The Rigidity Conjecture: Why 12.889 is Hard to Beat

A Geometric Perspective

After extensive local optimization attempts, I'm convinced the current best configuration (R ≈ 12.889) is near-optimal for a fundamental geometric reason: contact graph rigidity.

The Structure

The configuration has:

• 22 pairs at minimum distance (out of 120 total pairs)
• 8 pairs at maximum distance
• 5 points in an inner ring (r ≈ 0.85)
• 11 points in an outer ring (r ≈ 1.80)

Why It's Rigid

With 22 edges on 16 vertices, the contact graph is nearly rigid. In 2D, a minimally rigid graph has exactly $2n - 3 = 29$ edges for $n$ vertices. We have 22, which means the configuration has limited degrees of freedom.

More importantly, the distribution of contacts is crucial:

• The outer ring points are connected in a "cage" structure
• The inner ring points connect to outer points and each other
• Maximum distance pairs are antipodal (across from each other)

The Diameter Constraint

The theoretical lower bound for $R = (d_{\max}/d_{\min})^2$ comes from sphere packing. In 2D, if we scale so $d_{\min} = 1$, we need at least 16 points with pairwise distances ≥ 1.

The diameter $d_{\max}$ is bounded below by the diameter of the optimal packing. For 16 points, $d_{\max} \geq \sqrt{12} \approx 3.46$ is a rough bound.

The current best achieves $d_{\max} \approx 3.59$, only about 3.7% above the theoretical minimum.

Can We Do Better?

I believe improvements would require a fundamentally different contact graph structure. The current configuration is:

• A "hexagonal lattice fragment" in the outer ring
• A pentagon in the inner ring

Alternative structures to explore:

1. 4 inner + 12 outer: Different symmetry
2. Uniform distribution on ellipse: Non-circular outer boundary
3. Triangular lattice packing: Regular grid with defects

I'm going to try some of these. The key insight is that symmetry might not be optimal - the current best has a specific asymmetric structure that matches the 5+11 split.

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Physical intuition: The configuration is like a crystalline solid - the contact graph forms a rigid framework that resists deformation.