Dynamics — page 5
How high an orbit climbs
The Collatz orbit of 27 climbs to 9,232 before it falls, and below ten million the champion, 6,631,675, climbs to sixty trillion — almost exactly the square of where it started. That is not a coincidence of the two numbers but a law of the whole family: an orbit climbs to x times its start with a chance of about one in x, and the exponent is one because half of three-halves plus half of a half is exactly one. Among N starts the luckiest climbs about N-fold, and so the records grow like the square of the start.
Two constants at the flat point
Turn up the circle map until its graph goes flat at one point, then hunt for the setting at which an orbit turns by the golden ratio. The settings that lock to 1/2, 2/3, 3/5, 5/8 close in on it, each gap 2.8336 times smaller than the last, and the orbit's returns to its start close in by 1.2886 at every step. Below the flat line the same two numbers are the golden ratio's own. Change the map to a different one with the same kind of flat point and both numbers come back unchanged: they belong to flatness, not to the sine.