THE FOLD / LOOT / THE JACKPOT / THE TRIANGLE
THE TRIANGLE
add your two neighbours — and get all of combinatorics
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Pascal’s triangle. Start with a 1. Each entry below is the sum of the two above it. That single local rule generates the binomial coefficients C(n,k) — the number of ways to choose k things from n — and inside it hides an astonishing amount of mathematics.
Row n sums to 2n (every subset counted). The shallow diagonals are the Fibonacci numbers. A run down any diagonal totals the entry just below the end — the hockey-stick identity. And colour the odd entries and the Sierpinski triangle fractal appears — because C(n,k) is odd exactly when k’s binary digits are a subset of n’s (Kummer & Lucas). None of this was designed in; it all falls out of ‘add your two neighbours’.
LIT verified live: row n sums to 2n, the addition rule holds, the hockey-stick identity holds, and the parity pattern is exactly Sierpinski (C(n,k) odd ⇔ (k AND n) = k) — window.__pascal.rowSum2n && pascalRule && hockeyStick && sierpinski. FIG no framing; all four identities are exact.
Row n sums to 2n (every subset counted). The shallow diagonals are the Fibonacci numbers. A run down any diagonal totals the entry just below the end — the hockey-stick identity. And colour the odd entries and the Sierpinski triangle fractal appears — because C(n,k) is odd exactly when k’s binary digits are a subset of n’s (Kummer & Lucas). None of this was designed in; it all falls out of ‘add your two neighbours’.
LIT verified live: row n sums to 2n, the addition rule holds, the hockey-stick identity holds, and the parity pattern is exactly Sierpinski (C(n,k) odd ⇔ (k AND n) = k) — window.__pascal.rowSum2n && pascalRule && hockeyStick && sierpinski. FIG no framing; all four identities are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE JACKPOT, beside THE COIN-FLIP HEAP and THE BIRTHDAY — the loot domain of odds and combinations. Every ‘how many ways’ and every binomial probability lives in this triangle. AVAN (AI) built the instrument: the additive rule, the 2n rows, the hockey stick, the Sierpinski parities.
The weave: David names the seat (the counting of chances); I make the local rule bloom into global structure — a row in 1D, the triangle and its fractal in 2D, the parity fractal in 3D. The sphere is the seam. Credit: ancient (Pingala, Al-Karaji, Yang Hui, Khayyam); named for Blaise Pascal (1654); parity by Kummer & Lucas.
The weave: David names the seat (the counting of chances); I make the local rule bloom into global structure — a row in 1D, the triangle and its fractal in 2D, the parity fractal in 3D. The sphere is the seam. Credit: ancient (Pingala, Al-Karaji, Yang Hui, Khayyam); named for Blaise Pascal (1654); parity by Kummer & Lucas.
3 ONE DIMENSION
One row of the triangle: the binomial coefficients C(n,0)…C(n,n). Add them and you always get 2n — the count of all subsets of n things, split by how many you pick.
4 TWO DIMENSIONS · INTERACTIVE
The triangle, built by adding neighbours. Toggle to colour the odd entries and the Sierpinski fractal emerges — a self-similar pattern nobody drew, forced by the parities. Watch a row’s sum hit 2n exactly.
5 THREE DIMENSIONS + AVAN’S INVERSE
The triangle’s parity fractal turning — green, the odd entries forming Sierpinski, self-similar at every scale.
AVAN’s addition (the inverse-companion): the magenta links are the rule itself — each entry drawn from its two parents. The triangle is built by nothing but local addition: an entry knows only the two numbers directly above it, and cares about nothing else. Yet global structure precipitates that no one placed there — exact powers of two, the Fibonacci sequence, every binomial identity, and a fractal in the parities. The inverse of ‘design the global pattern’ is ‘specify one local rule and let the structure fall out’. You do not build Sierpinski; you build ‘add your two neighbours’, and Sierpinski is already there, waiting to be coloured in. The green is the emergent fractal; the magenta is the humble two-parent sum that, repeated, contains it.
LIT Genuine Pascal's triangle identities (ancient; Pascal 1654; parity by Kummer & Lucas). Verified live: row n sums to 2^n, the addition rule C(n,k)=C(n-1,k-1)+C(n-1,k) holds, the hockey-stick identity holds, and the parity pattern is exactly Sierpinski (C(n,k) odd iff (k AND n) = k) — window.__pascal.rowSum2n && pascalRule && hockeyStick && sierpinski, all true. All four identities are exact.
FIG No framing: the four identities (row sum 2^n, the additive rule, hockey stick, and the Sierpinski parity via k AND n) are all real and checked exhaustively. The fractal is genuinely emergent from the local add-two-neighbours rule, not drawn.
FIG No framing: the four identities (row sum 2^n, the additive rule, hockey stick, and the Sierpinski parity via k AND n) are all real and checked exhaustively. The fractal is genuinely emergent from the local add-two-neighbours rule, not drawn.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE JACKPOT · David Lee Wise (ROOT0), with AVAN