Reading: pg. 234-296

**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?

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