Put an amount on every dot. A network only allows certain patterns of amounts, called modes, and each mode has one number, its swing. In the Zeta Worlds lab every zero comes from one of these swings.
At every dot: the neighbours’ amounts add up to swing × the dot’s own amount.The same swing has to work at every dot at once. Find a pattern where it does, and you have found a mode.
Write each connection as two dot numbers joined by a dash, such as 1-2, 2-3, 3-1. A line like 4: 5 6 7 connects dot 4 to 5, 6 and 7. Bulk forms such as 1(primes) join one dot, or a whole set of dots, to a whole set at once (see below). Dot numbers start at 1, and the lab lays the dots out for you. A world written out from here also carries a layout: line and an amounts: line, so it comes back exactly as you left it. Lines starting with # are notes.
1(primes) joins dot 1 to every prime. primes(primes) joins all the primes to each other, even(odd) every even dot to every odd one, all(x+1) every dot to the next, all(2x) every dot to its double. Inside the brackets you can write a word (primes, composites, even, odd, all, squares, cubes, fib, tri, pow2, twins), a rule such as 3n or 3n+1, a range such as 5-12, a list such as 2,3,5, or x+1, 2x, x^2 for “relative to the dot on the left”; & filters, as in primes(x+2 & primes) for the twin primes. The left side takes the same words, so 3n+1(primes) works too. The words reach as far as the largest dot number in the list; add a line dots: 60 to set it.
| Dot | Amount | Neighbours add to | ÷ amount |
|---|
Pick one to load its amounts onto the dots, then check any dot by hand. A world with n dots has exactly n modes, counting a repeated swing as many times as it appears; a chip marked ×k stands for k modes, and clicking it again steps through them. Amounts are scaled so the largest is 1; doubling every amount gives the same mode.
Guess a swing. Put 1 on the first dot of the line. Each dot’s check then forces the next amount: next = swing × this one − the one before. Only a few guesses make the last dot’s check come out right.
Everything is computed in your browser. The modes are the eigenvectors of the table of connections, and the swing is the matching eigenvalue. The check counts as passing when every dot agrees with the same swing to within the rounding of the amounts typed in. When a swing appears more than once, any mix of its modes is also a mode, so the patterns shown for it are one choice among many.