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Rhombic Triacontahedron


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The rhombic triacontahedron is a zonohedron which is the dual polyhedron of the icosidodecahedron (Holden 1971, p. 55). It is composed of 30 golden rhombi joined at 32 vertices. It is a zonohedron and one of the five golden isozonohedra. It is illustrated above together with a wireframe version and a net that can be used for its construction.

It is Wenninger dual W_(12).

The intersecting edges of the dodecahedron-icosahedron compound form the diagonals of 30 rhombi which comprise the triacontahedron. The cube 5-compound has the 30 facial planes of the rhombic triacontahedron and its interior is a rhombic triacontahedron (Wenninger 1983, p. 36; Ball and Coxeter 1987).

Solids inscribed in a rhombic triacontahedron

More specifically, a tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed in the vertices of the rhombic triacontahedron (E. Weisstein, Dec. 25-27, 2009).

The rhombic triacontahedron is implemented in the Wolfram Language as PolyhedronData["RhombicTriacontahedron"].

RhombicTriacontahedronDodecahedronIcosahedron

The short diagonals of the faces of the rhombic triacontahedron give the edges of a dodecahedron, while the long diagonals give the edges of the icosahedron (Steinhaus 1999, pp. 209-210).

Taken together, the dodecahedron and icosahedron give a dodecahedron-icosahedron compound.

RhombicTriacontahedronConvexHulls

The rhombic triacontahedron is the convex hull of the dodecahedron-icosahedron compound and the hull of the first icosidodecahedron stellation.

The rhombs of the rhombic triacontahedron generated from an icosidodecahedron of unit edge lengths have dimensions

x=1/8(5+sqrt(5))
(1)
y=1/4sqrt(5),
(2)

giving a ratio

 x/y=phi,
(3)

where phi is the golden ratio, making them golden rhombi. The edge length are therefore

 s=1/4sqrt(5/2(5+sqrt(5))).
(4)

The rhombs are tangential quadrilaterals with inradius

 r^'=1/4sqrt(1/2(5+sqrt(5))).
(5)

The dihedral angle between adjacent faces is pi/5=36 degrees, while the dihedral angle between faces sharing only a single point in common is pi/3=60 degrees.

RhombicTriacontahedron12RhombicTriacontahedron12Wireframe

12 rhombic triacontahedra can be fit together about a central rhombic hexecontahedron, as illustrated above (Kabai 2002, p. 173).

The rhombic triacontahedron has inradius

 r=1/8(5+3sqrt(5)).
(6)

A rhombic triacontahedron with edge length q has surface area and volume given by

S=12sqrt(5)a^2
(7)
V=4sqrt(5+2sqrt(5))a^3
(8)

and inertia tensor given by

 I=[1/(75)(35+12sqrt(5))Ma^2 0 0; 0 1/(75)(35+12sqrt(5))Ma^2 0; 0 0 1/(75)(35+12sqrt(5))Ma^2].
(9)

See also

Archimedean Dual, Archimedean Solid, Cube 5-Compound, Dodecahedron, Dodecahedron-Icosahedron Compound, Golden Isozonohedron, Golden Rhombus, Great Rhombic Triacontahedron, Icosahedron, Icosidodecahedron, Rhombic Dodecahedron, Rhombic Triacontahedral Graph, Rhombic Triacontahedron Stellations, Rhombus, Triacontahedron, Zonohedron

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References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 137, 1987.Bulatov, V. "Stellations of Rhombic Triacontahedron." http://bulatov.org/polyhedra/rtc/.Cundy, H. and Rollett, A. "Rhombic Triacontahedron." §3.8.2 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 121-122 and 127, 1989.Holden, A. Shapes, Space, and Symmetry. New York: Columbia University Press, p. 55, 1971.Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, pp. 133, 173, and 177, 2002.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 207 and 209-210, 1999.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, p. 22, 1983.

Cite this as:

Weisstein, Eric W. "Rhombic Triacontahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RhombicTriacontahedron.html

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