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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 is not meant to be an exhaustive listing. It rather aims for some extremal ends: restricting to convex polytopes in general, and either to regular facets additionally or to non-isogonal results. – Note: this setup does not require the polytopes generally 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 (and thus are listed there).
It should be mentioned however, that already some results for non-convex deltahedra have been obtained. E.g. Rausenberger in 1915 and Cundy in 1952 were initiating a research which Olshevsky in 2006 brought to its avered end: the biform acoptic deltahedra (where acoptic refers to non-convex but still non-intersecting).
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