Deltahedra et al.

Convex polyhedra using only equilateral triangles for faces are called deltahedra. (Those are known such because of the shape of that upper-case Greek letter.) This kind of restriction could be considered a bit more generally, as well as for higher dimensions. The most general set-up there would be that of isohedral polytopes, being defined as having only a single shape for facets.

The list given here-below restricts to regular facets additionally.

Note: this setup does not require the polytopes to be isogonal in addition, i.e. having just a single type of vertices (vertex surroundings / vertex figures). Polytopes which are both, isogonal and isohedral, commonly are known under the attribute noble.

Facets being {3} (deltahedra)

•   4 {3} → tet tetrahedron (regular)
•   6 {3} → tridpy trigonal dipyramid (Johnson solid)
•   8 {3} → oct octahedron (regular)
• 10 {3} → pedpy pentagonal dipyramid (Johnson solid)
• 12 {3} → snadow snub disphenoid (Johnson solid)
• 14 {3} → tautip triaugmented trigonal prism (Johnson solid)
• 16 {3} → gyesqidpy gyroelongated square dipyramid (Johnson solid)
• 20 {3} → ike icosahedron (regular)

Facets being {4}

•   6 {4} → cube cube (regular)

Facets being {5}

• 12 {5} → doe dodecahedron (regular)

Facets being tetrahedra (tetrahedrochora)

•     5 tet → pen pentachoron (regular)
•     8 tet → tedpy tetrahedral dipyramid (Blind polytope)
•   16 tet → hex hexadecachoron (regular)
•   40 tet → ikedpy icosahedral dipyramid (Blind polytope)
• 600 tet → ex hexacosachoron (regular)

Facets being cubes

• 8 cube → tes tesseract (regular)

Facets being octahedra

• 24 oct → ico icositetrachoron (regular)

Facets being dodecahedra

• 120 doe → hi hecatonicosachoron (regular)

Facets being pentachora (pentachorotera)

•   6 pen → hix hexateron (regular)
• 10 pen → pendpy pentachoral dipyramid (Blind polytope)
• 32 pen → tac triacontiditeron (regular)

Facets being tesseracts

• 10 tes → pent penteract (regular)