Rhombohedron

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Rhombohedron
Rhombohedron
Type Prism
Faces 6 rhombi
Edges 12
Vertices 8
Symmetry group Ci, [2+,2+], (×), order 2
Properties convex, zonohedron

In geometry, a rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares but rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells.

In general the rhombohedron can have three types of rhombic faces in congruent opposite pairs, Ci symmetry, order 2.

Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an orthocentric tetrahedron, and all orthocentric tetrahedra can be formed in this way.[1]

Rhombohedral lattice system[edit]

The rhombohedral lattice system has rhombohedral cells, with 3 pairs of unique rhombic faces:

Rhombohedral.svg

Special cases[edit]

  • Cube: with Oh symmetry, order 48. All faces are squares. Hexahedron.png
  • Trigonal trapezohedron: with D3d symmetry, order 12. If all of the non-obtuse internal angles of the faces are equal (all faces are same). This can be see by stretching a cube on its body-digonal axis. For example a regular octahedron with two tetrahedra attached on opposite faces constructs a 60 degree trigonal trapezohedron: Gyroelongated triangular bipyramid.png
  • Right rhombic prism: with D2h symmetry, order 8. It constructed by two rhombi and 4 squares. This can be see by stretching a cube on its face-digonal axis. For example two triangular prisms attached together makes a 60 degree rhombic prism. Digonal orthobicupola.png
  • A general rhombic prism: With C2h symmetry, order 4. It has only one plane of symmetry through four vertices, and 6 rhombic faces.

References[edit]

  1. ^ Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron", American Mathematical Monthly: 499–502, JSTOR 2300415 .

External links[edit]