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 Möbius 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 Möbius 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.


Mobius Deltahedron '24a'

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.

No Vertex Name ...................................
07 6,6,3 Truncated Tetrahedron see Augmented Tetrahedron
13  6,6,4 Truncated Octahedron see Augmented Cube
14 8,8,3  Truncated Cube see Augmented Octahedron
16 8,6,4 Truncated Cuboctahedron see Deltified Augmented Rhombic Dodecahedron
21 8/3,6,8  Cubitruncated Cuboctahedron degenerate
see
Excavated Octahedron    
24 8/3,8/3,3 Stellatruncated Cube degenerate
see
Augmented Octahedron    
25 6,4,8/3 Great Truncated Cuboctahedron see Deltified Augmented Rhombic Dodecahedron
30 6,6,5 Truncated Icosahedron see Augmented Dodecahedron
31 10,10,3 Truncated Dodecahedron see Augmented Icosahedron
33 10,6,4 Truncated Icosidodecahedron see Deltified Augmented Rhombic Triacontahedron
42 10,10,5/2  Great Truncated Dodecahedron  see Augmented Small Stellated Dodecahedron
50 10,6,10/3  Icositruncated Dodecadodecahedron  see Deltified Excavated Ditrigonary Dodecadodecahedron Dual *
60 6,6,5/2  Great Truncated Icosahedron  (Dual)
Deltified Dual
**
63 10/3,10/3,5  Small Stellatruncated Dodecahedron see Excavated Icosahedron
64 10,10/3,4  Stellatruncated Dodecadodecahedron see Deltified Excavated Medial Rhombic Triacontahedron  *
71 3,10/3,10/3 Great Stellatruncated Dodecahedron see Excavated Great Icosahedron
73 6,10/3,4 Stellatruncated Icosidodecahedron  (Dual)
Deltified Dual

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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 Möbius Deltahedra, may not be complete.

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

 

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