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| |[[Image:Disphenoid tetrahedron.png|280px]]<BR>The '''tetragonal and digonal disphenoids''' can be positioned inside a [[cuboid]] bisecting two opposite faces. The tetragonal form has Coxeter diagram {{CDD|node_h|2|node_h|4|node}}. All four faces are isosceles triangles. Both have four equal edges going around the sides. The digonal has two sets of isosceles triangle faces, while the tetragonal form has four identical isosceles triangle faces.
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| |[[File:Rhombic disphenoid.png|280px]]<BR>A '''rhombic disphenoid''' has 4 congruent scalene triangle faces, and can fit diagonally inside of a [[cuboid]]. It has three sets of edge lengths, existing as opposite pairs.<BR>It can be given [[Coxeter diagram]] {{CDD|node_h|2|node_h|2|node_h}} as an [[Alternation (geometry)|alternation]] of a cuboid {{CDD|node_1|2|node_1|2|node_1}}.
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| In [[geometry]], a '''disphenoid''' (also '''bisphenoid''') (from Greek sphenoeides, "wedgelike" <ref>[http://www.merriam-webster.com/ Merriam-Webster Online Dictionary]''.</ref>) is a [[tetrahedron]] whose four faces are [[Congruence (geometry)|congruent]] acute-angled triangles.<ref>*[[H.S.M. Coxeter|Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. p. 15</ref> It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names are '''isosceles tetrahedron''' and '''equifacial tetrahedron'''. They can also be seen as digonal [[antiprism]]s as an [[Alternation (geometry)|alternated]] quadrilateral [[Prism (geometry)|prism]]. All the [[solid angle]]s and [[vertex figure]]s of a disphenoid are the same, and the sum of the face angles at each vertex is equal to two [[right angle]]s. However, a disphenoid is not a [[regular polyhedron]], because its faces are not [[regular polygon]]s.
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| == Special cases and Generalizations==
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| {{See|Tetrahedron#Isometries_of_irregular_tetrahedra}}
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| If the faces of a disphenoid are [[equilateral triangle]]s, it is a [[regular tetrahedron]] with '''T'''<SUB>d</SUB> [[tetrahedral symmetry]], although this is not normally called a disphenoid. The faces of a '''tetragonal disphenoid''' are identical [[isosceles]], and it has '''D'''<SUB>2d</SUB> dihedral symmetry. The faces of a '''rhombic disphenoid''' are [[scalene triangle|scalene]] and it has '''D'''<SUB>2</SUB> [[dihedral symmetry]]. Tetragonal disphenoids and rhombic disphenoids are [[isohedral figure|isohedra]].
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| The '''digonal disphenoid''' is not a disphenoid as defined above. It has two sets of isosceles triangles faces, and it has '''D'''<SUB>1h</SUB>. The most general disphenoid term is the '''phyllic disphenoid''' with only two types of scalene triangles.
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| [[File:Tetrahedral subgroup tree.png|320px|thumb|Tetrahedral subgroup relations]] | |
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| Tetrahedral diagrams are included for each type below, with edges colored by isometric equivalence, and are gray colored for unique edges.
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| {| class=wikitable
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| !rowspan=2|Name<BR>[[Coxeter diagram]]
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| !rowspan=2|Edge<BR>diagram
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| !rowspan=2|[[Face (geometry)|Faces]]
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| !colspan=4|[[List_of_spherical_symmetry_groups|Symmetry]]
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| |-
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| ![[Schönflies_notation|Schönflies]]
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| ![[Coxeter notation|Coxeter]]
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| ![[Orbifold notation|Orbifold]]
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| ![[Symmetry order|Order]]
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| |- align=center
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| |[[Regular tetrahedron]]
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| |[[File:Regular tetrahedron diagram.png|50px]]
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| |Identical<BR>[[equilateral triangle]]s
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| |'''T'''<SUB>d</SUB><BR>'''T'''||[3,3]<BR>[3,3]<sup>+</sup>||*332<BR>332||24<BR>12
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| |- align=center
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| |Tetragonal disphenoid
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| |[[File:Tetragonal disphenoid diagram.png|50px]]
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| |Identical<BR>[[isosceles triangle]]s
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| |'''D'''<SUB>2d</SUB><BR>'''S'''<SUB>4</SUB>||[2<sup>+</sup>,4]<BR>[2<sup>+</sup>,4<sup>+</sup>]||2*2<BR>2×||8<BR>4
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| |- align=center
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| |Rhombic disphenoid
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| |[[File:Rhombic disphenoid diagram.png|50px]]
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| |Identical<BR>[[scalene triangle]]s
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| |'''D'''<SUB>2</SUB>||[2,2]<sup>+</sup>||222||4
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| |- align=center
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| |Digonal disphenoid
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| |[[File:Digonal disphenoid diagram.png|50px]]
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| |Two types of<BR>isosceles triangles
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| |'''C'''<SUB>2v</SUB> or '''D'''<SUB>1h</SUB>||[2]||*22||4
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| |-align=center
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| |Phyllic disphenoid
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| |[[File:Half-turn tetrahedron diagram.png|50px]]
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| |Two types of<BR>scalene triangles.
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| |'''C'''<sub>2</sub> or '''D'''<sub>1</sub> ||[2]<sup>+</sup> ||22 ||2
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| |}
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| == Characterizations ==
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| A tetrahedron is a disphenoid [[if and only if]] its circumscribed [[parallelepiped]] is right-angled.<ref name=Andreescu/>
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| We also have that a tetrahedron is a disphenoid if and only if the [[Center (geometry)|center]] in the [[circumscribed sphere]] and the [[inscribed sphere]] coincide.<ref name=Brown>Brown, B. H., "Theorem of Bang. Isosceles tetrahedra", ''American Mathematical Monthly'', April 1926, pp. 224-226.</ref>
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| Another characterization states that if ''d<sub>1</sub>'', ''d<sub>2</sub>'' and ''d<sub>3</sub>'' are the common perpendiculars of ''AB'' and ''CD''; ''AC'' and ''BD''; and ''AD'' and ''BC'' respectively in a tetrahedron ''ABCD'', then the tetrahedron is a disphenoid if and only if ''d<sub>1</sub>'', ''d<sub>2</sub>'' and ''d<sub>3</sub>'' are pairwise [[perpendicular]].<ref name=Andreescu>Andreescu, Titu and Gelca, Razvan, "Mathematical Olympiad Challenges", Birkhäuser, second edition, 2009, pp. 30-31.</ref>
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| == Metric formulas ==
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| The [[volume]] of a disphenoid with opposite edges of length ''l'', ''m'' and ''n'' is given by<ref name=Leech>{{citation
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| | last = Leech | first = John
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| | journal = Mathematical Gazette | |
| | pages = 269–271 | |
| | title = Some properties of the isosceles tetrahedron
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| | volume = 34
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| | number = 310
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| | year = 1950}}.</ref>
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| :<math> V=\sqrt{\frac{(l^2+m^2-n^2)(l^2-m^2+n^2)(-l^2+m^2+n^2)}{72}}. </math>
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| The [[circumscribed sphere]] has radius<ref name=Leech/> (the circumradius)
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| :<math> R=\sqrt{\frac{l^2+m^2+n^2}{8}} </math> | |
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| and the [[inscribed sphere]] has radius<ref name=Leech/>
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| :<math> r=\frac{3V}{4T} </math>
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| where ''V'' is the volume of the disphenoid and ''T'' is the area of any face, which is given by [[Heron's formula]]. There is also the following interesting relation connecting the volume and the circumradius:<ref name=Leech/>
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| :<math>\displaystyle 16T^2R^2=l^2m^2n^2+9V^2. </math>
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| The square of the lengths of the [[Tetrahedron#Properties of a generalized tetrahedron|bimedians]] are<ref name=Leech/>
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| :<math> \tfrac{1}{2}(l^2+m^2-n^2),\quad \tfrac{1}{2}(l^2-m^2+n^2),\quad \tfrac{1}{2}(-l^2+m^2+n^2). </math>
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| == Other properties ==
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| If the four faces of a tetrahedron have the same perimeter, then the tetrahedron is a disphenoid.<ref name=Brown/>
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| If the four faces of a tetrahedron have the same area, then it is a disphenoid.<ref name=Andreescu/> <ref name=Brown/>
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| The centers in the [[circumscribed sphere |circumscribed]] and [[inscribed sphere]]s coincide with the [[centroid]] of the disphenoid.<ref name=Leech/>
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| The bimedians are [[perpendicular]] to the edges they connect and to each other.<ref name=Leech/>
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| == Honeycombs and crystals ==
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| Some tetragonal disphenoids will form [[honeycomb (geometry)|honeycomb]]s. The disphenoid whose four vertices are (-1, 0, 0), (1, 0, 0), (0, 1, 1), and (0, 1, -1) is such a disphenoid.<ref>Coxeter, pp. 71–72; {{cite journal
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| | last = Senechal | first = Marjorie | authorlink = Marjorie Senechal
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| | title = Which tetrahedra fill space?
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| | year = 1981
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| | journal = [[Mathematics Magazine]]
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| | volume = 54
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| | issue = 5
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| | pages = 227–243
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| | jstor = 2689983
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| | doi =10.2307/2689983 }}
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| </ref> Each of its four faces is an isosceles triangle with edges of lengths √3, √3, and 2. It can tesselate space to form the [[disphenoid tetrahedral honeycomb]]. As Gibb<ref>{{cite journal
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| | author = Gibb, William
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| | title = Paper patterns: solid shapes from metric paper
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| | year = 1990
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| | journal = Mathematics in School
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| | volume = 19
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| | issue = 3
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| | pages = 2–4}} Reprinted in {{cite book
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| | author = Pritchard, Chris, ed.
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| | year = 2003
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| | title = The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching
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| | publisher = Cambridge University Press
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| | isbn = 0-521-53162-4
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| | pages = 363–366}}</ref> describes, it can be folded without cutting or overlaps from a single sheet of [[a4 paper]].
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| "Disphenoid" is also used to describe two forms of [[Crystal system|crystal]]:
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| * A wedge-shaped crystal form of the [[Tetragonal crystal system|tetragonal]] or [[Orthorhombic crystal system|orthorhombic system]]. It has four triangular faces that are alike and that correspond in position to alternate faces of the tetragonal or orthorhombic [[Bipyramid|dipyramid]]. It is symmetrical about each of three mutually perpendicular diad axes of symmetry in all classes except the tetragonal-disphenoidal, in which the form is generated by an inverse tetrad axis of symmetry.
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| *A crystal form bounded by eight [[scalene triangle]]s arranged in pairs, constituting a tetragonal [[scalenohedron]].
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| == See also ==
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| * [[Orthocentric tetrahedron]]
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| * [[Snub disphenoid]] - A [[Johnson solid]] with 12 equilateral triangle faces and D<sub>2d</sub> symmetry.
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| * [[Trirectangular tetrahedron]]
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| ==References==
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| <references/>
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| == External links ==
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| *{{Mathworld | urlname=Disphenoid | title=Disphenoid }}
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| [[Category:Polyhedra]]
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