# Disphenoid

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Tetrahedron whose faces are all congruent

The **[tetragonal](/source/Tetragon) and [digonal](/source/Digon) disphenoids** can be positioned inside a [cuboid](/source/Cuboid) bisecting two opposite faces. Both have four equal edges going around the sides. The digonal has two pairs of congruent [isosceles triangle](/source/Isosceles_triangle) faces, while the tetragonal has four congruent isosceles triangle faces.

A **rhombic disphenoid** has congruent [scalene triangle](/source/Triangle#Types_of_triangle) faces, and can fit diagonally inside of a [cuboid](/source/Cuboid). It has three sets of edge lengths, existing as opposite pairs.

In [geometry](/source/Geometry), a **disphenoid** (from [Greek](/source/Greek_language) *sphenoeides* 'wedgelike') is a [tetrahedron](/source/Tetrahedron) whose four [faces](/source/Face_(geometry)) are [congruent](/source/Congruence_(geometry)) acute-angled triangles.[1] It can also be described as a tetrahedron in which every two [edges](/source/Edge_(geometry)) that are opposite each other have equal lengths. Other names for the same shape are **isotetrahedron**,[2] **sphenoid**,[3] **bisphenoid**,[3] **isosceles tetrahedron**,[4] **equifacial tetrahedron**,[5] **almost regular tetrahedron**,[6] and **tetramonohedron**.[7]

All the [solid angles](/source/Solid_angle) and [vertex figures](/source/Vertex_figure) of a disphenoid are the same, and the sum of the face angles at each vertex is equal to two [right angles](/source/Right_angle). However, a disphenoid is not a [regular polyhedron](/source/Regular_polyhedron), because, in general, its faces are not [regular polygons](/source/Regular_polygon), and its edges have three different lengths.

## Special cases and generalizations

Further information: [Tetrahedron § Isometries of irregular tetrahedra](/source/Tetrahedron#Isometries_of_irregular_tetrahedra)

If the faces of a disphenoid are [equilateral triangles](/source/Equilateral_triangle), it is a [regular tetrahedron](/source/Regular_tetrahedron) with T d {\displaystyle T_{d}} [tetrahedral symmetry](/source/Tetrahedral_symmetry), although this is not normally called a disphenoid. When the faces of a disphenoid are [isosceles triangles](/source/Isosceles_triangle), it is called a **tetragonal disphenoid**. In this case it has D 2 d {\displaystyle D_{2d}} [dihedral symmetry](/source/Dihedral_symmetry). A sphenoid with [scalene triangles](/source/Scalene_triangle) as its faces is called a **rhombic disphenoid** and it has D 2 {\displaystyle D_{2}} dihedral symmetry. Unlike the tetragonal disphenoid, the rhombic disphenoid has no [reflection symmetry](/source/Reflection_symmetry), so it is [chiral](/source/Chirality).[8] Both tetragonal disphenoids and rhombic disphenoids are [isohedra](/source/Isohedral_figure): as well as being congruent to each other, all of their faces are symmetric to each other.

It is not possible to construct a disphenoid with [right triangle](/source/Right_triangle) or [obtuse triangle](/source/Obtuse_triangle) faces.[4] When right triangles are glued together in the pattern of a disphenoid, they form a flat figure (a doubly-covered rectangle) that does not enclose any volume.[8] When obtuse triangles are glued in this way, the resulting surface can be folded to form a disphenoid (by [Alexandrov's uniqueness theorem](/source/Alexandrov's_uniqueness_theorem)) but one with acute triangle faces and with edges that in general do not lie along the edges of the given obtuse triangles.

Two more types of tetrahedron generalize the disphenoid and have similar names. The **digonal disphenoid** has faces with two different shapes, both isosceles triangles, with two faces of each shape. The **phyllic disphenoid** similarly has faces with two shapes of scalene triangles.

Disphenoids can also be seen as digonal [antiprisms](/source/Antiprism) or as [alternated](/source/Alternation_(geometry)) quadrilateral [prisms](/source/Prism_(geometry)).

## Characterizations

A tetrahedron is a disphenoid [if and only if](/source/If_and_only_if) its circumscribed [parallelepiped](/source/Parallelepiped) is right-angled.[9]

We also have that a tetrahedron is a disphenoid if and only if the [center](/source/Center_(geometry)) in the [circumscribed sphere](/source/Circumscribed_sphere) and the [inscribed sphere](/source/Inscribed_sphere) coincide.[10]

Another characterization states that if d 1 {\displaystyle d_{1}} , d 2 {\displaystyle d_{2}} and d 3 {\displaystyle d_{3}} are the common perpendiculars of A B {\displaystyle AB} and C D {\displaystyle CD} ; A C {\displaystyle AC} and B D {\displaystyle BD} ; and A D {\displaystyle AD} and B C {\displaystyle BC} respectively in a tetrahedron A B C D {\displaystyle ABCD} , then the tetrahedron is a disphenoid if and only if d 1 {\displaystyle d_{1}} , d 2 {\displaystyle d_{2}} and d 3 {\displaystyle d_{3}} are pairwise [perpendicular](/source/Perpendicular).[9]

The disphenoids are the only polyhedra having infinitely many non-self-intersecting [closed geodesics](/source/Closed_geodesic). On a disphenoid, all closed geodesics are non-self-intersecting.[11]

The disphenoids are the tetrahedra in which all four faces have the same [perimeter](/source/Perimeter),[10] the tetrahedra in which all four faces have the same area,[10][12] and the tetrahedra in which the [angular defects](/source/Angular_defect) of all four vertices equal π {\displaystyle \pi } . They are the polyhedra having a [net](/source/Net_(polyhedron)) in the shape of an acute triangle, divided into four [similar triangles](/source/Similarity_(geometry)) by segments connecting the edge midpoints.[6]

## Metric formulas

The [volume](/source/Volume) of a disphenoid with opposite edges of length l {\displaystyle l} , m {\displaystyle m} and n {\displaystyle n} is given by[13]

- V = ( l 2 + m 2 − n 2 ) ( l 2 − m 2 + n 2 ) ( − l 2 + m 2 + n 2 ) 72 . {\displaystyle V={\sqrt {\frac {(l^{2}+m^{2}-n^{2})(l^{2}-m^{2}+n^{2})(-l^{2}+m^{2}+n^{2})}{72}}}.}

The [circumscribed sphere](/source/Circumscribed_sphere) has radius[13] (the circumradius)

- R = l 2 + m 2 + n 2 8 , {\displaystyle R={\sqrt {\frac {l^{2}+m^{2}+n^{2}}{8}}},}

and the [inscribed sphere](/source/Inscribed_sphere) has radius[13]

- r = 3 V 4 T , {\displaystyle r={\frac {3V}{4T}},}

where V {\displaystyle V} is the volume of the disphenoid and T {\displaystyle T} is the area of any face, which is given by [Heron's formula](/source/Heron's_formula). There is also the following interesting relation connecting the volume and the circumradius:[13]

- 16 T 2 R 2 = l 2 m 2 n 2 + 9 V 2 . {\displaystyle 16T^{2}R^{2}=l^{2}m^{2}n^{2}+9V^{2}.}

The squares of the lengths of the [bimedians](/source/Tetrahedron#Properties_analogous_to_those_of_a_triangle) are[13]

- 1 2 ( l 2 + m 2 − n 2 ) , 1 2 ( l 2 − m 2 + n 2 ) , 1 2 ( − l 2 + m 2 + n 2 ) . {\displaystyle {\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}).}

## Other properties

If the four faces of a tetrahedron have the same perimeter, then the tetrahedron is a disphenoid.[10]

If the four faces of a tetrahedron have the same area, then it is a disphenoid.[9][10]

The centers in the [circumscribed](/source/Circumscribed_sphere) and [inscribed spheres](/source/Inscribed_sphere) coincide with the [centroid](/source/Centroid) of the disphenoid.[13]

The bimedians are [perpendicular](/source/Perpendicular) to the edges they connect and to each other.[13]

## Honeycombs and crystals

A space-filling tetrahedral disphenoid inside a cube. Two edges have [dihedral angles](/source/Dihedral_angle) of 90°, and four edges have dihedral angles of 60°.

Some tetragonal disphenoids will form [honeycombs](/source/Honeycomb_(geometry)). The disphenoid whose four vertices are ( − 1 , 0 , 0 ) {\displaystyle (-1,0,0)} , ( 1 , 0 , 0 ) {\displaystyle (1,0,0)} , ( 0 , 1 , 1 ) {\displaystyle (0,1,1)} , and ( 0 , 1 , − 1 ) {\displaystyle (0,1,-1)} is such a disphenoid.[14][15] Each of its four faces is an isosceles triangle with edges of lengths 3 {\displaystyle {\sqrt {3}}} , 3 {\displaystyle {\sqrt {3}}} , and 2 {\displaystyle 2} . It can [tessellate](/source/Tessellation) space to form the [disphenoid tetrahedral honeycomb](/source/Disphenoid_tetrahedral_honeycomb). As [Gibb (1990)](#CITEREFGibb1990) describes, it can be folded without cutting or overlaps from a single sheet of [A4 paper](/source/A4_paper).[16]

"Disphenoid" is also used to describe two forms of [crystal](/source/Crystal_system):

- A wedge-shaped crystal form of the [tetragonal](/source/Tetragonal_crystal_system) or [orthorhombic system](/source/Orthorhombic_crystal_system). It has four triangular faces that are alike and that correspond in position to alternate faces of the tetragonal or orthorhombic [dipyramid](/source/Bipyramid). 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.

- A crystal form bounded by eight [scalene triangles](/source/Scalene_triangle) arranged in pairs, constituting a tetragonal [scalenohedron](/source/Scalenohedron).

## Other uses

Six tetragonal disphenoids attached end-to-end in a ring construct a [kaleidocycle](/source/Kaleidocycle), a paper toy that can rotate on 4 sets of faces in a hexagon. The rotation of the six disphenoids with opposite edges of length l {\displaystyle l} , m {\displaystyle m} and n {\displaystyle n} ([without loss of generality](/source/Without_loss_of_generality) n ≤ l {\displaystyle n\leq l} , n ≤ m {\displaystyle n\leq m} ) is physically realizable if and only if[17]

- − 8 ( l 2 − m 2 ) 2 ( l 2 + m 2 ) − 5 n 6 + 11 ( l 2 − m 2 ) 2 n 2 + 2 ( l 2 + m 2 ) n 4 ≥ 0. {\displaystyle -8(l^{2}-m^{2})^{2}(l^{2}+m^{2})-5n^{6}+11(l^{2}-m^{2})^{2}n^{2}+2(l^{2}+m^{2})n^{4}\geq 0.}

## See also

- [Irregular tetrahedra](/source/Tetrahedron#Irregular_tetrahedra)

- [Orthocentric tetrahedron](/source/Orthocentric_tetrahedron)

- [Snub disphenoid](/source/Snub_disphenoid) - A [Johnson solid](/source/Johnson_solid) with 12 equilateral triangle faces and D 2 d {\displaystyle D_{2d}} symmetry.

- [Trirectangular tetrahedron](/source/Trirectangular_tetrahedron)

## References

1. **[^](#cite_ref-1)** [Coxeter, H. S. M.](/source/Harold_Scott_MacDonald_Coxeter) (1973), *[Regular Polytopes](/source/Regular_Polytopes_(book))* (3rd ed.), Dover Publications, p. [15](https://archive.org/details/regularpolytopes0000coxe/page/15), [ISBN](/source/ISBN_(identifier)) [0-486-61480-8](https://en.wikipedia.org/wiki/Special:BookSources/0-486-61480-8)

1. **[^](#cite_ref-akiyama2_2-0)** [Akiyama, Jin](/source/Jin_Akiyama); Matsunaga, Kiyoko (2020), "An Algorithm for Folding a Conway Tile into an Isotetrahedron or a Rectangle Dihedron", *[Journal of Information Processing](/source/Journal_of_Information_Processing)*, **28** (28): 750–758, [doi](/source/Doi_(identifier)):[10.2197/ipsjjip.28.750](https://doi.org/10.2197%2Fipsjjip.28.750), [S2CID](/source/S2CID_(identifier)) [230108666](https://api.semanticscholar.org/CorpusID:230108666).

1. ^ [***a***](#cite_ref-whittaker_3-0) [***b***](#cite_ref-whittaker_3-1) Whittaker, E. J. W. (2013), [*Crystallography: An Introduction for Earth Science (and other Solid State) Students*](https://books.google.com/books?id=aUUvBQAAQBAJ&pg=PA89), Elsevier, p. 89, [ISBN](/source/ISBN_(identifier)) [9781483285566](https://en.wikipedia.org/wiki/Special:BookSources/9781483285566).

1. ^ [***a***](#cite_ref-leech_4-0) [***b***](#cite_ref-leech_4-1) [Leech, John](/source/John_Leech_(mathematician)) (1950), "Some properties of the isosceles tetrahedron", *[The Mathematical Gazette](/source/The_Mathematical_Gazette)*, **34** (310): 269–271, [doi](/source/Doi_(identifier)):[10.2307/3611029](https://doi.org/10.2307%2F3611029), [JSTOR](/source/JSTOR_(identifier)) [3611029](https://www.jstor.org/stable/3611029), [MR](/source/MR_(identifier)) [0038667](https://mathscinet.ams.org/mathscinet-getitem?mr=0038667), [S2CID](/source/S2CID_(identifier)) [125145099](https://api.semanticscholar.org/CorpusID:125145099).

1. **[^](#cite_ref-5)** Hajja, Mowaffaq; Walker, Peter (2001), "Equifacial tetrahedra", *International Journal of Mathematical Education in Science and Technology*, **32** (4): 501–508, [doi](/source/Doi_(identifier)):[10.1080/00207390110038231](https://doi.org/10.1080%2F00207390110038231), [MR](/source/MR_(identifier)) [1847966](https://mathscinet.ams.org/mathscinet-getitem?mr=1847966), [S2CID](/source/S2CID_(identifier)) [218495301](https://api.semanticscholar.org/CorpusID:218495301).

1. ^ [***a***](#cite_ref-akiyama_6-0) [***b***](#cite_ref-akiyama_6-1) [Akiyama, Jin](/source/Jin_Akiyama) (2007), "Tile-makers and semi-tile-makers", *[American Mathematical Monthly](/source/American_Mathematical_Monthly)*, **114** (7): 602–609, [doi](/source/Doi_(identifier)):[10.1080/00029890.2007.11920450](https://doi.org/10.1080%2F00029890.2007.11920450), [JSTOR](/source/JSTOR_(identifier)) [27642275](https://www.jstor.org/stable/27642275), [MR](/source/MR_(identifier)) [2341323](https://mathscinet.ams.org/mathscinet-getitem?mr=2341323), [S2CID](/source/S2CID_(identifier)) [32897155](https://api.semanticscholar.org/CorpusID:32897155).

1. **[^](#cite_ref-7)** [Demaine, Erik](/source/Erik_Demaine); [O'Rourke, Joseph](/source/Joseph_O'Rourke_(professor)) (2007), [*Geometric Folding Algorithms*](/source/Geometric_Folding_Algorithms), Cambridge University Press, p. 424, [ISBN](/source/ISBN_(identifier)) [978-0-521-71522-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-71522-5).

1. ^ [***a***](#cite_ref-petitjean_8-0) [***b***](#cite_ref-petitjean_8-1) Petitjean, Michel (2015), ["The most chiral disphenoid"](http://match.pmf.kg.ac.rs/electronic_versions/Match73/n2/match73n2_375-384.pdf) (PDF), *MATCH Communications in Mathematical and in Computer Chemistry*, **73** (2): 375–384, [MR](/source/MR_(identifier)) [3242747](https://mathscinet.ams.org/mathscinet-getitem?mr=3242747).

1. ^ [***a***](#cite_ref-Andreescu_9-0) [***b***](#cite_ref-Andreescu_9-1) [***c***](#cite_ref-Andreescu_9-2) Andreescu, Titu; Gelca, Razvan (2009), *Mathematical Olympiad Challenges* (2nd ed.), Birkhäuser, pp. 30–31.

1. ^ [***a***](#cite_ref-Brown_10-0) [***b***](#cite_ref-Brown_10-1) [***c***](#cite_ref-Brown_10-2) [***d***](#cite_ref-Brown_10-3) [***e***](#cite_ref-Brown_10-4) Brown, B. H. (April 1926), "Theorem of Bang. Isosceles tetrahedra", Undergraduate Mathematics Clubs: Club Topics, *[American Mathematical Monthly](/source/American_Mathematical_Monthly)*, **33** (4): 224–226, [doi](/source/Doi_(identifier)):[10.1080/00029890.1926.11986564](https://doi.org/10.1080%2F00029890.1926.11986564), [JSTOR](/source/JSTOR_(identifier)) [2299548](https://www.jstor.org/stable/2299548).

1. **[^](#cite_ref-11)** [Fuchs, Dmitry](https://de.wikipedia.org/wiki/Dmitry_Fuchs) [in German]; Fuchs, Ekaterina (2007), ["Closed geodesics on regular polyhedra"](https://www.ams.org/distribution/mmj/vol7-2-2007/fuchs.pdf) (PDF), *Moscow Mathematical Journal*, **7** (2): 265–279, 350, [doi](/source/Doi_(identifier)):[10.17323/1609-4514-2007-7-2-265-279](https://doi.org/10.17323%2F1609-4514-2007-7-2-265-279), [MR](/source/MR_(identifier)) [2337883](https://mathscinet.ams.org/mathscinet-getitem?mr=2337883).

1. **[^](#cite_ref-12)** [Arnold, Vladimir I.](/source/Vladimir_Arnold) (2004), "Problem 1958-2", in Arnold, Vladimir I. (ed.), [*Arnold's Problems*](/source/Arnold's_Problems), Berlin: Springer-Verlag, p. 2, [doi](/source/Doi_(identifier)):[10.1007/b138219](https://doi.org/10.1007%2Fb138219), [ISBN](/source/ISBN_(identifier)) [3-540-20614-0](https://en.wikipedia.org/wiki/Special:BookSources/3-540-20614-0), [MR](/source/MR_(identifier)) [2078115](https://mathscinet.ams.org/mathscinet-getitem?mr=2078115)

1. ^ [***a***](#cite_ref-Leech_13-0) [***b***](#cite_ref-Leech_13-1) [***c***](#cite_ref-Leech_13-2) [***d***](#cite_ref-Leech_13-3) [***e***](#cite_ref-Leech_13-4) [***f***](#cite_ref-Leech_13-5) [***g***](#cite_ref-Leech_13-6) Leech, John (1950), "Some properties of the isosceles tetrahedron", *Mathematical Gazette*, **34** (310): 269–271, [doi](/source/Doi_(identifier)):[10.2307/3611029](https://doi.org/10.2307%2F3611029), [JSTOR](/source/JSTOR_(identifier)) [3611029](https://www.jstor.org/stable/3611029), [S2CID](/source/S2CID_(identifier)) [125145099](https://api.semanticscholar.org/CorpusID:125145099).

1. **[^](#cite_ref-14)** [Coxeter (1973](#CITEREFCoxeter1973), pp. 71–72).

1. **[^](#cite_ref-15)** [Senechal, Marjorie](/source/Marjorie_Senechal) (1981), "Which tetrahedra fill space?", *[Mathematics Magazine](/source/Mathematics_Magazine)*, **54** (5): 227–243, [doi](/source/Doi_(identifier)):[10.2307/2689983](https://doi.org/10.2307%2F2689983), [JSTOR](/source/JSTOR_(identifier)) [2689983](https://www.jstor.org/stable/2689983), [MR](/source/MR_(identifier)) [0644075](https://mathscinet.ams.org/mathscinet-getitem?mr=0644075)

1. **[^](#cite_ref-16)** Gibb, William (1990), "Paper patterns: solid shapes from metric paper", *Mathematics in School*, **19** (3): 2–4 Reprinted in Pritchard, Chris, ed. (2003), *The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching*, Cambridge University Press, pp. 363–366, [ISBN](/source/ISBN_(identifier)) [0-521-53162-4](https://en.wikipedia.org/wiki/Special:BookSources/0-521-53162-4).

1. **[^](#cite_ref-17)** [Sloane, N. J. A.](/source/Neil_Sloane) (ed.), ["Sequence A338336"](https://oeis.org/A338336), *The [On-Line Encyclopedia of Integer Sequences](/source/On-Line_Encyclopedia_of_Integer_Sequences)*, OEIS Foundation.

## External links

- [Mathematical Analysis of Disphenoid by H C Rajpoot](https://www.academia.edu/32874396/Mathematical_analysis_of_Disphenoid_isosceles_tetrahedron_Derivation_of_volume_surface_area_vertical_height_in_radius_circum_radius_coordinates_of_four_vertices_in_center_circum_center_and_centroid_for_optimal_configuration_of_a_disphenoid_in_3D_space_) from [Academia.edu](/source/Academia.edu)

- [Weisstein, Eric W.](/source/Eric_W._Weisstein), ["Disphenoid"](https://mathworld.wolfram.com/Disphenoid.html), *[MathWorld](/source/MathWorld)*

- [Weisstein, Eric W.](/source/Eric_W._Weisstein), ["Isosceles tetrahedron"](https://mathworld.wolfram.com/IsoscelesTetrahedron.html), *[MathWorld](/source/MathWorld)*

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Adapted from the Wikipedia article [Disphenoid](https://en.wikipedia.org/wiki/Disphenoid) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Disphenoid?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.