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Chapter 5 considers the metric spaces defined in the same way that are topologically a disk rather than a sphere, and studies the flexible polyhedral surfaces that result. [ 1 ] Chapters 6 through 8 of the book are related to a theorem of Hermann Minkowski that a convex polyhedron is uniquely determined by the areas and directions of its faces ...
The book covers both the mathematics of polyhedra and its historical development, limiting itself only to three-dimensional geometry. [2] [3] The notion of what it means to be a polyhedron has varied over the history of the subject, as have other related definitions, an issue that the book handles largely by keeping definitions informal and flexible, and by pointing out problematic examples ...
Although primarily a practical work, the Mishnat ha-Middot attempts to define terms and explain both geometric application and theory. [9] The book begins with a discussion that defines "aspects" for the different kinds of plane figures (quadrilateral, triangle, circle, and segment of a circle) in Chapter I (§1–5), and with the basic principles of measurement of areas (§6–9).
Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically. [23] Analytic geometry allows the study of curves unrelated to circles and lines.
Old axiom II.5 (Pasch's Axiom) is renumbered as II.4. V.2, the Axiom of Line Completeness, replaced: Axiom of completeness. To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms.
Chapter two covers more concepts in the geometry of the hyperbolic plane, connecting them to crocheted models of the plane. [ 4 ] The next three chapters take a step back to look at the broader history of the topics discussed in the book: geometry and its connection to human arts and architecture in chapter 3, crochet in chapter 4, and non ...
The book has 19 chapters. After two chapters introducing background material in linear algebra, topology, and convex geometry, two more chapters provide basic definitions of polyhedra, in their two dual versions (intersections of half-spaces and convex hulls of finite point sets), introduce Schlegel diagrams, and provide some basic examples including the cyclic polytopes.
Chapter five considers Monsky's theorem on the impossibility of partitioning a square into an odd number of equal-area triangles, and its proof using the 2-adic valuation, and chapter six applies Galois theory to more general problems of tiling polygons by congruent triangles, such as the impossibility of tiling a square with 30-60-90 right ...
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