# Talk:Disphenoid

## References

The Coxeter book is upstairs now. I will add a detailed citation tomorrow. -- Dominus 04:37, 17 December 2006 (UTC)

## Digonal disphenoid

A digonal disphenoid inside a general cuboid.

It says a disphenoid is a tetrahedron with 4 identical isosceles triangle faces. I'm wondering if there's a name for a tetrahedron with two sets of isosceles triangle faces, instead of fitting within a square cuboid, rather a general cuboid of all different lengths, so 4 edges are equal length and the remaining two edges can be different lengths. SockPuppetForTomruen (talk) 19:58, 2 February 2009 (UTC)

It looks like this shape is called a digonal disphenoid, as described in the vertex figures of the bitruncated tesseract and bitruncated_120-cell.
NOTE: If this is correct, it would contradict the definition here which says a disphenoid has 4 identical triangles. SockPuppetForTomruen (talk) 22:04, 2 February 2009 (UTC)
This contradicts the definition of disphenoid. This should not be called a special case of a disphenoid. If a disphenoid is defined to have 4 congruent faces, it makes no sense to say that a special case is one with two pairs of congruent faces which are not all congruent. Other characterizations of the disphenoid contradict the inclusion of "digonal disphenoids" as disphenoids. I was going to cite this page but I can't because of this error.50.177.1.160 (talk) 21:04, 13 December 2013 (UTC)

Definitions in relation to their symmetry:

A sphenoid tetrahedron has a single mirror symmetry (*11), , Cantitruncated 8-cell vertex figure
A (digonal) disphenoid tetrahedron has two normal mirror symmetry (*222), , Bitruncated tesseract vertex figure
A (tetragonal) disphenoid tetrahedron has 2 normal mirrors and 2-fold rotation symmetry (2*2), , Bitruncated 5-cell vertex figure

## rhombic disphenoid

It seems like a rhombic disphenoid would fit inside 4 vertices of a general cuboid, with its 6 edges being diagonals of the cuboid faces. If the cuboid had 3 different edge lengths, then you'd have scalene triangle faces in the disphenoid. I'll make a picture if I confirm this is true. SockPuppetForTomruen (talk) 22:05, 4 February 2009 (UTC)

(222) symmetry
Here's a not so great picture showing the rhombic dispenoid, apparently with exactly 3 sets of edge lengths, in upposite pairs. I labeled them with ABC, being the lengths of te cuboid sides, and AB implying length sqrt(A*A+B*B).SockPuppetForTomruen (talk) 23:14, 4 February 2009 (UTC)

Here's another attempt to show the varied forms, this time as 2D diagrams, edges colored by equal edge lengths. SockPuppetForTomruen (talk) 21:11, 5 February 2009 (UTC)