In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a
rhomboid) is a three-dimensional figure with six faces which are
rhombi
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
. It is a special case of a
parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclid ...
where all edges are the same length. It can be used to define the
rhombohedral lattice system, a
honeycomb
A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen.
Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of honey ...
with rhombohedral cells. A
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
is a special case of a rhombohedron with all sides
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
.
In general a ''rhombohedron'' can have up to three types of rhombic faces in congruent opposite pairs, ''C''
''i'' symmetry,
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
2.
Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an
orthocentric tetrahedron
In geometry, an orthocentric tetrahedron is a tetrahedron where all three pairs of opposite edges are perpendicular. It is also known as an orthogonal tetrahedron since orthogonal means perpendicular. It was first studied by Simon Lhuilier in 1782, ...
, and all orthocentric tetrahedra can be formed in this way.
Rhombohedral lattice system
The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a
trigonal trapezohedron
In geometry, a trigonal trapezohedron is a rhombohedron (a polyhedron with six rhombus-shaped faces) in which, additionally, all six faces are congruent. Alternative names for the same shape are the ''trigonal deltohedron'' or ''isohedral rh ...
:
:
Special cases by symmetry
*
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
: with
Oh symmetry, order 48. All faces are squares.
*
Trigonal trapezohedron
In geometry, a trigonal trapezohedron is a rhombohedron (a polyhedron with six rhombus-shaped faces) in which, additionally, all six faces are congruent. Alternative names for the same shape are the ''trigonal deltohedron'' or ''isohedral rh ...
(also called isohedral rhombohedron):
with D
3d symmetry, order 12. All non-obtuse internal angles of the faces are equal (all faces are congruent rhombi). This can be seen by stretching a cube on its body-diagonal axis. For example, a regular
octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
with two regular
tetrahedra
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
attached on opposite faces constructs a 60 degree ''trigonal trapezohedron''.
* Right
rhombic prism: with D
2h symmetry, order 8. It is constructed by two rhombi and four squares. This can be seen by stretching a cube on its face-diagonal axis. For example, two right
prisms with regular triangular bases attached together makes a 60 degree ''right rhombic prism''.
* Oblique rhombic prism: with
C2h symmetry, order 4. It has only one plane of symmetry, through four vertices, and six rhombic faces.
Solid geometry
For a unit (i.e.: with side length 1) isohedral rhombohedron,
with rhombic acute angle
, with one vertex at the origin (0, 0, 0), and with one edge lying along the x-axis, the three generating vectors are
:''e
1'' :
:''e
2'' :
:''e
3'' :
The other coordinates can be obtained from vector addition
of the 3 direction vectors: ''e
1'' + ''e
2'' , ''e
1'' + ''e
3'' , ''e
2'' + ''e
3'' , and ''e
1'' + ''e
2'' + ''e
3'' .
The volume
of an isohedral rhombohedron, in terms of its side length
and its rhombic acute angle
, is a simplification of the volume of a
parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclid ...
, and is given by
:
We can express the volume
another way :
:
As the area of the (rhombic) base is given by
, and as the height of a rhombohedron is given by its volume divided by the area of its base, the height
of an isohedral rhombohedron in terms of its side length
and its rhombic acute angle
is given by
:
Note:
:
''3'' , where
''3'' is the third coordinate of ''e
3'' .
The body diagonal between the acute-angled vertices is the longest. By rotational symmetry about that diagonal, the other three body diagonals, between the three pairs of opposite obtuse-angled vertices, are all the same length.
See also
*
Lists of shapes
References
External links
*
*
Volume Calculator https://rechneronline.de/pi/rhombohedron.php{{Polyhedron navigator
Prismatoid polyhedra
Space-filling polyhedra
Zonohedra