Arruated Polyhedra


           arruated icosahedron (4,2)-exo

Inspired by Dr Richard Klitzing's cuned twisters[1, Fn 1], I have explored further polyhedra using an icosahedron as the seed polyhedron. 

One method to generate the desired polyhedron is as follows:
1. Augment faces of the seed polyhedron with pyramids;
2. Join the peaks of egde adjacent pyramids with pairs of triangles; 'wedges' (effectively filling the valleys with distorted tetrahedra);
3. Relax the resulting polyhedron so that all faces are regular triangles.[Fn 2]

The effect is to place one of the above wedges over each edge of the original polyhedron. 

When George Olshevsky first described the above process in 2006[2], he used the term 'spheniated' [Fn 3, Fn 4].  Examples of Olshevsky's polyhedra can be found on Roger Kaufman's site[3].  Mason Green's cingulated antiprisms[4], discovered in 2005 turn out to be spheniated prisms but Green used a different method of generation. Green found these by dividing an n-gonal anti-prism into two equal sections each containing an n-gonal prismatic cap and its edge adjacent triangles, and inserting a 'cingulum' of 4n triangles. They were rediscovered in 2025 by discord user 'Harsin Sinquin' who used the approach of attaching six triangle complexes over triangle pairs of the antiprisms, and coming up with the term "cune".  Discussion between Klitzing and Sinquin led to the cuned twisters. 

Both Olshevsky and Klitzing's polyhedra only consider applying the spheniation/cuning process to a subset of the edges of the original polyhedron and some of the pyramidical faces and original faces of the seed polyhedron remain.  Because of these remaining faces, relaxation of the wedge faces to equilateral triangles involves a 'twist' of the remaining faces.

I decided to explore adding pyramids and wedges to all edges. This means all the original faces of the seed polyhedron are augmented and the resulting pyramidical faces have been also  been replaced.  This is a generalisation of 'spheniation/cuning' and as such needed a new term.  I felt a suitable verb would be 'arruo' (to cover with earth or to bury) hence we get the adjective 'arruated' .  The polyhedra consist only of the faces of the wedges.  This means that icosahedral symmetry is preserved and the 'twist' referred to above is replaced by the freedom to adjust the height of the pyramids to the point where an equilibrium is obtained between the edge height of the pyramids and the distance apart of two edge adjacent pyramids.

In similar fashion to the generation of the Edge Expanded Bi-Prisms and Bi-Antiprisms, it is possible to insert more than one wedge between each peak in step 2 above.  I term the group of wedges between two edge adjacent peaks a 'set'.  With more than one wedge in a set then different configurations can be obtained.  Again it is the height of the pyramids that is adjusted to obtain an equilibrium.  Solutions can also be obtained where the pyramids have a negative height, i.e. they are inverted. 

The naming convention is as follows:

- 'n' is the number of wedges in each set.
- 'exo' means the edge angle between the (now virtual) pyramids is less than 180 degrees, 'endo' means the edge angle is reflexive. 
- ''w' is the winding number, the number of times that the set of triangles winds around an axis set orthogonal to to plane containing the peaks of the edge adjacent pyramids and the origin.  The direction of winding is always outwards from the origin.  High winding numbers (> n/2) can give the appearance of being retrograde.
- The name is then given as (n,w)-exo/endo.

In generating the polyhedra below, I used a standard colouring format.

- If sets consist of one or two wedges they are coloured in yellow.
- For 3 or 4 wedges per set, the outermost ones are yellow, the inner ones orange.
- For 5 wedges per set the colours are as above but the central one is red.
- One set of wedges is highlighed by having the above colours replaced with green, cyan and blue respectively.  In the VRML files linked below, these highlighted wedges stay solid through the 'trans/frame/solid' view options. 
- This example has 5 wedges per set and shown before the relaxation step.

Below, with notes, is a list of those generated to date with n ≤ 5.  I make no claim that the list is complete, and I would welcome any further cases.  The degenerate cases may be specific to the seed polyhedron being the icosahedron. 

(1,0)-exo  These are equivalent to replacing the original polyhedra with its dual and triangulating the faces.
(1-0)-endo   The pyramids in this case do point outwards from the base icosahedron but are so low that the angle between them is > 180 degrees.

(2,0)-endo
(2,1)-exo

(3,0)-endo  This is a deltahedron.  All triangular faces are fully visible.
(3,1)-endo
(3,2)-exo

(4,0)-endo
(4,1)-exo
(4,1)-endo
(4,2)-exo
(4.2)-endo
(4,3)-exo - this is degenerate, the four wedges form faces of a pentagonal bipyramid (with one open pair) and then sets of wedges co-incide in pairs.

(5,0)-endo - this is degenerate, the five wedges form faces of a pentagonal bipyramid and the inner vertices are coincident
(5,1)-exo
(5,1)-endo
(5.2)-exo 
(5,3)-exo
(5,4)-exo

Other seed polyhedra

Numerous polyhedra can act as the seed for this process and I will only scratch the surface here.

Each n-goal face of the seed gets augmented with an n-gonal pyramid, sets of wedges are then inserted as above.

(4,0)-endo cases (coloured yellow and orange as above) are presented for the tetrahedron, octahedron (degenerate), cube and dodecahedron (the last two are deltahedra). 

As the (1,0) cases involve the dual of the seed, I also tried starting with a dual.  These are the (4,0)-endo cases for the rhombic dodecahedron (seed) and the  rhombic triacontahedron (seed) .  The red faces are adjacent to the 4 or 5-fold vertices, the hidden yellow faces are adjacent to the 3-fold vertices, central wedges are interpolated between them.  Here also is an n=4 case for the rhombic enneacontahedron (seed), wedges adjacent to 6 fold vertices are blue, 5 fold are red and 3-fold are yellow, central wedges are again interpolated.

With more than one type of pyramid present, the endo/exo categorisation gets more involved and can be mixed in a single polyhedron.  Examples are linked (with no attempt at categorisation) for the n=3 case for the icosidodecahedron (seed), the small rhombicosidodecahedron (seed) and the snub icosidodecahedron (seed)   In all cases the colouring convention is use the seed face for the pyramid as the colour for adjacent wedges and to use an interpolated colour for the centre wedge.  See the links to the seed polyhedra for the case colours. 

Further combinations are possible where there is more than one edge type to the seed polyhedron. An example was also generated (here) for the small rhombicosidodecahedron with sets of three wedges between the pentagonal and square pyramids, and sets of two wedges between the square and triangular pyramids.


Credits and Resources

My thanks to Dr Richard Klitzing for providing the inspiration for this page.

The augmentation of the seed polyhedra, creation of the nets and initial relaxation was performed using Great Stella.   Stella VRML files were then converted into HEDRON input files using Roger Kaufman's VRML2OFF utility.  Then final relaxation and VRML generation using HEDRON.

A zip file containing TXT and OFF files for all polyhedra on this page is here.

Footnotes

[Fn 1] In Latin, the word cuneus (plural cunei), means a wedge or a wedge-shaped object, area, or formation. This term is the source for words like "cuneiform" (wedge-shaped writing) and "cuneus" (a wedge-shaped part of the brain) and is a root of many English words describing wedge-shaped forms! (Google)

[Fn 2] Relaxing a polyhedron refers to the process of iteratively adjusting the locations of a polyhedron's vertices such that they meet a set of predefined criteria (e.g. all edge lengths are equal and they form regular faces).

[Fn 3] The prefix "spheno-" originates from the Greek word for "wedge" and refers to something wedge-shaped or of or relating to the sphenoid bone, a wedge-shaped bone at the base of the skull. Examples of its use include sphenogram, meaning a wedge-shaped character, and sphenopalatine, which refers to the sphenoid bone and the palate. (Google)

[Fn 4] Olshevsky [2] also describes a process he terms 'ambiation'.  To quote Olshevsky: "Ambiation is a generalization of spheniation to a patch of more than two faces.  Specifically, the patch is a regular n-gon surrounded on all sides by n equits [equilateral triangles]"[2].  This process is different from the process I am describing here.  

References

1  Klitzing, R. (2025) Cuned Twisters.  https://bendwavy.org/klitzing/explain/twisters.htm#cune
2. Olshevsky G. (c2006) Breaking Cundy's Deltahedron record (unpublished) linked from [3] below.
3. Kaufman R. (2008) The Cundy Deltahedra. https://www.interocitors.com/polyhedra/Deltahedra/Cundy/index.html
4. Green, M. & McNeill J.  (2005)  Cingulated Anti-prisms  https://www.orchidpalms.com/polyhedra/chiral-prisms/prisms9.html


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