You are the conductor of a trolley in the complex plane, surrounded on all sides by people, stretching out to infinity, covering the entire region | z | ≥ 5. The trolley will start at 0 and move iteratively, its next position calculated as f(z) = z^2 + c, where z is its current position. C is arbitrary. You may set it to any complex number, but cannot change it once the trolley starts moving. However, the people inside the trolley are freezing and rely on the heat generated from the trolley's movement to survive! The faster the trolley moves (on average over the entire trip), the more survive.What value of c will you choose?
it's not c = i btw
>>16966254I would guess it is a point on the edge of the mandelbrot set?
>>16966281It certainly needs to be a point on the set, but it also needs to be one that keeps the trolley moving forever and at a path whose points are maximally far from each other.
>>16966254-1+0.1iIt's not perfect but it's good enough. I'm not spending too much time trying to maximize how many "people" I save on the complex plane. Pretty sure they're demons anyway.
>>16966284c=100 kill em all v^v^v^v^v^v^v
You guys are all failing the challenge
Y'all really incapable of solving this?
2.3
>>16966564>>16966731The problem really just isnt that interesting. Basically just do what this anon did: >>16966295Except you make the real portion arbitrarily close to -sqrt(5-sqrt(5)) and the imaginary portion arbitrarily close to 0. Anywhere in that line emitting from the turtle's head bounces back and forth crossing both axes.
>>16966943>>16966943But you're both wrong, the goal of the problem is not to find a closed loop, it is to find the one with the maximum average speed, i.e. distance between points. And no, it's not c=-1+0.1i, nor is it c=-sqrt(5-sqrt(5)), nor is it c=i.Here's the problem properly restated so you will understand: Among all admissible parameters c, define the asymptotic average speed byA(c) = limsupₙ∞ (1/N) Σₙ=0ᴺ-1 |zₙ+1 − zₙ|Question:What is sup A(c) over all admissible c?Is this supremum actually attained by some parameter c? If so, for which value or values of c?Even the simplest class of possible parameters, which are postcritically finite Misiurewicz parameters where the critical orbit z0 = 0 eventually lands on a periodic cycle, easily yields a solution that beats all suggestions made in this thread so far:If the orbit eventually enters a period-p cycle w0 w1 ... wₚ-1 w0, the asymptotic average speed is justA(c) = (1/p) Σⱼ=0ᵖ-1 |wⱼ+1 − wⱼ|.Try the simplest nontrivial case: eventual period 2.Let a b be a genuine 2-cycle of f_c(z) = z2 + c. Thenb = a2 + ca = b2 + c.Subtracting givesb − a = a2 − b2 = (a − b)(a + b).Since a ≠ b, this impliesa + b = −1.Thus the two points of every genuine 2-cycle are the roots ofz2 + z + c + 1 = 0.Their separation is therefore|a − b| = sqrt(|4c + 3|).Hence, if the critical orbit eventually lands on this 2-cycle, thenA(c) = sqrt(|4c + 3|).The landing condition is:z_m2 + z_m + c + 1 = 0,where z0 = 0 and zₙ+1 = zₙ2 + c.For each m, this gives an algebraic equation in c. The early values include:m = 2: c = ±i, giving A(c) = sqrt(5) ≈ 2.2360679775.m = 3: a root of c3 + c2 − c + 1 = 0, giving A(c) ≈ 2.2955977425.m = 5: a root of the primitive period-2 landing factor of degree 15, givingc ≈ 0.3837394256 ± 0.6817392283i.
>>16966943By contrast, your suggestion of c=-sqrt(5-sqrt(5)), yields about A(c)≈1.52.Your trolley is poorly heated asf.
