Math 5392 Finite Geometries I Summer 2004
Dr. Cordero
Additional Problems
I. For Exercises 1-4, state whether a set of axioms could be as given:
- Complete but not independent
- Independent but not complete
- Independent but not consistent
- Consistent but not independent
II. For Exercises 5-8, state an axiom that could be added to the set of axioms given for each geometry so that the set would no longer be independent.
5. Three-point geometry
6. Four-line geometry
7. Fano’s geometry
8. Geometry of Pappus
III. Euclid’s Fifth Postulate
Through a given point C, not on a given line AB, passes exactly one line in the plane not intersecting the given line.
The Characteristic Postulate of Hyperbolic Geometry:
Through a given point C, not on a given line AB, passes more that one line in the plane not intersecting the given line.
The following finite geometry with 13 points and 26 lines has the characteristic postulate of Hyperbolic Geometry:
9. For the geometry given above:
a. Name all the lines through point A that do not have a point in common with line BDF.
b. Name all the lines through point F that do not have a point in common with line ABC.
c. Name all the lines through point B that do not have a point in common with line HJL.
10. Create another finite geometry that has the characteristic postulate of hyperbolic geometry, and prepare a table similar to the one given above.
11. For which of these geometries does Euclid’s fifth postulate always hold?
a. Geometry of Pappus
b. Fano’s geometry
c. Four-line geometry
d. Geometry of Desargues
12. Tell which property for a set of axioms is lacking in a set of axioms if the statement is true:
a. One “axiom” can be proved by using the other axioms.
b. Two contradictory statements can be proved from the axioms.
c. Neither of two contradictory statements involving concepts from the axioms can be proved.