Grünbaum's Skeletal Polyhedra

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This Demonstration shows Grünbaum's skeletal polyhedra.


Grünbaum [2] discovered nine regular skeletal polyhedra whose faces are regular skew polygons. Five of them are obtained from the Platonic solids and four from the Kepler–Poinsot polyhedra. To find a skew face in one of them, trace a path along its edges such that each pair of adjacent edges are sides of a face of the polyhedron but no three consecutive edges are sides of a single face. The polygon constructed is called a Petrie polygon [1, pp. 284–285].


Contributed by: Izidor Hafner (December 2017)
Open content licensed under CC BY-NC-SA



Branko Grünbaum published an influential book, called Convex Polytopes, on the subject in 1967 [3].


[1] P. R. Cromwell, Polyhedra, New York: Cambridge University Press, 1997.

[2] B. Grünbaum, "Regular polyhedra—Old and New," Aequationes Mathematicae, 16(1–2), 1977 pp. 1–20.

[3] B. Grünbaum, Convex Polytopes (Graduate Texts in Mathematics), 2nd ed., New York: Springer, 2003.

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