## 8 Polarity for Rigidity
### Sheet Structures and Polarity for Rigidity in \(\mathbb{R}^{3}\)  
We next introduce hinged sheetworks with planes and edges as the polars of bar-joint frameworks, preserving rigidity when we interpret the planes as statically rigid frameworks on the incident edges. These were mentioned, independently, in the work of a Danish Architectural Engineer Ture Wester, and were also mentioned in passing in [97]. The one published study of hinged sheetworks uses static rigidity [113], showing how the forces applied to a face in its plane transmits through the statically rigid framework of a face (a) and how a force applied to an edge splits into two forces in the two distinct faces at the edge (b), Figure 35. The equilibrium condition for forces at the original vertex becomes an equilibrium condition for the forces applied in the plane of the face. We can transfer all of the definitions for static rigidity to the hinged sheetworks following [113]. There is a companion infinitesimal rigidity version of this theory using plane centers of motion for vertices polarizing to point centers of the sheets. It has yet to receive a proper published exposition, but the gist can be seen by polarizing the theory of momenta and centers of motion of bar and joint frameworks. If the polarity is, then is the polar plane. The momentum of a vertex becomes a _weighted point center_ in the plane of the polar sheet. This point center of a sheet presents the component of the velocities of points in the sheet which lies within the plane of the plane-rigid sheet. The momentum of the hinge is the polar center of the hinge, a weighted line segment in the line joining the centers of motion of the two sheets. The hinge condition is the polar of the bar condition in terms of projective momenta:  becomes. There is a fully projective version of all this theory of sheets, but it has so far only been published in vocabulary of [113]. **Definition 1**. : _A hinged sheetwork in is an ordered pair consisting of a graph and an assignment of weighted plane sections to the vertices in projective 3-space such that if._  
**Definition 2**. : _An equilibrium load on a hinged sheetwork is an assignment of dual 2-extensors (forces),, to the sheets such that for each sheet we have and. A resolution of the load by a hinged sheetwork is an assignment of scalars to the edges if such that, for each sheet :_  
_A hinged sheetwork is statically rigid if every equilibrium load has a resolution. A static stress is a resolution of the zero load, i.e., a set of scalars for the edges such that at each sheet, sum over all edges attached to the sheet. A hinged sheetwork is independent if the only static stress is the trivial stress with all scalars zero (otherwise it is dependent) and is isostatic if it is statically rigid and independent._  
With a projective lens, it would be nice to have a good projective matrix for this. We propose that this should be the polar of the projective statics matrix for bar and joint frameworks. **Definition 3**. : _A hinged sheetwork and a bar-and-joint framework, both in, are polar if there is a non-singular linear transformation and a homogeneous multiplier (a set of scalars, is ) such that for each vertex of :_  
**Theorem 12** ([114]). : _A hinged sheetwork and any polar bar-and-joint framework in have the same static properties:_  
1. _is statically rigid if and only if_ _is statically rigid;_
2. _is independent if and only if_ _is independent;_
3. _isostatic if and only if_ _isostatic;_
4. _the spaces of equilibrium loads are isomorphic;__the spaces of resolved loads are isomorphic;_
6. _the spaces of stresses are isomorphic._  
**Example 25**. : _Figure 36 illustrates the polarity with a statically rigid octahedral framework going to a statically rigid hinged sheetwork on a cube, where each face is some statically rigid framework in the plane of the face. Note that we are really selecting from an equivalence class of all statically rigid sub-frameworks on vertices in the face.