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Putatively optimal projective spherical designs with little apparent symmetry

Alex Elzenaar, Shayne Waldron

TL;DR

The paper investigates putatively optimal projective spherical designs, exposing continuous families of designs with little apparent symmetry and developing new construction techniques beyond classical symmetric configurations. By combining numerical optimization on manifolds with algebraic-cubature reasoning, it uncovers explicit examples such as an $11$-point $(3,3)$-design in $\mathbb{R}^3$ and a $12$-point $(2,2)$-design in $\mathbb{R}^4$ built from four Mercedes-Benz frames in equi-isoclinic planes, while revealing that the algebraic variety of optimal designs often has positive dimension, i.e., a rich, high-dimensional landscape. The work also collects and analyzes number-theoretic and cubature-based constructions (Reznick, Stroud, Kempner, Kotelina–Pevnyi), highlighting regimes of infinite families versus highly symmetric finite designs and identifying exceptional cases. Overall, the paper advances understanding of when optimal designs are unique versus part of infinite families, and provides practical pathways to generate and certify putatively optimal designs for applications in cubature and frame theory. The results illuminate the geometry of sphere-based integration rules and suggest new directions for exploring the algebraic structure of design varieties.

Abstract

We give some new explicit examples of putatively optimal projective spherical designs. i.e., ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in general, which requires the introduction of new techniques for their construction. New examples of interest include an 11-point spherical (3, 3)-design for R 3 , and a 12-point spherical (2, 2)-design for R 4 given by four Mercedes-Benz frames that lie on equi-isoclinic planes. We also give results of an extensive numerical study to determine the nature of the real algebraic variety of optimal projective real spherical designs, and in particular when it is a single point (a unique design) or corresponds to an infinite family of designs.

Putatively optimal projective spherical designs with little apparent symmetry

TL;DR

The paper investigates putatively optimal projective spherical designs, exposing continuous families of designs with little apparent symmetry and developing new construction techniques beyond classical symmetric configurations. By combining numerical optimization on manifolds with algebraic-cubature reasoning, it uncovers explicit examples such as an -point -design in and a -point -design in built from four Mercedes-Benz frames in equi-isoclinic planes, while revealing that the algebraic variety of optimal designs often has positive dimension, i.e., a rich, high-dimensional landscape. The work also collects and analyzes number-theoretic and cubature-based constructions (Reznick, Stroud, Kempner, Kotelina–Pevnyi), highlighting regimes of infinite families versus highly symmetric finite designs and identifying exceptional cases. Overall, the paper advances understanding of when optimal designs are unique versus part of infinite families, and provides practical pathways to generate and certify putatively optimal designs for applications in cubature and frame theory. The results illuminate the geometry of sphere-based integration rules and suggest new directions for exploring the algebraic structure of design varieties.

Abstract

We give some new explicit examples of putatively optimal projective spherical designs. i.e., ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in general, which requires the introduction of new techniques for their construction. New examples of interest include an 11-point spherical (3, 3)-design for R 3 , and a 12-point spherical (2, 2)-design for R 4 given by four Mercedes-Benz frames that lie on equi-isoclinic planes. We also give results of an extensive numerical study to determine the nature of the real algebraic variety of optimal projective real spherical designs, and in particular when it is a single point (a unique design) or corresponds to an infinite family of designs.
Paper Structure (15 sections, 2 theorems, 67 equations, 4 figures, 1 table)

This paper contains 15 sections, 2 theorems, 67 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Numerics
  3. The overall picture
  4. A family of $12$-point spherical $(2,2)$-designs for $\mathbb{R}^4$
  5. Selected calculations
  6. A family of $24$-point spherical $(4,4)$-designs for $\mathbb{R}^3$
  7. Spherical $(t,t)$-designs with some structure
  8. Designs from number theory and cubature
  9. The Reznick $11$-point spherical $(3,3)$-design for $\mathbb{R}^3$
  10. A new $11$-point $(3,3)$-design for $\mathbb{R}^3$
  11. The Reznick/Stroud spherical $(2,2)$-designs for $\mathbb{R}^4,\mathbb{R}^5,\mathbb{R}^6$
  12. Kempner's $24$-point spherical $(3,3)$-design for $\mathbb{R}^4$
  13. Kotelina and Pevnyi's $120$-point $(3,3)$-design for $\mathbb{R}^8$
  14. Conclusion
  15. Acknowledgements

Key Result

Theorem 4.1

Let $(v_j)$ consist of four Mercedes-Benz frames that lie in four equi-isoclinic planes in $\mathbb{R}^4$. Then $(v_j)$ is a $12$-vector spherical $(2,2)$-design for $\mathbb{R}^4$.

Figures (4)

  • Figure 1: The graphs of $n\mapsto f_{t,d,n}$ and $n\mapsto \log_{10} f_{t,d,,n}$ for $t=2$, $d=6$, i.e., the error in numerical approximations to a unit-norm spherical $(2,2)$-design of $n$ vectors in $\mathbb{R}^6$.
  • Figure 2: The graphs of $n\mapsto f_{t,d,n}$ and $n\mapsto \log_{10} f_{t,d,,n}$ for $t=5$, $d=4$, i.e., the error in numerical approximations to a unit-norm spherical $(5,5)$-design of $n$ vectors in $\mathbb{R}^4$.
  • Figure 3: The graphs $n\mapsto f_{t,d,n}$ and $n\mapsto \log_{10} f_{t,d,,n}$ of the error in approximations to weighted designs with $t=6$ and $d=5$, i.e., $(6,6)$-designs of $n$ vectors in $\mathbb{R}^5$.
  • Figure 4: Special situation: the graphs $n\mapsto f_{t,d,n}$ and $n\mapsto \log_{10} f_{t,d,,n}$ for $t=3$, $d=8$.

Theorems & Definitions (15)

  • Example 3.1
  • Theorem 4.1
  • Theorem 5.1
  • Example 5.1
  • Example 5.2
  • Example 5.3
  • Example 5.4
  • Example 5.5
  • Example 5.6
  • Example 5.7
  • ...and 5 more