[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/rhombohedron-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki41\/rhombohedron-wikipedia\/","headline":"Rhombohedron – Wikipedia","name":"Rhombohedron – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 Polyhedron with six rhombi as faces In geometry, a rhombohedron (also called a","datePublished":"2021-12-04","dateModified":"2021-12-04","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki41\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/0\/03\/Rhombohedral.svg\/120px-Rhombohedral.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/0\/03\/Rhombohedral.svg\/120px-Rhombohedral.svg.png","height":"122","width":"120"},"url":"https:\/\/wiki.edu.vn\/en\/wiki41\/rhombohedron-wikipedia\/","about":["Wiki"],"wordCount":4132,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Polyhedron with six rhombi as facesIn geometry, a rhombohedron (also called a rhombic hexahedron[1] or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are 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. A cube is a special case of a rhombohedron with all sides square.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In general a rhombohedron can have up to 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.[2]Table of Contents       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Rhombohedral lattice system[edit]Special cases by symmetry[edit]Solid geometry[edit]See also[edit]References[edit]External links[edit]Rhombohedral lattice system[edit]The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a trigonal trapezohedron:Special cases by symmetry[edit]  Special cases of the rhombohedronFormCubeTrigonal trapezohedronRight rhombic prismOblique rhombic prismAngleconstraints\u03b1=\u03b2=\u03b3=90\u2218{displaystyle alpha =beta =gamma =90^{circ }}\u03b1=\u03b2=\u03b3{displaystyle alpha =beta =gamma }\u03b1=\u03b2=90\u2218{displaystyle alpha =beta =90^{circ }}\u03b1=\u03b2{displaystyle alpha =beta }SymmetryOhorder 48D3dorder 12D2horder 8C2horder 4Faces6 squares6 congruent rhombi2 rhombi, 4 squares6 rhombiCube: with Oh symmetry, order 48. All faces are squares.Trigonal trapezohedron (also called isohedral rhombohedron):[3] with D3d 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 with two regular tetrahedra attached on opposite faces constructs a 60 degree trigonal trapezohedron.Right rhombic prism: with D2h 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[edit]For a unit (i.e.: with side length 1) isohedral rhombohedron,[3] with rhombic acute angle \u03b8\u00a0{displaystyle theta ~}, with one vertex at the origin (0, 0, 0), and with one edge lying along the x-axis, the three generating vectors aree1\u00a0: (1,0,0),{displaystyle {biggl (}1,0,0{biggr )},}e2\u00a0: (cos\u2061\u03b8,sin\u2061\u03b8,0),{displaystyle {biggl (}cos theta ,sin theta ,0{biggr )},}e3\u00a0: (cos\u2061\u03b8,cos\u2061\u03b8\u2212cos2\u2061\u03b8sin\u2061\u03b8,1\u22123cos2\u2061\u03b8+2cos3\u2061\u03b8sin\u2061\u03b8).{displaystyle {biggl (}cos theta ,{cos theta -cos ^{2}theta  over sin theta },{{sqrt {1-3cos ^{2}theta +2cos ^{3}theta }} over sin theta }{biggr )}.}The other coordinates can be obtained from vector addition[4] of the 3 direction vectors: e1 + e2 , e1 + e3 , e2 + e3 , and e1 + e2 + e3 .The volume V{displaystyle V} of an isohedral rhombohedron, in terms of its side length a{displaystyle a} and its rhombic acute angle \u03b8\u00a0{displaystyle theta ~}, is a simplification of the volume of a parallelepiped, and is given byV=a3(1\u2212cos\u2061\u03b8)1+2cos\u2061\u03b8=a3(1\u2212cos\u2061\u03b8)2(1+2cos\u2061\u03b8)=a31\u22123cos2\u2061\u03b8+2cos3\u2061\u03b8\u00a0.{displaystyle V=a^{3}(1-cos theta ){sqrt {1+2cos theta }}=a^{3}{sqrt {(1-cos theta )^{2}(1+2cos theta )}}=a^{3}{sqrt {1-3cos ^{2}theta +2cos ^{3}theta }}~.}We can express the volume V{displaystyle V} another way\u00a0:V=23\u00a0a3sin2\u2061(\u03b82)1\u221243sin2\u2061(\u03b82)\u00a0.{displaystyle V=2{sqrt {3}}~a^{3}sin ^{2}left({frac {theta }{2}}right){sqrt {1-{frac {4}{3}}sin ^{2}left({frac {theta }{2}}right)}}~.}As the area of the (rhombic) base is given by a2sin\u2061\u03b8\u00a0{displaystyle a^{2}sin theta ~}, and as the height of a rhombohedron is given by its volume divided by the area of its base, the height h{displaystyle h} of an isohedral rhombohedron in terms of its side length a{displaystyle a} and its rhombic acute angle \u03b8{displaystyle theta } is given byh=a\u00a0(1\u2212cos\u2061\u03b8)1+2cos\u2061\u03b8sin\u2061\u03b8=a\u00a01\u22123cos2\u2061\u03b8+2cos3\u2061\u03b8sin\u2061\u03b8\u00a0.{displaystyle h=a~{(1-cos theta ){sqrt {1+2cos theta }} over sin theta }=a~{{sqrt {1-3cos ^{2}theta +2cos ^{3}theta }} over sin theta }~.}Note:h=a\u00a0z{displaystyle h=a~z}3 , where z{displaystyle z}3 is the third coordinate of e3 .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[edit]References[edit]External links[edit]     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/rhombohedron-wikipedia\/#breadcrumbitem","name":"Rhombohedron – Wikipedia"}}]}]