A shaky polyhedron constructed by replacing six pairs of adjacent triangles in an icosahedron (whose edges form a skew quadrilateral) with pairs of isosceles triangles sharing a common base. The polyhedron can be constructed by dividing the sides of the icosahedron in the golden ratio (as used in the construction of the icosahedron along the edges of the octahedron), but reversing the long and short segments.

The skeleton of Jessen's orthogonal icosahedron is the icosahedral graph.

The centers of the eight equilateral triangles which remain are then the vertices of a cube. The polyhedron can be deformed infinitesimally by pinching the angles between the isosceles triangles whose bases act as hinges. If the polyhedron is constructed using paper and tape instead of entirely rigid faces, it is possible to collapse the isosceles triangles onto one another, resulting in an octahedron.

Goldberg, M. "Unstable Polyhedral Structures." Math. Mag. 51, 165-170, 1978.

Jessen, B. "Orthogonal Icosahedron." Nordisk Mat. Tidskr. 15, 90-96, 1967.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 161, 1991.

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Weisstein, Eric W. "Jessen's Orthogonal Icosahedron." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/JessensOrthogonalIcosahedron.html