one of the oldest and most fascinating branches of mathematics. Some of the world’s leading geometers present a treasury of ideas, history, and culture to make the beauty of polyhedra accessible to students, teachers, polyhedra hobbyists, and professionals such as architects and designers, painters and sculptors, biologists and chemists, crystallographers, physicists and earth scientists, engineers and model builders, mathematicians and computer scientists. The creative chapters by more than 25 authors explore almost every imaginable side of polyhedra. From the beauty of natural forms to the monumental constructions made by man, there is something to fascinate every reader. The book is dedicated to the memory of the legendary geometer H. S. M. Coxeter and the multifaceted design scientist Arthur L. Loeb. Contributing Authors: P. Ash, T. F. Banchoff, J. Baracs, E. Bolker, C. Chieh, R. Connelly, H.S.M. Coxeter, H. Crapo, E. Demaine, M. Demaine, G. Fleck, B. Grünbaum, I. Hargittai, M. Hargittai, G. Hart, V. Hart, A. Loeb, J. Malkevitch, B. Monson, J. O'Rourke, J. Pedersen, D. Schattschneider, M. Schmitt, E. Schulte, M. Senechal, G.C. Shephard, I. Streinu, M. Walter, M. Wenninger, W. Whiteley, J. M. Wills, and G. M. Ziegler.
Polyhedra. --- Shapes. --- Forms (Shapes) --- Shape --- Polyhedral figures --- Polyhedrons --- Mathematics. --- Design. --- Geometry. --- Crystallography. --- Design, general. --- Geometry --- Surfaces --- Geometry, Solid --- Shapes --- Design and construction. --- Crystallography and Scattering Methods. --- Leptology --- Physical sciences --- Mineralogy --- Mathematics --- Euclid's Elements --- Creation (Literary, artistic, etc.)
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Complex cobordism and stable homotopy groups of spheres
Homotopy groups. --- Sphere. --- Spectral sequences (Mathematics) --- Cobordism theory. --- Differential topology --- Algebra, Homological --- Algebraic topology --- Sequences (Mathematics) --- Spectral theory (Mathematics) --- Geometry, Solid --- Shapes --- Orbs --- Group theory --- Homotopy theory --- Homotopy groups --- Sphere --- Cobordism theory
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Nanoparticles. --- Graphene. --- Plates (Engineering) --- Sphere. --- Geometry, Solid --- Shapes --- Orbs --- Disks (Mechanics) --- Panels --- Structural plates --- Elastic plates and shells --- Structural analysis (Engineering) --- Shells (Engineering) --- Polycyclic aromatic hydrocarbons --- Nano-particles --- NPs (Nanoparticles) --- Nanostructured materials --- Particles --- Nanoscale particles
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A flexagon is a motion structure that has the appearance of a ring of hinged polygons. It can be flexed to display different pairs of faces, usually in cyclic order. Flexagons can be appreciated as toys or puzzles, as a recreational mathematics topic, and as the subject of serious mathematical study. Workable paper models of flexagons are easy to make and entertaining to manipulate. The mathematics of flexagons is complex, and how a flexagon works is not immediately obvious on examination of a paper model. Recent geometric analysis, included in the book, has improved theoretical understanding of flexagons, especially relationships between different types. This profusely illustrated book is arranged in a logical order appropriate for a textbook on the geometry of flexagons. It is written so that it can be enjoyed at both the recreational mathematics level, and at the serious mathematics level. The only prerequisite is some knowledge of elementary geometry, including properties of polygons. A feature of the book is a compendium of over 100 nets for making paper models of some of the more interesting flexagons, chosen to complement the text. These are accurately drawn and reproduced at half full size. Many of the nets have not previously been published. Instructions for assembling and manipulating the flexagons are included. .
Geometry, Solid --Models. --- Mathematical recreations. --- Paper work. --- Polygons --Models. --- Polyhedra --Models. --- Geometry, Solid --- Polygons --- Polyhedra --- Paper work --- Mathematical recreations --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Models --- Mathematical puzzles --- Number games --- Recreational mathematics --- Recreations, Mathematical --- Paper craft --- Paper-cutting --- Paper folding (Handicraft) --- Papercraft --- Polygonal figures --- Solid geometry --- Mathematics. --- Geometry. --- History. --- Engineering. --- Engineering, general. --- Mathematics, general. --- History of Mathematical Sciences. --- Models. --- Puzzles --- Scientific recreations --- Games in mathematics education --- Magic squares --- Magic tricks in mathematics education --- Geometrical models --- Math --- Science --- Construction --- Industrial arts --- Technology --- Euclid's Elements --- Annals --- Auxiliary sciences of history
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