Rhombohedron: Difference between revisions
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{{Short description|Polyhedron with six rhombi as faces}} |
{{Short description|Polyhedron with six rhombi as faces}} |
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!bgcolor=#e7dcc3 colspan=2|Rhombohedron |
!bgcolor=#e7dcc3 colspan=2|Rhombohedron |
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In [[geometry]], a '''rhombohedron''' (also called a '''rhombic hexahedron'''<ref>{{Cite |
In [[geometry]], a '''rhombohedron''' (also called a '''rhombic hexahedron'''<ref>{{Cite journal|title=Maths Resource: Rhombic Dodecahedra Puzzles|first=William A.|last=Miller|journal=Mathematics in School|date=January 1989|volume=18|issue=1|pages=18–24|jstor=30214564}}</ref><ref>{{cite journal|last=Inchbald|first=Guy|date=July 1997|doi=10.2307/3619198|issue=491|journal=The Mathematical Gazette|jstor=3619198|pages=213–219|title=The Archimedean honeycomb duals|volume=81}}</ref> or, inaccurately, a '''rhomboid'''{{efn|More accurately, [[rhomboid]] is a two-dimensional figure.}}) is a special case of a [[parallelepiped]] in which all six faces are congruent [[rhombus|rhombi]].<ref>Coxeter, HSM. ''Regular Polytopes.'' Third Edition. Dover. p.26.</ref> It can be used to define the [[rhombohedral lattice system]], a [[Honeycomb (geometry)|honeycomb]] with rhombohedral cells. A [[cube]] is a special case of a rhombohedron with all sides [[square]]. |
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The rhombohedron has two opposite vertices at which all angles are equal. If this angle us acute then the rhombohedron is long and thin (prolate), if obtuse then it is low and wide (oblate). |
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In general a ''rhombohedron'' can have up to three types of rhombic faces in congruent opposite pairs, ''C''<sub>''i''</sub> symmetry, [[Order (group theory)|order]] 2. |
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⚫ | 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.<ref>{{citation|last=Court|first=N. A.|author-link=Nathan Altshiller Court|title=Notes on the orthocentric tetrahedron|journal=[[American Mathematical Monthly]]|date=October 1934|volume=41|issue=8|pages=499–502|jstor=2300415|doi=10.2307/2300415}}.</ref> |
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The common angle at the two apices is here given as <math>\theta</math>. |
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There are two general forms of the rhombohedron, oblate (flattened) and prolate (stretched. |
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{| style="margin:0pt auto 0pt auto" |
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| [[File:Rhombohedron-oblate.svg]] |
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| [[File:Prolate rhombohedron.svg]] |
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| align=center | ''Oblate rhombohedron'' |
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| align=center | ''Prolate rhombohedron'' |
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In the oblate case <math>\theta > 90^\circ</math> and in the prolate case <math>\theta < 90^\circ</math>. For <math>\theta = 90^\circ</math> the figure is a cube. |
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Some crystals are formed in rhombohedron shape; this solid is also sometimes called a ''rhombic prism''. |
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Certain proportions of the rhombs give rise to some well-known special cases. These typically occur in both prolate and oblate forms. |
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{{main|Rhombohedral lattice system}} |
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[[File:Special_cases_of_rhombohedron.svg|thumb|right|240px|Special cases of the rhombohedron]] |
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{| class=wikitable |
{| class=wikitable |
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!Form |
!Form |
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![[Cube]] |
![[Cube]] |
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!√2 Rhombohedron |
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![[Trigonal trapezohedron]] |
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!Golden Rhombohedron |
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!Right [[rhombic prism]] |
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!Oblique rhombic prism |
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|- align=center |
|- align=center |
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!Angle<br/>constraints |
!Angle<br/>constraints |
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|<math>\ |
|<math>\theta=90^\circ</math> |
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|<math>\ |
|<!-- <math>\theta=^\circ</math> TBA --> |
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|<math>\ |
|<!-- <math>\theta=^\circ</math> --> |
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|<math>\alpha=\beta</math> |
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|- align=center |
|- align=center |
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!Ratio of diagonals |
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!Symmetry |
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| 1 |
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|[[Octahedral symmetry|O<sub>h</sub>]]<br/>order 48 |
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| √2 |
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|D<sub>3d</sub><br/>order 12 |
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| [[Golden ratio]] |
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|D<sub>2h</sub><br/>order 8 |
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|[[Cyclic symmetries|C<sub>2h</sub>]]<br/>order 4 |
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|- align=center |
|- align=center |
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!Occurrence |
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!Faces |
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| [[Regular solid]] |
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|6 squares |
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| Dissection of the [[rhombic dodecahedron]] |
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|6 congruent rhombi |
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| Dissection of the [[rhombic triacontahedron]] |
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|2 rhombi, 4 squares |
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|6 rhombi |
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* '''[[Cube]]''': with [[Octahedral symmetry|O<sub>h</sub>]] symmetry, order 48. All faces are squares. |
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* '''[[Trigonal trapezohedron]]''' (also called '''isohedral rhombohedron'''):<ref name=":0">{{Cite book|title=Solid geometry: with chapters on space-lattices, sphere-packs and crystals|last=Lines|first=L|publisher=Dover Publications|year=1965}}</ref> with D<sub>3d</sub> 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''. |
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* '''Right [[rhombic prism]]''': with D<sub>2h</sub> 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 [[prism (geometry)|prism]]s with regular triangular bases attached together makes a 60 degree ''right rhombic prism''. |
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* '''Oblique rhombic prism''': with [[Cyclic symmetries|C<sub>2h</sub>]] symmetry, order 4. It has only one plane of symmetry, through four vertices, and six rhombic faces. |
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== Solid geometry == |
== Solid geometry == |
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For a unit (i.e.: with side length 1) |
For a unit (i.e.: with side length 1) rhombohedron,<ref name=":0">{{Cite book|title=Solid geometry: with chapters on space-lattices, sphere-packs and crystals|last=Lines|first=L|publisher=Dover Publications|year=1965}}</ref> with rhombic acute angle <math>\theta~</math>, with one vertex at the origin (0, 0, 0), and with one edge lying along the x-axis, the three generating vectors are |
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:'''''e<sub>1</sub>''''' : <math>\biggl(1, 0, 0\biggr),</math> |
:'''''e<sub>1</sub>''''' : <math>\biggl(1, 0, 0\biggr),</math> |
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The other coordinates can be obtained from vector addition<ref>{{Cite web|url=http://mathworld.wolfram.com/VectorAddition.html|title=Vector Addition|date=17 May 2016|publisher=Wolfram|access-date=17 May 2016}}</ref> of the 3 direction vectors: '''''e<sub>1</sub>''''' + '''''e<sub>2</sub>''''' , '''''e<sub>1</sub>''''' + '''''e<sub>3</sub>''''' , '''''e<sub>2</sub>''''' + '''''e<sub>3</sub>''''' , and '''''e<sub>1</sub>''''' + '''''e<sub>2</sub>''''' + '''''e<sub>3</sub>''''' . |
The other coordinates can be obtained from vector addition<ref>{{Cite web|url=http://mathworld.wolfram.com/VectorAddition.html|title=Vector Addition|date=17 May 2016|publisher=Wolfram|access-date=17 May 2016}}</ref> of the 3 direction vectors: '''''e<sub>1</sub>''''' + '''''e<sub>2</sub>''''' , '''''e<sub>1</sub>''''' + '''''e<sub>3</sub>''''' , '''''e<sub>2</sub>''''' + '''''e<sub>3</sub>''''' , and '''''e<sub>1</sub>''''' + '''''e<sub>2</sub>''''' + '''''e<sub>3</sub>''''' . |
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The volume <math>V</math> of |
The volume <math>V</math> of a rhombohedron, in terms of its side length <math>a</math> and its rhombic acute angle <math>\theta~</math>, is a simplification of the volume of a [[parallelepiped]], and is given by |
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:<math>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}~.</math> |
:<math>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}~.</math> |
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:<math>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)}~.</math> |
:<math>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)}~.</math> |
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As the area of the (rhombic) base is given by <math>a^2\sin\theta~</math>, and as the height of a rhombohedron is given by its volume divided by the area of its base, the height <math>h</math> of |
As the area of the (rhombic) base is given by <math>a^2\sin\theta~</math>, and as the height of a rhombohedron is given by its volume divided by the area of its base, the height <math>h</math> of a rhombohedron in terms of its side length <math>a</math> and its rhombic acute angle <math>\theta</math> is given by |
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:<math>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}~.</math> |
:<math>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}~.</math> |
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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. |
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. |
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===Relation to orthocentric tetrahedra=== |
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⚫ | 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.<ref>{{citation|last=Court|first=N. A.|author-link=Nathan Altshiller Court|title=Notes on the orthocentric tetrahedron|journal=[[American Mathematical Monthly]]|date=October 1934|volume=41|issue=8|pages=499–502|jstor=2300415|doi=10.2307/2300415}}.</ref> |
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==See also== |
==See also== |
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*[[Lists of shapes]] |
*[[Lists of shapes]] |
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==Notes== |
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{{notelist}} |
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==References== |
==References== |
Latest revision as of 17:17, 4 June 2024
Rhombohedron | |
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![]() | |
Type | prism |
Faces | 6 rhombi |
Edges | 12 |
Vertices | 8 |
Symmetry group | Ci , [2+,2+], (×), order 2 |
Properties | convex, equilateral, zonohedron, parallelohedron |
In geometry, a rhombohedron (also called a rhombic hexahedron[1][2] or, inaccurately, a rhomboid[a]) is a special case of a parallelepiped in which all six faces are congruent rhombi.[3] 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.
The rhombohedron has two opposite vertices at which all angles are equal. If this angle us acute then the rhombohedron is long and thin (prolate), if obtuse then it is low and wide (oblate).
Special cases[edit]
The common angle at the two apices is here given as . There are two general forms of the rhombohedron, oblate (flattened) and prolate (stretched.
![]() |
![]() |
Oblate rhombohedron | Prolate rhombohedron |
In the oblate case and in the prolate case . For the figure is a cube.
Certain proportions of the rhombs give rise to some well-known special cases. These typically occur in both prolate and oblate forms.
Form | Cube | √2 Rhombohedron | Golden Rhombohedron |
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Angle constraints |
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Ratio of diagonals | 1 | √2 | Golden ratio |
Occurrence | Regular solid | Dissection of the rhombic dodecahedron | Dissection of the rhombic triacontahedron |
Solid geometry[edit]
For a unit (i.e.: with side length 1) rhombohedron,[4] 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
- e1 :
- e2 :
- e3 :
The other coordinates can be obtained from vector addition[5] of the 3 direction vectors: e1 + e2 , e1 + e3 , e2 + e3 , and e1 + e2 + e3 .
The volume of a rhombohedron, in terms of its side length and its rhombic acute angle , is a simplification of the volume of a parallelepiped, 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 a rhombohedron in terms of its side length and its rhombic acute angle is given by
Note:
- 3 , where 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.
Relation to orthocentric tetrahedra[edit]
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.[6]
Rhombohedral lattice[edit]
The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a trigonal trapezohedron[citation needed]:
See also[edit]
Notes[edit]
References[edit]
- ^ Miller, William A. (January 1989). "Maths Resource: Rhombic Dodecahedra Puzzles". Mathematics in School. 18 (1): 18–24. JSTOR 30214564.
- ^ Inchbald, Guy (July 1997). "The Archimedean honeycomb duals". The Mathematical Gazette. 81 (491): 213–219. doi:10.2307/3619198. JSTOR 3619198.
- ^ Coxeter, HSM. Regular Polytopes. Third Edition. Dover. p.26.
- ^ Lines, L (1965). Solid geometry: with chapters on space-lattices, sphere-packs and crystals. Dover Publications.
- ^ "Vector Addition". Wolfram. 17 May 2016. Retrieved 17 May 2016.
- ^ Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron", American Mathematical Monthly, 41 (8): 499–502, doi:10.2307/2300415, JSTOR 2300415.