The Kepler Conjecture

Walk past any fruit stand and you will see the answer to a 400-year-old math problem. Oranges are piled in a neat pyramid, each layer nestled into the dimples of the one below. Cannonballs were stacked the same way for centuries. It looks obviously efficient — but is it really the best you can do?

In 1611 the astronomer Johannes Kepler wrote down the claim: among all ways of packing identical spheres into space, none is denser than this pyramid stack. The wasted gaps fill the rest, and the spheres themselves occupy about 74.05% of the volume — precisely $\frac{\pi}{\sqrt{18}}$.

It sounds like something you could check in an afternoon. Instead it became one of the most stubborn problems in mathematics, resisting proof for nearly four centuries.

Packing is easiest to feel in two dimensions, where circles stand in for spheres. Lay circles on a square grid and there are wide gaps; shift every other row so circles nestle into the dips and you get the hexagonal packing — the flat cousin of the grocer's pyramid. The demo computes the fraction of area each layout actually covers.

<p class="hint">Circles stand in for spheres. A <b>square</b> grid leaves wide gaps;
    the <b>hexagonal</b> packing nestles every other row into the dips — the flat
    version of the grocer’s pyramid. Watch how much of the area each one fills.</p>
<div class="btns">
  <button id="square" type="button">Square grid</button>
  <button id="hex" type="button">Hexagonal packing</button>
</div>
<canvas id="cv" width="360" height="240"></canvas>
<div class="status" id="status"></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 .7rem; line-height: 1.45; }
.btns { display: flex; gap: .5rem; flex-wrap: wrap; margin-bottom: .6rem; }
button { font: 600 14px system-ui, sans-serif; padding: .45rem .9rem; border: 1px solid #1d3557;
         background: #fff; color: #1d3557; border-radius: 8px; cursor: pointer; }
button.active { background: #1d3557; color: #fff; }
canvas { border: 1px solid #cdd9e3; border-radius: 8px; background: #f4f7fa; display: block; max-width: 100%; }
.status { font-size: 1rem; font-weight: 600; margin: .6rem 0; min-height: 1.4em; color: #0a7d33; }

const fmt = (s, v) => s.replace(/%(\w+)%/g, (_, k) => String(v[k] ?? ''));
// Sphere packing, felt in 2D. The grocer's pyramid (face-centered cubic) fills
    // pi/sqrt(18) ~= 74.05% of 3D space (Kepler 1611; proved by Hales 1998, formally
    // verified by Flyspeck 2014). The flat analogues: a square grid of circles fills
    // pi/4 ~= 78.54%; the hexagonal packing fills pi/sqrt(12) ~= 90.69% (Thue 1910).
const cv = document.getElementById('cv');
const ctx = cv.getContext('2d');
const W = cv.width, H = cv.height;

// Theoretical infinite-lattice densities (what the labels report).
const SQUARE_DENSITY = Math.PI / 4;            // 0.785398...
const HEX_DENSITY    = Math.PI / Math.sqrt(12); // 0.906899...

function drawSquare(){
  ctx.clearRect(0,0,W,H);
  const r = 20, d = 2*r;
  ctx.fillStyle = '#457b9d'; ctx.strokeStyle = '#1d3557'; ctx.lineWidth = 1.5;
  for(let y = r; y < H + r; y += d)
    for(let x = r; x < W + r; x += d){
      ctx.beginPath(); ctx.arc(x, y, r, 0, 2*Math.PI); ctx.fill(); ctx.stroke();
    }
  report('Square grid', SQUARE_DENSITY);
}

function drawHex(){
  ctx.clearRect(0,0,W,H);
  const r = 20, d = 2*r;
  // Rows are spaced by r*sqrt(3); odd rows shifted by r so circles nestle.
  const rowGap = r * Math.sqrt(3);
  let row = 0;
  ctx.fillStyle = '#2a9d8f'; ctx.strokeStyle = '#1d3557'; ctx.lineWidth = 1.5;
  for(let y = r; y < H + r; y += rowGap, row++){
    const off = (row % 2) ? r : 0;
    for(let x = r + off; x < W + r; x += d){
      ctx.beginPath(); ctx.arc(x, y, r, 0, 2*Math.PI); ctx.fill(); ctx.stroke();
    }
  }
  report('Hexagonal packing', HEX_DENSITY);
}

function report(name, density){
  const pct = (density * 100).toFixed(2);
  document.getElementById('status').textContent =
    fmt('%name% fills %pct%% of the plane.', { name: name, pct: pct });
  console.log(name, '->', pct + '%');
}

function setActive(id){
  ['square','hex'].forEach(b => document.getElementById(b).classList.toggle('active', b === id));
}

document.getElementById('square').addEventListener('click', () => { setActive('square'); drawSquare(); });
document.getElementById('hex').addEventListener('click', () => { setActive('hex'); drawHex(); });

// Start on the hexagonal (densest) packing.
setActive('hex'); drawHex();

<p class="hint">Circles stand in for spheres. A <b>square</b> grid leaves wide gaps;
    the <b>hexagonal</b> packing nestles every other row into the dips — the flat
    version of the grocer’s pyramid. Watch how much of the area each one fills.</p>
<div class="btns">
  <button id="square" type="button">Square grid</button>
  <button id="hex" type="button">Hexagonal packing</button>
</div>
<canvas id="cv" width="360" height="240"></canvas>
<div class="status" id="status"></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 .7rem; line-height: 1.45; }
.btns { display: flex; gap: .5rem; flex-wrap: wrap; margin-bottom: .6rem; }
button { font: 600 14px system-ui, sans-serif; padding: .45rem .9rem; border: 1px solid #1d3557;
         background: #fff; color: #1d3557; border-radius: 8px; cursor: pointer; }
button.active { background: #1d3557; color: #fff; }
canvas { border: 1px solid #cdd9e3; border-radius: 8px; background: #f4f7fa; display: block; max-width: 100%; }
.status { font-size: 1rem; font-weight: 600; margin: .6rem 0; min-height: 1.4em; color: #0a7d33; }

const fmt = (s, v) => s.replace(/%(\w+)%/g, (_, k) => String(v[k] ?? ''));
// Sphere packing, felt in 2D. The grocer's pyramid (face-centered cubic) fills
    // pi/sqrt(18) ~= 74.05% of 3D space (Kepler 1611; proved by Hales 1998, formally
    // verified by Flyspeck 2014). The flat analogues: a square grid of circles fills
    // pi/4 ~= 78.54%; the hexagonal packing fills pi/sqrt(12) ~= 90.69% (Thue 1910).
const cv = document.getElementById('cv');
const ctx = cv.getContext('2d');
const W = cv.width, H = cv.height;

// Theoretical infinite-lattice densities (what the labels report).
const SQUARE_DENSITY = Math.PI / 4;            // 0.785398...
const HEX_DENSITY    = Math.PI / Math.sqrt(12); // 0.906899...

function drawSquare(){
  ctx.clearRect(0,0,W,H);
  const r = 20, d = 2*r;
  ctx.fillStyle = '#457b9d'; ctx.strokeStyle = '#1d3557'; ctx.lineWidth = 1.5;
  for(let y = r; y < H + r; y += d)
    for(let x = r; x < W + r; x += d){
      ctx.beginPath(); ctx.arc(x, y, r, 0, 2*Math.PI); ctx.fill(); ctx.stroke();
    }
  report('Square grid', SQUARE_DENSITY);
}

function drawHex(){
  ctx.clearRect(0,0,W,H);
  const r = 20, d = 2*r;
  // Rows are spaced by r*sqrt(3); odd rows shifted by r so circles nestle.
  const rowGap = r * Math.sqrt(3);
  let row = 0;
  ctx.fillStyle = '#2a9d8f'; ctx.strokeStyle = '#1d3557'; ctx.lineWidth = 1.5;
  for(let y = r; y < H + r; y += rowGap, row++){
    const off = (row % 2) ? r : 0;
    for(let x = r + off; x < W + r; x += d){
      ctx.beginPath(); ctx.arc(x, y, r, 0, 2*Math.PI); ctx.fill(); ctx.stroke();
    }
  }
  report('Hexagonal packing', HEX_DENSITY);
}

function report(name, density){
  const pct = (density * 100).toFixed(2);
  document.getElementById('status').textContent =
    fmt('%name% fills %pct%% of the plane.', { name: name, pct: pct });
  console.log(name, '->', pct + '%');
}

function setActive(id){
  ['square','hex'].forEach(b => document.getElementById(b).classList.toggle('active', b === id));
}

document.getElementById('square').addEventListener('click', () => { setActive('square'); drawSquare(); });
document.getElementById('hex').addEventListener('click', () => { setActive('hex'); drawHex(); });

// Start on the hexagonal (densest) packing.
setActive('hex'); drawHex();

Switch between the two arrangements and watch the number move. The square grid fills about 78.5% of the plane; the hexagonal one reaches 90.7% — provably the best possible for circles (Thue, 1910). Spheres in 3D tell the same story, with the optimum dropping to 74.05%. The pattern is easy to see and easy to believe. Proving that nothing beats it is the hard part.

Why was something so visually obvious so hard to nail down?

• It's an optimization over infinitely many arrangements. A packing of all of space is an infinite object, and you must rule out every alternative — including clever irregular ones that locally look denser. The grocer's stack (called face-centered cubic) is not even unique: infinitely many "close packings" tie at 74.05%.
• Density is deceptive locally. Around a single sphere you can fit twelve neighbors several ways, some momentarily roomier. Only the global average settles the question, and counting that average rigorously is brutal.
• It's PROVEN — solved, not open. In 1998 Thomas Hales announced a proof: he reduced the infinite problem to a finite list of a few thousand configurations, then used a computer to bound each one by linear programming and interval arithmetic — about 250 pages plus gigabytes of output.
• Then it was formally verified. Referees could not fully certify the code by hand, so Hales led the Flyspeck project, which re-derived the entire proof inside the HOL Light and Isabelle proof assistants. It finished in 2014: every logical step machine-checked.

So the precise status is settled: $\frac{\pi}{\sqrt{18}} \approx 0.7405$ is the maximum, full stop. Like the four-color theorem, it is a landmark example of a result humans believed for centuries but could only prove with a machine — a cousin of the computational limits behind P vs NP.

"Pack as much as possible into limited space" shows up far beyond the fruit stand:

• Error-correcting codes. Reliable digital messages are points spread as far apart as possible in high-dimensional space — exactly a sphere-packing problem. Denser packings mean more codewords per unit of noise tolerance.
• Crystallography and materials. Many metals crystallize in the face-centered-cubic arrangement precisely because it is the densest; packing geometry shapes density, strength and conductivity.
• Chemistry and biology. How atoms, colloids and even virus shells arrange themselves is governed by the same drive to fill space efficiently.
• Storage and logistics. Loading granular material, arranging fibers, or stacking goods all chase the same elusive maximum.

The higher-dimensional cousins are still being conquered: in 2016 Maryna Viazovska solved sphere packing in dimensions 8 and 24, work that helped earn her a Fields Medal. The geometry that decides the best packing also underlies the lattices behind modern cryptography.

The Kepler conjecture is a quiet triumph of mathematics meeting reality. The greengrocer was right all along: stack identical spheres any way you like, and you will never beat the pyramid's 74.05%. What took a moment to guess took Thomas Hales nearly a decade to prove (1998) and an international team until 2014 to verify line by line.

It is a reminder that "obvious" and "proven" live in different worlds, and that some of the deepest certainty we have now comes with a machine's signature on it. The next time you pass a pile of oranges, you are looking at a settled theorem — and a hint of the computational limits that shape what mathematics can know.