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Goldberg Polyhedron


Goldberg polyhedra are convex polyhedra first described by Goldberg (1937) and classified in more detail by Hart (2013) for which each face is a regular pentagon or regular hexagon, exactly three faces meet at each vertex, and the rotational symmetry is that of a regular icosahedron.

Goldberg polyhedra can be constructed with planar equilateral (but not in general equiangular) faces, though in general the corresponding vertices do not lie on a sphere (i.e., the solid has no circumsphere).

In the GP(m,n) notation of Hart (2013, p. 126), the positive integers m and n indicate pentagon-to-pentagon "60-degree knights moves," meaning first take m steps in one direction, then turn 60 degrees to the left and takie n additional steps. Some special cases are summarized in the following table.

notationpolyhedron
GP(1,0)regular dodecahedron
GP(2,0)chamfered dodecahedron (truncated rhombic triacontahedron)
GP(1,1)truncated icosahedron

Fullerenes of type I (isomorphic to the skeletons of (n+1,0)-Goldberg polyhedra) and type II (isomorphic to the skeletons of (n,n)-Goldberg polyhedra) are implemented in the Wolfram Language as BuckyballGraph[n, "I"] and BuckyballGraph[n, "II"], respectively.


See also

Fullerene

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References

Goldberg, M. "A Class of Multi-Symmetric Polyhedra." Tôhoku Math. J. 43, 104-108, 1937.Hart, G. "Goldberg Polyhedra." Ch. 9 in Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, 2nd ed. (Ed. M. Senechal). New York: Springer, pp. 125-138, 2013.

Cite this as:

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

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