How many planes of symmetry does a dodecahedron?

Fifteen Planes
The Fifteen Planes of Symmetry of the Icosahedron and/or Dodecahedron. This paper model illustrates the fifteen planes of symmetry of the icosahedron and/or dodecahedron (which being mutually dual, share their symmetry group and symmetry planes). There is only one type of plane in this case.

How many sides does a rhombic dodecahedron have?

12
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron….

How do you make a rhombic dodecahedron?

The rhombic dodecahedron can be built up by a placing six cubes on the faces of a seventh, in the configuration of a metal “jack” (left figure). Joining the centers of the outer cubes with the vertices of the central cube then gives the rhombic dodecahedron (middle figure).

Is a dodecahedron symmetrical?

The symmetry group of the dodecahedron or icosahedron S(D). Because these two solids are dual to each other they have the same symmetry group. In fact one can embed five cubes in the dodecahedron which are permuted by each rotation.

How are the faces of a rhombic dodecahedron related?

This decomposition gives an interesting method for constructing the rhombic dodecahedron: cut a cube into six congruent square pyramids, and attach them to the faces of a second cube. The triangular faces of each pair of adjacent pyramids lie on the same plane, and so merge into rhombuses.

What kind of symmetry does the cuboctahedron have?

With O h symmetry, order 48, it is a rectified cube or rectified octahedron ( Norman Johnson) With T d symmetry, order 24, it is a cantellated tetrahedron or rhombitetratetrahedron. With D 3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are:

What are the acute angles of a rhombic dodecahedron?

The rhombic dodecahedron is a zonohedron. Its polyhedral dual is the cuboctahedron. The long face-diagonal length is exactly √ 2 times the short face-diagonal length; thus, the acute angles on each face measure arccos (1 3), or approximately 70.53°.

How many symmetry planes are there in the octahedron?

This paper model illustrates the nine planes of symmetry of the cube and/or octahedron. It contains two different types of symmetry planes: three of the planes are orthogonal to the three 4-fold symmetry axes; each such plane is parallel to, and halfway between, two opposite faces of the cube; these three planes are mutually orthogonal.