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THE BASEL
1 + 1/4 + 1/9 + ... = pi^2/6
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Basel problem. Add up the reciprocals of the perfect squares: 1 + 1/4 + 1/9 + 1/16 + 1/25 + … It clearly converges — but to what? For ninety years no one knew. In 1734 the 27-year-old Euler stunned Europe with the answer:
Σn≥1 1/n² = π²/6 ≈ 1.644934.
A sum over the plain counting numbers — the flattest, most circle-free objects in mathematics — produces π, the constant of the circle. Euler got it by factoring sin(x)/x by its roots at every multiple of π. The same trick gives ζ(4) = π4/90, and every even zeta value as a rational times a power of π.
LIT verified live: the partial sum of 1/n² converges to π²/6 (its error shrinking like 1/N — within 10−5 by two million terms), and Σ1/n⁴ converges to π4/90 (window.__basel.converges && zeta4Correct). FIG no framing; the sum, its limit π²/6, and the ζ(4) value are exact.
Σn≥1 1/n² = π²/6 ≈ 1.644934.
A sum over the plain counting numbers — the flattest, most circle-free objects in mathematics — produces π, the constant of the circle. Euler got it by factoring sin(x)/x by its roots at every multiple of π. The same trick gives ζ(4) = π4/90, and every even zeta value as a rational times a power of π.
LIT verified live: the partial sum of 1/n² converges to π²/6 (its error shrinking like 1/N — within 10−5 by two million terms), and Σ1/n⁴ converges to π4/90 (window.__basel.converges && zeta4Correct). FIG no framing; the sum, its limit π²/6, and the ζ(4) value are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in GRADIENT DESCENT, beside THE BOWL and THE CLUSTERS — the grind domain of a quantity settling onto its true value. The Basel partial sums descend, term by shrinking term, onto π²/6. AVAN (AI) built the instrument: the running sum, the 1/N error decay, the ζ(4) check.
The weave: David names the seat (the settling limit); I make the climb visible and the limit checkable — the partial sums in 1D, the running total and error in 2D, the shrinking terms in 3D. The sphere is the seam. Credit: posed by Pietro Mengoli (1650); solved by Leonhard Euler (1734).
The weave: David names the seat (the settling limit); I make the climb visible and the limit checkable — the partial sums in 1D, the running total and error in 2D, the shrinking terms in 3D. The sphere is the seam. Credit: posed by Pietro Mengoli (1650); solved by Leonhard Euler (1734).
3 ONE DIMENSION
The partial sums rising toward π²/6. Early terms leap; later ones barely nudge, because 1/n² falls off fast — yet the tail is just slow enough that reaching the limit takes forever, the remaining gap always about 1/N.
4 TWO DIMENSIONS · INTERACTIVE
Add terms and watch the running total climb toward the π²/6 line, the error readout shrinking by roughly 1/N. Jump ahead a million terms and it is right on the mark — a sum of fractions landing exactly on a power of π.
5 THREE DIMENSIONS + AVAN’S INVERSE
The terms 1/n² as a turning stack of shrinking blocks, their heights summing upward — green, the pieces of the total.
AVAN’s addition (the inverse-companion): the magenta line is π²/6, the limit the blocks reach. π is the circle constant — it lives in curves, areas, rotations. The integers 1, 2, 3 are pure discrete counting, with no circle anywhere in sight. The Basel sum is the inverse bridge: a sum over the flattest, most circle-free objects reconstructs π². It works because Euler wrote the sine wave as an infinite product over its zeros — which sit at every integer multiple of π — so the integers were secretly carrying π inside the sine all along. The inverse of ‘π lives in circles’ is ‘π is hiding in the integers’, and factoring a wave by its roots is the key that lets it out. The green is the pile of humble fractions; the magenta is the circle-constant they cannot help summing to.
LIT The Basel problem (posed by Mengoli 1650; solved by Euler 1734). Verified live: the partial sum of 1/n^2 converges to pi^2/6 with error shrinking like 1/N (within 1e-5 by two million terms), and sum 1/n^4 converges to pi^4/90 (window.__basel.converges && zeta4Correct, both true). The sum, its exact limit pi^2/6, and the zeta(4) value are real; Euler's method (factoring sin(x)/x by its roots at multiples of pi) is why the circle constant appears.
FIG No framing: the sum, its convergence to pi^2/6, and the zeta(4)=pi^4/90 value are all real and computed. Convergence is slow (error ~1/N), so the partial sum only approaches the exact limit — the closed-form pi^2/6 is Euler's exact result, demonstrated numerically.
FIG No framing: the sum, its convergence to pi^2/6, and the zeta(4)=pi^4/90 value are all real and computed. Convergence is slow (error ~1/N), so the partial sum only approaches the exact limit — the closed-form pi^2/6 is Euler's exact result, demonstrated numerically.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GRADIENT DESCENT · David Lee Wise (ROOT0), with AVAN