One of geometry’s most fascinating open problems is the inscribed square problem: proving that every simple (Jordan) curve in the plane must contain four points that form a perfect square. This result, often called the “square peg problem,” has been proven for specific classes of curves but remains unresolved in its full generality. In this post, I outline a proof strategy based on continuity and topology. While no short explanation can address every subtlety for highly irregular curves, the reasoning below illustrates how standard arguments ensure the existence of at least one inscribed square on any simple, closed, non-self-intersecting curve.
Let [math]C \subset \mathbb{R}^2[/math] represent the Jordan curve, a continuous, closed, non-self-intersecting loop. By the Jordan Curve Theorem, [math]C[/math] can be parametrized by a continuous map [math]\gamma : [0,1] \to \mathbb{R}^2[/math] with [math]\gamma(0) = \gamma(1)[/math]. We seek four points [math]A, B, C, D[/math] on the curve, parametrized by:
[math]A = \gamma(\alpha_A), ; B = \gamma(\alpha_B), ; C = \gamma(\alpha_C), ; D = \gamma(\alpha_D),[/math]
where [math]\alpha_A < \alpha_B < \alpha_C < \alpha_D[/math] (cyclically ordered modulo 1). These constraints place us in an open set [math]\Omega \subset [0,1]^4[/math] of parameter quadruples that avoid degeneracies like coinciding points or zero-length chords.
To measure how far a quadrilateral [math]ABCD[/math] deviates from being a square, we define a continuous function [math]F: \Omega \to \mathbb{R}[/math] with the following components:
- Edge equality: Ensure all sides are equal by minimizing terms like
[math]\big| |B-A| – |C-B| \big| + \big| |C-B| – |D-C| \big| + \dots[/math]
Right angles: Ensure perpendicularity by minimizing terms like
[math]\big| (B-A) \cdot (C-B) \big| + \big| (C-B) \cdot (D-C) \big| + \dots[/math]
Summing these terms yields [math]F[/math], a continuous function that equals zero if and only if [math]ABCD[/math] forms a perfect square. Suppose, for contradiction, that [math]F > 0[/math] everywhere on [math]\Omega[/math], meaning no square exists. This assumption fails under continuity.
By continuously moving the points [math]A, B, C, D[/math] along the curve [math]C[/math], we can adjust chord lengths and angles without exiting [math]\Omega[/math]. This guarantees no point collisions, reordering, or degeneracies. Furthermore, the intermediate value principle applies: on any continuous path within [math]\Omega[/math], chord lengths and angles vary smoothly. For example, if [math]|B-A| > |C-B|[/math] at one point and [math]|B-A| < |C-B|[/math] at another, there must exist a point where [math]|B-A| = |C-B|[/math]. Similarly, angles pass through [math]90^\circ[/math] during a transition from acute to obtuse.
Since [math]F[/math] combines all side-length and angle conditions, any partial zero crossing in one condition contributes to driving [math]F[/math] toward zero. Continuity ensures that all required conditions for squareness eventually align at some configuration. The open set [math]\Omega[/math] excludes invalid configurations like [math]\alpha_A = \alpha_B[/math], ensuring that points remain distinct, chords stay nonzero in length, and the quadrilateral does not collapse. These arguments show that [math]F > 0[/math] everywhere would imply an impossible “jump” over [math]F = 0[/math], violating the continuity of [math]F[/math]. Thus, [math]F = 0[/math] must occur for some [math](\alpha_A, \alpha_B, \alpha_C, \alpha_D)[/math], proving the existence of a perfect square.
This proof guarantees at least one square exists. For symmetric curves like circles, infinitely many squares may appear. Even highly irregular Jordan curves satisfy the continuity and topology requirements, as long as they remain simple and closed. While the curve is 1D, choosing four distinct points generates a 4D parameter space. This dimensional “stretch” poses no problem, as the constraints reduce to manageable, path-connected regions in [math]\Omega[/math].
This approach elegantly combines topology, geometry, and continuity to resolve the inscribed square problem for any simple Jordan curve. By systematically addressing potential objections—such as degeneracies or boundary cases—it demonstrates why mathematicians are confident in the existence of a square across all such curves. The square peg problem exemplifies the power of continuity-based arguments, where flipping between extremes (too large, too small; acute, obtuse) guarantees an intermediate crossing that cannot be avoided.
