Disphenoid
In geometry, a disphenoid (from Greek sphenoeides 'wedgelike') is a tetrahedron whose four faces are congruent acute-angled triangles.[1] 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,[2] sphenoid,[3] bisphenoid,[3] isosceles tetrahedron,[4] equifacial tetrahedron,[5] almost regular tetrahedron,[6] and tetramonohedron.[7]
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
If the faces of a disphenoid are
It is not possible to construct a disphenoid with
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.[9]
We also have that a tetrahedron is a disphenoid if and only if the
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.[9]
The disphenoids are the only polyhedra having infinitely many non-self-intersecting closed geodesics. 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, the tetrahedra in which all four faces have the same area,[10] and the tetrahedra in which the angular defects of all four vertices equal π. 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.[6]
Metric formulas
The volume of a disphenoid with opposite edges of length l, m and n is given by:[12]
The circumscribed sphere has radius[12] (the circumradius):
and the inscribed sphere has radius:[12]
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:[12]
The squares of the lengths of the bimedians are:[12]
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 and inscribed spheres coincide with the centroid of the disphenoid.[12]
The bimedians are perpendicular to the edges they connect and to each other.[12]
Honeycombs and crystals
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Oblate_tetrahedrille_cell.png/220px-Oblate_tetrahedrille_cell.png)
Some tetragonal disphenoids will form
"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 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 l, m and n (without loss of generality n≤l, n≤m) is physically realizable if and only if [16][17][18]
See also
- Irregular tetrahedra
- Orthocentric tetrahedron
- Snub disphenoid - A Johnson solid with 12 equilateral triangle faces and D2d symmetry.
- Trirectangular tetrahedron
References
- ISBN 0-486-61480-8
- S2CID 230108666.
- ^ ISBN 9781483285566.
- ^ S2CID 125145099.
- S2CID 218495301.
- ^ S2CID 32897155.
- ISBN 978-0-521-71522-5.
- ^ MR 3242747.
- ^ a b c Andreescu, Titu; Gelca, Razvan (2009), Mathematical Olympiad Challenges (2nd ed.), Birkhäuser, pp. 30–31.
- ^ JSTOR 2299548.
- MR 2337883.
- ^ S2CID 125145099.
- ^ Coxeter (1973, pp. 71–72).
- MR 0644075
- ISBN 0-521-53162-4
- ^ http://kociemba.org/themen/kaleidocycles/workingkaleidocycles.html
- ^ Interactive version of kaleidocycle
- ^ https://oeis.org/A338336