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Incidence Geometries Incidence Geometries Aaron Cinzori Fall 2012 Aaron Cinzori Incidence Geometries Incidence Geometries Incidence Geometries Undefined Terms: Point, Line, On. Aaron Cinzori Incidence Geometries Incidence Geometries Incidence Geometries Undefined Terms: Point, Line, On. Axiom 1: For each two distinct points there exists a unique line on both of them. Aaron Cinzori Incidence Geometries Incidence Geometries Incidence Geometries Undefined Terms: Point, Line, On. Axiom 1: For each two distinct points there exists a unique line on both of them. Axiom 2: For every line there exist at least two distinct points on it. Aaron Cinzori Incidence Geometries Incidence Geometries Incidence Geometries Undefined Terms: Point, Line, On. Axiom 1: For each two distinct points there exists a unique line on both of them. Axiom 2: For every line there exist at least two distinct points on it. Axiom 3: There exist at least three distinct points. Aaron Cinzori Incidence Geometries Incidence Geometries Incidence Geometries Undefined Terms: Point, Line, On. Axiom 1: For each two distinct points there exists a unique line on both of them. Axiom 2: For every line there exist at least two distinct points on it. Axiom 3: There exist at least three distinct points. Axiom 4: Not all points lie on the same line. Aaron Cinzori Incidence Geometries Incidence Geometries Incidence Geometries Undefined Terms: Point, Line, On. Axiom 1: For each two distinct points there exists a unique line on both of them. Axiom 2: For every line there exist at least two distinct points on it. Axiom 3: There exist at least three distinct points. Axiom 4: Not all points lie on the same line. Examples: Four-point geometry, Fano’s geometry, Young’s geometry. Aaron Cinzori Incidence Geometries Incidence Geometries Incidence Geometries Undefined Terms: Point, Line, On. Axiom 1: For each two distinct points there exists a unique line on both of them. Axiom 2: For every line there exist at least two distinct points on it. Axiom 3: There exist at least three distinct points. Axiom 4: Not all points lie on the same line. Examples: Four-point geometry, Fano’s geometry, Young’s geometry. Non-Example: Four-line geometry. Aaron Cinzori Incidence Geometries Incidence Geometries Incidence Geometries Undefined Terms: Point, Line, On. Axiom 1: For each two distinct points there exists a unique line on both of them. Axiom 2: For every line there exist at least two distinct points on it. Axiom 3: There exist at least three distinct points. Axiom 4: Not all points lie on the same line. Theorem 1: If two distinct lines intersect, then the intersection is exactly one point. Theorem 2: For each point there exist at least two lines containing it. Theorem 3: There exist three lines that do not share a common point. Aaron Cinzori Incidence Geometries Incidence Geometries Parallels? Given a line ` and a point P not on `, there are three possible states for lines through P. 1 There exist no lines through P parallel to `. Aaron Cinzori Incidence Geometries Incidence Geometries Parallels? Given a line ` and a point P not on `, there are three possible states for lines through P. 1 There exist no lines through P parallel to `. E.g., Fano’s geometry. Aaron Cinzori Incidence Geometries Incidence Geometries Parallels? Given a line ` and a point P not on `, there are three possible states for lines through P. 1 There exist no lines through P parallel to `. E.g., Fano’s geometry. 2 There exists exactly one line on P parallel to `. Aaron Cinzori Incidence Geometries Incidence Geometries Parallels? Given a line ` and a point P not on `, there are three possible states for lines through P. 1 There exist no lines through P parallel to `. E.g., Fano’s geometry. 2 There exists exactly one line on P parallel to `. E.g., Four-point geometry or Young’s geometry. Aaron Cinzori Incidence Geometries Incidence Geometries Parallels? Given a line ` and a point P not on `, there are three possible states for lines through P. 1 There exist no lines through P parallel to `. E.g., Fano’s geometry. 2 There exists exactly one line on P parallel to `. E.g., Four-point geometry or Young’s geometry. 3 There exists more than one line on P parallel to `. Aaron Cinzori Incidence Geometries Incidence Geometries Parallels? Given a line ` and a point P not on `, there are three possible states for lines through P. 1 There exist no lines through P parallel to `. E.g., Fano’s geometry. 2 There exists exactly one line on P parallel to `. E.g., Four-point geometry or Young’s geometry. 3 There exists more than one line on P parallel to `. E.g., Five-point geometry. Aaron Cinzori Incidence Geometries Incidence Geometries Parallels? Given a line ` and a point P not on `, there are three possible states for lines through P. 1 There exist no lines through P parallel to `. E.g., Fano’s geometry. 2 There exists exactly one line on P parallel to `. E.g., Four-point geometry or Young’s geometry. 3 There exists more than one line on P parallel to `. E.g., Five-point geometry. An incidence geometry satisfying 2. is called Euclidean or affine. An incidence geometry satisfying 1. or 3. is called non-Euclidean. A geometry satisfying 1. that has at least three points on each line is called projective. Aaron Cinzori Incidence Geometries