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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 **homohedral polytopes**, being defined as having only a single shape for facets.
(Note that *isohedral* then would add that those are required to be symmetry equivalent as well – which here is not required.
However obviously any isohedral polytope would also be homohedral.)

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 iso*gonal* in addition, i.e. having just a single type
of vertices (vertex surroundings / vertex figures), i.e. neither by equivalence nor be symmetry.
Polytopes which are both, isogonal *and* isohedral, commonly are known under the
attribute noble (and thus would be listed there).

As an aside it should be mentioned however, that already some results for

non-convexdeltahedra 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 (whereacopticrefers to non-convex but still non-intersecting).The right animated non-convex polyhedra then would be a triform acoptic deltahedron with 36 faces and a triform acoptic deltahedron with 120 faces respectively.

For all simplicial polytopes in general the set of Dehn-Summerville equations can be stated.

## ---- 2D ----

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

- 6 {4} → cube cube (regular)

- 12 {5} → doe dodecahedron (regular)

## ---- 3D ----

- 5 tet → pen pentachoron (regular)
- 8 tet → tete tetrahedral bipyramid (Blind polytope)
- 16 tet → hex hexadecachoron (regular)
- 40 tet → ite icosahedral bipyramid (Blind polytope)
- 600 tet → ex hexacosachoron (regular)

## ---- 4D ----

- 6 pen → hix hexateron (regular)
- 10 pen → penit pentachoral bipyramid (Blind polytope)
- 32 pen → tac triacontiditeron (regular)

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