Inscribe a square on a shape’s outline

Place four points on the shape’s outline so they form a square. As you move your cursor, the nearest outline point follows it — tap to drop a vertex. After four drops the quadrilateral is scored: closer to a perfect square (equal sides, 90° angles) means a higher score. You can drag any placed point afterwards to refine.

The Inscribed Square Problem (Toeplitz, 1911)

Otto Toeplitz asked a deceptively simple question in 1911: does every simple closed curve in the plane contain four points that form a square? More than a century later, the answer is probably yes — but it’s still one of the oldest open problems in geometry.

The problem is settled for nice classes of curves: polygons, convex curves, smooth curves, piecewise-smooth curves, and curves with bounded curvature all provably have an inscribed square. Recent work by Greene & Lobb (2020) settled the problem for smooth Jordan curves by connecting it to symplectic geometry. But for arbitrary Jordan curves — including pathological fractal ones — existence is still unproved.

On this page every generated shape is piecewise-smooth — a mix of straight segments, circular arcs, and Bezier curves. That’s well within the classes where the problem is proven, so a perfect inscribed square always exists on these shapes. Your job is to find one by eye.

Tips for spotting an inscribed square

  • Start with a diameter: pick two points on roughly opposite sides of the shape — they’re candidates for the square’s diagonal.
  • The two other vertices must sit on a line perpendicular to that diagonal, with the same length.
  • Convex regions usually have several inscribed squares; concave shapes sometimes have just one obvious one.
  • If three vertices already look good but the fourth is hard to place, adjust vertex two — the constraint often means the whole square has to rotate slightly.

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