The Collatz rule is one member of a family: n → qn + c when odd,
halve when even. Change one constant and the species changes completely. Some worlds funnel every
number into a single drain, some split the integers between rival whirlpools, and some fling almost
everything into the sky forever. Comparing them shows exactly how lucky 3n + 1 is.
A companion to the Hailstone Atlas.
The one number that decides a species' temperament: each tripling (or quintupling)
is followed on average by two halvings, so a typical flight gains log₂q − 2
bits per odd step. Negative drift and the storm settles; positive drift and it never comes back down.
One cell per starting number, 1 through 3,360, reading like text. The color is the seed's destiny: which cycle it eventually falls into, or escape if it climbs past 245 and never looks back. Click any cell to trace its full flight below.
The full orbit of one seed on a log scale, drawn with amber odd steps and cyan even steps. Once the orbit falls into its final cycle, the loop is tinted with that attractor's color. Escaping orbits are cut off when they pass 245, still climbing.