A polyhedron is truncated regular if it is vertex-transitive with isosceles
triangular vertex figures. Vertex transitivity means that for any two
vertices of the polyhedron, there exists a translation, rotation, and/or
reflection that leaves the outward appearance of the polyhedron unchanged
yet moves one vertex to the other. A vertex figure is the polygon produced
by connecting the midpoints of the edges meeting at the vertex in the same
order that the edges appear around the vertex. There are five truncated
regular polyhedra that are not self-intersecting, namely the
Truncated Tetrahedron, the
Truncated Octahedron, the
Truncated Cube, the
Truncated Icosahedron, and the
Truncated Dodecahedron.
When self-intersection is allowed, there are five other truncated regular
polyhedra.

Johann Pitsch described all five of these polyhedra in 1881
[2].
Albert Badoureau described the Stellated Truncated Hexahedron and
the Small Stellated Truncated Dodecahedron in 1881
[1].

Jean Paul Albert Badoureau, Mémoire sur les Figures Isocèles, Journal de l'École polytechnique49 (1881), 47-172.

[2]

Johann Pitsch, Über Halbreguläre Sternpolyeder, Zeitschrift für das Realschulwesen6 (1881), 9-24, 64-65, 72-89, 216.

[3]

H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller, Uniform Polyhedra, Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences246 (1954), 401-450.