Axiomatic Geometry - Mathematics 360 - Spring 2009

For Homework 8

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  1. There's something wrong with the definition of invariant on pages 40-41. Find it!
  2. For every pair of points z, w in the complex plane, define d(z,w) = |z - w|. We proved in Problem 1 on page 46 that d is an invariant function for Euclidean geometry. What is the "implicitly understood" set of figures on which we're defining d? (That is, for this function d, what is the "D" that's mentioned in the definition of "invariant function" on p. 40-41?)
  3. For every pair of points z, w in the complex plane, define d(z,w) = |z + w|. Answer each of the following questions. (Hint: first think pictorially, then algebraically.)
    1. Is d an invariant function in Euclidean geometry? Prove your answer.
    2. Is d an invariant function in translational geometry? Prove your answer.
    3. Is d an invariant function in rotational geometry? Prove your answer.