__Isogonal Deltahedra__

This page contains an exploration of the possibilities for isogonal deltahedra, i.e. polyhedra which are face transitive and consist of equilateral triangular faces. A total of 33 isogonal deltahedra are presented, of which 10 are thought to be new discoveries.

__Deltahedra from Uniform Polyhedra__

Four isogonal deltahedra appear amongst the platonic and Kepler-Poinsot polyhedra

Tetrahedron |
Octahedron |

Icosahedron |
Great Icosahedron |

__Deltahedra From Johnson Solids__

Two further deltahedra appear amongst the Johnson Solids:

Triangular Bipyramid |
Pentagonal Bipyramid |

Extending these figures to consider re-entant polyhedra, the family of {n/d}-bipyramids consists of isogonal deltahedra within the range 2 < n/d < 6. As examples, see this {7/2}-bipyramid and {7/3}-bipyramid.

__Deltahedra from Augmented Uniform Polyhedra__

Each of the platonic and Kepler-Poinsot polyhedra
can potentially be augmented or excavated with pyramids. A number of the figures are degenerate. Overall **
12** isogonal deltahedra are formed.

Tetrahedron | Augmented |
degenerate see Tetrahedron |

Octahedron | Augmented (Stellar Octangula) |
Excavated |

Cube | Augmented |
Excavated |

Icosahedron | Augmented |
Excavated |

Dodecahedron | Augmented |
Excavated |

Great Icosahedron | Augmented |
Excavated |

Great Dodecahedron | degenerate see Icosahedron |
see Excavated Icosahedron |

Small Stellated Dodecahedron | Augmented |
degenerate see Great Icosahedron |

Great Stellated Dodecahedron | degenerate see Great Icosahedron |
see Excavated Icosahedron |

__Deltahedra from Deformed Augmented Duals of
Tetra-valent Semi-regular Polyhedra - and the ____M°bius
Deltahedra__

Apart from the cube,
which is discussed above, there are four other rhombic
polyhedra which have the property of being edge-transitive. These are
all duals of tetra-valent semi-regular polyhedra. These can each be
augmented or excavated with pyramids to form non-regular deltahedra. These
can all then be deformed such that the faces become equilateral triangles.
Face transitivity is retained in all cases. The ** 8** isogonal
deltahedra formed in this fashion are listed below.

Rhombic Dodecahedron | Augmented |
Excavated |

Rhombic Triacontahedron | Augmented |
Excavated |

Medial Rhombic Triacontahedron | Augmented Vert |
Excavated Vert |

Great Rhombic Triacontahedron | Augmented |
Excavated |

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 isogonal deltahedral
figures known as M°bius Deltahedra. Melinda Green discusses these figures
at http://www.superliminal.com/geometry/deltahedra/deltahedra.htm.
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 derived from the medial
rhombic triacontahedron and great
rhombic triacontahedron are thought to be new.

__Deltahedra from Deformed Augmented Duals of
Hexa-valent Semi-regular Polyhedra__

The semi-regular 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 **4** new deltahedra that result are included in the
table below.

Uniform Polyhedron | ||

Ditrigonary
Dodecadodecahedron
(Dual) |
Augmented |
Excavated Vert |

Small
Ditrigonary Icosidodecahedron
(Dual) |
Degenerate see Augmented Dodecahedron |
Excavated |

Great
Ditrigonary Icosidodecahedron
(Dual) |
Degenerate see Great Icosahedron |
Excavated (partially degenerate) |

__Deltahedra from Deformed Duals of Trivalent
Uniform Polyhedra__

A number of the uniform
polyhedra are tri-valent, i.e. they have three faces meeting at each
vertex. If the platonic and Kepler-Poinsot polyhedra are excluded (as they
are considered above), the remainder have duals with non-regular triangular faces.
These duals can be deformed such that the faces become equilateral
triangles. Face transitivity is retained in all cases. The **2**
new deltahedra that result are included in the table below.

Notes:

* The deltahedra derived from the stellatruncated dodecahedron and icositruncated 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.

** The deltahedron derived from the great truncated icosahedron dual superficially resembles great icosahedron but each icosahedral face contains three coplanar triangles.

__Credits__

I am grateful to Mason Green for inspiring this page by suggesting that the list of isogonal deltahedra, consisting of the top three sections above plus the M°bius Deltahedra, may not be complete.

A number of the original dual figures were produced using Great Stella and relaxed using HEDRON.