A history of non-euclidean geometry: evolution of the by Boris A. Rosenfeld, Abe Shenitzer, Hardy Grant

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.

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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.

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