Disphenoid

{{Further|Tetrahedron#Isometries of irregular tetrahedra}}

If the faces of a disphenoid are equilateral triangles, it is a regular tetrahedron with T_d 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 D_2d dihedral symmetry.

A sphenoid with scalene triangles as its faces is called a rhombic disphenoid and it has D₂ dihedral symmetry. Unlike the tetragonal disphenoid, the rhombic disphenoid has no reflection symmetry, so it is chiral.

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. 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.{{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}}. 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.

A tetrahedron is a disphenoid if and only if its circumscribed parallelepiped is right-angled.

We also have that a tetrahedron is a disphenoid if and only if the center in the circumscribed sphere and the inscribed sphere coincide.{{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}}.

Another characterization states that if d₁, d₂ and d₃ 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 d₁, d₂ and d₃ are pairwise perpendicular.{{citation |last1=Andreescu |first1=Titu |last2=Gelca |first2=Razvan |title=Mathematical Olympiad Challenges |publisher=Birkhäuser |edition=2nd |year=2009 |pages=30–31}}.

The disphenoids are the only polyhedra having infinitely many non-self-intersecting closed geodesics. On a disphenoid, all closed geodesics are non-self-intersecting.{{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

}}.

The disphenoids are the tetrahedra in which all four faces have the same perimeter, the tetrahedra in which all four faces have the same area, and the tetrahedra in which the angular defects of all four vertices equal {{pi}}. 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.{{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 }}.

The volume of a disphenoid with opposite edges of length l, m and n is given by{{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

}}.

: $V\; =\; \backslash sqrt\{\backslash frac\{(l^2\; +\; m^2\; -\; n^2)(l^2\; -\; m^2\; +\; n^2)(-l^2\; +\; m^2\; +\; n^2)\}\{72\}\}.$

The circumscribed sphere has radius (the circumradius)

: $R\; =\; \backslash sqrt\{\backslash frac\{l^2\; +\; m^2\; +\; n^2\}\{8\}\},$

and the inscribed sphere has radius

: $r\; =\; \backslash frac\{3V\}\{4T\},$

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:

: $16T^2\; R^2\; =\; l^2\; m^2\; n^2\; +\; 9V^2.$

The squares of the lengths of the bimedians are

: $\backslash tfrac\{1\}\{2\}(l^2\; +\; m^2\; -\; n^2),\backslash quad\; \backslash tfrac\{1\}\{2\}(l^2\; -\; m^2\; +\; n^2),\backslash quad\; \backslash tfrac\{1\}\{2\}(-l^2\; +\; m^2\; +\; n^2).$

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

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

The centers in the circumscribed and inscribed spheres coincide with the centroid of the disphenoid.

The bimedians are perpendicular to the edges they connect and to each other.

File:Oblate tetrahedrille cell.pngs of 90°, and four edges have dihedral angles of 60°.]]

Some tetragonal disphenoids will form honeycombs. The disphenoid whose four vertices are (−1, 0, 0), (1, 0, 0), (0, 1, 1), and (0, 1, −1) is such a disphenoid.{{harvtxt|Coxeter|1973|pages=71–72}}.{{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 }}

Each of its four faces is an isosceles triangle with edges of lengths {{radic|3}}, {{radic|3}}, and 2. 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.{{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}}.

"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.

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 l, m and n (without loss of generality n ≤ l, n ≤ m) is physically realizable if and only if{{cite OEIS|A338336}}.

: $-8(l^2\; -\; m^2)^2\; (l^2\; +\; m^2)\; -\; 5n^6\; +\; 11(l^2\; -\; m^2)^2\; n^2\; +\; 2(l^2\; +\; m^2)\; n^4\; \backslash ge\; 0.$

• Irregular tetrahedra
• Orthocentric tetrahedron
• Snub disphenoid - A Johnson solid with 12 equilateral triangle faces and D_2d symmetry.
• Trirectangular tetrahedron

• [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