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Rhombohedron
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a rhombohedron is a three-dimensional figure like a cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
, except that its faces are not squares but rhombi
Rhombus

In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
. It is a special case of a parallelepiped
Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds....
 where all edges are the same length.

In general the rhombohedron can have three types of rhombus faces in congruent opposite pairs.

If all of the non-obtuse internal angles of the faces are equal (all faces are same), it can be called a trigonal trapezohedron
Trigonal trapezohedron

The trigonal trapezohedron or deltohedron is the first in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms....
.

Another special case is that, where there is a plane of symmetry through four vertices (with symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
 C2h
Cyclic symmetries

This article deals with the four infinite series of point groups in three dimensions with n-fold rotational symmetry about one axis , and no other rotational symmetry :...
), and a special case of that, where there is another plane of symmetry through the other four vertices (with symmetry group D2h
Dihedral symmetry in three dimensions

This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ....
).

The cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
 combines these special properties, and so is a special case of the rhombohedron.








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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a rhombohedron is a three-dimensional figure like a cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
, except that its faces are not squares but rhombi
Rhombus

In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
. It is a special case of a parallelepiped
Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds....
 where all edges are the same length.

In general the rhombohedron can have three types of rhombus faces in congruent opposite pairs.

If all of the non-obtuse internal angles of the faces are equal (all faces are same), it can be called a trigonal trapezohedron
Trigonal trapezohedron

The trigonal trapezohedron or deltohedron is the first in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms....
.

Another special case is that, where there is a plane of symmetry through four vertices (with symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
 C2h
Cyclic symmetries

This article deals with the four infinite series of point groups in three dimensions with n-fold rotational symmetry about one axis , and no other rotational symmetry :...
), and a special case of that, where there is another plane of symmetry through the other four vertices (with symmetry group D2h
Dihedral symmetry in three dimensions

This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ....
).

The cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
 combines these special properties, and so is a special case of the rhombohedron.

See also

  • Rhombohedral - crystal system
    Crystal system

    A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete class of Point groups in three dimensions....


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