Euclidean and Non-Euclidean Geometry – Fall 2007
Dr. Hamblin
Modeling the Incidence Axioms
The smallest possible model for all five incidence axioms has 4 points, 6 lines, and 4 planes:
Points: {A, B, C, D}
Lines: {A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D}
Planes: {A, B, C}, {A, B, D}, {A, C, D}, {B, C, D}
Axiom I-1: Each two distinct points determine a line.
We have to check every possible pair of points to make sure that there is one and only one line
containing that pair.
Pair of points
A, B
A, C
A, D
B, C
B, D
C, D
Line(s) containing that pair
{A, B}
{A, C}
{A, D}
{B, C}
{B, D}
{C, D}
This axiom is satisfied because each pair of points is contained in one and only one line.
Axiom I-2: Three noncollinear points determine a plane.
We have to check every possible to trio of points to make sure that if the points are noncollinear, then
there is one and only one plane containing them.
Trio of points
A, B, C
A, B, D
A, C, D
B, C, D
Collinear?
no
no
no
no
Plane(s) containing those points
{A, B, C}
{A, B, D}
{A, C, D}
{B, C, D}
Axiom I-2 is satisfied because all four trios of noncollinear points determine a plane.
Axiom I-3: If two points lie in a plane, then any line containing those two points lies in that
plane.
For this axiom, we consider one plane at a time. In each plane, we look at all the possible pairs of
points. Then we consider all lines containing those points, and check to see that those lines are
contained in the plane.
As we can see from the table on the next page, Axiom I-3 is satisfied.
Euclidean and Non-Euclidean Geometry – Fall 2007
Plane
{A, B, C}
{A, B, D}
{A, C, D}
{B, C, D}
Pair of
points
A, B
A, C
B, C
A, B
A, D
B, D
A, C
A, D
C, D
B, C
B, D
C, D
Dr. Hamblin
Line(s) containing those points
Line(s) contained in plane?
{A, B}
{A, C}
{B, C}
{A, B}
{A, D}
{B, D}
{A, C}
{A, D}
{C, D}
{B, C}
{B, D}
{C, D}
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
Axiom I-4: If two distinct planes meet, their intersection is a line.
For this axiom, we take every possible pair of planes and find their intersection. If the two planes do not
intersect, then there is nothing to check. If they do intersect, then their intersection must be a line.
As we can see from the table below, Axiom I-4 is satisfied.
Intersection of planes
Meet?
{A, B, C} ∩ {A, B, D} = {A, B}
{A, B, C} ∩ {A, C, D} = {A, C}
{A, B, C} ∩ {B, C, D} = {B, C}
{A, B, D} ∩ {A, C, D} = {A, D}
{A, B, D} ∩ {B, C, D} = {B, D}
{A, C, D} ∩ {B, C, D} = {C, D}
yes
yes
yes
yes
yes
yes
If planes meet, is
intersection a line?
yes
yes
yes
yes
yes
yes
Axiom I-5: Space consists of at least four noncoplanar points, and contains three
noncollinear points. Each plane is a set of points of which at least three are noncollinear,
and each line is a set of at least two distinct points.
Space consists of at least four noncoplanar points. Space has the points A, B, C, and D, and
there is no plane that contains all four.
Space contains three noncollinear points. There are many examples, including A, B, and C.
Each plane is a set of points of which at least three are noncollinear. Each plane has exactly
three points, and for each the three points are noncollinear.
Each line is a set of at least two distinct points. This is clear from the definitions of our model.
We have checked that all four parts are satisfied, therefore Axiom I-5 is satisfied in this model.