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The raspberries in three dimensions with at most two sizes of berry

Miek Messerschmidt

TL;DR

We address the problem of classifying three-dimensional raspberries—configurations of spheres tangent to a central unit sphere with interiors disjoint and whose berry contact graph triangulates the central pit. The authors encode the combinatorics via $01$- and $02$-necklaces and derive algebraic constraints using Gröbner-basis methods to compute necklace polynomials, enabling a finite, verifiable catalog under dihedral-angle and pit-area constraints. The main results include a complete catalog of raspberries with at most two berry sizes, notably thirty non-flexible raspberries and a set of flexible (and one icosahedral) special cases, all supported by publicly available data. This work provides a concrete, computationally realizable classification in higher-dimensional compact sphere packings and demonstrates a scalable approach for enumerating intricate geometric configurations guided by combinatorial encodings.

Abstract

In three dimensional Euclidean space, a raspberry is defined to be an arrangement of spheres with pairwise disjoint interiors, where all spheres are tangent to a central unit sphere and such that the contact graph of the non-central spheres triangulates the central sphere. We discuss the relevance of these structures in related work. We present a catalog of all configurations of radii that permit the formation of raspberries that have at most two sizes of non-central spheres. Throughout, we discuss the construction of this catalog.

The raspberries in three dimensions with at most two sizes of berry

TL;DR

We address the problem of classifying three-dimensional raspberries—configurations of spheres tangent to a central unit sphere with interiors disjoint and whose berry contact graph triangulates the central pit. The authors encode the combinatorics via - and -necklaces and derive algebraic constraints using Gröbner-basis methods to compute necklace polynomials, enabling a finite, verifiable catalog under dihedral-angle and pit-area constraints. The main results include a complete catalog of raspberries with at most two berry sizes, notably thirty non-flexible raspberries and a set of flexible (and one icosahedral) special cases, all supported by publicly available data. This work provides a concrete, computationally realizable classification in higher-dimensional compact sphere packings and demonstrates a scalable approach for enumerating intricate geometric configurations guided by combinatorial encodings.

Abstract

In three dimensional Euclidean space, a raspberry is defined to be an arrangement of spheres with pairwise disjoint interiors, where all spheres are tangent to a central unit sphere and such that the contact graph of the non-central spheres triangulates the central sphere. We discuss the relevance of these structures in related work. We present a catalog of all configurations of radii that permit the formation of raspberries that have at most two sizes of non-central spheres. Throughout, we discuss the construction of this catalog.
Paper Structure (13 sections, 103 equations, 5 figures, 4 tables)

This paper contains 13 sections, 103 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Computation of necklace polynomials
  4. Computing all raspberries with at most two sizes of berry
  5. The 01-necklace codes and necklace polynomials
  6. The raspberries with one size of berry
  7. The flexible raspberries with at most two sizes of berry
  8. Bounds on the dihedral angle $\widehat{02/xy}$ in a raspberry
  9. Computing compatible necklace code pairs
  10. Excluding necklace code pairs
  11. Analyzing the numerical solutions to systems of necklace dihedral angle constraints
  12. The thirty non-flexible raspberries with two sizes of berry
  13. Data availability statement

Figures (5)

  • Figure 1.1: Examples of raspberries with three sizes of berry.
  • Figure 2.1: A necklace with necklace code $01{:}33121$ from a raspberry with three sizes of berry.
  • Figure 4.1: The raspberries with one size of berry.
  • Figure 4.2: The flexible raspberries with at most two sizes of berry.
  • Figure 4.3: The geometric failure in attempting to form a raspberry with necklace codes $01{:}22111$ and $02{:}211211$ and $(r_{1},r_{2})\approx(0.6105,1.1754)$. The large berries must occur tangent to an equator of the pit. However these berries are too small for five of them to fit around the pit's equator, and too large for six of them to fit around the pit's equator.

Theorems & Definitions (3)

  • Definition 1.1
  • Definition 2.1
  • Remark 4.1