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Non-Euclidean Crystallographic Rigidity

Jack Esson, Eleftherios Kastis, Bernd Schulze

TL;DR

This work extends rigidity theory to non-Euclidean normed planes by providing combinatorial characterisations for forced-symmetric and forced-periodic rigidity under a fixed lattice. It develops orbit rigidity matrices for $oldsymbol{ mo}_q$ $(q eq 2)$ and polytopic planes, and proves minimal rigidity via inductive constructions built from Henneberg-type extensions, yielding precise tightness conditions: $(2,2)$-tight for periodic rigidity, $(2,2,1)$-gain-tight for reflectional symmetry, and $(oldsymbol{b Z}^2 timesoldsymbol{ m C}_s)_q$-tight for wallpaper-symmetric cases. Parallel results are obtained for the $oldsymbol{ mo}_1$/$oldsymbol{ mo}_ ty$ planes using monochrome-framework decompositions and analogous extension moves. Collectively, the paper delivers a unified inductive framework to characterise minimal rigidity in diverse non-Euclidean settings, enabling design and analysis of symmetric crystalline frameworks in these geometries. It also lays groundwork for future exploration of other symmetries, flexible lattices, and higher dimensions.

Abstract

This paper establishes combinatorial characterisations of forced-symmetric and forced-periodic rigidity (under a fixed lattice) of bar-joint frameworks in non-Euclidean normed planes. In $\ell_q$-planes for $q\in(1,\infty)\backslash\{2\}$, we prove characterisations for periodic rigidity and finite reflectionally-symmetric rigidity. We also characterise symmetric rigidity in this space with respect to the orientation-reversing wallpaper group $\mathbb{Z}^2\rtimes\mathcal{C}_s$, otherwise known as $pm$ in crystallography. In the $\ell_1$ and $\ell_\infty$-planes, we provide characterisations for periodic rigidity and $\mathbb{Z}^2\rtimes\mathcal{C}_s$-symmetric rigidity. All of these characterisations are proved by inductive constructions involving Henneberg-type graph operations.

Non-Euclidean Crystallographic Rigidity

TL;DR

This work extends rigidity theory to non-Euclidean normed planes by providing combinatorial characterisations for forced-symmetric and forced-periodic rigidity under a fixed lattice. It develops orbit rigidity matrices for and polytopic planes, and proves minimal rigidity via inductive constructions built from Henneberg-type extensions, yielding precise tightness conditions: -tight for periodic rigidity, -gain-tight for reflectional symmetry, and -tight for wallpaper-symmetric cases. Parallel results are obtained for the / planes using monochrome-framework decompositions and analogous extension moves. Collectively, the paper delivers a unified inductive framework to characterise minimal rigidity in diverse non-Euclidean settings, enabling design and analysis of symmetric crystalline frameworks in these geometries. It also lays groundwork for future exploration of other symmetries, flexible lattices, and higher dimensions.

Abstract

This paper establishes combinatorial characterisations of forced-symmetric and forced-periodic rigidity (under a fixed lattice) of bar-joint frameworks in non-Euclidean normed planes. In -planes for , we prove characterisations for periodic rigidity and finite reflectionally-symmetric rigidity. We also characterise symmetric rigidity in this space with respect to the orientation-reversing wallpaper group , otherwise known as in crystallography. In the and -planes, we provide characterisations for periodic rigidity and -symmetric rigidity. All of these characterisations are proved by inductive constructions involving Henneberg-type graph operations.

Paper Structure

This paper contains 14 sections, 34 theorems, 35 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background on symmetric frameworks
  3. Extensions
  4. Basic Rigidity in $\ell_q$-planes for $q\in(1,\infty)\backslash\{2\}$
  5. Periodic Rigidity in $\ell_q$-planes for $q\in(1,\infty)\backslash\{2\}$
  6. Reflective Symmetric Rigidity in $\ell_q$-planes for $q\in(1,\infty)\backslash\{2\}$
  7. $\mathbb{Z}^2\rtimes\mathcal{C}_s$-Symmetric Rigidity in $\ell_q$-planes for $q\in(1,\infty)\backslash\{2\}$
  8. Basic Rigidity in the $\ell_1$ and $\ell_\infty$-planes
  9. Periodic Rigidity in the $\ell_1$ and $\ell_\infty$-planes
  10. $\mathbb{Z}^2\rtimes\mathcal{C}_s$-symmetric Rigidity in the $\ell_1$ and $\ell_\infty$-Planes
  11. Further Work
  12. Other Symmetries
  13. Flexible Lattice Representations
  14. Higher Dimensions

Key Result

Lemma 2.1

Let $(G,m)$ be a $\Gamma$-gain graph such that, for every vertex $v$ of $G$, the gain space $\langle(G,m)\rangle_v$ is contained in some subgroup $\Gamma'\leq \Gamma$. Then there is an equivalent gain graph $(G,m')$ in which every gain is an element of $\Gamma'$.

Figures (13)

  • Figure 3.1: The two variations of the gained $0$-extension.
  • Figure 3.2: The four variations of the gained $1$-extension.
  • Figure 3.3: The gained loop-$1$-extension. Note that the gain $m(l)$ must have a non-trivial linear component.
  • Figure 3.4: The gained vertex-to-$4$-cycle move.
  • Figure 3.5: The gained vertex-to-$K_4$ move.
  • ...and 8 more figures

Theorems & Definitions (63)

  • Lemma 2.1
  • Definition 3.1
  • Theorem 5.1
  • Proposition 5.2
  • proof
  • Theorem 5.3
  • proof : Proof of sufficiency for Theorem \ref{['lqPeriodic']}
  • Theorem 6.1
  • Proposition 6.2
  • Proposition 6.3
  • ...and 53 more