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The applicability of equal area partitions of the unit sphere

Paul Leopardi

TL;DR

This study uses equal-area partitioning of the unit sphere as a case study to examine the practical applicability of mathematical constructions. It outlines the EQ$(d,N)$ partition and EQP$(d,N)$ codes, traces their historical development, and surveys how they are used and evaluated across disciplines, from energy and covering to quadrature. While demonstrating broad applicability, it also highlights limitations such as spherical-harmonic reconstruction accuracy, and discusses refinements like the diamond ensemble and norming-set approaches that improve performance. The work emphasizes that the partitions are broadly useful but domain-specific, offering actionable guidance for applying sphere partitions in astronomy, geophysics, imaging, and climate science. Overall, the paper provides a structured synthesis of the state of the art and points to directions for integrating EQ-based partitions with other numerical methods for robust real-world use.

Abstract

This paper addresses the idea of the applicability of mathematics, using, as a case study, a construction and software package that partition the unit sphere into regions of equal area. The paper assesses the applicability of this construction and software by examining citing works, including papers, dissertations and software.

The applicability of equal area partitions of the unit sphere

TL;DR

This study uses equal-area partitioning of the unit sphere as a case study to examine the practical applicability of mathematical constructions. It outlines the EQ partition and EQP codes, traces their historical development, and surveys how they are used and evaluated across disciplines, from energy and covering to quadrature. While demonstrating broad applicability, it also highlights limitations such as spherical-harmonic reconstruction accuracy, and discusses refinements like the diamond ensemble and norming-set approaches that improve performance. The work emphasizes that the partitions are broadly useful but domain-specific, offering actionable guidance for applying sphere partitions in astronomy, geophysics, imaging, and climate science. Overall, the paper provides a structured synthesis of the state of the art and points to directions for integrating EQ-based partitions with other numerical methods for robust real-world use.

Abstract

This paper addresses the idea of the applicability of mathematics, using, as a case study, a construction and software package that partition the unit sphere into regions of equal area. The paper assesses the applicability of this construction and software by examining citing works, including papers, dissertations and software.
Paper Structure (12 sections, 1 equation, 24 figures)

This paper contains 12 sections, 1 equation, 24 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some related problems
  4. Some history
  5. The partition algorithm.
  6. Spherical codes from equal area partitions.
  7. Follow-up papers and generalizations
  8. Evaluations and improvements
  9. Evaluations
  10. Improvements
  11. Some applications
  12. Conclusion

Figures (24)

  • Figure 1: Brauchart, "Spherical Fibonacci lattices," brauchart2012low.
  • Figure 2: Themistoclakis and Van Barel, "Examples of the tensor product Gauss–Legendre quadrature nodes related to degrees of precision 31 and 51, i.e., having $N = 512$ (left) and $N = 1352$ (right) points," themistoclakis2018optimal.
  • Figure 3: Atschuler et al. "Lattice configurations for 132 (a) and 1032 (b) charges," altschuler1997possible.
  • Figure 4: Dartmouth College Electron Microscope Facility, "A grain of pollen from Morning Glory flowers," dartmouth2024electron.
  • Figure 5: Saff and Womersley, "Covering of a sphere with 169 equal spherical caps," saff2013optimal.
  • ...and 19 more figures