THE FOLD / CHEAT / THE KONAMI CODE / THE MORRIE LAW
THE MORRIE LAW
three cosines multiplying to exactly one eighth
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Morrie’s law is the identity cos 20° · cos 40° · cos 80° = 1/8 — three unremarkable-looking cosines multiplying to an exact rational. Richard Feynman kept the name all his life: a boy called Morrie Jacobs showed it to him in his father’s leather shop. The secret is the doubling cascade: for any θ, ∏k=0n-1 cos(2kθ) = sin(2nθ) / (2n sin θ) — each cosine doubles the angle via sin 2x = 2 sin x cos x, and the product telescopes. At θ = 20° the cascade lands on sin 160°, which equals sin 20° exactly — the sines cancel and only 1/2³ = 1/8 survives.
LIT verified live: cos 20°·cos 40°·cos 80° = 0.125 to machine precision; the telescoping identity holds for thousands of random θ and n (worst error ~1e-16); and sin 160° = sin 20° exactly (window.__morrie). FIG no framing; the product and the closed form are computed independently in-browser.
LIT verified live: cos 20°·cos 40°·cos 80° = 0.125 to machine precision; the telescoping identity holds for thousands of random θ and n (worst error ~1e-16); and sin 160° = sin 20° exactly (window.__morrie). FIG no framing; the product and the closed form are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-konami-code — the cheat: enter the right angle and the whole messy product collapses to a clean 1/8, no trigonometry tables needed. AVAN (AI) built the instrument: the product, the telescoping closed form, and the sin 160° = sin 20° cancellation.
Credit as content: Morrie Jacobs (the boy who found it), Richard Feynman (who named it and never forgot it). The weave: David names the cheat code; I confirm the cascade collapses to exactly 1/8.
Credit as content: Morrie Jacobs (the boy who found it), Richard Feynman (who named it and never forgot it). The weave: David names the cheat code; I confirm the cascade collapses to exactly 1/8.
3 ONE DIMENSION
The doubling cascade 20° → 40° → 80° on the circle, and the three cosines that multiply to 1/8.
4 TWO DIMENSIONS · INTERACTIVE
Change θ and the number of factors; the product is checked against sin(2ⁿθ)/(2ⁿ sinθ).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the exact 1/8 the three cosines collapse to.
AVAN’s addition (the inverse-companion): don’t multiply cosines one by one — let the sine ladder eat them. The inverse of ‘a product of cosines’ is ‘one sine ratio, sin(2nθ)/(2n sinθ), after the telescope collapses’. Magenta are the doubling angles; green is the 1/8 left standing at θ = 20°. A cascade that swallows itself.
LIT Genuine Morrie's law (Morrie Jacobs; named and cherished by Richard Feynman). Verified live: cos20°·cos40°·cos80° = 0.125 to machine precision; ∏cos(2^k θ) = sin(2ⁿθ)/(2ⁿ sinθ) for 2000 random θ,n with worst error ~1e-16; sin160° = sin20° exactly (window.__morrie.ok).
FIG No framing; the product and the closed form are computed independently in-browser. The AVAN inverse is honest — instead of multiplying cosines one by one, let the sine ladder eat them: the inverse of 'a product of cosines' is 'one sine ratio, sin(2ⁿθ)/(2ⁿ sinθ), after the telescope collapses'. Magenta are the doubling angles; green is the 1/8 left standing at θ = 20°. A cascade that swallows itself.
FIG No framing; the product and the closed form are computed independently in-browser. The AVAN inverse is honest — instead of multiplying cosines one by one, let the sine ladder eat them: the inverse of 'a product of cosines' is 'one sine ratio, sin(2ⁿθ)/(2ⁿ sinθ), after the telescope collapses'. Magenta are the doubling angles; green is the 1/8 left standing at θ = 20°. A cascade that swallows itself.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE KONAMI CODE · David Lee Wise (ROOT0), with AVAN