Equivariant Cosheaves and Finite Group Representations in Graphic Statics

TL;DR

The paper develops an equivariant cosheaf framework to extend graphic statics from generic planar frameworks to those with finite-group symmetry. By endowing force and position data with a G-action and decomposing the resulting chain complexes into irreducible representations, it yields a symmetry-resolved Maxwell counting rule via a symmetric Euler characteristic hat{X}^{(j)} and proves that the Maxwell-Cremona correspondence respects symmetry. The approach provides a principled classification of symmetric self-stresses and reciprocal diagrams by irreducible representations, and delivers open-source software to compute cosheaf homologies for concrete examples. This framework enables symmetry-aware design and analysis of structurally efficient gridshells and related systems, with potential extensions to periodic and higher-dimensional settings.

Abstract

This work extends the theory of reciprocal diagrams in graphic statics to frameworks that are invariant under finite group actions by utilizing the homology and representation theory of cellular cosheaves, recent tools from applied algebraic topology. By introducing the structure of an equivariant cellular cosheaf, we prove that pairs of self-stresses and reciprocal diagrams of symmetric frameworks are classified by the irreducible representations of the underlying group. We further derive the symmetry-aligned Euler characteristics of a finite dimensional equivariant chain complex, which for the force cosheaf yields a new formulation of the symmetry-adapted Maxwell counting rule for detecting symmetric self-stresses and kinematic degrees of freedom in frameworks. A freely available program is used to implement the relevant cosheaf homologies and illustrate the theory with examples.

Figures (11)

• Figure 1: A planar framework with $D_6$-symmetry is pictured (left). There are six dimensions of reciprocal diagrams (right), including two trivial translation dimensions. Each reciprocal diagram is an assignment of geometric coordinates to the dual graph, constrained such that dual edges must be parallel to their primal counterparts. Dual nodes may be drawn overlapping, as is exemplified in the trivial reciprocal diagram (bottom right). The space of reciprocal diagrams is subdivided by symmetry (corresponding to irreducible representations of $D_6$).
• Figure 2: A planar framework (geometric graph) with $D_4$ symmetry is pictured (a). This framework may be considered as a form diagram of a small gridshell roof, as it has no edge crossings and exhibits some other key desirable features, such as quad-dominance, aligned vertices and dihedral symmetry. Every reciprocal framework can be subdivided into a linear combination of component frameworks exhibiting symmetries of different irreducible representations of $D_4$, in the sense that adding the point coordinates of the diagrams yields the original reciprocal diagram (b).
• Figure 3: A framework in Desargues configuration with a self-stress and vertical mirror-symmetry (a) and its corresponding parallel reciprocal diagram with horizontal mirror-symmetry (b). The "transformation" of the mirror from vertical to horizontal is a consequence of Remark \ref{['rem:sign_flip']}.
• Figure 4: A sketch of the trivial representation $\iota$ over the constant cosheaf $\overline{\mathbb{R}}$, satisfying the constraint equation \ref{['eq:Gchain_complex']}. The edge $e$ changes orientation under the group action $g$, meaning $[ge,e] = -1$. The trivial representation $\iota$ does not detect cell geometry or embedding, only orientations.
• Figure 5: The $D_8$-constant cosheaf $(\overline{\mathbb{R}^2}, \eta_1)$ is pictured over a square cell complex, with 1 and 0 dimensional data drawn. Here we examine the commutativity condition (i) of Definition \ref{['def:Gcosheaf']} over edges and vertices. To the left the group element is a $\pi/2$ rotation counter-clockwise, and to the right the group element is a reflection about the horizontal axis. Take note of the sign alignment $[ge,e]=\pm 1$ between an edge $e$ and its permutation.

Theorems & Definitions (44)

• Theorem 1.1: \ref{['thm:G_GS']}) (Symmetric Planar Graphic Statics
• Definition 2.1: Cellular Cosheaf
• Definition 2.2: Chain complex
• Example 2.3: Constant cosheaves
• Example 2.4: The force cosheaf
• Definition 2.5
• Example 2.6: Planar graphic statics
• Example 3.1: Standard representation of cyclic and dihedral groups
• Example 3.2: Irreducible representations of common cyclic and dihedral groups
• Example 3.3: Permutation Representation