Polyhedron with 30 faces
3D model of a great rhombic triacontahedron In geometry , the great rhombic triacontahedron is a nonconvex isohedral , isotoxal polyhedron . It is the dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombic faces, 60 edges and 32 vertices (also 20 on 3-fold and 12 on 5-fold axes).
It can be constructed from the convex solid by expanding the faces by factor of
φ φ -->
3
≈ ≈ -->
4.236
{\displaystyle \varphi ^{3}\approx 4.236}
φ φ -->
{\displaystyle \varphi \!}
golden ratio .
This solid is to the compound of great icosahedron and great stellated dodecahedron what the convex one is to the compound of dodecahedron and icosahedron :
The crossing edges in the dual compound are the diagonals of the rhombs.
What resembles an "excavated" rhombic triacontahedron (compare excavated dodecahedron and excavated icosahedron ) can be seen within the middle of this compound. The rest of the polyhedron strikingly resembles a rhombic hexecontahedron .
The rhombs have two angles of
arccos
-->
(
1
5
5
)
≈ ≈ -->
63.434
948
822
92
∘ ∘ -->
{\displaystyle \arccos({\frac {1}{5}}{\sqrt {5}})\approx 63.434\,948\,822\,92^{\circ }}
arccos
-->
(
− − -->
1
5
5
)
≈ ≈ -->
116.565
051
177
08
∘ ∘ -->
{\displaystyle \arccos(-{\frac {1}{5}}{\sqrt {5}})\approx 116.565\,051\,177\,08^{\circ }}
dihedral angles equal
arccos
-->
(
− − -->
1
4
+
1
4
5
)
=
72
∘ ∘ -->
{\displaystyle \arccos(-{\frac {1}{4}}+{\frac {1}{4}}{\sqrt {5}})=72^{\circ }}
φ φ -->
{\displaystyle \varphi }
The face diagonal lengths of the three rhombic triacontahedra are powers of
φ φ -->
{\displaystyle \varphi }
Orthographic projections from 2-, 3- and 5-fold axes
References
External links
Kepler-Poinsot (nonconvex Uniform truncations Nonconvex uniform hemipolyhedra Duals of nonconvex Duals of nonconvex
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