Interactive · Prime differences
The gaps, the gaps between gaps, and on down.
Start with the primes. The first bar row below them is the difference between each pair — the prime gaps. The next row is the difference between those. The next, the difference between those. Stack as many levels as you like and watch whether the pattern ever calms down. (It doesn't.)
20 primes40 primes80 primes160 primes300 primes
primes
Δ gaps between primes
ΔΔ 2nd difference
ΔΔΔ 3rd difference
The top row is the primes themselves. Each row below is the difference between neighbours in the row above: row 2 is the prime gaps, row 3 the gaps-between-gaps, and so on. Gold bars are positive, red bars negative. Notice the gaps are all even after the first (2→3), and the higher differences oscillate around zero without ever settling — primes have no polynomial rule, so the differences never flatten to a constant the way they would for, say, squares.
For a polynomial sequence like n² the differences flatten to a constant after a few levels. The primes never do — every level keeps jittering, a small visual echo of the fact that primes obey no simple closed-form rule. See primes for the raw dots.