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Flexible Polyhedron


Although the rigidity theorem states that if the faces of a convex polyhedron are made of metal plates and the polyhedron edges are replaced by hinges, the polyhedron would be rigid, concave polyhedra need not be rigid. A nonrigid polyhedron may be "shaky" (infinitesimally movable) or flexible (continuously movable; Wells 1991).

FlexiblePolyhedron

In 1897, Bricard constructed several self-intersecting flexible octahedra (Cromwell 1997, p. 239). Connelly (1978) found the first example of a true flexible polyhedron, consisting of 18 triangular faces (Cromwell 1997, pp. 242-244). Mason discovered a 34-sided flexible polyhedron constructed by erecting a pyramid on each face of a cube adjoined square antiprism (Cromwell 1997). Kuiper and Deligne modified Connelly's polyhedron to create a flexible polyhedron having 18 faces and 11 vertices (Cromwell 1997, p. 245), and Steffen found a flexible polyhedron with only 14 triangular faces and 9 vertices (shown above; Cromwell 1997, pp. 244-247; Mackenzie 1998). Maksimov (1995) proved that Steffen's is the simplest possible flexible polyhedron composed of only triangles (Cromwell 1997, p. 245).

Connelly et al. (1997) proved that a flexible polyhedron must keep its volume constant, confirming the so-called bellows conjecture (Mackenzie 1998).


See also

Bellows Conjecture, Polyhedron, Quadricorn, Rigid Polyhedron, Rigidity Theorem, Shaky Polyhedron

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References

Cauchy, A. L. "Sur les polygones et les polyèdres." XVIe Cahier IX, 87-89, 1813.Connelly, R. "A Flexible Sphere." Math. Intel. 1, 130-131, 1978.Connelly, R.; Sabitov, I.; and Walz, A. "The Bellows Conjecture." Contrib. Algebra Geom. 38, 1-10, 1997.Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp. 222, 224, and 239-247, 1997.Mackenzie, D. "Polyhedra Can Bend But Not Breathe." Science 279, 1637, 1998.Maksimov, I. G. "Polyhedra with Bendings and Riemann Surfaces." Uspekhi Matemat. Nauk 50, 821-823, 1995.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 161-162, 1991.

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Flexible Polyhedron

Cite this as:

Weisstein, Eric W. "Flexible Polyhedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FlexiblePolyhedron.html

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