Disphenoid

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.
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 (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.

  1. ^ Coxeter, H. S. M. (1973), Regular Polytopes (3rd ed.), Dover Publications, p. 15, ISBN 0-486-61480-8
  2. ^ Akiyama, Jin; Matsunaga, Kiyoko (2020), "An Algorithm for Folding a Conway Tile into an Isotetrahedron or a Rectangle Dihedron", Journal of Information Processing, 28 (28): 750–758, doi:10.2197/ipsjjip.28.750, S2CID 230108666.
  3. ^ a b Whittaker, E. J. W. (2013), Crystallography: An Introduction for Earth Science (and other Solid State) Students, Elsevier, p. 89, ISBN 9781483285566.
  4. ^ Leech, John (1950), "Some properties of the isosceles tetrahedron", The Mathematical Gazette, 34 (310): 269–271, doi:10.2307/3611029, JSTOR 3611029, MR 0038667, S2CID 125145099.
  5. ^ Hajja, Mowaffaq; Walker, Peter (2001), "Equifacial tetrahedra", International Journal of Mathematical Education in Science and Technology, 32 (4): 501–508, doi:10.1080/00207390110038231, MR 1847966, S2CID 218495301.
  6. ^ Cite error: The named reference akiyama was invoked but never defined (see the help page).
  7. ^ Demaine, Erik; O'Rourke, Joseph (2007), Geometric Folding Algorithms, Cambridge University Press, p. 424, ISBN 978-0-521-71522-5.