TOPICS
Search

Multistable Polyhedron

A multistable polyhedron is a polyhedron that can change form from one stable configuration to another with only a slight transient nondestructive elastic stretch (Goldberg 1978). The simplest example of a polyhedron having multistable forms is Wunderlich's bistable jumping octahedron (Cromwell 1997, pp. 222-223).

SiameseDipyramidSchematic
SiameseDipyramid

Goldberg (1978) give two tristable polyhedra: one having 12 faces and one having 20. Goldberg's bistable icosahedron, illustrated above, consists of two adjoined pentagonal dipyramids, each with two adjacent triangles (one on top and one on bottom) omitted (Goldberg 1978; Wells 1991; Cromwell 1997, pp. 222 and 224). The variables in the schematic above are connected by the equations

sintheta=1/(2r)
(1)
x^2=1-r^2
(2)
y=rsin(5theta)=r(5sintheta-20sin^3theta+16sin^5theta)
(3)
=rsintheta(5-20sin^2theta+16sin^4theta)
(4)
=1/2(5-5/(r^2)+1/(r^4)).
(5)

Plugging in r^2=1-x^2 and setting y=x gives the quintic equation

2x^5-4x^4-4x^3+5x^2+2x-1=0,
(6)

which has smallest positive solution x approx 0.327267. Goldberg gives (x,y)=(0.071,0.49) and (0.49,0.071) as other solutions, although it is not clear where these come from.

See also

Jumping Octahedron, Unistable Polyhedron

Explore with Wolfram|Alpha

References

Cromwell, P. R. Polyhedra. New York: Cambridge University Press, 1997.Efimow, N. W. "Flachenverbiegung im Grossen." Berlin: Akademie-Verlag, p. 130, 1957.Goldberg, M. "Unstable Polyhedral Structures." Math. Mag. 51, 165-170, 1978.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991.Wunderlich, W. "Starre, kippende, wackelige und bewegliche Achtflache." Elem. Math. 20, 25-32, 1965.

Referenced on Wolfram|Alpha

Multistable Polyhedron

Cite this as:

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

Subject classifications