Coptic Biform Deltahedra
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. 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).
Links below display various models
- Clicking on pictures opens an Interactive OFF Viewer
- OFF files open in an online browser. Downloaded files may be viewed with an application such as Antiprism
- STEL files may be saved and opened with Stella
- VRML files (wrl,switch,cyl) open in an online browser. Downloaded files may be viewed with a VRML browser such as FreeWRL
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 |
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| Cundy's Excluded Coptic Biform Deltahedra |
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| Cundy's Excluded Coptic Biform Deltahedra |
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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 |
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Kepler-Poinsot 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 Kepler-Poinsot Polyhedra |
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| Coptic Biform Deltahedra Generated from Kepler-Poinsot Polyhedra |
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| Coptic Biform Deltahedra Generated from Kepler-Poinsot Polyhedra |
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| Coptic Biform Deltahedra Generated from Kepler-Poinsot Polyhedra |
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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 |
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| Coptic Biform Deltahedra Generated from Uniform Nonconvex Snubs |
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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 |
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| Coptic Biform Deltahedra Generated from other Uniform Polyhedra |
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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 Tetra-excambiated_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 |
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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 di-augmented or di-excavated 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 |
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Question or comments about the web page should be directed to PolyhedraSmith@gmail.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
POV-ray.
History:
2024-06-17 Use base.css
2024-06-13 Switch to Multi OFF Viewer
2023-09-04 Switch to Simple OFF Viewer and X_Ite VRML Viewer
2019-03-12 Changed email address from defunct bigfoot.com
2012-02-22 Removed Dodecagyrexcavated Icosahedron, Dodecagyraugmented Gike and Dodecagyrexcavated Gike as these were found to be tri-forms
2009-09-26 Fixed link to Stella
2008-02-08 Changed Bi- to Di- in keeping with greek prefixes
2008-02-02 Quitsissid changed to Quitgissid
2008-02-01 Added Hexexcaspheniated Icosahedron2
2008-01-31 Added Tetra-excambiated Icosahedron4, Nonconvex 48-deltahedron1 & 2, New name: 2 Unit Blended Antiprismatic Tower
2008-01-30 Tetra-excambiated Icosahedron3
2008-01-29 Added Dodecaugmented Quitgissid, Dodekexcavated Quitgissid, Hexexcaspheniated Icosahedron1, Tetra-excambiated Icosahedron1 & 2
2008-01-28 Changed "Nonconvex Nonacoptic" to simply "Coptic". Added Di-excavated Square Antiprism, and Star Dipyramids
2008-01-27 Initial Release
2008-01-26 Added the exotic and infinite sets. Added two prsimatic forms in the Cundy exclusion table
2008-01-24 Added the Uniform Nonconvex Snubs and others that can be generated from Uniform Polyhedra
2008-01-23 Added Kepler-Poinsot forms. Added missing D3 Cundy excluded forms found by Jim McNeill
2008-01-21 Alpha
Back to the main Polyhedron Page.
Link to this page as http://www.interocitors.com/polyhedra/Deltahedra/Cundy/Coptic
