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arxiv: 2604.21451 · v1 · submitted 2026-04-23 · 🧮 math.MG · math.OC

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Bounding the density of spherical polygon packings

Fernando M\'ario de Oliveira Filho , Andreas Spomer , Frank Vallentin
Authors on Pith no claims yet

Pith reviewed 2026-05-08 13:00 UTC · model grok-4.3

classification 🧮 math.MG math.OC
keywords spherical polygonspacking densityLovász theta numberCayley graphsSO(3)semidefinite programmingharmonic analysismanifold optimization
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The pith

Regular spherical polygons achieve maximum packing densities that can be bounded above using an extended Lovász theta number on rotation-group graphs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper finds candidate best arrangements of identical regular spherical polygons on the sphere by running optimization over smooth manifolds. It introduces an algebraic test that decides whether any two such polygons have disjoint interiors, which turns all packing constraints into a single uniform form. For several concrete cases the authors prove these arrangements are optimal by extending the Lovász theta number to Cayley graphs whose vertices are rotations in SO(3). Harmonic analysis on the rotation group then converts the theta number into a trigonometric sum-of-squares problem that is solved by semidefinite programming, producing explicit numerical upper bounds on density.

Core claim

Putative optimal packings of regular spherical polygons are located by manifold optimization; maximality for several families is established by the Lovász theta number of the associated Cayley graph on SO(3). The key step is an algebraic criterion that exactly characterizes when two congruent regular spherical polygons have disjoint interiors, allowing every packing constraint to be written uniformly. Harmonic analysis reduces the resulting theta-number computation to a trigonometric sum-of-squares problem that is solved via semidefinite programming.

What carries the argument

Algebraic criterion for disjoint interiors of congruent regular spherical polygons, which unifies the packing constraints and permits the Lovász theta number to be defined on Cayley graphs over SO(3).

If this is right

  • For the polygon types treated, the manifold-optimized arrangements attain the highest possible density.
  • The method supplies explicit, computable upper bounds on packing density that can be checked numerically.
  • The same algebraic criterion and reduction apply uniformly across different numbers of sides and spherical excesses.
  • The resulting semidefinite programs can be re-solved for additional polygon families to generate further bounds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same extension of the theta number could be attempted for packings of non-regular spherical tiles or for objects on higher-dimensional spheres.
  • The trigonometric sum-of-squares formulation might be tightened by adding further symmetry constraints or by using higher-order harmonics.
  • Numerical comparison of the obtained bounds against known Euclidean or hyperbolic polygon packings could reveal cross-geometry patterns.
  • The manifold-optimization stage could be replaced by a purely discrete search once the algebraic criterion is in hand.

Load-bearing premise

The algebraic criterion correctly identifies every pair of congruent regular spherical polygons whose interiors are disjoint.

What would settle it

A concrete pair of regular spherical polygons that the algebraic criterion declares non-overlapping yet whose interiors intersect, or a spherical packing whose measured density exceeds the SDP-computed theta bound for the same polygon type.

Figures

Figures reproduced from arXiv: 2604.21451 by Andreas Spomer, Fernando M\'ario de Oliveira Filho, Frank Vallentin.

Figure 1
Figure 1. Figure 1: On the left, the spherical cap packing corresponding to the maximal fcc kissing configuration in R 3 . On the right, a packing of 20 spherical triangles corresponding to the bases of reg￾ular tetrahedra with a vertex at 0; these triangles are obtained by shrinking the 20 faces of a regular icosahedron. fill a container without leaving gaps. This led to the question whether the space around a point can be f… view at source ↗
Figure 2
Figure 2. Figure 2: Some of the configurations from view at source ↗
Figure 3
Figure 3. Figure 3: Some of the larger configurations from view at source ↗
read the original abstract

We determine putative optimal packings of regular spherical polygons via optimization on smooth manifolds. For several cases, we establish maximality by extending the Lov\'asz theta number to Cayley graphs on the special orthogonal group ${\rm SO}(3)$. To this end, we introduce an algebraic criterion characterizing when congruent regular spherical polygons have disjoint interiors, leading to a unified formulation of the packing constraints. Using harmonic analysis on ${\rm SO}(3)$, we reduce the theta number to a trigonometric sum-of-squares problem, which can be solved via semidefinite programming.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims to determine putative optimal packings of regular spherical polygons via optimization on smooth manifolds. For several cases, maximality is established by extending the Lovász theta number to Cayley graphs on SO(3), using a new algebraic criterion that characterizes when congruent regular spherical polygons have disjoint interiors; this leads to a unified packing constraint that is reduced, via harmonic analysis on SO(3), to a trigonometric sum-of-squares problem solvable by semidefinite programming.

Significance. If the algebraic criterion is shown to be exact and the SDP relaxations are tight, the work supplies rigorous upper bounds on spherical polygon densities together with a general technique for applying Lovász theta numbers to continuous symmetry groups. The combination of manifold optimization, harmonic analysis, and SDP is a concrete methodological advance that could be reused for other packing problems on homogeneous spaces.

major comments (1)
  1. [Algebraic criterion for disjoint interiors (used to define the Cayley-graph edges)] The maximality claims rest on the algebraic criterion (introduced after the abstract and used to define the adjacency relation of the Cayley graph) being both necessary and sufficient for two congruent regular spherical polygons to have disjoint interiors. If any pair whose interiors intersect is not flagged by the criterion, the resulting graph is a proper subgraph of the true conflict graph; independent sets in the computed graph may therefore contain geometrically invalid configurations, and the theta value upper-bounds only a relaxation. The manuscript must supply a complete proof that the criterion detects every overlap, together with explicit verification against known intersecting configurations or a geometric argument showing completeness.
minor comments (2)
  1. [Reduction to SDP via harmonic analysis] The abstract states that the theta number is reduced to a trigonometric sum-of-squares SDP, but the precise change of variables, the choice of basis for the harmonic expansion, and the numerical tolerances used to certify tightness should be stated explicitly in the main text.
  2. A short table or paragraph comparing the new bounds with previously known spherical-packing densities (e.g., for spherical triangles or squares) would help readers assess the improvement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive feedback on the algebraic criterion. We address the major comment below and will incorporate the requested clarifications into the revised version.

read point-by-point responses

  1. Referee: [Algebraic criterion for disjoint interiors (used to define the Cayley-graph edges)] The maximality claims rest on the algebraic criterion (introduced after the abstract and used to define the adjacency relation of the Cayley graph) being both necessary and sufficient for two congruent regular spherical polygons to have disjoint interiors. If any pair whose interiors intersect is not flagged by the criterion, the resulting graph is a proper subgraph of the true conflict graph; independent sets in the computed graph may therefore contain geometrically invalid configurations, and the theta value upper-bounds only a relaxation. The manuscript must supply a complete proof that the criterion detects every overlap, together with explicit verification against known intersecting configurations or a geometric argument showing completeness.

    Authors: We agree that a self-contained proof of necessity and sufficiency is essential for the validity of the upper bounds. The criterion is obtained by expressing the condition for interior overlap in terms of the relative orientation g in SO(3) and reducing it, via the rotational symmetry of the regular polygon, to a finite set of algebraic inequalities on the entries of the rotation matrix (specifically, on the spherical distances between corresponding vertices after rotation). In the current draft this derivation is sketched but not expanded into a full if-and-only-if argument. In the revision we will add a dedicated subsection that (i) proves sufficiency by showing that satisfaction of the algebraic inequalities implies the minimal geodesic distance between any pair of points in the two polygons is positive, and (ii) proves necessity by exhibiting an explicit intersecting configuration whenever any inequality is violated. We will also include direct numerical verification on a collection of known overlapping and non-overlapping pairs (including the equatorial great-circle case and the standard spherical triangle overlaps) to confirm that the criterion flags precisely the intersecting instances. With these additions the Cayley graph coincides with the true conflict graph, so the SDP relaxation of the Lovász theta number yields a rigorous upper bound on the packing density. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external Lovász theta, harmonic analysis, and SDP on independently introduced algebraic criterion

full rationale

The paper introduces an algebraic criterion for disjoint interiors of spherical polygons, constructs the corresponding Cayley graph on SO(3), and computes its Lovász theta number via standard harmonic analysis reduced to a trigonometric sum-of-squares SDP. None of these steps define a quantity in terms of itself or rename a fitted parameter as a prediction; the theta bound is obtained from the graph defined by the criterion, which is external to the final density value. The approach relies on established tools (Lovász theta, representation theory of SO(3), semidefinite programming) whose validity does not presuppose the target packing densities. No self-citation chain or ansatz smuggling is load-bearing for the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard representation theory of SO(3) and the correctness of the newly introduced algebraic non-overlap criterion; no free parameters or invented entities are visible from the abstract.

axioms (1)
  • standard math Harmonic analysis on SO(3) reduces the theta number on the Cayley graph to a trigonometric sum-of-squares problem
    Invoked to obtain an SDP formulation from the graph-theoretic relaxation.

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