Introduction

In 1924, mathematicians Stefan Banach and Alfred Tarski proved one of the most disorienting theorems in all of mathematics: a solid ball can be decomposed into a finite number of pieces and those pieces reassembled — using only rigid rotations and translations — into two balls identical to the original.

No stretching. No overlap. No extra material. Just five carefully chosen subsets, rearranged.

This is not a paradox in the sense of a contradiction. It is a rigorously proved theorem. What it reveals is that infinite sets behave in ways that defy the intuition we build from physical objects. The pieces involved are so wildly intricate — so-called non-measurable sets — that you cannot assign them a meaningful volume. And that is precisely what makes the duplication possible.

The engine behind the whole construction is the Axiom of Choice: the set-theoretic principle that says you can always pick one element from each set in any collection of non-empty sets, even an infinite collection with no rule for choosing. Comfortable for finite problems, it opens a door to behavior that looks like magic when applied to infinite ones.

See It Happen

The pieces in the real Banach-Tarski construction are non-constructive — they exist by the Axiom of Choice and cannot be drawn explicitly. What we can do is visualize the idea: decompose a ball into labeled regions and show how they reassemble into two copies.

<!-- Banach-Tarski schematic demo -->
<p class="hint">This schematic shows the Banach-Tarski idea: one ball split into 5 labeled pieces, then reassembled into two complete balls. The real pieces are non-measurable — they cannot be drawn — but the logic holds exactly.</p>
<div class="stage-label" id="stage-label">Original ball (5 non-measurable pieces)</div>
<div class="canvas-wrap">
  <canvas id="cvs" width="340" height="180"></canvas>
</div>
<div class="status" id="status">One ball — ready to split.</div>
<div class="btns">
  <button id="btn-split" type="button">Split into pieces</button>
  <button id="btn-reassemble" type="button" disabled>Reassemble into two balls</button>
  <button id="btn-reset" type="button" class="ghost">Reset</button>
</div>
<p class="footnote">Note: the real Banach-Tarski pieces cannot be visualized — they are built from the Axiom of Choice and are non-measurable. This diagram is a schematic only.</p>

/* Styles for the Banach-Tarski schematic demo */
* { box-sizing: border-box; }
body { font-family: system-ui, sans-serif; color: #222; margin: 0; padding: .5rem; }
.hint { font-size: .88rem; color: #444; margin: 0 0 .6rem; line-height: 1.5; }
.stage-label { font-size: .8rem; font-weight: 700; letter-spacing: .06em;
               text-transform: uppercase; color: #5a7088; margin-bottom: .3rem; }
.canvas-wrap { border: 1px solid #cdd9e3; border-radius: 10px; overflow: hidden;
               background: #f4f7fa; display: inline-block; }
canvas { display: block; }
.status { font-size: .95rem; font-weight: 600; margin: .55rem 0 .4rem; min-height: 1.4em; color: #1d3557; }
.status.ok { color: #0a7d33; }
.status.info { color: #1d3557; }
.btns { display: flex; gap: .5rem; flex-wrap: wrap; }
button { font: 600 14px system-ui, sans-serif; padding: .45rem .9rem; border: 1px solid #1d3557;
         background: #1d3557; color: #fff; border-radius: 8px; cursor: pointer; transition: opacity .15s; }
button:disabled { opacity: .38; cursor: default; }
button.ghost { background: #fff; color: #1d3557; }
.footnote { font-size: .78rem; color: #666; margin: .6rem 0 0; line-height: 1.5; }

// Banach-Tarski paradox — schematic visualization
// The real pieces are non-measurable sets — they have no well-defined Lebesgue measure.
// We label them A-E and show a schematic split and reassembly into two balls.

const canvas = document.getElementById('cvs');
const ctx = canvas.getContext('2d');
const W = canvas.width, H = canvas.height;
const CX = W / 2, CY = H / 2;
const R = 72; // radius of the original ball on canvas

// Five distinct colors for the five schematic pieces
const COLORS = ['#e63946','#2a9d8f','#e9c46a','#457b9d','#a8dadc'];
const PIECE_LABELS = ['A','B','C','D','E'];

// Demo phase state
let phase = 'original'; // original | split | two

// Draw the single original ball divided into 5 sectors
function drawOriginal() {
  ctx.clearRect(0, 0, W, H);
  // Draw five equal sectors, each representing one non-measurable piece
  const sectors = 5;
  for (let i = 0; i < sectors; i++) {
    const start = (i / sectors) * 2 * Math.PI - Math.PI / 2;
    const end = ((i + 1) / sectors) * 2 * Math.PI - Math.PI / 2;
    ctx.beginPath();
    ctx.moveTo(CX, CY);
    ctx.arc(CX, CY, R, start, end);
    ctx.closePath();
    ctx.fillStyle = COLORS[i];
    ctx.globalAlpha = 0.85;
    ctx.fill();
    ctx.globalAlpha = 1;
    ctx.strokeStyle = '#fff';
    ctx.lineWidth = 2;
    ctx.stroke();
    // Label each sector with its piece letter
    const midAngle = (start + end) / 2;
    const lx = CX + (R * 0.62) * Math.cos(midAngle);
    const ly = CY + (R * 0.62) * Math.sin(midAngle);
    ctx.fillStyle = '#fff';
    ctx.font = 'bold 15px system-ui, sans-serif';
    ctx.textAlign = 'center';
    ctx.textBaseline = 'middle';
    ctx.fillText(PIECE_LABELS[i], lx, ly);
  }
  // Draw the outer circle boundary
  ctx.beginPath();
  ctx.arc(CX, CY, R, 0, 2 * Math.PI);
  ctx.strokeStyle = '#1d3557';
  ctx.lineWidth = 2.2;
  ctx.stroke();
}

// Draw the five pieces scattered apart
function drawSplit() {
  ctx.clearRect(0, 0, W, H);
  // Each piece shown as a small circle, scattered across the canvas
  const positions = [
    { x: 80,  y: 55  },
    { x: 200, y: 45  },
    { x: 310, y: 80  },
    { x: 110, y: 145 },
    { x: 255, y: 148 }
  ];
  const r = 28; // radius of scattered piece circles
  for (let i = 0; i < 5; i++) {
    const px = positions[i].x, py = positions[i].y;
    ctx.beginPath();
    ctx.arc(px, py, r, 0, 2 * Math.PI);
    ctx.fillStyle = COLORS[i];
    ctx.globalAlpha = 0.85;
    ctx.fill();
    ctx.globalAlpha = 1;
    ctx.strokeStyle = '#1d3557';
    ctx.lineWidth = 1.6;
    ctx.stroke();
    ctx.fillStyle = '#fff';
    ctx.font = 'bold 15px system-ui, sans-serif';
    ctx.textAlign = 'center';
    ctx.textBaseline = 'middle';
    ctx.fillText(PIECE_LABELS[i], px, py);
  }
}

// Draw two complete balls formed from the five pieces
function drawTwo() {
  ctx.clearRect(0, 0, W, H);
  const R2 = 62; // radius of each reassembled ball
  // First ball uses pieces pieces A, C, E
  const b1x = W / 4, b1y = H / 2;
  const b1pieces = [0, 2, 4]; // indices for pieces A, C, E
  const total1 = b1pieces.length;
  for (let j = 0; j < total1; j++) {
    const i = b1pieces[j];
    const start = (j / total1) * 2 * Math.PI - Math.PI / 2;
    const end = ((j + 1) / total1) * 2 * Math.PI - Math.PI / 2;
    ctx.beginPath();
    ctx.moveTo(b1x, b1y);
    ctx.arc(b1x, b1y, R2, start, end);
    ctx.closePath();
    ctx.fillStyle = COLORS[i];
    ctx.globalAlpha = 0.85;
    ctx.fill();
    ctx.globalAlpha = 1;
    ctx.strokeStyle = '#fff';
    ctx.lineWidth = 2;
    ctx.stroke();
    const midAngle = (start + end) / 2;
    const lx = b1x + (R2 * 0.6) * Math.cos(midAngle);
    const ly = b1y + (R2 * 0.6) * Math.sin(midAngle);
    ctx.fillStyle = '#fff';
    ctx.font = 'bold 14px system-ui, sans-serif';
    ctx.textAlign = 'center';
    ctx.textBaseline = 'middle';
    ctx.fillText(PIECE_LABELS[i], lx, ly);
  }
  ctx.beginPath();
  ctx.arc(b1x, b1y, R2, 0, 2 * Math.PI);
  ctx.strokeStyle = '#1d3557';
  ctx.lineWidth = 2.2;
  ctx.stroke();
  // Second ball uses pieces pieces B, D
  const b2x = (3 * W) / 4, b2y = H / 2;
  const b2pieces = [1, 3]; // indices for pieces B, D
  const total2 = b2pieces.length;
  for (let j = 0; j < total2; j++) {
    const i = b2pieces[j];
    const start = (j / total2) * 2 * Math.PI - Math.PI / 2;
    const end = ((j + 1) / total2) * 2 * Math.PI - Math.PI / 2;
    ctx.beginPath();
    ctx.moveTo(b2x, b2y);
    ctx.arc(b2x, b2y, R2, start, end);
    ctx.closePath();
    ctx.fillStyle = COLORS[i];
    ctx.globalAlpha = 0.85;
    ctx.fill();
    ctx.globalAlpha = 1;
    ctx.strokeStyle = '#fff';
    ctx.lineWidth = 2;
    ctx.stroke();
    const midAngle = (start + end) / 2;
    const lx = b2x + (R2 * 0.6) * Math.cos(midAngle);
    const ly = b2y + (R2 * 0.6) * Math.sin(midAngle);
    ctx.fillStyle = '#fff';
    ctx.font = 'bold 14px system-ui, sans-serif';
    ctx.textAlign = 'center';
    ctx.textBaseline = 'middle';
    ctx.fillText(PIECE_LABELS[i], lx, ly);
  }
  ctx.beginPath();
  ctx.arc(b2x, b2y, R2, 0, 2 * Math.PI);
  ctx.strokeStyle = '#1d3557';
  ctx.lineWidth = 2.2;
  ctx.stroke();
  // Label the two balls below
  ctx.fillStyle = '#1d3557';
  ctx.font = '700 13px system-ui, sans-serif';
  ctx.textAlign = 'center';
  ctx.textBaseline = 'top';
  ctx.fillText('Ball 1 (A, C, E)', b1x, b1y + R2 + 6);
  ctx.fillText('Ball 2 (B, D)', b2x, b2y + R2 + 6);
}

function setStatus(t, cls) {
  const s = document.getElementById('status');
  s.textContent = t;
  s.className = 'status ' + cls;
}

function setStageLabel(t) {
  document.getElementById('stage-label').textContent = t;
}

document.getElementById('btn-split').addEventListener('click', function () {
  phase = 'split';
  drawSplit();
  setStageLabel('Pieces separated');
  setStatus('5 pieces, scattered. Each has no well-defined volume.', 'info');
  this.disabled = true;
  document.getElementById('btn-reassemble').disabled = false;
});

document.getElementById('btn-reassemble').addEventListener('click', function () {
  phase = 'two';
  drawTwo();
  setStageLabel('Two complete balls');
  setStatus('Two balls — each the same size as the original. Volume is not conserved because the pieces are non-measurable.', 'ok');
  this.disabled = true;
});

document.getElementById('btn-reset').addEventListener('click', function () {
  phase = 'original';
  drawOriginal();
  setStageLabel('Original ball (5 non-measurable pieces)');
  setStatus('One ball — ready to split.', 'info');
  document.getElementById('btn-split').disabled = false;
  document.getElementById('btn-reassemble').disabled = true;
});

// Initial render
drawOriginal();
setStatus('One ball — ready to split.', 'info');
Result
Console

Notice that the five colored regions together cover the original ball exactly once. After reassembly, each copy is built from a disjoint subset of those regions — and yet both copies are the same size as the original. The key insight is that non-measurable pieces have no well-defined volume, so "size is conserved" simply does not apply.

The Real Math

How does this actually work? The argument has three layers.

Layer 1 — The Hausdorff paradox (1914). Felix Hausdorff showed that the sphere S2S^2 can be split into four parts AA, BB, CC, DD where DD is countable and the remaining three satisfy A≅B≅C≅B∪CA \cong B \cong C \cong B \cup C (congruent under rotations). This already breaks volume intuition on the sphere surface.

Layer 2 — The free group F2F_2. The proof uses two specific rotations of three-dimensional space, say ρ\rho and σ\sigma, that together generate a free group of rank 2: every non-identity word in ρ\rho, ρ−1\rho^{-1}, σ\sigma, σ−1\sigma^{-1} gives a distinct rotation, and there are no accidental cancellations. The orbit of almost any point on the sphere under this group is an infinite, intricately tangled set.

Layer 3 — The Axiom of Choice picks representatives. From each orbit we choose exactly one point. This "choice set" is non-measurable: it has no well-defined Lebesgue measure. The five pieces of the ball are assembled from cosets of this choice set under F2F_2. Because the pieces are non-measurable, the usual conservation law volume(A∪B)=volume(A)+volume(B)\text{volume}(A \cup B) = \text{volume}(A) + \text{volume}(B) simply does not apply — and that is the entire loophole.

The result is sharp: the theorem requires three or more dimensions. In one or two dimensions an analogous decomposition is impossible — measurable sets behave normally there, and the relevant groups are amenable (they admit an invariant mean). The connection to the halting problem and undecidability is philosophical rather than formal, but both share the same moral: invoking an existence guarantee (the Axiom of Choice; an oracle) without a constructive procedure leads to objects that break our everyday rules.

Where It Matters

The paradox is not physically realizable — atoms are not points, and non-measurable sets cannot be cut with scissors. But its mathematical fallout is enormous:

  • Measure theory: Banach-Tarski was a major reason why Lebesgue measure is defined only on measurable sets. The paradox makes precise what "having a volume" really means and why some sets do not qualify.
  • Foundations of set theory: The theorem is a concrete demonstration of what the Axiom of Choice can do. Mathematicians who find the result unacceptable sometimes work in axiom systems that deny Choice (such as ZF without AC, or certain constructive frameworks) where Banach-Tarski fails.
  • Amenable groups: The impossibility of the paradox in low dimensions led to the study of amenable groups — groups that do admit an invariant finitely-additive measure. This connects directly to areas of geometric group theory and ergodic theory.
  • Philosophy of mathematics: The paradox is one of the clearest illustrations that formal truth and physical intuition can diverge. A statement can be a theorem and still feel impossible — because mathematics is not constrained to model the physical world.

It also connects to the broader family of "non-constructive existence" results: like the halting problem, it shows that knowing something exists (a decomposition, a solution) does not mean you can find or describe it.

Conclusion

The Banach-Tarski paradox is real — in the only sense that matters for mathematics. The proof is valid, the five pieces exist (by the Axiom of Choice), and two balls genuinely do emerge from one. There is no error, no trick, no sleight of hand.

What the paradox teaches is that "size" is a gift we grant to sets, not a property they inherently carry. The non-measurable pieces in the decomposition simply fall outside the scope of any consistent notion of volume. Remove the Axiom of Choice and the paradox vanishes — which tells you exactly which axiom is doing the work.

If that feels uncomfortable, it is supposed to. Mathematics is not a description of the physical world; it is a formal game with rules, and some of those rules — like Choice — have consequences that no physical experiment could ever demonstrate. Banach-Tarski is the clearest reminder that a theorem can be both rigorously true and completely impossible to perform.

Share this article

Pick a channel — or use your device's native share sheet.

Comments

Loading comments...

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