Finite Geometry

1. Charlene P. Aposaga
MA. Ed Mathematics

2. Pappus of Alexandria (340 A.D.)

3. If A , B and C are three distinct
points on one line and if A’, B’ an C’ are
three different distinct points on a
second line, then the intersections of
𝐴𝐶′and 𝐶𝐴′ , 𝐴𝐵′ and 𝐵𝐴′, and 𝐵𝐶′
and 𝐶𝐵′ are collinear.
Theorem 1.9. Theorem of Pappus

4. A
B
C
A'
B'
C'
If points A,B and C are on one line and
A', B' and C' are on another line then
the points of intersection of the lines
AB' and BA', AC' and CA', and BC' and
CB' lie on a common line called the
Pappus line of the configuration.
Pappus’
Theorem

5. Construction

6. A’
B’ C’
A
B
C

8. How many points and lines
in the Pappus’ geometry
configuration?

9. The Pappus’ geometry
configuration has 9 points and
9 lines.

10. Axioms for Finite Geometry of Pappus
1. There exists at least one line.
2. Every line has exactly three points.
3. Not all points are on the same line.
4. There exists exactly one line through a point not on a
line that is parallel to the given line.
5. If P is a point not on a line, there exists exactly one
point P’ on the line such that no line joins P and P’.
6. With the exception in Axiom 5, if P and Q are distinct
points , then exactly one line contains both of them

11. Theorem 1.10
Each point in the geometry of
Pappus lies on exactly three lines.

12. Simple proof in Euclidean geometry.

13. Point-Line Duality

14. Point/Line duality
If you have any diagram of points and lines, you
can replace every point with coordinates <a,b,c>
with the line of coordinates <a,b,c> and vice
versa, and you still have a valid diagram.
If you do this to Pappus’ theorem, you get
another version (called the “dual” version) of
Pappus’ theorem.

15. Pappus’ theorem: Dual formulation
Pick any two points. Through each, draw a red
line, a blue line, and a green line.

16. Find the intersection of the lines of different
color.

17. Draw the lines that connects the two red-blue
crossings, the two red-green crossings, and the
two blue-green crossings.

18. These lines are coincident

19. Pappus’ theorem: Original and dual
Draw two lines with red,
blue and green points.
Draw the lines connecting
points of different colors.
Find the intersections of
the two red-blue, the two
red-green, and the two
blue-green lines.
These points are collinear.
Draw two points with red,
blue, and green lines.
Find the intersection of
lines of different colors.
Draw the lines connecting
the two red-blue, the two
red-green, and the two
blue-green points.
These lines are coincident.