Locally Convex Octahedral and Icosahedral Polyhedra

Robert Webb, the author of the Great Stella polyhedral program, contacted me regarding a search he had conducted for polyhedra, in addition to the uniform polyhedra, which meet the following criteria:

(1) Regular faces.
(2) Locally-convex, i.e. none of the dihedral angles are reflex.
(3) All vertices are vertices of the model's convex hull.
(4) Octahedral or icosahedral symmetry.

I refer to polyhedra meeting these criteria as Type I polyhedra.

I relaxed Robert's criterion (3) slightly to allow further polyhedra to be generated.  Criterion (3) became:

(3) All vertices lie on the exterior of the polyhedron.

I refer to polyhedra meeting these criteria as Type II polyhedra.  A Type I polyhedron is automatically also a Type II polyhedron.  I am still finding additional examples, and would be very surprised if the current list were to be complete.

It should be stressed that condition (4) is an arbitrary limitation added only to confine the resulting list to manageable proportions.

The uniform polyhedra themselves meet Type I criteria.  All further polyhedra discovered to date have been formed by augmenting or excavating uniform polyhedra or augmented uniform polyhedra.  The process of augmenting or excavating a polyhedron means to replace one or more sides of the original polyhedron with a further polyhedron having one side co-incident with a side of the original polyhedron.  I define augmentation as outward (away from the centroid of the vertex figure), excavation as inwards (towards the centroid of the vertex figure).  Note that this definition is slightly different to that used by Great Stella.

For each of the uniform polyhedra, I have defined a 'convexity'.  This is normally considered to be; global (i.e. the polyhedron is globally convex - the Platonic and Archimedean polyhedra fall into this category), local (the polyhedron is locally convex - i.e. the polygons at each vertex all cycle the vertex in the same direction) or non-convex (the vertex is crossed).  In the latter case I have distinguished between non-convex vertices of the type m-n-m'-n' or m-n-m'-n (where m' is a retrograde m) and other vertices having retrograde polygons.  In certain cases the latter can be augmented by excavating the retrograde polygon with a suitable polyhedron and restoring the local convexity of the vertex.  (Care must be taken in the selection of the excavating polyhedron, it must itself be locally convex, it must have the requisite pyramidical symmetry (3-, 4- or 5-fold) and it must be sized such that the new vertices are visible on the exterior of the resulting polyhedron.)  I term the convexity of these polyhedra 'potential'.  Polyhedra with vertices of the forms first referred to above I designate as having a convexity of 'none'.

Augmentation can be performed in one of five (or six) ways (diagrams):

(a) one or more of the faces adjoining a locally convex vertex can be augmented, the limiting factor is to retain the local convexity of the vertex.

(b) all of the faces adjoining a locally convex vertex can be excavated.  The excavation must be deep enough to re-create a (highly wound) locally convex vertex (with opposite polarity to the original).

(b') a variation on case (b) where one of the original faces is left un-excavated.  The remaining excavations are deep enough to re-create a (highly wound) locally convex vertex (with opposite polarity to the original).

(c) one or more of the faces adjoining a locally convex vertex can be excavated, the excavating polyhedron must have one (and only one) retrograde face.  This face is then the face used for excavation.  The excavation must be shallow enough to retain the local convexity of the vertex.  Such an excavation would normally add to the 'winding number' of the vertex figure.

(d) the retrograde face of certain polyhedra with crossed vertices can also have its vertices uncrossed by excavating the retrograde polygon with a suitable locally convex polyhedron.  Again such an excavation would normally add to the 'winding number' of the vertex figure.

(e) all of the prograde faces of certain polyhedra with crossed vertices can also have its vertices uncrossed by excavating the prograde polygons with suitable locally convex polyhedra.

Combinations of the above are also possible in a single augmentation, e.g.(ac), (cd) or (ad).

Second order augmentations and beyond can be generated by augmenting one of the new faces added during an augmentation or an excavation.  Care must be taken that all vertices remain locally convex.

In the table below, the description of the augmented forms is given as 'Augment/Excavate {m},{n} with X,Y'. {m} and {n} are the faces of the original polyhedron, X and Y are the additional polyhedra attached to {m} and {n} respectively.  A short table of abbreviations is given below:

Pn : n-gonal prism
Qn: n-gonal cupola
Sn: n-gonal antiprism
Yn : n-gonal pyramid
Unn : Uniform polyhedron no. nn from the list below.
Jnn : Johnson Solid no. nn
GJnn : 'Great' Isomorph to Johnson Solid no. nn
RK'...'  :  Faceting of Uniform Polyhedron (as discovered by Dr Richard Klitzing)

At times I also refer to individual augmentations as 10a, 10b etc.  10a is simply the first listed augmentation of polyhedron 10 (the octahedron).  The form 10aa would be the first listed augmented form of 10a. etc.

Clicking on the uniform polyhedron name links to a page describing the augmented forms in detail.  For the augmented forms, clicking on the description links directly to a VRML model.  Clicking on the number in the left column links to a VRML model of the uniform polyhedron.

A gallery of all discovered augmented forms is linked here.

Credits and Great Stella Files

All of the augmented polyhedra were generated using Robert Webb's Great Stella program, except for the occasional framework models which were generated using Hedron.  Thanks to Robert for introducing the idea for this search and for providing the initial examples 19a, 41a, 45a and 47a.

For users of Great Stella:  Stella files of all the augmented polyhedra are included in these archives: augment1.zip  (octahedral: polyhedra 1-26),  augment2.zip  (icosahedral part 1: polyhedra 27-59),  augment3.zip  (icosahedral part 2: polyhedra 60-71),   augment4.zip  (icosahedral part 3: polyhedra 72-80).