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.

No Vertex Name Convexity Augmented Forms
01 n,4,4 Pentagonal Prism Global
02 n,3,3,3 Pentagonal Anti-prism Global  
03 n/d,4,4 Pentagrammic Prism Global  
04 n/d,3,3,3 Pentagrammic Anti-Prism Global  
05 n/(n-d),3,3,3 Pentagrammic Crossed Anti-Prism Potential  
06 3,3,3 Tetrahedron Global  
07 6,6,3 Truncated Tetrahedron  Global  
08 3,6,3/2,6 Octahemioctahedron None  
09 3,4,3/2,4 Tetrahemihexahedron None  
10 3,3,3,3 Octahedron  Global Type I(b): Excavate {3} with P3 ("10a")
     Type II(a): Augment {4} in 10a above with Y4
Type II(b): Excavate {3} with Y3
11 4,4,4  Cube Global Type I(b): Excavate {4} with S4 
Type II(b): Excavate {4} with Y4
12 4,3,4,3 Cuboctahedron Global  
13 6,6,4 Truncated Octahedron Global  
14 8,8,3  Truncated Cube Global  
15 4,4,4,3 Rhombicuboctahedron Global
16 8,6,4 Truncated Cuboctahedron Global
17 4,3,3,3,3  Snub Cuboctahedron Global
18 4,8,3/2,8 Small Cubicuboctahedron Potential Type II(d): Excavate {3/2} with U17 
19 4,8/3,3,8/3 Great Cubicuboctahedron Local Type I(a): Augment {4} with Y4
Type II(b): Excavate {8/3},{4},{3} with S8/3,Y4,S3
Type II(c): Excavate {8/3} with Q8/5
20 4,6,4/3,6  Cubohemioctahedron None  
21 8/3,6,8  Cubitruncated Cuboctahedron Local Type II(a): Augment {8} with Q4
22 4,4,3/2,4  Great Rhombicuboctahedron Potential Type I(d): Excavate  {3/2} with P3
     Type II(a): Augment {3} in P3 in 22a above with Y3
Type II(d): Excavate  {3/2} with S3  
Type II(ad): Excavate {3/2} with P3 & Augment {2nd & 3rd 4} with Y4 
     Type II(a): Augment {3} in P3 in 22c above with Y3 
Type II(ad): Excavate {3/2} with S3 & Augment {2nd & 3rd 4} with Y4 
23 8,4,8/7,4/3  Small Rhombicube None
24 8/3,8/3,3 Stellatruncated Cube Potential Type I(b'): Excavate {8/3} with Y8/3
Type II(b): Excavate {8/3},{3} with Y8/3,Y3
Type I(c): Excavate {8/3} with S8/5(Y8/3)
25 6,4,8/3 Great Truncated Cuboctahedron Local Type I(a): Augment {6} with S6
26 8/3,4,8/5,4/3 Great Rhombicube None
27 3,3,3,3,3  Icosahedron Global Type I(b): Excavate {3} with Q3 
     Type II(a): Augment {6} in Q3 in 27a above with ortho-Q3
Type II(b): Excavate {3} with Y3
28 5,5,5 Dodecahedron Global Note
29 5,3,5,3 Icosidodecahedron Global  
30 6,6,5 Truncated Icosahedron Global  
31 10,10,3 Truncated Dodecahedron Global
32 5,4,3,4 Rhombicosidodecahedron (small) Global  
33 10,6,4 Truncated Icosidodecahedron Global
34 5,3,3,3,3 Snub Icosidodecahedron Global
35 3,5/2,3,5/2,3,5/2 Small Ditrigonal Icosidodecahedron Local Type II(b): Excavate {5/2},(3) with P5/2,P3
36 3,6,5/2,6 Small Icosicosidodecahedron Local
37 5/2,3,3,3,3,3 Snub Disicosidodecahedron Local
38 5,10,3/2,10 Small Dodecicosidodecahedron Potential  
39 5/2,5/2,5/2,5/2,5/2 Small Stellated Dodecahedron Local Type II(b): Excavate {5/2} with S5/2 
     Type II(c): Excavate {5/2} in S5/2 above with Q5/3
Type II(b): Excavate {5/2} with U59
Type II(b): Excavate {5/2} with U79 
Type II(c): Excavate {5/2} with GJ06
40 (5,5,5,5,5)/2 Great Dodecahedron Local Type I(b): Excavate {5} with P5 
41 5/2,5,5/2, Dodecadodecahedron Local Type I(a): Augment {5} with Y5
42 10,10,5/2  Great Truncated Dodecahedron  Local
43 5,4,5/2, Rhombidodecadodecahedron Local  
44 4,10,4/3,10/9  Small Rhombidodecahedron  None  
45 5/2,3,5,3,3  Snub Dodecadodecahedron  Local Type I(a): Augment {5} with Y5
46  5/3,5,5/3,5,5/3, Ditrigonary Dodecadodecahedron  Potential Type II(d): Excavate {5/3} with P5/2 
     Type II(c): Excavate {5/2} in P5/2 in 46a above with Q5/3 
          Type II(a): Augment {4} in Q5/3 in 46aa above with Y4
47 5,10/3,3,10/3  Great Dodekified Icosidodecahedron  Local Type I(a): Augment {5} with Y5
48 3,10,5/3,10  Small Dodekified Icosidodecahedron Potential
49 5,6,5/3, Icosified Dodecadodecahedron  Potential
50 10,6,10/3  Icositruncated Dodecadodecahedron  Local Type II(a): Augment {10} with Q5
51  5/3,3,5,3,3,3 Snub Icosidodecadodecahedron Potential
52 (5,3,5,3,5,3)/2  Great Ditrigonary Icosidodecahedron  Local Type II(a): Augment {5} with Y5
53 5,6,3/2,6  Great Icosified Icosidodecahedron  Potential Type II(d): Excavate {3/2} with U07
54 3,10,3/2,10 Small Icosihemidodecahedron  None  
55 6,10,6/5,10/9  Small Dodekicosahedron  None  
56 5,10,5/4,10 Small Dodecahemidodecahedron None  
57 5/2,5/2,5/2 Great Stellated Dodecahedron Local Type II(b): Excavate {5/2} with Y5/2  
Type II(b): Excavate {5/2} with P5/2
58 (3,3,3,3,3)/2 Great Icosahedron  Local Type I(b): Excavate {3} with Y3 
59 5/2,3,5/2,3 Great Icosidodecahedron Local Type II(a): Augment {3} with Y3
Type I(b): Excavate {5/2},{3} with Y5/2,Y3
Type II(b): Excavate {5/2},{3} with S5/2,Y3
Type II(c): Excavate {5/2} with S5/3(Y5/2)
60 6,6,5/2  Great Truncated Icosahedron Local
61 6,4,6/5,4/3  Rhombicosahedron None  
62 5/2,3,3,3,3 Great Snub Icosidodecahedron Local
63 10/3,10/3,5  Small Stellatruncated Dodecahedron Local
64 10,10/3,4  Stellatruncated Dodecadodecahedron Local Type I(a): Augment {10} with S10
65 5/3,3,5,3,3 Vertisnub Dodecadodecahedron Potential Type I(d): Excavate {5/3} with U35 
Type I(d): Excavate {5/3} with RK'sid-6-10-0-2'
66 10/3,5/2,10/3, Great Dodekicosidodecahedron  Local Type II(a): Augment {3} with Y3
Type II(b): Excavate {10/3},{5/2},{3} with S10/3,P5/2,P3
     Type II(c): Excavate {5/2} in P5/2 in 66b above with Q5/3
67 5/2,6,5/3, Small Dodecahemicosahedron None  
68 10/3,6,10/7,6/5  Great Dodekicosahedron None  
69 5/3,3,5/2,3,3,3  Great Snub Icosidisdodecahedron Potential Type II(d): Excavate {5/3} with P5/2  
     Type II(a): Augment {4} in P5/2 in 69a above with Y4 
     Type II(c): Excavate {5/2} in P5/2 in 69a above with Q5/3 
Type II(e): Excavate {5/3} with S5/2 
Type II(cd): Excavate {5/3},{5/2} with P5/2,Q5/3 
     Type II(c): Excavate {5/2} in P5/2 in 69c above with Q5/3 
     Type II(a): Augment {4} in Q5/3 in 69c above with Y4
     Type II(ac): Augment {4} in Q5/3, excavate {5/2} in P5/2 in in 69c above with Y4,Q5/3
Type II(cd): Excavate {5/3},{5/2} with S5/2,Q5/3 
70 6,5,6,5/ Great Dodecahemicosahedron None  
71 3,10/3,10/3 Great Stellatruncated Dodecahedron Local Type II(a): Augment {3} with Y3 
Type II(b): Excavate {10/3},{3} with Y10/3,S3 
Type II(b): Excavate {10/3},{3} with Q5/3(P5/2),S3
72 4,5/3,4,3  Great Rhombicosidodecahedron  Potential Type I(d): Excavate {5/3} with P5/2 
     Type II(a): Augment {5/2} in P5/2 in 72a above with Y5/2
Type II(d): Excavate {5/3} with S5/2 
Type II(e): Excavate {4},{3} with Y4,S3 
Type II(ad): Excavate {5/3} with P5/2 & Augment {4} with Y4  
     Type II(a): Augment {5/2} in P5/2 in 72d above with Y5/2
73 6,10/3,4 Stellatruncated Icosidodecahedron Local
74 5/3,3,3,3,3  Great Vertisnub Icosidodecahedron Potential Type I(d): Excavate {5/3} with S5/2 
Type II(d): Excavate {5/3} with U59
Type II(d): Excavate {5/3} with GJ06 
     Type II(a): Augment {10/3} in GJ06 in 74c above with Y10/3
75 5/2,10/3,5/3,10/3 Great Dodecahemidodecahedron None  
76 3,10/3,3/2,10/3  Great Icosihemidodecahedron None  
77 (5/3,3,3,3,3,3)/2 Small Retrosnub Icosicosidodecahedron Potential Type I(d): Excavate {5/3} with Y5/2 
Type II(e): Excavate {3} with Y3 
Type II(d): Excavate {5/3} with U79 
Type II(da): Excavate {5/3},augment 2nd {3} with Y5/2,Y3
Type II(da): Excavate {5/3},augment 2nd&3rd {3} wuth Y5/2,Y3
78 10/3,4,10/7,4/3  Great Rhombidodecahedron None  
79 (5/2,3,3,3,3)/2  Great Retrosnub Icosidodecahedron Local Type II(a): Augment 2nd {3} with Y3
Type II(a): Augment {5/2} with Y5/2
Type II(b): Excavate {5/2},{3} with Y5/2,U24
Type II(c): Excavate {5/2} with modified 77a 

Type I(c): Excavate {5/2} with S5/3(Y5/2)
Type II(ca): Excavate {5/2},augment 2nd{3} with S5/3(Y5/2),Y3 
Type II(c): Excavate {5/2} with S5/3(S5/2)
Type II(ca): Excavate {5/2},augment 2nd{3} with S5/3(S5/2),Y3
80 (5/2,4,3,4, 5/3,4,3/2,4)/2  Great Disnub Disicosidisdodecahedron Potential

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


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). 

Updates

27-Jan-03: Set of VRML files completed.

3-Dec-02: 24c, 59d, 77d,77e, 79e,79f,79g,79h

25-Nov-02: 19c, 39aa, 46aa,46aaa, 66ba, 69cb,69cc, 71c, 79d

7-Nov-02: 27a,27aa, 28a, 39d, 53a, 64a, 65b, 74c,74ca, 79c