Trirectangular tetrahedron

A trirectangular tetrahedron with its base shown in green and its apex as a solid black disk. It can be constructed by a coordinate octant and a plane crossing all 3 axes away from the origin (x>0; y>0; z>0) and x/a+y/b+z/c<1

In geometry, a trirectangular tetrahedron is a tetrahedron where all three face angles at one vertex are right angles. That vertex is called the right angle or apex of the trirectangular tetrahedron and the face opposite it is called the base. The three edges that meet at the right angle are called the legs and the perpendicular from the right angle to the base is called the altitude of the tetrahedron (analogous to the altitude of a triangle).

Kepler's drawing of a regular tetrahedron inscribed in a cube (on the left), and one of the four trirectangular tetrahedra that surround it (on the right), filling the cube.

An example of a trirectangular tetrahedron is a truncated solid figure near the corner of a cube or an octant at the origin of Euclidean space. Kepler discovered the relationship between the cube, regular tetrahedron and trirectangular tetrahedron.[1]

Only the bifurcating graph of the Affine Coxeter group has a Trirectangular tetrahedron fundamental domain.

  1. ^ Kepler 1619, p. 181.