Math 455
Test I
Name_______________________
I. Consider the axioms below for an finite geometry.
Axiom 1.
Axiom 2.
Axiom 3.
There exist exactly three points.
Each two distinct points share at least one line
Each point is on exactly two distinct lines.
1. Prove or disprove the geometry is consistent.
2. Is Axiom 1 independent of the others? Could a different geometry be constructed for every natural
number in place of “three” in Axiom 1? Prove or disprove.
3. Do any of the geometries in question 2 have parallel lines if parallel lines are defined as those lines
with no point in common?
4. Prove or disprove: Each two distinct points share exactly one line.
5. Prove or disprove: Given a line L and a point P not on L, there is a unique line parallel to l.
6. Definition: An incidence geometry that has no parallel lines is called a projective geometry.
Provided Axiom 1 could specify any number of points, which resulting geometries would be
projective?
II. Suppose we remove the finiteness Axiom above. The remaining two axioms qualify for an incidence
geometry (one where an infinite number of points are possible).
1. Are the remaining two axioms independent? Prove or disprove.
2. Definition: An incidence geometry that exhibits Euclid’s Parallel Postulate—see problem 5 in part
I—is called an affine geometry. Is this new two-axiom geometry an affine geometry? Prove or
disprove.
3. A geometry with no parallel lines is called projective. Give an example of a geometry that is
a) non-trivial, has a finite number of points and is projective;
b) is non-trivial, has an infinite number of points and is projective.
III. Find a formula for the volume of a portion of a Ball Br of radius r bounded by two planes through the center
of Br and the luna between the great circles generated by the planes. Your formula should be based on the angle
between the great circles.
IV. Derive the volume for a sphere of radius r by either method of calculus: slicing or revolution.
V. Use integration and substitution methods to find the area under the semicircle of radius r. SHOW ALL
STEPS!