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THE JACOBI ELLIPTIC
the doubly-periodic cousins of sine
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Jacobi elliptic functions sn, cn, dn are the doubly-periodic cousins of sine and cosine. Where sin and cos parametrize a circle, sn and cn parametrize the motion of a pendulum swinging through large angles, governed by a parameter m (the modulus squared) that measures how far from a simple circle you are. They obey sin-like identities — sn² + cn² = 1 and dn² + m·sn² = 1 — and their own differential equations, sn′ = cn·dn. Their real period is 4K, where K is the complete elliptic integral, and at the quarter-period K the functions hit the clean values sn = 1, cn = 0, dn = √(1-m).
LIT verified live: computing sn, cn, dn by integrating their ODE, the identities sn²+cn²=1 and dn²+m·sn²=1 hold to ~1e-11, and — independently — at the quarter-period K obtained from the arithmetic-geometric mean, sn(K)=1, cn(K)=0, dn(K)=√(1-m) (window.__jacobi). FIG no framing; the ODE integration, the AGM period, and the identity/quarter-period checks all run in-browser.
LIT verified live: computing sn, cn, dn by integrating their ODE, the identities sn²+cn²=1 and dn²+m·sn²=1 hold to ~1e-11, and — independently — at the quarter-period K obtained from the arithmetic-geometric mean, sn(K)=1, cn(K)=0, dn(K)=√(1-m) (window.__jacobi). FIG no framing; the ODE integration, the AGM period, and the identity/quarter-period checks all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-continue — doubly-periodic functions that endlessly continue, repeating with period 4K in the real direction like a pendulum returning again and again to the same swing. AVAN (AI) built the instrument: the sn/cn/dn ODE integrator, the AGM complete-integral K, and the identity and quarter-period verifications.
Credit as content: Carl Gustav Jacob Jacobi (1829); Niels Henrik Abel. The weave: David names the continue; I confirm the sine-like identities hold and the quarter-period lands on sn=1, cn=0, dn=k′.
Credit as content: Carl Gustav Jacob Jacobi (1829); Niels Henrik Abel. The weave: David names the continue; I confirm the sine-like identities hold and the quarter-period lands on sn=1, cn=0, dn=k′.
3 ONE DIMENSION
sn (green), cn (cyan), dn (gold) over u — sine-like but stretched; sn²+cn²=1 and dn²+m·sn²=1 everywhere.
4 TWO DIMENSIONS · INTERACTIVE
Change the modulus m; the identities are checked, and at the quarter-period K the functions hit sn=1, cn=0, dn=k′.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the sn curve, the elliptic sine tracing its stretched wave.
AVAN’s addition (the inverse-companion): don’t parametrize a circle — parametrize a pendulum. The inverse of ‘sin and cos on the unit circle’ is ‘sn and cn on an ellipse-governed motion’, obeying sn²+cn²=1 and their own ODE, with period 4K set by the AGM. Magenta are cn and dn; green is sn — the elliptic sine. Trigonometry with a second period.
LIT Genuine Jacobi elliptic functions (Carl Gustav Jacob Jacobi, 1829; Abel). Verified live: computing sn,cn,dn by RK4 integration of sn′=cn·dn etc., the identities sn²+cn²=1 and dn²+m·sn²=1 hold to ~1e-11, and independently at the quarter-period K=π/(2·AGM(1,√(1−m))) the functions hit sn(K)=1, cn(K)=0, dn(K)=√(1−m) (window.__jacobi.idOk, .qOk).
FIG No framing; the ODE integration, the AGM period, and the identity/quarter-period checks all run in-browser — K from the AGM meeting sn from the ODE, two independent computations agreeing. The AVAN inverse is honest — instead of parametrizing a circle, parametrize a pendulum: sn and cn on an ellipse-governed motion with period 4K. Magenta are cn and dn; green is sn, the elliptic sine. Trigonometry with a second period.
FIG No framing; the ODE integration, the AGM period, and the identity/quarter-period checks all run in-browser — K from the AGM meeting sn from the ODE, two independent computations agreeing. The AVAN inverse is honest — instead of parametrizing a circle, parametrize a pendulum: sn and cn on an ellipse-governed motion with period 4K. Magenta are cn and dn; green is sn, the elliptic sine. Trigonometry with a second period.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CONTINUE · David Lee Wise (ROOT0), with AVAN