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# Rigid Polyhedron

A polyhedron is rigid if it cannot be continuously deformed into another configuration. A rigid polyhedron may have two or more stable forms which cannot be continuously deformed into each other without bending or tearing (Wells 1991).

A polyhedron that can change form from one stable configuration to another with only a slight transient nondestructive elastic stretch is called a multistable polyhedron (Goldberg 1978).

A non-rigid polyhedron may be "shaky" (infinitesimally movable) or flexible. An example of a concave flexible polyhedron with 18 triangular faces was given by Connelly (1978), and a flexible polyhedron with only 14 triangular faces was subsequently found by Steffen (Mackenzie 1998).

Jessen's orthogonal icosahedron is an example of a shaky polyhedron.

Flexible Polyhedron, Jessen's Orthogonal Icosahedron, Jumping Octahedron, Multistable Polyhedron, Pentagonal Dipyramid, Rigid Graph, Shaky Polyhedron

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## References

Cauchy, A. L. "Sur les polygons et le polyhéders." XVIe Cahier IX, 87-89, 1813.Connelly, R. "A Flexible Sphere." Math. Intel. 1, 130-131, 1978.Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Rigidity of Polyhedra." §B13 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 61-63, 1991.Cromwell, P. R. "Equality, Rigidity, and Flexibility." Ch. 6 in Polyhedra. New York: Cambridge University Press, pp. 219-247, 1997.Gluck, H. Almost All Simply Connected Closed Surfaces are Rigid. Heidelberg, Germany: Springer-Verlag, pp. 225-239, 1975.Goldberg, M. "Unstable Polyhedral Structures." Math. Mag. 51, 165-170, 1978.Graver, J.; Servatius, B.; and Servatius, H. Combinatorial Rigidity. Providence, RI: Amer. Math. Soc., 1993.Mackenzie, D. "Polyhedra Can Bend But Not Breathe." Science 279, 1637, 1998.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 161-162, 1991.Wunderlich, W. "Starre, kippende, wackelige und bewegliche Achtflache." Elem. Math. 20, 25-32, 1965.

Rigid Polyhedron

## Cite this as:

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