31. (c) The distance between the first period-doubling bifurcation (the appearance of the 2-cycle) and the second period-doubling bifurcation (the appearance of the 4-cycle) is 34.5 squares. Between the second and the third, the distance is 7.5 squares; between the third and the fourth, the distance is 1.5 squares. Using this data, we approximate the Feigenbaum constant by (34.5 - 7.5)/(7.5 - 1.5), approximatley 4.5

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