Plane Geometry - Bruce E. Shapiro
Plane Geometry - Bruce E. Shapiro Plane Geometry - Bruce E. Shapiro
54 SECTION 13. INCIDENCE GEOMETRYFigure 13.1: Graph diagram illustrating three-point geometry. Parallel linesdo not exist in this geometry.There are three possible lines in this geometry:{A, B}, {B, C}, {A, C}We will describe finite geometries with graph-diagrams consisting of nodesand line segments (figure 13.1). The nodes represent the points, and theline segments connecting the nodes represent the sets that represent lines.This is a representation of a finite geometry, not a picture of the points oflines in the usual sense. In other words, the 3-point geometry does not looklike a triangle; we just represent it by a graph that looks like a triangle.Example 13.2 Four-point Geometry. Here we define (see figure 13.2):ˆ A point is an element of the set {A, B, C, D}ˆ A line is a pair of points such as l = {A, B}ˆ A point P lies on a line l if P ∈ l.There are six lines in this example: {A, B}, {A, C}, {A, D}, {B, C}, {B, D},and {C, D}.Example 13.3 Fano’s Geometry. Here we have seven points given bythe set {A, B, C, D, E, F, G} and we define lines as any of the followingseven specific subsets:as illustrated in figure 13.3{A, B, C}, {C, D, E}, {E, F, A}, {A, G, D},{C, G, F }, {E, G, B}, {B, D, F }« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 13. INCIDENCE GEOMETRY 55Figure 13.2: Graph diagram illustrating four-point geometry. Every linehas precisely one other line that is parallel to it, and and there is preciselyone parallel line through each point that is not on a given line, and hencefour point geometry obeys the Euclidean parallel postulate.Figure 13.3: Graph diagram illustrating Fano’s geometry. Each line segmentand the circle represents a line in this geometry.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
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SECTION 13. INCIDENCE GEOMETRY 55Figure 13.2: Graph diagram illustrating four-point geometry. Every linehas precisely one other line that is parallel to it, and and there is preciselyone parallel line through each point that is not on a given line, and hencefour point geometry obeys the Euclidean parallel postulate.Figure 13.3: Graph diagram illustrating Fano’s geometry. Each line segmentand the circle represents a line in this geometry.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
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