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Equilibrium stressability of multidimensional frameworks

Oleg Karpenkov, Christian Müller, Gaiane Panina, Brigitte Servatius, Herman Servatius, Dirk Siersma

Abstract

We prove an equilibrium stressability criterium for trivalent multidimensional tensegrities. The criterium appears in different languages: (1) in terms of stress monodromies, (2) in terms of surgeries, (3) in terms of exact discrete 1-forms, and (4) in Cayley algebra terms.

Equilibrium stressability of multidimensional frameworks

Abstract

We prove an equilibrium stressability criterium for trivalent multidimensional tensegrities. The criterium appears in different languages: (1) in terms of stress monodromies, (2) in terms of surgeries, (3) in terms of exact discrete 1-forms, and (4) in Cayley algebra terms.

Paper Structure

This paper contains 19 sections, 19 theorems, 35 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Main definitions and constructions
  3. Self-stressability of frameworks.
  4. Self-stressability of face-paths and face-cycles
  5. Face-path and face-cycle
  6. Self-stressability of face-path $d$-frameworks
  7. Edge-orientation transition
  8. Self-stressability of face-cycles of length 3
  9. $\mathop{\mathrm{H\Phi}}\nolimits$-surgeries
  10. Stress transition and stress monodromy
  11. Face-path equivalence
  12. Face-path $d$-frameworks in $d$-frameworks
  13. Geometric Characterizations of self-stressability for Trivalent $d$-Frameworks
  14. Ratio condition for self-stressable multidimensional trivalent frameworks
  15. Cayley algebra conditions
  16. ...and 4 more sections

Key Result

Proposition 3.1

Any generic face-path $d$-framework which contains no face-cycle has a one-dimen-sional space of self-stresses. All the stresses for all the planes of $F$ and $\hat{F}$ are either simultaneously zero, or simultaneously non-zero.

Figures (13)

  • Figure 1: A $3$-framework based on the cube with three types of faces.
  • Figure 2: Three face types: triangles, squares, or faces with just two edges.
  • Figure 3: A face path contained in the framework described in Example \ref{['K5stresssEx']} with $f_{0,1}= \{1,2,4\}$, $f_{1,2}= \{2,3,4\}$, $f_{2,3}= \{3, 4, 5\}$, $f_{3,4}= \{1,3,5\}$.
  • Figure 4: A face path contained in the framework described in Example \ref{['K5stresssEx']}, with labeling as before and $f_{4,0}= \{1,2,5\}$.
  • Figure 5: A face-cycle contained in $K_5$ which is not edge-orientable. For this particular example faces correspond to triangles, and we choose all the normals to point inward.
  • ...and 8 more figures

Theorems & Definitions (64)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 54 more