Arruated Polyhedra

3. Partial Arruation of the Platonic and Archimedean polyhedra


Partially arruated (3-s & s-s) snub icosidodecahedron

Where a polyhedron has more than one edge type, there is the possibililty of an intermediate process between Olshevsky's spheniation and arruation.  I term this 'partial arruation'.

In a partially arruated polyhedron, some but not necessarily all of the edges have been augmented with wedges.  Unlike spheniation, the wedges need not be distinct, and the face pyramids can form part of multiple wedges.  

The level-1 partial arruations have been explored for all of the Platonic and Archimedean polyhedra.  The endo/exo distinction is not so clear in some cases as the distortion caused in the relaxation process can at times be significant.  For all cases below, exo/endo notation refers to the start point where the exo mode has all of the pyramids everted (outwards) from the seed polyhedron (for faces with 6 or more edges the pyraimds are elongated), endo mode has them inverted.  The edges are identified by their adjacent faces, so a 3-5 edge would be between a triangle and a pentagon.  For the snub polyhedra, the snub triangles are represented with 's'.  The colours are taken from the seed polyhedron (red, blue and yellow) with wedges being an intermediate colour (green, cyan and magenta).  Where the two adjacent faces are alike, the wedges are grey.  Where the text is struck out, the example was either degenerate or could not be formed.

Level 1 arruation where all edges have been replaced can also be seen as the dual of the seed polygon which has then been triangulated and relaxed, partial arruations do not have such a simple alternative.  As a result, for seed polyhedra with one edge type the results are fairly trivial.  For two or three edge types some interesting results are obtained but as mentioned above the relaxation stage can introduce significant distortion.   There is also the possibility of further isomorphs being generated from different start points where there is a mixture of everted and inverted pyramids.  These have not yet been explored here.

In a number of cases, the anticipated result was not achieved.  This may be due to any of (i) the case does not exist, (ii) the case exists but the chosen starting point is not one that relaxes to it.

 - Where the relaxation stage failed, the text is struck out.  It is my opinion that these cases do not exist.
 - Where the endo mode says 'see exo', both starting points resulted in the same polyhedron. 
 - Where the result has coplanar faces it is marked with a (C), these involve three coplanar triangular faces forming a trapezoid.
 - Some cases resulted in degerate solutions with multiply coincident faces  These are marked with a (D)

One edge type:

4. Higher Order Spheniated Polyhedra


(3,1)-endo spheniated snub icosidodecahedron

Olshevsky's spheniated polyhedra can be regarded as an order 1 spheniation, with one wedge inserted between adjacent pyramids.  In similar fashion to the arruated polyhedra, higher order spheniations can be obtained by inserting more than one wedge between the pyramids. 

Using the snub icosidodecahedron as an example, spheniations up to order 3 have been explored.  In the examples below, the remaining pyramidical faces on the snub triangles are coloured pale yellow.  Inserted wedge faces are coloured yellow and orange as in the arruated polyhedra above.

(1,0)-endo

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

(3,1)-endo
(3,1)-exo

Attempts to generate (3,0) examples resulted in degenerate polyhedra with coplanar faces. 

5. Antiprismatically Arruated Polyhedra


Antiprismatically arruated dodecahedron (5,0) exo

A second family of arruated polyhedra exists.  Rather than augmenting the faces of the seed polyhedron with pyramids, augment instead with antiprisms.  Then connect the outer vertices of the edge connected antiprisms with wedges and relax them.  Olshevsky ^[2] uses the term 'ambiation' to mean this process applied to a single face.  The process of antiprismatic arruation is thus a generalisation of Olshevsky's ambiated polyhedra in the same way that the (pyramidically) arruated polyhedra are a generalisation of his spheniated polyhedra.

Applying one set of wedges does not result in any polyhedra of particular interest.  The remining triangular faces of the antiprisms and the wedge faces become complanar and form hexagons.  As an an example, the (1,0) anitiprismatically arruated dodecahedron then becomes a partially triangulated truncated icosahedron.  The level-1 examples are somewhat related to the monorhombal edge rhombified polyhedra with the rhombi being split into two equliateral triangles.

With two or more sets of wedges the results become more interesting, I have explored these for levels 2 and 3 for the platonic polyhedra, with some additional focus on the dodecahedron where I have explored up to level 5.  

The naming convention is consistent with that above.  The colour convention is that the upper faces of the antiprisms are green, the remaining lateral antiprismatic faces are cyan, the wedge faces are yellow, orange and red as in the (pyramidically) arruated polyhedra.  Where the text is shown in red and struck out, the example is either degenerate or does not exist.  Where the text is black and unlinked, I have not been able to generate the example, but cannot rule out that it exists.

My thanks to Dr Richard Klitzing for providing the inspiration for this page and to Don Romano for his boundless enthusiasm and modelling skills.

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.  Some of the cases were generated using an excel spreadsheet to spherically invert the vertex locations.  The spreadsheet is included in the zip file.

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)