Collection of the couples of archetypal compound forms which are different in the 2 3 5 and m 3 5 icosahedral point groups

Livio Zefiro
Dip.Te.Ris., Universita' di Genova, Italy
(E-mail address: livio.zefiro@fastwebnet.it)

Notes
  • Tested with Internet Explorer 7, Mozilla Firefox 3.5, Opera 10 and Google Chrome at 1024x768 and 1280x1024 pixels
  • All the images have been generated by SHAPE 7.0, a software program for drawing the morphology of crystals, distributed by Shape Software
  • Hovering with the mouse on each image, the symbol of a pointer appears: if your web browser is set up to visualize VRML files, by clicking with the left button of the mouse it should be possible to visualize in a new window the corresponding dynamic image, that can be enlarged, rotated, shifted... Cortona 3D Viewer is a good Web3D visualizer that can be downloaded from here: at present it is working also with the updated versions of the main browsers.
 

In two previous works [1,2] it has been shown that all the icosahedral compound forms belonging to the holohedral m 3 5 point group can be derived from the combination of seven single forms, corresponding to two Platonic and five Catalan polyhedra; the same thing happens also in case of 235, the other icosahedral point group.
The only difference between the two point groups concerns the {hkl} general form relative to each point group: in m 3 5 point group it is a hexakis-icosahedron, consisting of 120 faces, whereas in 235 point group it is a pentagonal hexecontahedron, consisting of 60 faces.
The different multiplicity of the {hkl} general forms depends on the lack, in the 235 point group, of both inversion centre and mirror planes.
The other six forms (two Platonic and four Catalan polyhedra), coincide in the two point groups: as a matter of fact, the number of faces belonging to each form does not vary, even in the presence of the fifteen mirror planes characterizing the m 3 5  point group, since all the faces lie just in correspondence to the mirror planes.
 
Platonic and Catalan single forms with icosahedral symmetry
1) dodecahedron 2) icosahedron 3) rhomb-triacontahedron 4) triakis-icosahedron
5) deltoid-hexecontahedron 6) pentakis-dodecahedron 7') hexakis-icosahedron
 (only in m 3 5  point group)		 7") pentagonal
 hexecontahedron
(only in 235 point group)
Clicking by the left button of the mouse on the image of each polyhedron, one can visualize the corresponding VRML dynamic image; analogously, the StereoNet and the relative view along [001] can be visualized clicking on the name of each polyhedron.

The entire collection of the 120 compound forms, belonging to m 3 5  point group and derived from the two Platonic and the five Catalan polyhedra, was reported in a recent work [3], where all the faces of each compound form are equidistant from the centre of the polyhedron.
The present work shows the sixty-three couples of compound forms which are different in m 3 5  and 235 point groups, since they include, in combination with the other single forms, either the hexakis-icosahedron or the pentagonal hexecontahedron (which in turn give rise to a sixty-fourth couple, this time consisting in two single forms).
The other fifty-seven compound forms which do not include either the hexakis-icosahedron or the pentagonal hexecontahedron are identical in the two icosahedral point groups; therefore, taking into account also the six single forms in common, the forms shared by the two point groups are sixty-three altogether.
In this work the compound forms shared by the two point groups are not reported, since they were already shown in [3].

Sixty-three couples of compound forms, belonging to m 3 5 or 235 icosahedral point groups, which include either the hexakis-icosahedron or the pentagonal hexecontahedron

(Clicking by the left button of the mouse on the image of each polyhedron, one can visualize the corresponding VRML dynamic image)
m 3 5  point group 235 point group m 3 5  point group 235 point group

6 couples of compound forms made of two single forms
1+7' 1+7" 2+7' 2+7"
3+7' 3+7" 4+7' 4+7"
5+7' 5+7" 6+7' 6+7"

15 couples of compound forms made of three single forms
1+6+7' 1+6+7" 2+6+7' 2+6+7"
3+6+7' 3+6+7" 4+6+7' 4+6+7"
5+6+7' 5+6+7" 1+5+7' 1+5+7"
2+5+7' 2+5+7" 3+5+7' 3+5+7"
4+5+7' 4+5+7" 1+4+7' 1+4+7"
2+4+7' 2+4+7" 3+4+7' 3+4+7"
1+3+7' 1+3+7" 2+3+7' 2+3+7"
1+2+7' 1+2+7"

20 couples of compound forms made of four single forms
1+5+6+7' 1+5+6+7" 2+5+6+7' 2+5+6+7"
3+5+6+7' 3+5+6+7" 4+5+6+7' 4+5+6+7"
1+4+6+7' 1+4+6+7" 2+4+6+7' 2+4+6+7"
3+4+6+7' 3+4+6+7" 1+3+6+7' 1+3+6+7"
2+3+6+7' 2+3+6+7" 1+2+6+7' 1+2+6+7"
1+4+5+7' 1+4+5+7" 2+4+5+7' 2+4+5+7"
3+4+5+7' 3+4+5+7" 1+3+5+7' 1+3+5+7"
2+3+5+7' 2+3+5+7" 1+2+5+7' 1+2+5+7"
1+3+4+7' 1+3+4+7" 2+3+4+7' 2+3+4+7"
1+2+4+7' 1+2+4+7" 1+2+3+7' 1+2+3+7"

15 couples of compound forms made of five single forms
1+4+5+6+7' 1+4+5+6+7" 2+4+5+6+7' 2+4+5+6+7"
3+4+5+6+7' 3+4+5+6+7" 1+3+5+6+7' 1+3+5+6+7"
2+3+5+6+7' 2+3+5+6+7" 1+2+5+6+7' 1+2+5+6+7"
1+3+4+6+7' 1+3+4+6+7" 2+3+4+6+7' 2+3+4+6+7"
1+2+4+6+7' 1+2+4+6+7" 1+2+3+6+7' 1+2+3+6+7"
1+3+4+5+7' 1+3+4+5+7" 2+3+4+5+7' 2+3+4+5+7"
1+2+4+5+7' 1+2+4+5+7" 1+2+3+5+7' 1+2+3+5+7"
1+2+3+4+7' 1+2+3+4+7"

6 couples of compound forms made of six single forms
1+2+3+4+5+7' 1+2+3+4+5+7" 1+2+3+4+6+7' 1+2+3+4+6+7"
1+2+3+5+6+7' 1+2+3+5+6+7" 1+2+4+5+6+7' 1+2+4+5+6+7"
1+3+4+5+6+7' 1+3+4+5+6+7" 2+3+4+5+6+7' 2+3+4+5+6+7"

couple of compound forms
made of seven single forms
1+2+3+4+5+6+7' 1+2+3+4+5+6+7"

As already pointed out in [3], in order to obtain whichever icosahedral form starting from the previous archetypal polyhedra, one can vary:
As an example, in the next image one can see the compound forms resulting from the combination of the rhomb-triacontahedron {100} with three non Catalan polyhedra: two different deltoid-hexecontahedra, having indices {110} and {210} and a third polyhedron which, depending on the icosahedral point group, consists in a hexakis-icosahedron {211} or in a pentagonal hexecontahedron {211},
In order to obtain these particular compound forms, the relative distances of the forms from the centre of the solid have been properly set.


 Links
  1. Zefiro L., Ardigo' M.R.
    Description of the forms belonging to the 235 and m35 icosahedral point groups starting from the pairs of dual polyhedra: icosahedron-dodecahedron and Archimedean polyhedra-Catalan polyhedra
    VisMath, volume 9, No. 4, 2007
     
  2. Zefiro L., Ardigo' M.R.
    Platonic and Catalan polyhedra as archetypes of forms belonging to the cubic and icosahedral systems
    VisMath, volume 11, No. 2, 2009


  3. Zefiro L.
    Collection of forms resulting from all the combinations of the Platonic and Catalan polyhedra which characterize the m 3 5 icosahedral point group
    VisMath, volume 11, No. 3, 2009