Progress Report 1:

 

Fixed points and fixed lines

 

Introduction and instructions: This project will investigate the relation between the properties of reflections and the parallel postulate assumed. As a warm-up activity, this progress report asks you to examine reflections on the Lenart sphere with respect to fixed points and lines (defined below).

Definition: Let M be a mapping from the plane to itself. A point P  is called a fixed point of M if M(P ) = P  and a line b is called a fixed line of M if M(b) = b.

Questions

  1. Define a mapping on the Lenart sphere that satisfies all of the criteria for a reflection and M.1 and M.2. (Hint: Try to mimic the action of the MIRA. How would you do a reflection with just a ruler and protractor?)

  2. Is every point of a fixed line a fixed point? Justify your answer.

  3. Find all of the fixed points and fixed lines for Mira/GSP reflections and for reflections on the Lenart sphere.

  4. Using only the provided axioms and definitions, give complete proofs of the following propositions:

    1. If P and Q are fixed points of Ra, then the line determined by P and Q is a fixed line of Ra.

    2. If l and m intersect at a point P and are different fixed lines of a reflection Ra, then P is a fixed point of Ra .

    3. Every perpendicular of a line a is fixed by the reflection Ra.

  5. Which member of your group travelled the farthest from Athens over Spring break? How far did they go?