View source for Disphenoid

{{Short description|Tetrahedron whose faces are all congruent}}
{{multiple images
|image1=Disphenoid tetrahedron.png
|caption1=The '''tetragonal and [[Digon|digon]]al disphenoids''' can be positioned inside a [[Cuboid|cuboid]] bisecting two opposite faces. Both have four equal edges going around the sides. The digonal has two pairs of congruent [[Isosceles triangle|isosceles triangle]] faces, while the tetragonal has four congruent isosceles triangle faces.
|image2=Rhombic disphenoid.png
|caption2=A '''rhombic disphenoid''' has congruent [[Triangle#Types of triangle|scalene triangle]] faces, and can fit diagonally inside of a [[Cuboid|cuboid]]. It has three sets of edge lengths, existing as opposite pairs.}}

In [[Geometry|geometry]], a '''disphenoid''' ({{ety|el|sphenoeides|wedgelike}}) is a [[Tetrahedron|tetrahedron]] whose four [[Face (geometry)|faces]] are [[Congruence (geometry)|congruent]] acute-angled triangles.<ref>{{citation|last=Coxeter|first=H. S. M.|title=Regular Polytopes|edition=3rd|publisher=Dover Publications|year=1973|isbn=0-486-61480-8|page=[https://archive.org/details/regularpolytopes0000coxe/page/15 15]}}</ref> It can also be described as a tetrahedron in which every two [[Edge (geometry)|edges]] that are opposite each other have equal lengths. Other names for the same shape are '''isotetrahedron''',<ref name="akiyama2">{{citation
 | issue = 28
 | journal = Journal of Information Processing
 | mr = 
 | pages =  750–758
 | title = An Algorithm for Folding a Conway Tile into an Isotetrahedron or a Rectangle Dihedron
 | volume = 28
 | year = 2020
 | doi = 10.2197/ipsjjip.28.750
 | jstor =  | s2cid =  230108666| doi-access = free
 }}.</ref> '''sphenoid''',<ref name="whittaker">{{citation|title=Crystallography: An Introduction for Earth Science (and other Solid State) Students|first=E. J. W.|last=Whittaker|publisher=Elsevier|year=2013|isbn=9781483285566|page=89|url=https://books.google.com/books?id=aUUvBQAAQBAJ&pg=PA89}}.</ref> '''bisphenoid''',<ref name="whittaker"/> '''isosceles tetrahedron''',<ref name="leech">{{citation
 | doi = 10.2307/3611029
 | journal = The Mathematical Gazette
 | mr = 0038667
 | pages = 269–271
 | title = Some properties of the isosceles tetrahedron
 | volume = 34
 | year = 1950| issue = 310 | jstor = 3611029 | s2cid = 125145099 }}.</ref> '''equifacial tetrahedron''',<ref>{{citation
 | last1 = Hajja | first1 = Mowaffaq
 | last2 = Walker | first2 = Peter
 | doi = 10.1080/00207390110038231
 | issue = 4
 | journal = International Journal of Mathematical Education in Science and Technology
 | mr = 1847966
 | pages = 501–508
 | title = Equifacial tetrahedra
 | volume = 32
 | year = 2001| s2cid = 218495301
 }}.</ref> '''almost regular tetrahedron''',<ref name="akiyama"/> and '''tetramonohedron'''.<ref>{{citation|title=Geometric Folding Algorithms|title-link=Geometric Folding Algorithms|last1=Demaine|first1=Erik|last2=O'Rourke|first2=Joseph|publisher=Cambridge University Press|isbn=978-0-521-71522-5|year=2007|page=424}}.</ref>

All the [[Solid angle|solid angle]]s and [[Vertex figure|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|right angle]]s. However, a disphenoid is not a [[Regular polyhedron|regular polyhedron]], because, in general, its faces are not [[Regular polygon|regular polygon]]s, and its edges have three different lengths.

== Special cases and generalizations ==
If the faces of a disphenoid are [[Equilateral triangle|equilateral triangle]]s, it is a regular tetrahedron with '''T'''<SUB>d</SUB> [[Tetrahedral symmetry|tetrahedral symmetry]], although this is not normally called a disphenoid. When the faces of a disphenoid are [[Isosceles triangle|isosceles triangle]]s, it is called a '''tetragonal disphenoid'''. In this case it has '''D'''<SUB>2d</SUB> dihedral symmetry.
A sphenoid with [[Scalene triangle|scalene triangle]]s as its faces is called a '''rhombic disphenoid''' and it has '''D'''<SUB>2</SUB> dihedral symmetry. Unlike the tetragonal disphenoid, the rhombic disphenoid has no [[Reflection symmetry|reflection symmetry]], so it is [[Chirality|chiral]].<ref name="petitjean"/>
Both tetragonal disphenoids and rhombic disphenoids are [[Isohedral figure|isohedra]]: 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|right triangle]] or obtuse triangle faces.<ref name="leech"/> 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.<ref name="petitjean">{{citation
 | last = Petitjean | first = Michel
 | issue = 2
 | journal = MATCH Communications in Mathematical and in Computer Chemistry
 | mr = 3242747
 | pages = 375–384
 | title = The most chiral disphenoid
 | url = http://match.pmf.kg.ac.rs/electronic_versions/Match73/n2/match73n2_375-384.pdf
 | volume = 73
 | year = 2015}}.</ref> When obtuse triangles are glued in this way, the resulting surface can be folded to form a disphenoid (by [[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 [[Antiprism|antiprism]]s or as [[Alternation (geometry)|alternated]] quadrilateral [[Prism (geometry)|prism]]s.

== Characterizations ==
A tetrahedron is a disphenoid [[If and only if|if and only if]] its circumscribed [[Parallelepiped|parallelepiped]] is right-angled.<ref name=Andreescu/>

We also have that a tetrahedron is a disphenoid if and only if the center in the [[Circumscribed sphere|circumscribed sphere]] and the [[Inscribed sphere|inscribed sphere]] coincide.<ref name=Brown>{{citation|last=Brown|first=B. H.|title=Theorem of Bang. Isosceles tetrahedra|journal=[[American Mathematical Monthly]]|date=April 1926|pages=224–226|volume=33|issue=4|jstor= 2299548|department=Undergraduate Mathematics Clubs: Club Topics|doi=10.1080/00029890.1926.11986564}}.</ref>

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|perpendicular]].<ref name=Andreescu>{{citation|last1=Andreescu|first1=Titu|last2=Gelca|first2= Razvan|title=Mathematical Olympiad Challenges|publisher= Birkhäuser|edition=2nd|year= 2009|pages= 30–31}}.</ref>

The disphenoids are the only polyhedra having infinitely many non-self-intersecting [[Closed geodesic|closed geodesic]]s. On a disphenoid, all closed geodesics are non-self-intersecting.<ref>{{citation
 | last1 = Fuchs | first1 = Dmitry
 | last2 = Fuchs | first2 = Ekaterina
 | issue = 2
 | journal = Moscow Mathematical Journal
 | mr = 2337883
 | pages = 265–279, 350
 | title = Closed geodesics on regular polyhedra
 | url = https://www.ams.org/distribution/mmj/vol7-2-2007/fuchs.pdf
 | volume = 7
 | year = 2007| doi = 10.17323/1609-4514-2007-7-2-265-279
 }}.</ref>

The disphenoids are the tetrahedra in which all four faces have the same [[Perimeter|perimeter]], the tetrahedra in which all four faces have the same area,<ref name="Brown"/> and the tetrahedra in which the [[Angular defect|angular defect]]s of all four vertices equal {{pi}}. They are the polyhedra having a [[Net (polyhedron)|net]] in the shape of an acute triangle, divided into four similar triangles by segments connecting the edge midpoints.<ref name="akiyama">{{citation
 | issue = 7
 | journal = [[American Mathematical Monthly]]
 | mr = 2341323
 | pages = 602–609
 | title = Tile-makers and semi-tile-makers
 | volume = 114
 | year = 2007
 | doi = 10.1080/00029890.2007.11920450
 | jstor = 27642275 | s2cid = 32897155 }}.</ref>

== Metric formulas ==
The [[Physics:Volume|volume]] of a disphenoid with opposite edges of length ''l'', ''m'' and ''n'' is given by:<ref name=Leech>{{citation
 | last = Leech | first = John
 | journal = Mathematical Gazette
 | pages = 269–271
 | title = Some properties of the isosceles tetrahedron
 | volume = 34
 | number = 310
 | year = 1950
 | doi=10.2307/3611029| jstor = 3611029
 | s2cid = 125145099
 }}.</ref>

:<math> V=\sqrt{\frac{(l^2+m^2-n^2)(l^2-m^2+n^2)(-l^2+m^2+n^2)}{72}}. </math>

The [[Circumscribed sphere|circumscribed sphere]] has radius<ref name=Leech/> (the circumradius):

:<math> R=\sqrt{\frac{l^2+m^2+n^2}{8}} </math>

and the [[Inscribed sphere|inscribed sphere]] has radius:<ref name=Leech/>

:<math> r=\frac{3V}{4T} </math>

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/>

:<math>\displaystyle 16T^2R^2=l^2m^2n^2+9V^2. </math>

The squares of the lengths of the [[Tetrahedron#Properties analogous to those of a triangle|bimedians]] are:<ref name=Leech/>

:<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>

== Other properties ==
If the four faces of a tetrahedron have the same perimeter, then the tetrahedron is a disphenoid.<ref name=Brown/>

If the four faces of a tetrahedron have the same area, then it is a disphenoid.<ref name=Andreescu/><ref name=Brown/>

The centers in the [[Circumscribed sphere|circumscribed]] and [[Inscribed sphere|inscribed sphere]]s coincide with the [[Centroid|centroid]] of the disphenoid.<ref name=Leech/>

The bimedians are [[Perpendicular|perpendicular]] to the edges they connect and to each other.<ref name=Leech/>

== Honeycombs and crystals ==
[[File:Oblate tetrahedrille cell.png|thumb|A space-filling tetrahedral disphenoid inside a cube. Two edges have [[Dihedral angle|dihedral angle]]s of 90°, and four edges have dihedral angles of 60°.]]
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>{{harvtxt|Coxeter|1973|pages=71–72}}.</ref><ref>{{citation
 | title = Which tetrahedra fill space?
 | year = 1981
 | journal = Mathematics Magazine
 | mr = 0644075
 | volume = 54
 | issue = 5
 | pages = 227–243
 | jstor = 2689983
 | doi =10.2307/2689983 }}
</ref> Each of its four faces is an isosceles triangle with edges of lengths {{radic|3}}, {{radic|3}}, and 2. It can [[Tessellation|tessellate]] space to form the disphenoid tetrahedral honeycomb. As {{harvtxt|Gibb|1990}} describes, it can be folded without cutting or overlaps from a single sheet of a4 paper.<ref>{{citation
 | last = Gibb | first = William
 | title = Paper patterns: solid shapes from metric paper
 | year = 1990
 | journal = Mathematics in School
 | volume = 19
 | issue = 3
 | pages = 2–4}} Reprinted in {{citation
 | editor-last = Pritchard | editor-first = Chris
 | year = 2003
 | title = The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching
 | publisher = Cambridge University Press
 | isbn = 0-521-53162-4
 | pages = 363–366}}</ref>

"Disphenoid" is also used to describe two forms of [[Crystal system|crystal]]:

* A wedge-shaped crystal form of the [[Physics:Tetragonal crystal system|tetragonal]] or [[Physics: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.
*A crystal form bounded by eight [[Scalene triangle|scalene triangle]]s arranged in pairs, constituting a tetragonal scalenohedron.

==Other uses==
Six tetragonal disphenoids attached end-to-end in a ring construct a [[Kaleidocycle|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, m and n (without loss of generality n≤l, n≤m) is physically realizable if and only if <ref>http://kociemba.org/themen/kaleidocycles/workingkaleidocycles.html</ref><ref>[http://kociemba.org/themen/kaleidocycles/interactive.html Interactive version of kaleidocycle]</ref><ref>https://oeis.org/A338336</ref>
:<math>-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 </math>

== See also ==
* [[Tetrahedron#Irregular tetrahedra|Irregular tetrahedra]]
* [[Orthocentric tetrahedron]]
* [[Snub disphenoid]] - A [[Johnson solid]] with 12 equilateral triangle faces and D<sub>2d</sub> symmetry.
* [[Trirectangular tetrahedron]]

==References==
<references/>

== External links ==
*[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_ Mathematical Analysis of Disphenoid by H C Rajpoot] from [[Company:Academia.edu|Academia.edu]]
*{{Mathworld | urlname=Disphenoid | title=Disphenoid }}
*{{Mathworld | urlname=IsoscelesTetrahedron | title=Isosceles tetrahedron }}

[[Category:Polyhedra]]

{{Sourceattribution|Disphenoid}}