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From the separable Tammes problem to extremal distributions of great circles in the unit sphere

Károly Bezdek, Zsolt Lángi

TL;DR

This work extends the Tammes problem to spherical spaces via totally separable packings and coverings of congruent spherical caps on $S^2$, deriving exact and bound results for the separable Tammes problem up to eight caps and establishing a tight link to the centered separable kissing number in 3D. It introduces a spherical Molnár decomposition to analyze extremal distributions generated by great-circle tilings, and proves sharp bounds on the inradius and circumradius of cells in tilings, including characterizations of equality cases with octahedral and cuboctahedral symmetry. The paper further develops tight density-type inequalities for TS-packings and TS-coverings, proves thinnest-TS-covering bounds, and extends several results to higher dimensions, using a blend of geometric, combinatorial, and projection techniques. Collectively, the results advance understanding of how symmetry and separability constrain packing, covering, and tiling in spherical and high-dimensional spaces, with implications for related kissing-number problems and spherical tessellations.

Abstract

A family of spherical caps of the 2-dimensional unit sphere $\mathbb{S}^2$ is called a totally separable packing in short, a TS-packing if any two spherical caps can be separated by a great circle which is disjoint from the interior of each spherical cap in the packing. The separable Tammes problem asks for the largest density of given number of congruent spherical caps forming a TS-packing in $\mathbb{S}^2$. We solve this problem up to $8$ spherical caps and upper bound the density of any TS-packing of congruent spherical caps in terms of their angular radius. Based on this, we show that the centered separable kissing number of $3$-dimensional Euclidean balls is $8$. Furthermore, we prove bounds for the maximum of the smallest inradius of the cells of the tilings generated by $n>1$ great circles in $\mathbb{S}^2$. Next, we prove dual bounds for TS-coverings of $\mathbb{S}^2$ by congruent spherical caps. Here a covering of $\mathbb{S}^2$ by spherical caps is called a totally separable covering in short, a TS-covering if there exists a tiling generated by finitely many great circles of $\mathbb{S}^2$ such that the cells of the tiling are covered by pairwise distinct spherical caps of the covering. Finally, we extend some of our bounds on TS-coverings to spherical spaces of dimension $>2$.

From the separable Tammes problem to extremal distributions of great circles in the unit sphere

TL;DR

This work extends the Tammes problem to spherical spaces via totally separable packings and coverings of congruent spherical caps on , deriving exact and bound results for the separable Tammes problem up to eight caps and establishing a tight link to the centered separable kissing number in 3D. It introduces a spherical Molnár decomposition to analyze extremal distributions generated by great-circle tilings, and proves sharp bounds on the inradius and circumradius of cells in tilings, including characterizations of equality cases with octahedral and cuboctahedral symmetry. The paper further develops tight density-type inequalities for TS-packings and TS-coverings, proves thinnest-TS-covering bounds, and extends several results to higher dimensions, using a blend of geometric, combinatorial, and projection techniques. Collectively, the results advance understanding of how symmetry and separability constrain packing, covering, and tiling in spherical and high-dimensional spaces, with implications for related kissing-number problems and spherical tessellations.

Abstract

A family of spherical caps of the 2-dimensional unit sphere is called a totally separable packing in short, a TS-packing if any two spherical caps can be separated by a great circle which is disjoint from the interior of each spherical cap in the packing. The separable Tammes problem asks for the largest density of given number of congruent spherical caps forming a TS-packing in . We solve this problem up to spherical caps and upper bound the density of any TS-packing of congruent spherical caps in terms of their angular radius. Based on this, we show that the centered separable kissing number of -dimensional Euclidean balls is . Furthermore, we prove bounds for the maximum of the smallest inradius of the cells of the tilings generated by great circles in . Next, we prove dual bounds for TS-coverings of by congruent spherical caps. Here a covering of by spherical caps is called a totally separable covering in short, a TS-covering if there exists a tiling generated by finitely many great circles of such that the cells of the tiling are covered by pairwise distinct spherical caps of the covering. Finally, we extend some of our bounds on TS-coverings to spherical spaces of dimension .
Paper Structure (27 sections, 20 theorems, 43 equations, 22 figures)

This paper contains 27 sections, 20 theorems, 43 equations, 22 figures.

Table of Contents

  1. Introduction
  2. On the densest TS-packings by congruent spherical caps in $\mathbb{S}^2$
  3. Maximizing the smallest angular inradius of the cells of the tilings generated by $n$ great circles in $\mathbb{S}^2$.
  4. Minimizing the largest circumradius of the cells of the tilings generated by $n$ great circles in $\mathbb{S}^2$
  5. On the thinnest TS-coverings by congruent spherical caps in $\mathbb{S}^2$
  6. On the analogue of Problem \ref{['separable-coverings']} in high dimensions
  7. Proof of Theorem \ref{['thm:few_caps']}
  8. Spherical Molnár decomposition
  9. Proof of Theorem \ref{['main2']}
  10. Proof of Part (i)
  11. Proof of Part (ii)
  12. Proof of Theorem \ref{['sub-main2']}
  13. Proof of Theorem \ref{['main3']}
  14. Proof of Part (i)
  15. Proof of Part (ii)
  16. ...and 12 more sections

Key Result

Theorem 1

For $5 \leq k \leq 6$ we have $r_{\rm STam}(k, \mathbb{S}^2)= \arctan \frac{3}{4}$, and any $k$-optimal TS-packing is a subfamily of a cuboctahedral TS-packing. Furthermore, for $7 \leq k \leq 8$ we have $r_{\rm STam}(k, \mathbb{S}^2)= \arcsin \frac{1}{\sqrt{3}}$, and any $k$-optimal TS-packing is a

Figures (22)

  • Figure 1: An octahedral TS-packing in $\mathbb{S}^2$.
  • Figure 2: A cuboctahedral TS-packing in $\mathbb{S}^2$.
  • Figure 3: The graph of the function $\delta(\rho)$ defined in Definition \ref{['defn:density_TSpacking']}.
  • Figure 4: A $4$-optimal arrangement of great circles in $\mathbb{S}^2$.
  • Figure 5: $r_{\rm gc}(6, \mathbb{S}^2)\geq\rho_6$
  • ...and 17 more figures

Theorems & Definitions (48)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • Definition 3
  • Definition 4
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Definition 5
  • ...and 38 more