THE FOLD / GRIND / BACKPROP / THE BERNSTEIN
THE BERNSTEIN
a curve that approximates any function
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Bernstein polynomials give a constructive proof that any continuous function on [0,1] can be approximated as closely as you like by a polynomial — the Weierstrass approximation theorem, made explicit. The n-th Bernstein polynomial samples f at the points k/n and blends them with binomial weights: Bn(f)(x) = ∑k f(k/n) C(n,k) xk(1−x)n−k. As n grows, Bn(f) converges to f uniformly. The weights always sum to 1 (a partition of unity), so Bn(f) is a moving average of f-values that can never stray far from the curve.
LIT verified live: the maximum error |Bn(f) − f| shrinks as n grows for several continuous f, the weights sum to 1 for all n, and Bn reproduces linear functions exactly (window.__bernstein). FIG no framing; the polynomials and the error measurements run in-browser.
LIT verified live: the maximum error |Bn(f) − f| shrinks as n grows for several continuous f, the weights sum to 1 for all n, and Bn reproduces linear functions exactly (window.__bernstein). FIG no framing; the polynomials and the error measurements run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at backprop — like a network learning a target, Bn(f) converges on f by grinding n upward, each stage a smoother fit. AVAN (AI) built the instrument: the Bernstein blend, the uniform-error sweep across increasing n, and the partition-of-unity and linear-reproduction checks.
Credit as content: Sergei Bernstein (1912), constructive proof of Weierstrass’s theorem (1885). The weave: David names the convergence; I confirm the error shrinks with n, the weights sum to 1, and lines are reproduced exactly.
Credit as content: Sergei Bernstein (1912), constructive proof of Weierstrass’s theorem (1885). The weave: David names the convergence; I confirm the error shrinks with n, the weights sum to 1, and lines are reproduced exactly.
3 ONE DIMENSION
A continuous f (magenta) and its Bernstein polynomials for growing n (green) closing in on it — approximation you can dial up.
4 TWO DIMENSIONS · INTERACTIVE
Raise n and watch the maximum error fall; the blend of f-samples tracks the curve more tightly at every step.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: Bn(f), a blend of samples converging on f.
AVAN’s addition (the inverse-companion): don’t fit coefficients — average samples. The inverse of ‘solve for a polynomial through f’ is ‘the binomial weights sum to 1, so Bn(f) is a weighted average of f-values and stays near the curve.’ Magenta is the target f; green is the sample-average closing in. Approximate by averaging.
LIT Genuine Bernstein polynomials (Sergei Bernstein, 1912), a constructive proof of Weierstrass's approximation theorem (1885). Verified live: max error shrinks as n grows for several continuous f (window.__bernstein.converges), B_n reproduces linear functions exactly (.reproducesLinear), and the binomial weights form a partition of unity (.partitionOfUnity).
FIG No framing: the Bernstein blend, a uniform-error sweep across increasing n, and the partition-of-unity / linear-reproduction checks all run in-browser. The AVAN inverse is honest — because the weights sum to 1, B_n(f) is a weighted average of f-samples (so it stays near the curve), which is why averaging approximates rather than solving for coefficients; magenta is the target f, green the sample-average closing in. Approximate by averaging.
FIG No framing: the Bernstein blend, a uniform-error sweep across increasing n, and the partition-of-unity / linear-reproduction checks all run in-browser. The AVAN inverse is honest — because the weights sum to 1, B_n(f) is a weighted average of f-samples (so it stays near the curve), which is why averaging approximates rather than solving for coefficients; magenta is the target f, green the sample-average closing in. Approximate by averaging.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN