A deltahedron is a polyhedron (formally, a finite abstract polytope of rank 3 embedded in R^3 such that no two vertices map to the same point) whose faces are all equilateral triangles. An elementary deltahedron is a deltahedron that cannot be formed from two smaller deltahedra via the process of blending or augmentation; that is, placing one deltahedron next to another such that one triangular face overlaps, and removing said triangular face as well as identifying the three overlapping edges and vertices. I exclude the triangular dihedron, which can be considered the identity of the blending operation.
Here is a list of small non-self-intersecting elementary deltahedra by face count.
Tetrahedron - convex, regular
Octahedron - convex, regular
Pentagonal bipyramid - convex, isohedral, Johnson (J13)
Snub disphenoid - convex, Johnson (J84)
Triaugmented triangular prism - convex, Johnson (J51)
Crownacorn polyhedron - 2 concave ridges
Hexagonal sphenohedron - 2 concave ridges
Gyroelongated square bipyramid - Johnson (J17)
Tapeenra polyhedron - 1 concave ridge
Chord of icosahedron - 3 concave ridges
Extended hexagonal sphenohedron - 2 concave ridges
Hemi-tapeenra polyhedron - 4 concave ridges
Trapezoidally subdivided sphenocorona - This one is not concave, but not strictly convex because it has 4 flat ridges.
Inverted trapezoidally subdivided sphenocorona - 4 concave ridges. This is an alternative realization of the trapezoidally subdivided sphenocorona.
Sliverette tapeenra polyhedron - 4 concave ridges. Magnet model.
Inverted tapeenra polyhedron - 4 concave ridges. This is an alternative realization of the sliverette tapeenra polyhedron.
18-4 - This one is completely asymmetrical.
18-5 - 2 concave ridges
Augmented semi-extended hexagonal sphenohedron - 4 concave ridges
18-10 - 6 concave ridges. This one has triangular prismatic symmetry and looks really cool.
