Math 3210 Higher Geometry I Fall 2006

Exam I

    Part A : Choose 2 problems (10 points apiece)

  1. Young's geometry (see axioms in problem 8) is not self-dual. Let D denote the geometry dual to Young's geometry. State any theorem that is valid in D which does not require a proof that utilizes the axioms that define D.

    Solution:
    Young's geometry has 9 points and 12 lines. The dual statement, i.e., there exist 12 points and 9 lines, will be a valid theorem in D.

  2. Can the composition of a rotation and a reflection be a translation? Explain your answer.

    Solution:
    No. A rotation is a direct motion and a reflection an opposite motion. The composition of a direct and opposite motion is an opposite motion, i.e., a reflection or a glide reflection.

  3. Find the image of the line 2x - y = 5 under a rotation by 90o about the origin.

    Solution:
    A rotation by 90o about the origin is given by the equations: x' = xCos 90o - ySin 90o = - y and y' = xSin 90o + yCos 90o = x. The image of 2x - y = 5 is therefore 2(y') - (-x') = 5, i.e., x + 2y = 5.

  4. In the geometry PG(2,4) there are 5 points on each line. How would you use this to prove that in this geometry there exist 5 lines through each point?

    Solution:
    PG(2,4) is a self-dual geometry. Since the statements "there are 5 points on each line" and "there are 5 lines through each point" are dual statements, if one is true in PG(2,4) then the other is true as well.

Part B: Choose 4 problems (20 points apiece)

5. In the 4-line geometry, whose axioms are:

  1. There exist exactly four lines.
  2. Any two distinct lines have exactly one point on both of them.
  3. Each point is on exactly two lines.

Prove that there is a pair of points that are not joined by a line.

Solution 1:
By Axiom a, there exist 4 lines. Call these lines, m, n, o and p. By Axiom b, m and n meet at a point which we will call A, and o and p meet at a point which we will call B. Suppose that there was a line joining A and B. By Axiom c, it would have to be either m or n since it contains A, and either o or p since it contains B. But since these lines are distinct, this is impossible. So, A and B are not joined by a line, proving the statement.

Solution 2:
Since the 4-point geometry is the dual geometry of the 4-line geometry, we can prove this statement by proving its dual in the 4-point geometry. The dual statement is, there exists a pair of lines which have no point in common. This was proved in question 1.

6. (a) Write the equations for the glide reflection which consists of a reflection about the line y = x followed by a translation by the vector (-2, -2).
(b) Find the inverse of this glide reflection and write it's equations.

Solution:
(a)Reflection about the line y = x is given by (x,y)(x',y'), where x' = y and y' = x. Translation by (-2,-2) is given by (x',y')(x", y"), where x" = x'-2 and y" = y'-2. So, the equations for the slide reflection (x,y)(x",y") are x" = y -2 and y" = x-2.
(b) The inverse glide reflection sends (x",y")(x,y) and so has equations given by y = x" + 2 and x = y" + 2. Written as (x,y)(x''', y''') we would have x''' = y + 2 and y''' = x + 2.

7. Let f: Z arrowZ be given by f(x) = 5x - 3. (Z is the set of all integers, positive, negative and zero).

  1. Is f an onto function? Prove or give a counterexample.
    Solution:
    No. 1 is in the codomain (Z), but there is no integer x with f(x) = 1.

  2. Is f a one-one function? Prove or give a counterexample.
    Solution:
    Yes. If f(x) = f(y), then 5x - 3 = 5y - 3, so 5x = 5y and therefore x = y. Thus f is a one-one function.

8. In Young's geometry, whose axioms are:

  1. There exists at least one line.
  2. Every line of the geometry has exactly three points on it.
  3. Not all the points are on the same line.
  4. For each two distinct points, there exists exactly one line on both of them.
  5. If a point does not lie on a given line, then there exists exactly one line on that point that does not intersect the given line (Playfair's Axiom).

Prove that two lines parallel to a third line are parallel to each other.

Solution:
Assume that, for distinct lines l, m, n, we have l || m and m || n. If l || n, there is nothing to prove, so assume that l and n meet at a point P. Since P is on l and l || m, P is not on m. We now have a contradiction to Axiom 5 since there are two lines through P that do not intersect m. Thus, we must have l || n.

9. Sketch, if possible, an example of each of the following. If no example exists, give a reason.
(a)Two congruent rectangles such that one can be mapped to the other by both a rotation and a translation.
(b)Two congruent squares such that one cannot be mapped to the other by a translation.
(c)Two congruent circles such that one cannot be mapped to the other by a reflection or a translation.

Solution:

(a)

(b)

(c) No possible example, two congruent circles can always be mapped to each other by a translation that sends the center of one to the center of the other or a reflection through the line which is the perpendicular bisector of the line segment which joins the centers of the circles.

10. The group of symmetries of a rectangle which is not a square has four elements. Describe these elements in cyclic notation and write out the operation table for this group.

Solution: