The rhombic icosahedron is a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on th ...
shaped like an
oblate sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. Its 20 faces are
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
golden rhombi;
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,
+,10 (2*5) symmetry group, of order 20; thus it has a center of symmetry (since 5 is odd).
Even though all its faces are congruent, the rhombic icosahedron is not
face-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
, 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
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...
, that is dual to a
pentagonal gyrobicupola
In geometry, the pentagonal gyrobicupola is one of the Johnson solids (). Like the pentagonal orthobicupola (), it can be obtained by joining two pentagonal cupolae () along their bases. The difference is that in this solid, the two halves are ...
with regular triangular, regular pentagonal, but irregular quadrilateral faces.
The rhombic icosahedron has 5 sets of 8 parallel edges, described as 8
5 belts.
The rhombic icosahedron forms the convex hull of the vertex-first projection of a
5-cube to 3 dimensions. 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 geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for ...
.
In the same way, one can obtain a
Bilinski dodecahedron
In geometry, the Bilinski dodecahedron is a Convex set, convex polyhedron with twelve Congruence (geometry), congruent golden rhombus faces. It has the same topology but a different geometry than the face-transitive rhombic dodecahedron. It is a ...
from a
4-cube, and a
rhombic triacontahedron
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Cata ...
from a
6-cube
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.
It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It ...
.
Related polyhedra
The rhombic icosahedron can be derived from the
rhombic triacontahedron
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Cata ...
by removing a belt of 10 middle faces.
(*) (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.
References
External links
*
* http://www.georgehart.com/virtual-polyhedra/zonohedra-info.html
**
VRML
VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graphi ...
Mode
Zonohedra
{{Polyhedron-stub