By Boris A. Rosenfeld, Abe Shenitzer, Hardy Grant
This ebook is an research of the mathematical and philosophical elements underlying the invention of the concept that of noneuclidean geometries, and the next extension of the concept that of house. Chapters one via 5 are dedicated to the evolution of the concept that of house, best as much as bankruptcy six which describes the invention of noneuclidean geometry, and the corresponding broadening of the concept that of area. the writer is going directly to speak about suggestions similar to multidimensional areas and curvature, and transformation teams. The ebook ends with a bankruptcy describing the functions of nonassociative algebras to geometry.
Read or Download A history of non-euclidean geometry: evolution of the concept of a geometric space PDF
Similar geometry and topology books
Quantum Geometry - A Statistical Field Theory Approach
This graduate point textual content describes in a unified style the statistical mechanics of random walks, random surfaces and random better dimensional manifolds with an emphasis at the geometrical facets of the speculation and purposes to the quantization of strings, gravity and topological box conception. With chapters on random walks, random surfaces, two-and higher-dimensional quantum gravity, topological quantum box theories and Monte Carlo simulations of random geometries, the textual content presents a self-contained account of quantum geometry from a statistical box thought perspective.
The Geometry of Ecological Interactions: Simplifying Spatial Complexity
The sector of theoretical ecology has elevated dramatically some time past few years, whereas one of the most fascinating paintings has been performed utilizing spatial versions with stochasticity. This well timed quantity brings jointly the paintings of best researchers operating with this version and explores its function within the learn of atmosphere dynamics.
- New Scientific Applications of Geometry and Topology (Proceedings of Symposia in Applied Mathematics, V. 45)
- Semi-Riemannian geometry: with applications to relativity
- Integral Formulas in Riemannian Geometry
- Differentialgeometrie: Kurven - Flachen - Mannigfaltigkeiten
- Proceedings of the 14th Gokova Geometry-Topology Conference 2007
Additional resources for A history of non-euclidean geometry: evolution of the concept of a geometric space
Example text
15 This plane of seven points and seven lines is called the Fano plane. Either of the following two diagrams represents the Fano plane! A Z A Z D D Y Y X B C B X C The Fano plane has exactly three points on each line, and three lines through each point. We now prove that this is a consequence of the Fano plane being a projective plane with a finite number of points. 16 Let π be a projective plane which has a finite number N2 of points. Then 1. Every line of π has the same number N1 = q + 1 of points.
4. Every point of the Euclidean plane R2 has well-defined homogeneous coordinates, namely the point (x, y) has homogeneous coordinates (x, y, 1). For the points at infinity, proceed as follows. Consider two parallel lines of R2 , ax + by + c = 0, ax + by + c = 0, c=c. Writing x = X/Z, y = Y /Z, and then multiplying by Z gives aX + bY + cZ = 0, aX + bY + c Z = 0, which are the homogeneous equations of these two lines. By solving the above two linearly independent homogeneous linear equations, the solution set is {ρ(a, b, 0) | ρ ∈ R\{0}}.
Parallel lines non-parallel lines Proof. 5, a line ∈ R2 when extended is denoted by ∗ . 1. Let A and B be two distinct points of the EEP. (a) If A and B are two distinct points of R2 , then (AB)∗ is the unique line of the EEP which contains them. (b) If A and B are two distinct points at ∞, then ∞ is the unique line of the EEP which contains them. (c) Suppose A ∈ R2 , and B ∈ ∞ . Then B is the point at infinity of a unique pencil of parallel lines. There is a unique line of that pencil passing through A.