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