{{Short description|Tetrahedron whose faces are all congruent}} {{CS1 config|mode=cs2}} {{multiple images |image1=Disphenoid tetrahedron.png |caption1=The '''tetragonal and digonal disphenoids''' can be positioned inside a cuboid bisecting two opposite faces. Both have four equal edges going around the sides. The digonal has two pairs of congruent isosceles triangle faces, while the tetragonal has four congruent isosceles triangle faces. |image2=Rhombic disphenoid.png |caption2=A '''rhombic disphenoid''' has congruent scalene triangle faces, and can fit diagonally inside of a cuboid. It has three sets of edge lengths, existing as opposite pairs.}}
In geometry, a '''disphenoid''' ({{ety|el|sphenoeides|wedgelike}}) is a tetrahedron whose four faces are congruent acute-angled triangles.<ref>{{citation|authorlink=Harold Scott MacDonald Coxeter|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 edges that are opposite each other have equal lengths. Other names for the same shape are '''isotetrahedron''',<ref name="akiyama2">{{citation | last1 = Akiyama | first1 = Jin | authorlink1 = Jin Akiyama | last2 = Matsunaga | first2 = Kiyoko | authorlink2 = | 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 | last = Leech | first = John | authorlink = John Leech (mathematician) | 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 |authorlink1=Erik Demaine |last2=O'Rourke |first2=Joseph |authorlink2=Joseph O'Rourke (professor) |publisher=Cambridge University Press |isbn=978-0-521-71522-5 |year=2007 |page=424}}.</ref>
All the solid angles and vertex figures of a disphenoid are the same, and the sum of the face angles at each vertex is equal to two right angles. However, a disphenoid is not a regular polyhedron, because, in general, its faces are not regular polygons, and its edges have three different lengths.
== Special cases and generalizations == {{Further|Tetrahedron#Isometries of irregular tetrahedra}} If the faces of a disphenoid are equilateral triangles, it is a regular tetrahedron with <math>T_d</math> tetrahedral symmetry, although this is not normally called a disphenoid. When the faces of a disphenoid are isosceles triangles, it is called a '''tetragonal disphenoid'''. In this case it has <math>D_{2d}</math> dihedral symmetry. A sphenoid with scalene triangles as its faces is called a '''rhombic disphenoid''' and it has <math>D_2</math> dihedral symmetry. Unlike the tetragonal disphenoid, the rhombic disphenoid has no reflection symmetry, so it is chiral.<ref name="petitjean"/> Both tetragonal disphenoids and rhombic disphenoids are 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 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 antiprisms or as alternated quadrilateral prisms.
== Characterizations == A tetrahedron is a disphenoid if and only if its circumscribed 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 and the 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 <math>d_1</math>, <math>d_2</math> and <math>d_3</math> are the common perpendiculars of <math>AB</math> and <math>CD</math>; <math>AC</math> and <math>BD</math>; and <math>AD</math> and <math>BC</math> respectively in a tetrahedron <math>ABCD</math>, then the tetrahedron is a disphenoid if and only if <math>d_1</math>, <math>d_2</math> and <math>d_3</math> are pairwise 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 geodesics. On a disphenoid, all closed geodesics are non-self-intersecting.<ref>{{citation | last1 = Fuchs | first1 = Dmitry | authorlink1= :de:Dmitry Fuchs | 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,<ref name="Brown"/> the tetrahedra in which all four faces have the same area,<ref name="Brown"/><ref>{{citation | last = Arnold | first = Vladimir I. | editor-first1 = Vladimir I. | editor-last1 = Arnold | author-link = Vladimir Arnold | contribution = Problem 1958-2 | doi = 10.1007/b138219 | isbn = 3-540-20614-0 | mr = 2078115 | page = 2 | publisher = Springer-Verlag | location = Berlin | title = Arnold's Problems | title-link = Arnold's Problems | year = 2004}}</ref> and the tetrahedra in which the angular defects of all four vertices equal <math>\pi</math>. They are the polyhedra having a net in the shape of an acute triangle, divided into four similar triangles by segments connecting the edge midpoints.<ref name="akiyama">{{citation | last = Akiyama | first = Jin | authorlink = Jin Akiyama | 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 volume of a disphenoid with opposite edges of length <math>l</math>, <math>m</math> and <math>n</math> 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 has radius<ref name=Leech/> (the circumradius) : <math>R = \sqrt{\frac{l^2 + m^2 + n^2}{8}},</math> and the inscribed sphere has radius<ref name=Leech/> : <math> r = \frac{3V}{4T},</math> where <math>V</math> is the volume of the disphenoid and <math>T</math> 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>16T^2 R^2 = l^2 m^2 n^2 + 9V^2.</math>
The squares of the lengths of the 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 and inscribed spheres coincide with the centroid of the disphenoid.<ref name=Leech/>
The bimedians are 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 angles of 90°, and four edges have dihedral angles of 60°.]] Some tetragonal disphenoids will form honeycombs. The disphenoid whose four vertices are <math>(-1,0,0)</math>, <math>(1,0,0)</math>, <math>(0,1,1)</math>, and <math>(0,1,-1)</math> is such a disphenoid.<ref>{{harvtxt|Coxeter|1973|pages=71–72}}.</ref><ref>{{citation | last = Senechal | first = Marjorie | authorlink = Marjorie Senechal | 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 <math>\sqrt{3}</math>, <math>\sqrt{3}</math>, and <math>2</math>. It can 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: * A wedge-shaped crystal form of the tetragonal or orthorhombic system. It has four triangular faces that are alike and that correspond in position to alternate faces of the tetragonal or orthorhombic 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 triangles arranged in pairs, constituting a tetragonal scalenohedron.
==Other uses== Six tetragonal disphenoids attached end-to-end in a ring construct a 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 <math>l</math>, <math>m</math> and <math>n</math> (without loss of generality <math>n\leq l</math>, <math>n\leq m</math>) is physically realizable if and only if<ref>{{cite OEIS|A338336}}.</ref> : <math>-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 \ge 0.</math>
== See also == * Irregular tetrahedra * Orthocentric tetrahedron * Snub disphenoid - A Johnson solid with 12 equilateral triangle faces and <math>D_{2d}</math> 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 Academia.edu * {{Mathworld | urlname=Disphenoid | title=Disphenoid }} * {{Mathworld | urlname=IsoscelesTetrahedron | title=Isosceles tetrahedron }}
Category:Tetrahedra