Klein-Arnold tensegrities

Abstract

In this paper, we introduce new classes of infinite and combinatorially periodic tensegrities, derived from algebraic multidimensional continued fractions in the sense of F. Klein. We describe the stress coefficients on edges through integer invariants of these continued fractions, as initiated by V.I. Arnold, thereby creating a novel connection between geometric rigidity theory and the geometry of continued fractions. Remarkably, the new classes of tensegrities possess rational self-stress coefficients. To establish the self-stressability of the frameworks, we present a projective version of the classical Maxwell-Cremona lifting principle, a result of independent interest.

Figures (7)

• Figure 1: Left: Central projection of a polyhedral surface $S(p)$ with vertices $p_i$ to a proper projection plane $\pi$. The projected framework $G_S(\bar{p})$ with vertices $\bar{p}_i$ is a tensegrity. Right: Two faces adjacent to an oriented edge $\overrightarrow{p_i p_j}$. The given orientation of the surface specifies a left and right face with respect to an oriented edge. The projective lifting coefficient $\omega_{ij}$ is determined per edge independent from the orientation of the edge.
• Figure 2: Left: Integer plane $\pi$ in $\mathbb{Z}^3$. It containes a full rank integer sub-lattice (bigger red dots). centre-top: Integer length between points: ${\rm I}\ell(p_1, p_2) = 1$ and ${\rm I}\ell(q_1, q_2) = 3$. Centre-bottom: Integer distance from point $p$ to line $\pi$: ${\rm Id}(p, \pi) = 5$. Right: Integer area of a parallelogram spanned by two vectors: ${\rm IA}(v_1, v_2) = 5$. The five cosets are illustrated in different colors and different line-styles.
• Figure 3: An illustration of a sail. The simplicial cone $\mathcal{C}(x_1, x_2, x_3)$ has three faces and three edges (illustrated in white). The vertices of $\mathbb{Z}^3$ on the A-hull, the convex hull of $\mathbb{Z}^3 \cap \mathcal{C}\setminus\{O\}$, are illustrated in red.
• Figure 4: A sail $S(p)$ and its projection $G_S(\bar{p})$ on the proper projection plane $\pi_1$ given by equation $z = 1$(top-left). The centre of projection is $O = (0,0,0)$ and the sail is enclosed within the cone generated by the eigenspaces ${\rm span}(x_1), {\rm span}(x_2), {\rm span}(x_3)$. The resulting framework $G_S(\bar{p})$ without projective distortion is depicted at bottom-left. The triangle shape of the boundary is the intersection of the corresponding simplicial cone with $\pi_1$. Most edges of the framework $G_S(\bar{p})$ accumulate along these triangle edges. Projecting the framework to another plane $\pi_2$, which is given by the normal vector $x_1/\|x_1\| + x_2/\|x_2\| + x_3/\|x_3\|$ (average of eigenvectors), is depicted at top-right. Projecting to the same plane $\pi_2$ but from centre $(0,0,-2)$ results in the framework at bottom-right. Here more of the edges that are accumulating along the triangle edges are visible.
• Figure 5: Combinatorial view of Example \ref{['example1']} (the actual geometry is depicted in Figure \ref{['fig:sail-framework']}). The projection-coefficients $\beta_i$ are on the left; the projective lifting coefficients $\omega_{ij}$ for a fundamental domain are in the centre; the projection-stresses $\bar{\omega}_{ij}$ are on the right.

Theorems & Definitions (40)

• Theorem 1.1: Maxwell-Cremona correspondence
• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Remark 2.7
• Theorem 2.8
• Remark 2.9