Talk:Flexible polyhedron

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[edit] Physical realizability of flexible polyhedra

Polyhedron rigidity can be assessed analytically using a pin-bar model (the edges, modeled as bars, are connected at the ends by spherical hinges). This model leads to a system of simultaneous quadratic equations representing the invariance of the bar lengths expressed in terms of the nodal coordinates. Six linear equations are added to the system to prevent the rigid translations and rotations in 3D space. The set of the nodal coordinate values of the assembly in its reference configuration is an obvious solution for the resulting system of equations.

For the polyhedron to allow non-rigid deformations, the solution must be non-unique, which requires singularity of the Jacobian matrix. However, any small imperfection, say, a deviation from the nominal values of the bar lengths, destroys singularity and restores the polyhedron rigidity. A physically realized "flexible" polyhedron (e.g., one built using any published cutout pattern), in contrast to its idealized geometric counterpart, is always imperfect. As a result, such a near-by physical polyhedron is rigid (any non-rigid deformations require material pliability). The polyhedron resists flexing and, upon release, reverts to the original configuration.

Finally, flexible polyhedra are noncomputable in the sense that evaluating numerically the combination of the bar lengths and nodal coordinates necessary for flexibility would require computing with infinite precision. Since perfect precision is attainable only in symbolic (e.g., algebraic or integer) calculations or in description-geometric (as opposed to analytical-geometric) operations. {helpme} 128.174.192.194 (talk) 17:53, 2 January 2008 (UTC) {helpme}