Disphenoid tetrahedron inside a square cuboid

A disphenoid is a polyhedron whose four faces are identical isosceles or scalene triangles.^[1] (A regular tetrahedron has four identical triangular faces, but is not normally considered a disphenoid.) The faces of a tetragonal disphenoid are isosceles; the faces of a rhombic disphenoid are scalene. In anatomy, the cuboid bone is a bone in the foot. ... For the game magazine, see Polyhedron (magazine). ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... A triangle. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

All the solid angles and vertex figures of a disphenoid are the same. However, a disphenoid is not a regular polyhedron, because its faces are not regular polygons. A solid angle is the three dimensional analog of the ordinary angle. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ...

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 a such a disphenoid.^[2] Each of its four faces is an isosceles triangle with edges of lengths √3, √3, and 2. It can tesselate space to form the disphenoid tetrahedral honeycomb. As Gibb^[3] describes, it can be folded without cutting or overlaps from a single sheet of a4 paper. In geometry, a honeycomb is a name for a space-filling tessellation, just as a tiling is a tessellation of a plane or 2-dimensional surface. ... The Disphenoid tetrahedral honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of identical nonregular tetrahedral cells. ... A comparison of different paper sizes A4 is a standard paper size, defined by the international standard ISO 216 as 210×297 mm (roughly 8. ...

References

1. ^ *Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. p. 15
2. ^ Coxeter, pp. 71–72; Senechal, Marjorie (1981). "Which tetrahedra fill space?". Mathematics Magazine 54 (5): 227–243. 
3. ^ 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, 363–366. ISBN 0-521-53162-4.