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Rhombohedron
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<h2 id="special-cases">Special cases</h2> <p>The common angle at the two apices is here given as θ {\displaystyle \theta } . There are two general forms of the rhombohedron: oblate (flattened) and prolate (stretched). </p> <table><tbody><tr><td></td><td></td></tr><tr><td><i>Oblate rhombohedron, note there is a mistake in the labelling of angles here. All angles labeled theta should be on the acute angles. Here, two are on the obtuse and one is on the acute.</i></td><td><i>Prolate rhombohedron</i></td></tr></tbody></table> <p>In the oblate case θ > 90 ∘ {\displaystyle \theta >90^{\circ }} and in the prolate case θ < 90 ∘ {\displaystyle \theta <90^{\circ }} . For θ = 90 ∘ {\displaystyle \theta =90^{\circ }} the figure is a cube. </p><p>Certain proportions of the rhombs give rise to some well-known special cases. These typically occur in both prolate and oblate forms. </p> <table><tbody><tr><th>Form</th><th><a href="/facts/Cube/65lUaAqN">Cube</a></th><th>√2 Rhombohedron</th><th>Golden Rhombohedron</th></tr><tr><th>Angleconstraints</th><td> θ = 90 ∘ {\displaystyle \theta =90^{\circ }} </td><td></td><td></td></tr><tr><th>Ratio of diagonals</th><td>1</td><td>√2</td><td><a href="/facts/Golden_ratio/anloSRmD">Golden ratio</a></td></tr><tr><th>Occurrence</th><td><a href="/facts/Regular_solid/jraREXFD">Regular solid</a></td><td>Dissection of the <a href="/facts/Rhombic_dodecahedron/X3npAlmr">rhombic dodecahedron</a></td><td>Dissection of the <a href="/facts/Rhombic_triacontahedron/HmHWG8JY">rhombic triacontahedron</a></td></tr></tbody></table> <h2 id="solid-geometry">Solid geometry</h2> <p>For a unit (i.e.: with side length 1) rhombohedron,<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> with rhombic acute angle θ {\displaystyle \theta ~} , with one vertex at the origin (0, 0, 0), and with one edge lying along the x-axis, the three generating vectors are </p> <i>e1</i> : ( 1 , 0 , 0 ) , {\displaystyle {\biggl (}1,0,0{\biggr )},} <i>e2</i> : ( cos θ , sin θ , 0 ) , {\displaystyle {\biggl (}\cos \theta ,\sin \theta ,0{\biggr )},} <i>e3</i> : ( cos θ , cos θ − cos 2 θ sin θ , 1 − 3 cos 2 θ + 2 cos 3 θ sin θ ) . {\displaystyle {\biggl (}\cos \theta ,{\cos \theta -\cos ^{2}\theta \over \sin \theta },{{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }} \over \sin \theta }{\biggr )}.} <p>The other coordinates can be obtained from vector addition<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> of the 3 direction vectors: <i>e1</i> + <i>e2</i> , <i>e1</i> + <i>e3</i> , <i>e2</i> + <i>e3</i> , and <i>e1</i> + <i>e2</i> + <i>e3</i> . </p><p>The volume V {\displaystyle V} of a rhombohedron, in terms of its side length a {\displaystyle a} and its rhombic acute angle θ {\displaystyle \theta ~} , is a simplification of the volume of a <a href="/facts/Parallelepiped/Qn7cDgSr">parallelepiped</a>, and is given by </p> V = a 3 ( 1 − cos θ ) 1 + 2 cos θ = a 3 ( 1 − cos θ ) 2 ( 1 + 2 cos θ ) = a 3 1 − 3 cos 2 θ + 2 cos 3 θ . {\displaystyle V=a^{3}(1-\cos \theta ){\sqrt {1+2\cos \theta }}=a^{3}{\sqrt {(1-\cos \theta )^{2}(1+2\cos \theta )}}=a^{3}{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }}~.} <p>We can express the volume V {\displaystyle V} another way : </p> V = 2 3 a 3 sin 2 ( θ 2 ) 1 − 4 3 sin 2 ( θ 2 ) . {\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)}}~.} <p>As the area of the (rhombic) base is given by a 2 sin θ {\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 a rhombohedron in terms of its side length a {\displaystyle a} and its rhombic acute angle θ {\displaystyle \theta } is given by </p> h = a ( 1 − cos θ ) 1 + 2 cos θ sin θ = a 1 − 3 cos 2 θ + 2 cos 3 θ sin θ . {\displaystyle h=a~{(1-\cos \theta ){\sqrt {1+2\cos \theta }} \over \sin \theta }=a~{{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }} \over \sin \theta }~.} <p>Note: </p> h = a z {\displaystyle h=a~z} <i>3</i> , where z {\displaystyle z} <i>3</i> is the third coordinate of <i>e3</i> . <p>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. </p> <h3>Relation to orthocentric tetrahedra</h3> <p>Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an <a href="/facts/Orthocentric_tetrahedron/hMeLotdH">orthocentric tetrahedron</a>, and all orthocentric tetrahedra can be formed in this way.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <h2 id="rhombohedral-lattice">Rhombohedral lattice</h2> <p class="note">Main article: <a href="/facts/Rhombohedral_lattice/sSSok8wo">Rhombohedral lattice</a></p> <p>The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a <a href="/facts/Trigonal_trapezohedron/w5qPRnR3">trigonal trapezohedron</a>: </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Lists_of_shapes/lP1HBYsV">Lists of shapes</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Rhombohedron.html">"Rhombohedron"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="https://rechneronline.de/pi/truncated-rhombohedron.php">Volume Calculator https://rechneronline.de/pi/rhombohedron.php</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Miller, William A. (January 1989). "Maths Resource: Rhombic Dodecahedra Puzzles". Mathematics in School. 18 (1): 18–24. JSTOR 30214564. <a href="/wiki/JSTOR_(identifier)" target="_blank">/wiki/JSTOR_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Inchbald, Guy (July 1997). "The Archimedean honeycomb duals". The Mathematical Gazette. 81 (491): 213–219. doi:10.2307/3619198. JSTOR 3619198. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>More accurately, rhomboid is a two-dimensional figure. <a href="/wiki/Rhomboid" target="_blank">/wiki/Rhomboid</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Coxeter, HSM. Regular Polytopes. Third Edition. Dover. p.26. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Lines, L (1965). Solid geometry: with chapters on space-lattices, sphere-packs and crystals. Dover Publications. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"Vector Addition". Wolfram. 17 May 2016. Retrieved 17 May 2016. <a href="http://mathworld.wolfram.com/VectorAddition.html" target="_blank">http://mathworld.wolfram.com/VectorAddition.html</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron", American Mathematical Monthly, 41 (8): 499–502, doi:10.2307/2300415, JSTOR 2300415. <a href="/wiki/Nathan_Altshiller_Court" target="_blank">/wiki/Nathan_Altshiller_Court</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>