Summary
This work studies how to fit rings—like donut shapes—inside a bigger circle, allowing smaller rings to nest inside bigger ones. The researchers show that for certain nicely ordered ring sizes, a simple greedy approach always finds the best way to pack them. However, when ring sizes do not follow this order, the problem becomes much more complex and depends on subtle placement choices. They identify exact mathematical thresholds where easy solutions stop working, involving special numbers like the Tribonacci constant and the golden ratio. These findings clarify when simple packing rules succeed and when the problem gets much harder.
What this means in practice
- •For industrial designers: Designers can precisely predict when simple packing strategies for ring-shaped parts guarantee optimal layouts versus when complex placement rules become necessary.
- •For manufacturing process planners: Planners can identify thresholds where standard nesting approaches for circular components fail, informing tooling and layout decisions for efficient material usage.
A theory result. No direct application yet.
Abstract
We study packings of annuli ("rings") of a common width into a disk, where a ring may nest inside the hole of a strictly larger one, a selection-oriented relative of the Recursive Circle Packing Problem. The two natural objectives, cardinality and contact area, genuinely diverge. For superincreasing radii (each exceeding the sum of all smaller ones) we prove that the descending greedy maximizes every positive, increasing, superadditive objective. Our main structural theorem shows more: the placement rule is irrelevant - any choice among feasible containers yields the lexicographically maximal feasible set, for containers of arbitrary shape and in every dimension. Both hypotheses are sharp: placement irrelevance holds for at most three rings and fails at four, and twin instances rule out every rule that is a function of the observable state. Write $ρ=\max_i(\sum_{j>i}r_j)/r_i$ for the violation of superincreasingness. The additive relaxation has universal threshold exactly $ρ=1$. In the geometric model we prove, with no tangency idealization, that the rigid four-ring family has infimum exactly the Tribonacci constant $T\approx1.83929$. Yet $T$ is not the global threshold: an explicit golden family breaks placement obliviousness at $ρ=\varphi+3\varepsilon$ for every small $\varepsilon>0$, proving $τ\le\varphi<T$ for the geometric threshold $τ$ and refuting the natural Tribonacci-threshold conjecture. The matching bound $τ\ge\varphi$ remains conjectural; we prove it for pair profiles and outside an explicit heavy region. We also give a phase diagram for this divergence and split hardness into a geometric layer and a combinatorial (subset-sum) layer, of which superincreasingness eliminates exactly the latter. The main theorems carry complete written proofs; every computer-assisted closure carries an epistemic label and a script in the verification map.