When Cundy proposed to enumerate nonconvex biform deltahedra (see The Cundy Deltahedra page), he noted that his table included "only those solids in which the triangles are totally on the outside". This excluded coptic deltahedra, or those with intersecting faces.
If this restriction is relaxed, there can be found other interesting polyhedra, but not so many for the enumeration to be unmanageable. Coincident vertex, edge and faces forms will continue to be excluded. This would exclude the Hexexcavated Cuboctahedron as it has six vertices coincident at the center and 24 edges coincident in twelve pairs. Also those with coplanar faces meeting at an edge will be excluded. This excluded Dodecaugmented Gid. But Dodecaugmented Sirsid is included because while some triangles are on the same plane they don't meet on edge.
This page will be devoted to finding these forms, hopefully in an orderly way. So far 67 distinct forms have been found, 16 of which are chiral. There are also four infinite series. If anyone finds any others not listed on this page please send them in.
First it is worth mentioning that there is only one coptic uniform deltahedron. The Great Icosahedron is isogonal which means it has one form of vertex. While polyhedra that have one form of vertex and regular faces are called uniform, those with two forms of vertices are sometimes termed biform.
Note that the terms gyrexcavated and gyraugmented indicate excavation or augmentation with octahedra.
Each figure in the following tables lists the symmetry (S) Dn  Dihedral, T  Tetrahedral, O  Octahedral, I  Icosahedral. The total Face, Edge and Vertex counts are given. A (C) after the name denotes that the polyhedron is chiral. An (I) after the name denotes that the polyhedron is isohedral (literally meaning all the faces are the same).
The following 16 models of biform deltahedra are those which would have been excluded from Cundy's table simply because they are coptic.
Cundy's Excluded Coptic Biform Deltahedra  





 
Cundy's Excluded Coptic Biform Deltahedra  




 
Cundy's Excluded Coptic Biform Deltahedra  


Excavating convex prisms and antiprisms would probably have been tried by Cundy but for all cases the results would have been coptic. There are only four cases. The triangular prism can be excavated by three square pyramids. The other cases are the triangular antiprism (a.k.a. the octahedron), the square antiprism, and pentagonal antiprism.
More Cundy's Excluded Coptic Biform Deltahedra
Triexcavated Triangular Prism S=D3 F=14 E=21 V=9 off wrl switch
Diexcavated Octahedron S=D3 F=12 E=18 V=8 off wrl switch
Diexcavated Square Antiprism S=D4 F=16 E=24 V=10 off wrl switch
Diexcavated Pentagonal Antiprism S=D5 F=20 E=30 V=12 off wrl switch
KeplerPoinsot Polyhedra can be augmented or excavated to create 18 biform deltahedra. In the Great Icosahedron (Gike), the triangles can be augmented or excavated with either the tetrahedron or octahedron. A pentagonal pyramid can be used in the case of the Great Dodecahedron (Gad). A pentagrammic pyramid can be used in the case of the Small Stellated Dodecahedron (Sissid) and Great Stellated Dodecahedron (Gissid).
Coptic Biform Deltahedra Generated from KeplerPoinsot Polyhedra
Tetraugmented Gike (C) S=T F=28 E=42 V=16 off wrl switch
Tetraexcavated Gike (C) S=T F=28 E=42 V=16 off wrl switch
Octaugmented Gike S=T F=36 E=54 V=20 off wrl switch
Octaexcavated Gike S=T F=36 E=54 V=20 off wrl switch
Dodecaugmented Gike S=T F=44 E=66 V=24 off wrl switch Coptic Biform Deltahedra Generated from KeplerPoinsot Polyhedra
Dodekexcavated Gike S=T F=44 E=66 V=24 off wrl switch
Icosaugmented Gike (I) S=I F=60 E=90 V=32 off wrl switch
Icosaexcavated Gike (I) S=I F=60 E=90 V=32 off wrl switch
Tetragyraugmented Gike (C) S=T F=44 E=66 V=24 off wrl switch
Tetragyrexcavated Gike (C) S=T F=44 E=66 V=24 off wrl switch Coptic Biform Deltahedra Generated from KeplerPoinsot Polyhedra
Octagyraugmented Gike S=T F=68 E=102 V=36 off wrl switch
Octagyrexcavated Gike S=T F=68 E=102 V=36 off wrl switch
Icosagyraugmented Gike S=I F=140 E=210 V=72 off wrl switch
Icosagyrexcavated Gike S=I F=140 E=210 V=72 off wrl switch
Dodekexcavated Gad (I) S=I F=60 E=90 V=32 off wrl switch Coptic Biform Deltahedra Generated from KeplerPoinsot Polyhedra
Dodecaugmented Sissid (I) S=I F=60 E=90 V=32 off wrl switch
Dodecaugmented Gissid (I) S=I F=60 E=90 V=32 off wrl switch
Dodekexcavated Gissid (I) S=I F=60 E=90 V=32 off wrl switch
The nonconvex snub uniform polyhedra can be augmented or excavated using pentagrammic pyramids. 10 biform deltahedra can be created this way. The Small Inverted Retrosnub Icosicosidodecahedron (Sirsid) yields the most complex higher symmetry biform deltahedra known with 160 faces.
Coptic Biform Deltahedra Generated from Uniform Nonconvex Snubs
Dodecaugmented Seside S=I F=160 E=240 V=82 off wrl switch
Dodekexcavated Seside S=I F=160 E=240 V=82 off wrl switch
Dodecaugmented Gosid (C) S=I F=140 E=210 V=72 off wrl switch
Dodekexcavated Gosid (C) S=I F=140 E=210 V=72 off wrl switch
Dodecaugmented Gisid (C) S=I F=140 E=210 V=72 off wrl switch Coptic Biform Deltahedra Generated from Uniform Nonconvex Snubs
Dodekexcavated Gisid (C) S=I F=140 E=210 V=72 off wrl switch
Dodecaugmented Girsid (C) S=I F=140 E=210 V=72 off wrl switch
Dodekexcavated Girsid (C) S=I F=140 E=210 V=72 off wrl switch
Dodecaugmented Sirsid S=I F=160 E=240 V=82 off wrl switch
Dodekexcavated Sirsid S=I F=160 E=240 V=82 off wrl switch
Other uniform polyhedra can be used to create biform deltahedra. This produces 9 more. The Stellated Truncated Hexahedron (Quith) is the only one generating those in octahedral symmetry. It is the only case in the uniform polyhedra which can be augmented with a octagrammic pyramid to yield biform deltahedra. Similarly the Great Stellated Truncated Dodecahedron (Quitgissid) is the only uniform polyhedron that can be augmented by the decagrammic pyramid.
Coptic Biform Deltahedra Generated from other Uniform Polyhedra
Octaugmented Quith S=O F=56 E=84 V=30 off wrl switch
Octaexcavated Quith S=O F=56 E=84 V=30 off wrl switch
Dodekexcavated Gid S=I F=80 E=120 V=42 off wrl switch
Dodecaugmented Gidtid S=I F=80 E=120 V=42 off wrl switch
Dodekexcavated Gidtid S=I F=80 E=120 V=42 off wrl switch Coptic Biform Deltahedra Generated from other Uniform Polyhedra
Dodecaugmented Sidtid S=I F=80 E=120 V=42 off wrl switch
Dodekexcavated Sidtid S=I F=80 E=120 V=42 off wrl switch
Dodecaugmented_Quitgissid S=I F=140 E=210 V=72 off wrl switch
Dodekexcavated_Quitgissid S=I F=140 E=210 V=72 off wrl switch
There are only a few exotic Coptic Biform Deltahedra which have been discovered so far. Two are the 3/2 Snub, and the Great 3/2 Snub Antiprisms  see here. Isomers of nonconvex forms of #24 and #25 on the The Cundy Deltahedra page have been discovered by George Olshevsky, Jim McNeill and myself. At first Tetraexcambiated_Icosahedron1 seems to look like Tetragyrexcavated Gike and the even have the same number of faces. But looking at the interiors confirms the difference.
Exotic Coptic Biform Deltahedra
SAP 32 S=D3 F=20 E=30 V=12 off wrl switch
Great SAP 32 S=D3 F=20 E=30 V=12 off wrl switch
Hexexcaspheniated Icosahedron1 S=T F=44 E=66 V=24 off wrl switch
Hexexcaspheniated Icosahedron2 S=T F=44 E=66 V=24 off wrl switch
Tetraexcambiated Icosahedron1 (C) S=T F=44 E=66 V=24 off wrl switch Exotic Coptic Biform Deltahedra
Tetraexcambiated Icosahedron2 (C) S=T F=44 E=66 V=24 off wrl switch
Tetraexcambiated Icosahedron3 (C) S=T F=44 E=66 V=24 off wrl switch
Tetraexcambiated Icosahedron4 (C) S=T F=44 E=66 V=24 off wrl switch
Nonconvex 48deltahedron1 S=T F=48 E=72 V=26 off wrl switch
Nonconvex 48deltahedron2 S=T F=48 E=72 V=26 off wrl switch
Finally there are four known infinite series of Coptic Biform Deltahedra. One is any dipyramid of n/m where m is greater than 1. These star dipyramids, as they are sometimes referred, are also isohedral. Pictured is a 7/3 Star Dipyramid. Two others are the diaugmented or diexcavated star antiprism of n/m where m is greater than 1. Pictured are a pair of 7/2 of this type. There is also an interesting form called 2 Unit Blended Antiprismatic Tower discribed on this page. The blended 7/3 and 7/4 antiprism is shown below. Blended antiprisms are actually toroids with genus = 1.
Infinite Series of Coptic Biform Deltahedra
Star Dipyramid (I) S=Dn F= E= V= off wrl switch
Diaugmented Star Antiprism S=Dn F= E= V= off wrl switch
Diexcavated Star Antiprism S=Dn F= E= V= off wrl switch
2 Unit Blended Antiprismatic Tower S=Dn F= E= V= off wrl switch
Question or comments about the web page should be directed to polyhedra@bigfoot.com.
Special thanks to Jim McNeill for finding the snub antiprism forms and information on the antiprism blends.
The deltahedra were created in Robert Webb's Stella application and Antiprism. The generation of OFF and VRML files were processed with Antiprism. The Hedron application by Jim McNeill was used to generate switch files. The image files were created with off2pov and MegaPOV (POVray).
History:
20120222 Removed Dodecagyrexcavated Icosahedron, Dodecagyraugmented Gike and Dodecagyrexcavated Gike as these were found to be triforms
20090926 Fixed link to Stella
20080208 Changed Bi to Di in keeping with greek prefixes
20080202 Quitsissid changed to Quitgissid
20080201 Added Hexexcaspheniated Icosahedron2
20080131 Added Tetraexcambiated Icosahedron4, Nonconvex 48deltahedron1 & 2, New name: 2 Unit Blended Antiprismatic Tower
20080130 Tetraexcambiated Icosahedron3
20080129 Added Dodecaugmented Quitgissid, Dodekexcavated Quitgissid, Hexexcaspheniated Icosahedron1, Tetraexcambiated Icosahedron1 & 2
20080128 Changed "Nonconvex Nonacoptic" to simply "Coptic". Added Diexcavated Square Antiprism, and Star Dipyramids
20080127 Initial Release
20080126 Added the exotic and infinite sets. Added two prsimatic forms in the Cundy exclusion table
20080124 Added the Uniform Nonconvex Snubs and others that can be generated from Uniform Polyhedra
20080123 Added KeplerPoinsot forms. Added missing D3 Cundy excluded forms found by Jim McNeill
20080121 Alpha