Examples of Finite Affine and Projective Planes

(Geometric Structures)

prepared by:

Joseph Malkevitch
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451

Finite Affine planes

Affine planes are ones for which each pair of points there is a unique line which contains these points, and for each point P, not on a line l, there is a unique parallel line m, to l, which contains P.

Points = { 1, 2, 3, 4 }

Lines:

1. 1 , 2

2. 1, 3

3, 1, 4

4, 3, 4

5, 2, 4

6. 2, 3

Points = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Lines:

1. 1, 2, 3

2. 4, 5, 6

3. 7, 8, 9

4. 1, 4, 7

5, 2, 5, 8

6. 3, 6, 9

7. 1, 5, 9

8. 2, 6, 7

9. 3, 4, 8

10. 1, 6, 8

11. 2, 4, 9

12. 3, 5, 7

Points = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 }

Lines:

1. 1, 2, 3, 4

2. 1, 5, 9, 14

3. 4, 5, 10, 13

4. 2, 6, 10, 14

5. 2, 7, 12, 13

6, 3, 8, 10, 16

7. 3, 5, 12, 15

8. 5, 6, 7, 8

9. 4, 7, 9, 16

10. 4, 8, 12, 14

11. 1, 6, 12, 16

12. 1, 7, 10, 15

13. 3, 6, 9, 13

14. 2, 8, 9, 15

15. 9, 10, 11, 12

16. 3, 7, 11, 14

17. 4, 6, 11, 15

18. 13, 14, 15, 16

19. 2, 5, 11, 16

20. 1, 8, 11, 13

Note: A projective plane can be created by adding points 17, 18, 19, 20, and 21 and one of each these points to the lines that form parallel classes in the affine plane above.

Projective Planes

Projective planes are ones for which each pair of points there is a unique line which contains these points, and where every pair of lines intersect in a unique point. In a projective plane, no lines are parallel.

Points = { 1, 2, 3, 4, 5, 6, 7 }

Lines:

1. 1, 2, 5

2. 1, 3, 6

3. 1, 4, 7

4. 5, 6, 7

5. 3, 4, 5

6. 2, 4, 6

7. 2, 3, 7

Exercise: Choose two different ways of selecting three points on two lines and verify Pappus' Theorem for these cases and select two different sets of triangles in perspective from a point and verify Desargues' Theorem holds for both of the the two projective planes below.

Points = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 }

Lines:

1. 1, 2, 3, 10

2. 1, 4, 7, 11

3. 1, 5, 9, 12

4. 1, 6, 8, 13

5. 4, 5, 6, 10

6. 2, 5, 8, 11

7. 2, 6, 7, 12

8. 2, 4, 9, 13

9. 7, 8, 9, 10

10. 3, 6, 9, 11

11. 3, 4, 8, 12

12. 3, 5, 7, 13

13. 10, 11, 12, 13

Points = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 }

Lines:

1. 1, 2, 3, 4, 5

2. 5, 9, 13, 17, 18

3. 2, 6, 10, 14, 18

4. 2, 9, 11, 16, 20

5. 2, 7, 12, 17, 19

6. 1, 10, 11, 12, 13

7. 1, 18, 19, 20, 21

8. 3, 7, 11, 15, 18

9. 4, 9, 10, 15, 19

10, 5, 6, 12, 15, 20

11. 5, 8, 11, 14, 19

12, 1, 6, 7, 8, 9

13, 4, 7, 13, 14, 20

14, 2, 8, 13, 15, 21

15. 3, 8, 10, 17, 20

16, 1, 14, 15, 16, 17

17. 3, 6, 13, 16, 19

18. 5, 7, 10, 16, 21

19. 4, 8, 12, 16, 18

20. 3, 9, 12, 14, 21

21. 4, 6, 11, 17, 21

In general, the finite affine plane of order n satisfies:

1. n points per line

2. Each point lies on exactly n + 1 lines

3. There are exactly n2 points

4. There are exactly n2 + n lines

5. There are n + 1 parallel classes each of which consists of n parallel lines.

In general the finite projective plane of order n satisfies:

1. There are n + 1 points per lines

2. There are n + 1 lines through every point

3. There are exactly n2 + n + 1 points

4. There are exactly n2 + n + 1 lines

Starting with a finite projective plane of order n one can construct a finite affine plane of the same order by deleting a single line m and all of the points on it, and altering the other lines to delete the one point on the line where the line in question used to meet line m. Starting with a finite affine plane this process can be reversed to obtain a finite projective plane.

It is know that for every prime power there exists a finite affine plane of order n and a finite projective plane of order n.

There are infinitely many additional values of n which can not occur as shown by the Bruck-Ryser Theorem. There can be no finite projective plane of order 10 or finite affine plane of order 10 as shown by a computer search coupled with theoretical work. All other integers that are not prime powers, 10, nor ruled out by the Bruck-Ryser Theorem are open, in the sense that it is not know if there is or is not a finite projective (affine) plane of order n. It is widely thought that there can not exist any planes for orders other than the prime powers.

Finite Bolyai-Lobachevsky Planes

These are planes where given a point P not on a line l there are more than one parallel through P to l.

Here are two different "geometries" with 13 points and 26 lines. Do you think they deserve to be called finite Bolyai-Lobachevsky Planes?

Points = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }

Lines:

1. 0, 1, 2

2. 0, 3, 4

3. 0, 5, 6

4. 0, 7, 8

5. 0, 9, 10

6. 0, 11, 12

7. 1, 3, 5

8. 1, 4, 7

9. 1, 6, 8

10. 1, 9, 11

11. 1. 10, 12

12. 2, 3, 9

13. 2, 4, 5

14. 2, 6, 10

15. 2, 7, 12

16. 2, 8, 11

17. 3, 6, 11

18, 3, 7, 10

19. 3, 8, 12

20. 4, 6, 12

21. 4, 8, 9

22. 4, 10, 11

23. 5, 7, 11

24. 5, 8, 10

25. 5, 9, 12

26. 6, 7, 9

Points = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }

Lines:

1. 0, 1, 2

2. 0, 3, 4

3. 0, 5, 6

4. 0, 7, 8

5. 0, 9, 10

6. 0, 11, 12

7. 1, 3, 5

8. 1, 4, 7

9. 1, 6, 8

10. 1, 9, 11

11. 1. 10, 12

12. 2, 3, 9

13. 2, 4, 5

14. 2, 6, 10

15. 2, 7, 11

16. 2, 8, 12

17. 3, 6, 11

18, 3, 7, 12

19. 3, 8, 10

20. 4, 6, 12

21. 4, 8, 9

22. 4, 10, 11

23. 5, 7, 10

24. 5, 8, 11

25. 5, 9, 12

26. 6, 7, 9

Research Exercise:

Can you find a way to relate the two planes above to either an affine or projective plane?