When z is a nonzero complex number z = r exp(i*theta) and alpha is any complex number, then classically: z^alpha is defined to be the set of complex values given by (*) {exp(alpha * log(z)) where log(z) takes exactly all the values of the form {c + 2K*pi*i | K in Z} for some complex number c with exp(c) = z, e.g., c = ln(r) + i*theta . Now (*) exp(alpha * (ln(r) + i*theta). Let alpha = a + bi for a, b real. Then we have z^alpha = {exp((a + bi)*(ln(r) + i*theta)) = {exp(a*ln(r) - b*theta) + i(a*theta + b*ln(r))) = {exp(a*ln(r) - b*theta_0) i(a*theta_0 + b*ln(r))) * exp(2pi*(-b + a*i))^K where (say) -pi < theta_0 <= pi defines theta_0 uniquely, so the last equation reads: z^alpha = {u * v^K | K in Z} for complex numbers u, v with u = exp(a*ln(r) - b*theta_0) i(a*theta_0 + b*ln(r))), and v = exp(2pi*(-b + a*i)). Now, when people talk about i^i^i^i^.... they usually mean by taking the parentheses from the top down — and using the principal logarithm: i^z = exp((pi*i/2)*z) and iterating: i^...^z = i^(i^(...^(i^z)...)) = exp((pi*i/2)*(exp((pi*i/2)*...*exp((pi*i/2)z)...)) But what if we *didn't* restrict to using the principal logarithm, but instead considered *all* the determinations of i^previous stuff each time we exponentiated. Then at each stage there will be countably many determinations. Now (as usual) let the *number of exponentiations* approach oo. QUESTION: What are the limit points of the determinations: -------- Those points z of C that are *limits* of any sequence of points of the form z_n = something born at exactly n exponentiations, i.e., such that for every epsilon > 0 there exists N such that there is a point z_n that is a result of n > N exponentiations with |z - z_n| < epsilon. —Dan