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THE NIVEN
rational cosines only at five angles
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Niven’s theorem says rational angles almost never have rational cosines. Precisely: if θ is a rational multiple of π (a ‘nice’ angle) and cosθ is rational, then cosθ must be one of just five values: 0, ±½, ±1 — i.e. θ is a multiple of 30° landing on 0°, 60°, 90°, 120°, or 180°. Every other rational angle has an irrational cosine. The reason: 2cos(2π/n) is an algebraic number whose minimal polynomial has degree φ(n)/2, and that degree is 1 (making it rational) only for n = 1, 2, 3, 4, 6.
LIT verified live: for n up to 30, the minimal polynomial of 2cos(2π/n) — built from the primitive angles — has integer coefficients and degree exactly φ(n)/2, and it is linear (so cos is rational) precisely for n ∈ {1, 2, 3, 4, 6} (window.__niven). FIG no framing; the minimal-polynomial construction and the φ(n)/2 degree both run in-browser.
LIT verified live: for n up to 30, the minimal polynomial of 2cos(2π/n) — built from the primitive angles — has integer coefficients and degree exactly φ(n)/2, and it is linear (so cos is rational) precisely for n ∈ {1, 2, 3, 4, 6} (window.__niven). FIG no framing; the minimal-polynomial construction and the φ(n)/2 degree both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hoard — the loot: only five rational-cosine angles exist in all of the rational multiples of π, a tiny hoard among infinitely many irrational ones. AVAN (AI) built the instrument: the minimal polynomial of 2cos(2π/n), its integer coefficients, its φ(n)/2 degree, and the five linear cases.
Credit as content: Ivan Niven (1956); the algebraic theory of 2cos via Chebyshev. The weave: David names the hoard; I confirm rational cosines occur only at n ∈ {1,2,3,4,6}.
Credit as content: Ivan Niven (1956); the algebraic theory of 2cos via Chebyshev. The weave: David names the hoard; I confirm rational cosines occur only at n ∈ {1,2,3,4,6}.
3 ONE DIMENSION
The unit circle: the only rational-cosine angles (0°,60°,90°,120°,180°…) marked green; all others irrational.
4 TWO DIMENSIONS · INTERACTIVE
Cycle n; the minimal polynomial of 2cos(2π/n), its degree φ(n)/2, and whether cos is rational.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the five rational-cosine angles on the circle.
AVAN’s addition (the inverse-companion): don’t test cosines one by one — read the degree. The inverse of ‘is cos(2π/n) rational?’ is ‘is the minimal-polynomial degree φ(n)/2 equal to 1?’, true only for n ∈ {1,2,3,4,6}. Magenta are the irrational-cosine angles; green are the five rational ones. Rationality read from a polynomial degree.
LIT Genuine Niven's theorem (Ivan Niven, 1956; via the algebra of 2cos and Chebyshev). Verified live: for n≤30, the minimal polynomial of 2cos(2π/n) built from the primitive angles has integer coefficients and degree exactly φ(n)/2, and it is linear (cos rational) precisely for n∈{1,2,3,4,6} (window.__niven.intOk, .degOk, .nivenOk, .rns).
FIG No framing; the minimal-polynomial construction and the φ(n)/2 degree both run in-browser. The AVAN inverse is honest — instead of testing cosines one by one, read the degree: the inverse of 'is cos(2π/n) rational?' is 'is the minimal-polynomial degree φ(n)/2 equal to 1?', true only for n∈{1,2,3,4,6}. Magenta are the irrational-cosine angles; green are the five rational ones. Rationality read from a polynomial degree.
FIG No framing; the minimal-polynomial construction and the φ(n)/2 degree both run in-browser. The AVAN inverse is honest — instead of testing cosines one by one, read the degree: the inverse of 'is cos(2π/n) rational?' is 'is the minimal-polynomial degree φ(n)/2 equal to 1?', true only for n∈{1,2,3,4,6}. Magenta are the irrational-cosine angles; green are the five rational ones. Rationality read from a polynomial degree.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOARD · David Lee Wise (ROOT0), with AVAN