Rhombic icosahedron

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Rhombic icosahedron
Rhombic icosahedron.png
Type Zonohedron
Faces 20 congruent golden rhombi
Edges 40
Vertices 22
Symmetry group D5d = D5v, [2+,10], (2*5)
Properties convex
A rhombic icosahedron Rhombic icosahedron.stl
A rhombic icosahedron

The rhombic icosahedron is a polyhedron shaped like an oblate sphere. Its 20 faces are congruent golden rhombi; [1] 3, 4, or 5 faces meet at each vertex. It has 5 faces (green on top figure) meeting at each of its 2 poles; these 2 vertices lie on its axis of 5-fold symmetry, which is perpendicular to 5 axes of 2-fold symmetry through the midpoints of opposite equatorial edges (example on top figure: most left-hand and most right-hand mid-edges). Its other 10 faces follow its equator, 5 above and 5 below it; each of these 10 rhombi has 2 of its 4 sides lying on this zig-zag skew decagon equator. The rhombic icosahedron has 22 vertices. It has D5d, [2+,10], (2*5) symmetry group, of order 20; thus it has a center of symmetry (since 5 is odd).

Contents

Even though all its faces are congruent, the rhombic icosahedron is not face-transitive, since one can distinguish whether a particular face is near the equator or near a pole by examining the types of vertices surrounding this face.

Zonohedron

The rhombic icosahedron is a zonohedron.

The rhombic icosahedron has 5 sets of 8 parallel edges, described as 85 belts.

Rhombic icosahedron 5-color-paralleledges.png The edges of the rhombic icosahedron can be grouped in 5 parallel-sets, seen in this wireframe orthogonal projection.

The rhombic icosahedron forms the convex hull of the vertex-first[ clarification needed ] projection of a 5-cube to 3 dimensions[ citation needed ]. The 32 vertices of a 5-cube map into the 22 exterior vertices of the rhombic icosahedron, with the remaining 10 interior vertices forming a pentagonal antiprism.

In the same way, one can obtain a Rhombic dodecahedron from a 4-cube, and a rhombic triacontahedron from a 6-cube.

The rhombic icosahedron can be derived from the rhombic triacontahedron by removing a belt of 10 middle faces with parallel edges.

Rhombic triacontahedron middle colored.png
A rhombic triacontahedron can be seen as an elongated rhombic icosahedron.
Dual dodecahedron t1 H3.png
The rhombic icosahedron and the rhombic triacontahedron have the same 10-fold symmetric orthogonal projection. (*)

(*) (For example, on the left-hand figure):

The orthogonal projection of the (vertical) belt of 10 middle faces of the rhombic triacontahedron is just the (horizontal) exterior regular decagon of the common orthogonal projection.

Removal of a further belt of 8 faces with parallel edges from the icosahedron results in the Bilinski dodecahedron, which is topologically equivalent but not congruent to the regular rhombic dodecahedron.

References

  1. Weisstein, Eric W. "Rhombic Icosahedron". mathworld.wolfram.com. Retrieved 2019-12-20.