Introduction

With a compass and an unmarked straightedge you can do a surprising amount. You can copy a length, drop a perpendicular, and — in a few clean strokes — split any angle into two equal halves. Bisection is so easy it feels like the universe owes you the next step.

So here is the next step: split an angle into three equal parts. Trisection. The Greeks asked for it more than two thousand years ago, and for two millennia people kept trying, certain that a clever enough sequence of arcs and lines had to exist.

It does not. In 1837, Pierre Wantzel proved that trisecting an arbitrary angle with compass and straightedge is impossible — not "hard," not "unsolved," but provably, permanently out of reach. The strange twist: for some special angles, like a right angle, trisection works perfectly. The line between "works" and "impossible" is the whole story.

Try It

Pick an angle and ask the demo to trisect it with the only two tools the rules allow: a compass and a straightedge. The target rays sit at exactly one-third and two-thirds of the angle.

<p class="hint">A right angle (90°) <b>can</b> be trisected with compass and straightedge. A 60° angle <b>cannot</b> — the part you would need is 20°, and no finite construction can ever produce it (Wantzel, 1837). Pick an angle and press <b>Trisect</b>.</p>
<div class="picker">
  <label>Angle:</label>
  <button class="ang sel" data-a="90" type="button">90°</button>
  <button class="ang" data-a="60" type="button">60°</button>
  <button class="ang" data-a="120" type="button">120°</button>
  <button class="ang" data-a="180" type="button">180°</button>
</div>
<canvas id="cv" width="340" height="240"></canvas>
<div class="status" id="status">90° is constructible — press Trisect.</div>
<div class="btns">
  <button id="trisect" type="button">Trisect</button>
  <button id="reset" type="button" class="ghost">Reset</button>
</div>

* { box-sizing: border-box; }
body { font-family: system-ui, sans-serif; color: #222; margin: 0; }
.hint { font-size: .9rem; color: #444; margin: 0 0 .6rem; line-height: 1.45; }
.picker { display: flex; align-items: center; gap: .4rem; flex-wrap: wrap; margin: .2rem 0 .5rem; }
.picker label { font-weight: 600; font-size: .9rem; }
.ang { font: 600 14px system-ui, sans-serif; padding: .35rem .7rem; border: 1px solid #adb1b8;
       background: #fff; color: #1d3557; border-radius: 8px; cursor: pointer; }
.ang.sel { background: #1d3557; color: #fff; border-color: #1d3557; }
canvas { background: #f4f7fa; border: 1px solid #cdd9e3; border-radius: 8px; display: block; }
.status { font-size: 1rem; font-weight: 600; margin: .55rem 0; min-height: 1.4em; }
.status.ok { color: #0a7d33; }
.status.bad { color: #c92f3c; }
.btns { display: flex; gap: .5rem; flex-wrap: wrap; }
button#trisect, button#reset { font: 600 14px system-ui, sans-serif; padding: .45rem .9rem;
         border: 1px solid #1d3557; background: #1d3557; color: #fff; border-radius: 8px; cursor: pointer; }
button.ghost { background: #fff; color: #1d3557; }

const fmt = (s, v) => s.replace(/%(\w+)%/g, (_, k) => String(v[k] ?? ''));
// Trisecting an angle θ means constructing cos(θ/3). The triple-angle identity
//   4·cos³(θ/3) − 3·cos(θ/3) = cos θ
// turns this into a CUBIC. The needed length is constructible (compass+straightedge)
// only when that cubic's root lies in a tower of square-root extensions — degree a
// power of 2. For θ=90°, cos θ=0 → x(4x²−3)=0, root cos30°=√3/2 (degree 2): OK.
// For θ=60°, cos θ=1/2 → 8x³−6x−1=0, no rational root → irreducible (degree 3):
// cos20° is NOT constructible. Wantzel proved this in 1837.
const cv = document.getElementById('cv'), ctx = cv.getContext('2d');
const OX = 60, OY = 190, R = 150; // vertex (apex) and ray length
let angle = 90, done = false;

// Is the cubic 4x³ − 3x − cosθ = 0 reducible over Q (→ constructible)?
// The picker uses only angles whose cosine is rational (0, ±1/2, ±1), so the
// rational-root test is valid: a rational root means the cubic factors → the
// needed length sits in a degree-2 tower → constructible. No rational root means
// the cubic is irreducible (degree 3) → not constructible (Wantzel, 1837).
function constructible(deg) {
  const c = Math.cos(deg * Math.PI / 180);
  const f = x => 8*x*x*x - 6*x - 2*c;        // = 2·(4x³ − 3x − cosθ)
  // rational-root candidates p/q with p|const, q|8 (const term = −2c, scaled cases)
  const cands = [];
  for (let p = -8; p <= 8; p++) for (const q of [1,2,4,8]) cands.push(p/q);
  // de-dup and test
  const seen = new Set();
  for (const x of cands) {
    const key = x.toFixed(6); if (seen.has(key)) continue; seen.add(key);
    if (Math.abs(f(x)) < 1e-9) return true; // rational root → reducible → constructible
  }
  return false;
}

function ray(deg) { // canvas point at angle `deg` above the +x axis from vertex
  const a = -deg * Math.PI / 180; // canvas y grows downward
  return [OX + R * Math.cos(a), OY + R * Math.sin(a)];
}
function line(p, col, w) {
  ctx.strokeStyle = col; ctx.lineWidth = w || 2;
  ctx.beginPath(); ctx.moveTo(OX, OY); ctx.lineTo(p[0], p[1]); ctx.stroke();
}
function arc() {
  ctx.strokeStyle = '#9fb2c8'; ctx.lineWidth = 1.5;
  ctx.beginPath(); ctx.arc(OX, OY, 60, -angle*Math.PI/180, 0); ctx.stroke();
}
function draw() {
  ctx.clearRect(0,0,cv.width,cv.height);
  arc();
  line(ray(0), '#1d3557', 2.5);        // base ray
  line(ray(angle), '#1d3557', 2.5);    // upper ray
  // label
  ctx.fillStyle = '#1d3557'; ctx.font = '600 14px system-ui';
  ctx.fillText(angle + '°', OX + 70, OY - 12);
  if (done) {
    const ok = constructible(angle);
    if (ok) { // exact thirds reachable
      line(ray(angle/3), '#0a7d33', 2);
      line(ray(2*angle/3), '#0a7d33', 2);
      ctx.fillStyle = '#0a7d33';
      ctx.fillText((angle/3) + '°', ray(angle/6)[0]+8, ray(angle/6)[1]);
    } else { // honest construction falls short — draw a near-miss bisector-of-bisector
          // What you CAN build easily is the bisector (angle/2). Show it overshooting the
          // first third to make the failure visible; the true 1/3 ray is dashed & unreachable.
      line(ray(angle/2), '#e08a00', 2);          // a constructible ray (the bisector)
      ctx.setLineDash([5,4]);
      line(ray(angle/3), '#c92f3c', 1.6);        // the UNREACHABLE true third
      line(ray(2*angle/3), '#c92f3c', 1.6);
      ctx.setLineDash([]);
      ctx.fillStyle = '#c92f3c';
      ctx.fillText('?', ray(angle/6)[0]+8, ray(angle/6)[1]);
    }
  }
}
function setStatus(t, cls){ const s=document.getElementById('status'); s.textContent=t; s.className='status '+cls; }
function selBtn(a){ document.querySelectorAll('.ang').forEach(b=>b.classList.toggle('sel', +b.dataset.a===a)); }

document.querySelectorAll('.ang').forEach(b => b.addEventListener('click', () => {
  angle = +b.dataset.a; done = false; selBtn(angle); draw();
  setStatus(fmt('%angle%° — press Trisect.', { angle: angle }), '');
}));
document.getElementById('trisect').addEventListener('click', () => {
  done = true; draw();
  const third = angle/3;
  if (constructible(angle)) {
    setStatus(fmt('✓ %angle%° trisects: %third%° is constructible (degree a power of 2).', { angle: angle, third: third }), 'ok');
    console.log(fmt('Trisect %angle%° → %third%°: constructible. cos%third%° =', { angle: angle, third: third }), Math.cos(third*Math.PI/180));
  } else {
    setStatus(fmt('✗ %angle%° cannot be trisected — %third%° needs an irreducible cubic. (Wantzel, 1837)', { angle: angle, third: third }), 'bad');
    console.log(fmt('Trisect %angle%° → %third%°: 8x³−6x−%v%=0 has no rational root → not constructible.', { angle: angle, third: third, v: 2*Math.cos(angle*Math.PI/180) }));
  }
});
document.getElementById('reset').addEventListener('click', () => { done=false; draw(); setStatus(fmt('%angle%° — press Trisect.', { angle: angle }),''); });
draw();
Result
Console

Choose 90° and the construction lands dead on the thirds — 30° and 60° are both reachable. Choose 60° and the honest compass-and-straightedge attempt misses: the angle you would need is 20°, and no finite sequence of arcs and lines can ever produce it. Not because we are not clever enough — because it is impossible.

The Real Complexity

Why does bisection work but trisection fail? The answer is algebra, not draughtsmanship.

  • Constructible numbers. A point you can reach with compass and straightedge has coordinates built from the starting lengths using +, −, ×, ÷ and square roots only. Every constructible number lives in a tower of field extensions whose degree is a power of 2.
  • Trisection is a cubic. Trisecting an angle θ\theta means constructing cos⁡(θ/3)\cos(\theta/3). The triple-angle identity ties them together:

    4cos⁡3(θ/3)−3cos⁡(θ/3)=cos⁡θ4\cos^3(\theta/3) - 3\cos(\theta/3) = \cos\theta

    So trisection asks you to solve a cubic equation.
  • The 60° case. Here cos⁡θ=1/2\cos\theta = 1/2, giving

    8x3−6x−1=08x^3 - 6x - 1 = 0

    This cubic has no rational root, so it is irreducible over the rationals — cos⁡20∘\cos 20^\circ generates a degree-3 extension. Three is not a power of two, so cos⁡20∘\cos 20^\circ is not constructible. Trisecting 60° is impossible.
  • The 90° case. Here cos⁡θ=0\cos\theta = 0, and the cubic factors as x(4x2−3)=0x(4x^2 - 3) = 0. The needed root cos⁡30∘=32\cos 30^\circ = \frac{\sqrt{3}}{2} uses only a square root, degree 2. Constructible. Trisecting 90° works.
  • The verdict. Pierre Wantzel (1837) turned "no one has managed it" into "no one ever can." Trisection of a general angle is proven impossible — a closed question, not an open one.

This is a different flavour of hardness from P vs NP: not "we don't know if it's efficient," but "we have a proof it can't be done at all," the same kind of certainty behind the halting problem.

Where It Matters

A puzzle about drawing might seem far from anything useful. The opposite is true — the method of the proof became one of the pillars of modern mathematics.

  • Field theory. Reducing geometry to "which numbers can you build" gave a template for measuring impossibility with algebra, the heart of Galois theory.
  • The other ancient impossibles. The same machinery proves you cannot double the cube (you'd need 23\sqrt[3]{2}, degree 3) or square the circle (π\pi is transcendental). One idea closes three legendary problems.
  • Coding and cryptography. Finite fields — the algebraic cousins of constructible-number towers — are the backbone of error-correcting codes and much of modern cryptography.
  • Knowing when to stop. An impossibility proof is a gift: it tells engineers and mathematicians to stop hunting for a construction that cannot exist and to change tools instead — to a marked ruler, a paper fold, or a curve.

Learn why trisection fails and you have met the idea that some tasks are not merely difficult but categorically out of reach — the same theme behind every undecidable problem.

Conclusion

Bisection and trisection sound like neighbours, but a cubic equation stands between them. Two equal parts you can always draw; three equal parts you usually cannot — and Pierre Wantzel (1837) proved it once and for all by showing the needed lengths simply are not constructible.

The lesson outlives the puzzle. Some impossibilities are not gaps in our cleverness but features of the rules we chose, and a single algebraic insight can settle a question that resisted two thousand years of brilliant attempts. The next time something feels like it should be one more easy step, remember 20° — and that the answer can be a proven, permanent no.

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https://www.kipuhub.com/en/article/angle-trisection/Content licensed under CC BY-NC 4.0.