The Collatz orbit of 27
collatz is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
With nothing chosen
The share that provably comes down
The first 5 steps, decided by the start
The tree above one, 9 levels up
Every parity pattern of length up to 12, and each occurring exactly once
64 patterns, 64 rational cycles, 2 of them whole
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- step 1 follows the rule ×178
- every one of the 2 patterns of length 1 occurs exactly once ×12
- the orbit of 27 reaches one ×7
- −1 is a fixed point of T ×1
- a positive starting number ×1
- a power of two is never a power of three ×1
- a starting number is a whole number between 3 and 1000000 ×1
- across a large sample of starts, half the steps are odd ones ×1
- an integer agreeing with −1 to k digits takes k odd steps first ×1
- and 27 is the famous long one ×1
- and by the largest run drawn it is above nine tenths ×1
- and it is the one at one ×1
- and it then comes down to 1 all the same ×1
- and no two residues give the same parity string ×1
- and reaches a few thousandths within the range drawn ×1
- and the average fall per step is the value the pairing predicts ×1
- at least one pattern gives a whole number ×1
- between two and four odd offsets, each at most eleven ×1
- each convergent beats every fraction with a smaller denominator ×1
- each new digit of x is decided by the next parity ×1
- each row's parity string is the row above shifted left by one ×1
- every number in the program factors over the first ten primes ×1
- every number the reverse tree reaches does come back to one ×1
- every number up to 2,000 is hit ×1
- every orbit trends downwards in the logarithm ×1
- every pattern of the length was solved ×1
- every peak is held exactly in double precision ×1
- every positive whole number has exactly one preimage ×1
- in parity coordinates T is the doubling map ×1
- it halts at 5 to the power a·b and nothing else ×1
- its size after k steps is 3^(odd steps) / 2^k times the start, to within rounding ×1
- most parity strings shrink the number by the end of the run ×1
- no pattern with a negative denominator gives a positive whole solution ×1
- no two numbers go to the same place ×1
- one is already at one ×1
- one length per start ×1
- past a hundred thousand every record's exponent lies between 1.7 and 2.1 ×1
- path records arrive at a steady rate per decade ×1
- Q sends the numbers below 2^K to all 2^K parity strings, each once ×1
- several cycles are found ×1
- six to thirteen binary digits ×1
- small factors ×1
- so the residues and the parity strings match one for one ×1
- the average of the γ-th power of the step is one at γ = 0 and γ = 1 ×1
- the chance of climbing to x times the start falls like 1/x ×1
- the continued fraction of log₂3 starts 1, 1, 1, 2 ×1
- the denominator is always odd, so the solution stays where the map is defined ×1
- the depth of the tree is a whole number between 3 and 12 ×1
- the exponents of the powers of two are the primes, in order ×1
- the first path records are 3, 7, 15, 27, 255, 447, 639, 703, … ×1
- the integer follows its pattern for all k steps ×1
- the largest number of steps is a whole number between 4 and 15 ×1
- the largest start is a whole number between 20 and 4000 ×1
- the longest pattern length checked is a whole number between 6 and 14 ×1
- the longest pattern searched is a whole number between 8 and 20 ×1
- the map is a bijection at every length checked ×1
- the multiplication program halts ×1
- the multiplier condition predicts the drop, on a large member of the class ×1
- the number of convergents listed is a whole number between 5 and 10 ×1
- the number of steps is a whole number between 3 and 7 ×1
- the orbit of 8 has not closed up within the steps drawn ×1
- the orbit's peak is the record ×1
- the ordinary rule has exactly one cycle at these lengths ×1
- the ordinary rule is among those searched ×1
- the parity of the first steps depends only on the start modulo two to the k ×1
- the parity vector of T(x) is the parity vector of x shifted by one ×1
- the pattern length is a whole number between 4 and 10 ×1
- the relative gap shrinks along the list ×1
- the rule with minus one has at least three, which no drift argument distinguishes ×1
- the run reaches several powers of two ×1
- the share provably driven down grows with the number of steps allowed ×1
- the starting number is a whole number between 1 and 100000 ×1
- the tree is still growing at the depth drawn ×1
- the view is one the family draws ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
Almost every number comes down
The Collatz conjecture is open and a great deal about it is not. Whether a number falls below its own start in the first few steps is decided entirely by its remainder on division by a power of two, and the share of numbers for which it happens can be counted exactly.
DynamicsEvery pattern happens exactly once
Choose any sequence of odds and evens and there is exactly one residue class whose orbit follows it, and exactly one fraction that cycles through it forever. The Collatz conjecture is then the statement that only one of those infinitely many cycles is made of whole numbers.
DynamicsFourteen fractions that list the primes
Change the Collatz rule so that the multiplier depends on the remainder modulo some other number than two, and the resulting maps can compute anything a computer can. John Conway proved it in 1972, which means no method can decide, for every such map, whether its orbits come down — and fourteen fractions, applied in order, turn out to be enough to print every prime.
DynamicsHow 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.
DynamicsHow short a cycle could be
The drift argument cannot see cycles at all, which is why it is not a proof. What can see them is arithmetic — a cycle's shape has to be a fraction that approximates the logarithm of three to base two extraordinarily well, and there are very few such fractions.
DynamicsThe heuristic that cannot be a proof
There is a two-line argument that the Collatz conjecture is true, it is convincing, and everybody who works on the problem believes it. It also cannot be turned into a proof, and understanding exactly where it fails is more instructive than the argument itself.
DynamicsThe question nobody can answer
Halve it if it is even, triple it and add one if it is odd. Every number anyone has tried comes down to one. Nobody can prove they all do, and the reason is not that the problem is hard to state.
DynamicsWhere the Collatz map is a coin
Extend the Collatz map from the whole numbers to the 2-adic integers — binary strings that run on for ever to the left — and it stops being mysterious. It becomes, after a change of coordinates, the simplest chaotic system there is: shifting a string of coin tosses one place. Everything about it is then known, and none of it says anything about the whole numbers, which is the most instructive failure in the whole story.