Show that the four similitudes T1 x y = 3 4 1 0 0 1 x y T2 x y = 3 4 1 0 0 1 x y + 1 4 0

Chapter 10, Problem 8

(choose chapter or problem)

Show that the four similitudes T1 x y = 3 4 1 0 0 1 x y T2 x y = 3 4 1 0 0 1 x y + 1 4 0 T3 x y = 3 4 1 0 0 1 x y + 0 1 4 T4 x y = 3 4 1 0 0 1 x y + 1 4 1 4 express the unit squar express the unit square as the union of four overlapping squares. Evaluate the right-hand side of Equation (2) for the values of k and s determined by these similitudes, and show that the result is not the correct value of the Hausdorff dimension of the unit square. [Note: This exercise shows the necessity of the nonoverlapping condition in the definition of a self-similar set and its Hausdorff dimension.]