# Homework #3

1. (Refer to pg. 175 of the text.) Verify that both binary expansions given for .5 are correct. Give an example of a point with rational coordinates which is in the Sierpinski gasket (but which is not on one of the three bounding line segments), and an example of a point with irrational coordinates which is in the Sierpinski gasket (but which is not on one of the three bounding line segments).

2. (Refer to pg. 175 of the text.) Consider the Hutchinson operator for the Sierpinski gasket with vertices (0,0), (1,0), and (0,1). Apply it to the point (0,0); apply it again; apply it once more. Plot the results.

3. Sketch a typical ellipse in the real plane with the taxicab metric. (An ellipse is the set of all points the sum of whose distances from two given points, its foci, is a constant.) Are all circles ellipses in this metric space?

4. Prove that the sup (or maximum) metric is a metric on the real plane.

5. Find all fixed points of the sine and cosine functions. What happens for the tangent function?