A ball bounces inside a circle, going in chords of 2pi/phi per leg, i.e., golden ratio of the circumference.
The path never repeats because of irrationality; however, to accurately depict a large number of bounces requires a high precision calculation of phi. Create an algorithm which uses a spigot algorithm, so the simulation can run indefinitely without having to calculate all the digits initially. Fibonacci numbers might be involved. Can the running time remain constant per bounce?
Draw the path in the style of pi paths, with paths going under. It will be a space-filling curve (with intersections).
The ball never hits the same point on the circumference twice. An irrational number can be completely specified by a discrete sequence of old points each new bounce falls in between, specified by index number. How computationally difficult is it to compute this sequence?
How can we reconstruct the number from a sequence? Maintain a bounds from a whole bunch of inequalities. Knowing bounds, perhaps some bounces may be losslessly left unspecified as data compression.
No comments :
Post a Comment