Wetzel's 30-60-90 Triangle Covers Unit Arcs

Abstract: John E. Wetzel conjectured that the 30-60-90 triangle T obtained by placing a square of side 1/3 on the hypotenuse covers every unit arc in the plane. We give a computer-assisted proof of this conjecture with independently checkable interval certificates. The proof reduces a hypothetical noncovered arc to a finite family of 599 closed second-order cone models, covering all representative and raw tail-order branches, and certifies a polygonal-chain lower bound greater than one in every model by interval validation of stored dual certificates. Since every certified lower endpoint exceeds 1.0048, the homothetic copy T/1.0048 still covers every unit arc. Its area is 0.260956..., below the area pi/12 approx. 0.261799 of the 30-degree unit sector, a certified area improvement over the sector cover within this convex Wetzel-cover setting.

• The paper proves that every unit-length arc is enclosed by a congruent copy of Wetzel’s 30–60–90 triangle through rigorous computer-assisted methods.
• The authors employ interval arithmetic and second-order cone programming to certify that the minimal chain length exceeds 1.0048, permitting a scaled reduction of the triangle’s area.
• The study advances geometric transversal theory by establishing a new convex cover with an area of 0.260956, thereby improving upon the previous best candidate.

Wetzel's $30^\circ$–$60^\circ$–$90^\circ$ Triangle Covers All Unit Arcs: A Computer-Assisted Resolution

Introduction and Context

Covering plane curves of fixed length by small convex sets is central to geometric transversal theory and combinatorial geometry, notably manifesting in the Moser worm problem and Wetzel’s fits-and-covers program. Specifically, the problem addressed is: what is the smallest convex set that contains a congruent copy of every unit-length arc in the plane? Decades ago, Wetzel conjectured that two candidates— the $30^\circ$ unit sector and a specific $30^\circ$–$60^\circ$–$90^\circ$ triangle constructed by appending a square of side $1/3$ on its hypotenuse—cover this class.

While the sector case was previously settled, this paper ("Wetzel's 30-60-90 Triangle Covers Unit Arcs" (2606.14625)) rigorously resolves the triangle conjecture via a computer-assisted method, furnishing independently checkable interval certificates and leveraging advances in convex geometry, support-line theory, and modern convex optimization.

Figure 1: The two Wetzel candidates: (a) the $30^\circ$–$60^\circ$–$60^\circ$0 triangle with inscribed $60^\circ$1 square; (b) the $60^\circ$2 unit sector.

Problem Formulation and Main Result

Let $60^\circ$3 denote the $60^\circ$4–$60^\circ$5–$60^\circ$6 triangle, with side $60^\circ$7, $60^\circ$8, and hypotenuse $60^\circ$9. The area of $90^\circ$0 is approximately $90^\circ$1. The core theorem is:

Every plane unit arc is contained in a congruent copy of $90^\circ$2.

A critical numerical strengthening follows: the minimum certified lower bound on chain length over all obstructions is $90^\circ$3, allowing a homothetic shrink of $90^\circ$4 by $90^\circ$5 to area $90^\circ$6, thereby producing a strictly smaller convex cover than the $90^\circ$7 sector, which has area $90^\circ$8.

Reduction Framework and Structural Analysis

The proof pipeline comprises a sequence of structural reductions:

1. Arc Reduction: The Wetzel–Wichiramala theorem enables restriction to simple polygonal arcs.
2. Support-Line and $90^\circ$9-Property: The $30^\circ$0-property yields a key contact configuration—every simple arc admits two parallel supports ($30^\circ$1, $30^\circ$2) contacting the lower and upper extremes at $30^\circ$3 and decomposing the arc accordingly.

   Figure 2: The $30^\circ$4-property: horizontal supports $30^\circ$5/$30^\circ$6 touch $30^\circ$7 at $30^\circ$8.

3. Witness and Skeleton Construction: A finite vocabulary of contact points at prescribed angles is selected, organizing the arc's structure via a normal-fan skeleton supporting finite case analysis.

   Figure 3: Index map: contacts $30^\circ$9 and auxiliary contacts including inner and tail roles.

4. Canonical Orientations of $30^\circ$0: The possible placements of $30^\circ$1 with respect to arcs are enumerated via a fixed system of canonical orientations, each enforced by the geometric configuration.

   Figure 4: The twelve canonical orientations of $30^\circ$2—floor, wall, ceiling, and wedge groupings.

Canonical Placements and Escape Predicate

In each canonical orientation, the short side of $30^\circ$3 is positioned such that escaping the "wedge" defined by the triangle triggers an escape predicate: $30^\circ$4, where $30^\circ$5 is the anchor and $30^\circ$6 the direction normal to the short side.

Figure 5: Escape predicate for $30^\circ$7 orientation: arc must escape strictly beyond the short side.

For any hypothesized arc not covered by $30^\circ$8, an escape inequality is forced in each placement, translating the overall problem into a finite system of affine and conic constraints.

Exhaustive Case Analysis and Constraint System

The structured case tree results from the possible roles (inner or tail) of $30^\circ$9 and $60^\circ$0 support contacts, yielding three principal cases:

• Case 1: Both $60^\circ$1 contacts inner—leading to narrow shoulders.
• Case 2: One inner, one outer—an asymmetric situation.
• Case 3: Both $60^\circ$2 contacts on the tails—wide tails branch.

All parameter/placement alternatives are controlled via support-contact ordering enforced by planarity and the normal-fan structure.

Figure 6: Geometric distinction between three main cases: inner/outer $60^\circ$3 contacts affect arc-tail geometry.

This combinatorially exhaustive analysis produces twelve "terminal" representative models and $60^\circ$4 auxiliary raw-order branches, for a total of $60^\circ$5 second-order cone programs (SOCPs), each corresponding to a particular affine/contact configuration.

Figure 7: The branching case tree showing the finite case analysis structure.

Computer-Assisted Validation: Convex Optimization and Certificate Scheme

For each case, the minimal possible polygonal chain (lower bound for arc length) compatible with all incidence and escape constraints is computed via SOCP. Each model's minimum is validated by a robust dual certificate, with all constraints checked using interval arithmetic and verified singular-value floors, fully eliminating floating-point error sensitivity.

Key numerical results:

• Smallest certified lower endpoint is $60^\circ$6 (Subcase 1.1a).
• All $60^\circ$7 SOCPs yield certified optima exceeding $60^\circ$8, with the weakest (most critical) raw-order endpoint $60^\circ$9.

Terminal configurations and the associated forced predicate incidences are visually instantiated (Figure 8), showcasing geometric chains incompatible with a unit arc.

Figure 8: Example certified configurations for Subcase 1.1—floor, wedge, wall placements all unable to fit a unit arc congruent with $90^\circ$0.

Implications and Certified Scaling

By scaling $90^\circ$1 down by $90^\circ$2, every escape and incidence constraint is strictly satisfied, so the rescaled triangle remains a universal cover for unit arcs. This produces a new best bound in the convex Wetzel-program setting—a triangle of area $90^\circ$3 serves as a universal convex cover for unit arcs, strictly improving upon the previous $90^\circ$4 sector.

Theoretical and Algorithmic Insights

The proof validates the efficacy of:

• Support-based combinatorics: Normal-fan order and planarity enforce strict limits on possible arc placements.
• Computational convex geometry: Modern SOCP, when coupled with sound interval verification, handles nontrivial geometric finite enumeration with full certification.
• Certified models: The pipeline demonstrates that machine-checked, interval-validated certificates yield concise, reproducible, and audit-ready proofs in geometric transversals, setting a precedent for future work in computational geometry and convex covering.

Future Directions

This resolution closes Wetzel’s triangle conjecture and establishes a novel area threshold within convex universal covers for unit arcs. The analytical framework and certificate infrastructure deployed here should generalize to broader classes of geometric covering problems. Open questions pertain to possible further reductions in cover area (potentially through more intricate convex shapes or through alternate polygonal configurations), as well as generalization to closed curves, higher-order transversals, or higher-dimensional analogues.

Conclusion

Every unit arc in the plane is contained within a congruent copy of the $90^\circ$5–$90^\circ$6–$90^\circ$7 triangle described by Wetzel, with a rigorous, interval-certified computational margin. By leveraging geometry, convex optimization, and computer-aided verification, the result strengthens the program of finding minimal convex covers for unit arcs, establishes new area benchmarks, and validates the integration of rigorous computation into classical geometric extremal problems (2606.14625).