Isohedral Deltahedra
This page contains an exploration of the possibilities for isohedral deltahedra, i.e. polyhedra which are face transitive and consist of equilateral triangular faces. A total of 34 isohedral deltahedra and one infinite family of deltahedra are presented. A systematic treatment of isohedral deltahedra can be found in Shephard [1], to which cross references are provided, there are 6 further examples on this page that are not included in Shephard.
Deltahedra from Uniform Polyhedra
Four isohedral deltahedra appear amongst the platonic and KeplerPoinsot polyhedra
Tetrahedron Shephard:T2(2) 
Octahedron Shephard:O7 
Icosahedron Shephard:I11(1) 
Great Icosahedron Shephard:I10(1) 
Deltahedra From Johnson Solids
Two further isohedral deltahedra appear amongst the Johnson Solids:
Triangular Bipyramid Shephard:D3(1) 
Pentagonal Bipyramid Shephard:D5(1) 
Extending these figures to consider reentant polyhedra, the family of {n/d}bipyramids consists of isohedral deltahedra within the range 2 < n/d < 6. As examples, see this {7/2}bipyramid and {7/3}bipyramid. In Shephard's notation, these are case 1, Dn(d)
Deltahedra from Augmented Uniform Polyhedra
Each of the platonic and KeplerPoinsot polyhedra can potentially be augmented or excavated with pyramids. A number of the figures are degenerate. Overall 15 isohedral deltahedra are formed.
Tetrahedron  Augmented Triakis Tetrahedron Shephard:T2(1) 
degenerate see Tetrahedron 
Octahedron
Triakis Octahedra 
Augmented Stellar Octangula Shephard: O6(1) 
Excavated Shephard: O6(2)

Cube
Tetrakis Hexahedra 
Augmented Shephard: T1(1) & O5(1) 
Excavated Shephard: T1(2) & O5(2) 
Icosahedron
Triakis Icosahedra 
Augmented Shephard: I6(1) 
Excavated Shephard: I6(2) 
Dodecahedron
Pentakis Dodecahedra 
Augmented Shephard: I9(1) 
Excavated Shephard: I9(2) 
Great
Icosahedron
Triakis Great Icosahedra 
Augmented Shephard: I7(2) 
Excavated Shephard: I7(1) 
Great
Dodecahedron

degenerate see Icosahedron 
Excavated Pentakis Great Dodecahedron Shephard: I11(2) 
Small Stellated Dodecahedron  Augmented Triakis Small Stellated Dodecahedron Shephard: I10(2) 
degenerate see Great Icosahedron 
Great
Stellated Dodecahedron
Pentakis Great Stellated Dodecahedra 
Augmented ** Shephard: I8(1) 
Excavated Shephard: I8(2) 
** The deltahedron derived from augmenting the great stellated dodecahedron superficially resembles great icosahedron but each icosahedral face contains three coplanar triangles.
A number of the above deltahedra are also Cundy deltahedra, being concave acoptic (non selfintersecting) deltahedra with two kinds of polyhedron vertices. These have been enumerated by Roger Kaufman, who has also investigated their coptic (selfintersecting) variants. All of the above polyhedra fall into one of these two classes. The remaining deltahedra listed below all have three types of vertex.
Deltahedra from Deformed Augmented Duals of Tetravalent Semiregular Polyhedra  and the Möbius Deltahedra
Apart from the cube, which is discussed above, there are four other rhombic polyhedra which have the property of being edgetransitive. These are all duals of tetravalent semiregular polyhedra. These can each be augmented or excavated with pyramids to form nonregular deltahedra. These can all then be deformed such that the faces become equilateral triangles. Face transitivity is retained in all cases. The 8 isohedral deltahedra formed in this fashion are listed below, four of which are not included in Shephard.
Rhombic Dodecahedron  Augmented Octakis Hexahedron Möbius 48A Shephard: O2(2) 
Excavated Hexakis Octahedron Möbius 48B Shephard: O2(1) 
Rhombic Triacontahedron  Augmented Hexakis Icosahedron Möbius 120A Shephard: I1(2) 
Excavated 10akis Dodecahedron Möbius 120B Shephard: I1(1) 
Medial Rhombic Triacontahedron  Augmented Vert Shephard: not present see [1] 
Excavated Vert Shephard: not present see [2] 
Great Rhombic Triacontahedron  Augmented Shephard: not present see [3] 
Excavated Shephard: not present see [4] 
The text 'Vert' next to the models of the medial rhombic triacontahedron leads to VRML files with the vertices highlighted. Switching to frame mode and zooming into the centre of the polyhedra shows the configurations of the inner vertices.
There is a set of five isohedral deltahedral figures known as Möbius Deltahedra. Melinda Green discusses these figures at http://www.superliminal.com/geometry/deltahedra/deltahedra.htm and Roger Kaufman at http://www.interocitors.com/polyhedra/Deltahedra/Mobius/index.html. Four of these figures can be obtained by augmenting or excavating the rhombic dodecahedron and rhombic triacontahedron as above. The fifth is a 24 faced figure isomorphous to the augmented cube (image below), it is not derived from any rhombic figure. The remaining 4 deltahedra are derived from the medial rhombic triacontahedron and great rhombic triacontahedron.
Mobius 24A
Shephard: T1(3)
Deltahedra from Deformed Augmented Duals of Hexavalent Semiregular Polyhedra
The semiregular ditrigonary polyhedra  with vertex figures of (m,n,m,n,m,n)  have duals which consist of hexagonal faces and are also edge transitive. The above process of augmentation/excavation and deltification can also be applied to these figures. Some of the resulting figures are degenerate. The 3 further deltahedra that result are included in the table below.
Uniform Polyhedron  
Ditrigonary
Dodecadodecahedron
(Dual) 
Augmented Shephard: I5(1) 
Excavated Vert Shephard: I5(2) 
Small
Ditrigonary Icosidodecahedron
(Dual) 
Degenerate see Augmented Dodecahedron 
Excavated Shephard: I9(3) 
Great
Ditrigonary Icosidodecahedron
(Dual) 
Degenerate see Augmented Great Stellated Dodecahedron 
Degenerate see Excavated Great Stellated Dodecahedron 
Deltahedra from Deformed Duals of Trivalent Uniform Polyhedra
A number of the uniform polyhedra are trivalent, i.e. they have three faces meeting at each vertex. If the platonic and KeplerPoinsot polyhedra are excluded (as they are considered above), the remainder have duals with nonregular triangular faces. These duals can be deformed such that the faces become equilateral triangles. Face transitivity is retained in all cases. The 1 new isohedral deltahedron that results is included in the table below.
Notes:
* The deltahedra derived from excavating the medial rhombic triacontahedron and dual of the ditrigonary dodecadodecahedron duals are similar in appearance. They are distinct and have different nets. The distinction is best seen by following the above links to 'Vert', switching to 'frame' view and zooming on the centres of the polyhedra.
Further Examples not in Shephard
The above deltahedron is noted by Shephard as case I7(3). It was not generated by any of the methods described above.
In February 2008, Adrian Rossiter conducted an exercise using his Antiprism software which effectively performed a systematic repeat of Shephard's calculations . Adrian discovered all of the above figures (including those absent from Shephard) plus one additional figure, which in Shephard fashion he denotes I3(4) (see [6]).
Examples not included in Shephard
Shephard style notation for these figures:
No.  Symbol  Vectors (t = tau)  Constants 
[1] I2(1)  [1/2, 2/5, 4/5]  a = (t, t^{2}, 1) / 2t b = (1, 0 t) / sqrt(t).5^{1/4} c = (0, t, 1) / sqrt(t).5^{1/4} 
alpha = 1.035310786 beta = 1.017991959 gamma = 0.041794129 
[2] I2(2)  [1/2, 3/5, 4/5]  a = (t, t^{2}, 1) / 2t b = (1, 0 t) / sqrt(t).5^{1/4} c = (0, t, 1) / sqrt(t).5^{1/4} 
alpha = 0.06959024 beta = 0.961660566 gamma = 0.940133523 
[3] I3(1)  [1/2, 1/3, 2/5]  a = (t, t^{2}, 1) / 2t b = (1, 1, 1) / sqrt(3) c = (0, t, 1) / sqrt(t).5^{1/4} 
alpha = 1.067540005 beta = 0.454812301 gamma = 0.979984199 
[4] I3(2)  [1/2, 2/3, 3/5]  a = (t, t^{2}, 1) / 2t b = (1, 1, 1) / sqrt(3) c = (0, t, 1) / sqrt(t).5^{1/4} 
alpha = 0.365481362 beta = 0.809498076 gamma = 0.758298682 
[5] I3(3)  [1/2, 1/3, 2/5]  a = (t, t^{2}, 1) / 2t b = (1, 1, 1) / sqrt(3) c = (0, t, 1) / sqrt(t).5^{1/4} 
alpha = 1.03739492 beta = 0.123497623 gamma = 1.015782486 
[6] I3(4)  [1/2, 2/3, 2/5]  a = (t, t^{2}, 1) / 2t b = (1, 1, 1) / sqrt(3) c = (0, t, 1) / sqrt(t).5^{1/4} 
alpha = 0.840436103 beta = 0.919239757 gamma = 0.257365255 
The Excel spreadsheet linked here contains a full suite of data in the above style for all isohedral deltahedra.
References
[1] Shephard G.C. (1999) Isohedral Deltahedra, Periodica Mathematica Hungarica Vol. 39 (13), 83106
Credits
I am grateful to Professor G.C. Shephard for kindly providing a copy of his paper [1] above, and to Mason Green for inspiring this page by suggesting that the list of isohedral deltahedra, consisting of the top three sections above plus the Möbius Deltahedra, may not be complete.
I am also grateful to Adrian Rossiter and Roger Kaufman for an enlightening exchange of correspondence leading up to and during Adrian's recent exercise (Feb 2008)
A number of the original dual figures were produced using Great Stella and relaxed using HEDRON.