Mathematics has a bad hair day – New Scientist

Seaweed? Worms? Bad hair day? The animation below is a new way to look at the Collatz conjecture, one of the most high-profile open problems in mathematics.

GrowCollatzs

Edmund Harriss

The conjecture is easy to explain, given that all you need to know is how to add 1, how to divide by 2, and how to multiply by 3. Such simplicity, however, stands in striking contrast to the difficulty of proving the conjecture itself. “Mathematics is not yet ready for such problems,” the late Paul Erdös said, and the animation gives an insight into why it is such a tough nut to crack.


To begin: take a number, any number, and apply this rule:

  • If the number is even, halve it;
  • If the number is odd, triple it, add one, and halve it;
  • Repeat.

Let’s get going with 13:

  • 13 is odd, so we triple it to get 39, add one to get 40, and halve it to get 20;
  • Now we repeat with 20, which is even, so we halve it to get 10;
  • 10 is even, so we halve it again to get 5;
  • 5 is odd, so we triple it and add one, halve the result, and get 8.

The full sequence from 13 looks like this:

13 → 20 → 10 → 5 → 8 → 4 → 2 → 1

The Collatz conjecture is the statement, first made by the German mathematician Lothar Collatz in 1937, that whatever number you start with, by applying these rules you will always, eventually, end up at 1.

Most mathematicians believe he was correct. Computers have checked it out up to pretty high numbers and found no counterexamples – but no one has a clue how to prove it.

The left-hand image below shows the sort of tree diagram you get in books or papers that discuss the conjecture. For each number in the tree – look at the 13, for example – the numbers along the branch leading downwards are those in its Collatz sequence. Since all numbers investigated so far reach 2 (and then 1), all the numbers are branches in the same tree.

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Edmund Harriss

To illustrate why the problem is so complicated, mathematical artist Edmund Harriss at the University of Arkansas in Fayetteville took the standard diagram, but added a rule for the direction of the branch (illustrated above, centre), to reflect which half of the algorithm was operating to generate the number.

This wrinkle reveals how unpredictable the path can be from a given number to 1. Whichever number you are at, if the one above it in the tree is double that value, the direction of the branch turns at a fixed clockwise angle – and if not, it turns at a fixed anticlockwise angle. In the right-hand image, the numbers have been taken out and the lines thickened.

The animation shows how the tree grows when it includes every number under 10,000. It displays brilliantly how a simple, orderly rule can create havoc: the structure looks organic and wild. Harriss’s image makes us understand instantly why solving the Collatz conjecture is so hard.

The image of the Collatz conjecture comes from Visions of Numberland: A colouring journey through the mysteries of maths, by Alex Bellos and Edmund Harriss (published in the US as Visions of the Universe)

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