Talk:Flexible polyhedron

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}

Scientific American article

I recall there was an article in Scientific American, sometime between 1975 and 1985, that described a flexible polyhedron. It was composed of two flexible halves that were joined together to form a non-convex polyhedron which could flex. (The author used a simple paper cut-out template, which was provided in the article.) If anyone could locate the article, it would be a good reference for this subject. — Loadmaster (talk) 16:33, 15 May 2012 (UTC)

I believe that a "flexible polyhedron" is also known as a "flexahedron". A google search turns up a few interesting links. I've added a redirect for "Flexahedron" to this page. — Loadmaster (talk) 16:20, 23 May 2012 (UTC)

I also found the page "Flexible Polyhedron" at MathWorld (Wolfram.com). No mention of Scientific American, though. — Loadmaster (talk) 19:03, 8 August 2016 (UTC)

Also Bob Connelly's page at Cornell.edu. Footnote 53 there references a one-page description of the construction of a 14-face flexible 3D polyhedron by Kaluse Steffen, which is at www.math.cornell.edu/~connelly/Steffen.pdf. — Loadmaster (talk) 19:13, 8 August 2016 (UTC)

I just ran across this: in the column "Mathematical Recreations," in the May, 1991 issue of Scientific American, the topical subheading was "The theory of rigidity, or how to brace yourself against unlikely accidents." Rather than being a treatise on safety (the columnist, A. K. Dewdney, liked to be amusing at times), the column addressed, non-exhaustively, the rigidity of two- and three-dimensional assemblages of struts and panels. Mentioned, along with instructions for its construction, was the "Connelly-Steffen surface," which might be what Loadmaster is referring to above; Dewdney wrote that the surface was the first triangle-faced polyhedron found, convex or not, that was not rigid. — Preceding unsigned comment added by Engrj (talk • contribs) 16:37, 25 July 2020 (UTC)

References

Description of Connelly sphere

Can we add a description &/or diagram of the Connelly sphere (since it redirects here) ? "Flexible Polyhedron"(@MathWorld) just says "Connelly (1978) found the first example of a true flexible polyhedron, consisting of 18 triangular faces (Cromwell 1997, pp. 242-244)." (Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp. 222, 224, and 239-247, 1997) - Rod57 (talk) 11:25, 8 November 2017 (UTC)