The Triakis Tetrahedron
The triakis tetrahedron is a 3D Catalan solid bounded by 12 isosceles triangles, 18 edges (6 long edges, 12 short edges), and 8 vertices. It is the dual of the truncated tetrahedron.
The triakis in the name means to erect pyramids on triangular faces; triakis tetrahedron means the polyhedron obtained by erecting pyramids on the triangular faces of the regular tetrahedron.
The triangular faces are transitive, consisting of 2 short edges and 1 long edge, with a 3 : 5 length ratio. The 8 vertices are of two kinds: 4 order6 vertices corresponding to an inscribed tetrahedron, and 4 order3 vertices corresponding to the apices of the triangular pyramids erected on this inscribed tetrahedron.
The triakis tetrahedron is the envelope of the vertexfirst projection of the joined pentachoron, a 4D Catalan polychoron, into 3D.
Projections
The following are images of the triakis tetrahedron from various viewpoints:
Projection  Description 

Parallel projection centered on top vertex. 

Side view looking at vertex with short edges. 

Front view looking at long edge. 

Oblique side view with many coinciding edges. 
Animation
Here's an animation of a triakis tetrahedron rotating around the vertical axis:
Coordinates
The Cartesian coordinates for the triakis tetrahedron, obtained by inverting a truncated tetrahedron centered on the origin with edge length 2, are:
 (2/√6, 0, 0)
 (−6/(5√6), 0, 0)
 (2/(5√6), −4/(5√3), 0)
 (−2/(3√6), 4/(3√3), 0)
 (2/(5√6), 2/(5√3), ±2/5)
 (−2/(3√6), −2/(3√3), ±2/3)
These coordinates give an “upright” triakis tetrahedron, in the sense that the inscribed tetrahedron outlined by its long edges has one face parallel to the horizon.
Algebraically nicer
coordinates may be obtained in a different
orientation by constructing the triakis tetrahedron from the truncated
tetrahedron as the truncated alternation of the cube. The resulting coordinates
are all even changes of sign of:
 (1, 1, −1)
 (5/3, 5/3, 5/3)
Credits
The coordinates derived from the truncated alternation of the cube are obtained from the Wikipedia article on the triakis tetrahedron.