Collection of forms resulting from all the combinations of the Platonic and Catalan polyhedra which characterize the m 3 5 icosahedral point group
Livio Zefiro
Dip.Te.Ris., Universita' di Genova, Italy
(Email address: livio.zefiro@fastwebnet.it)
Notes

As already pointed out in two previous papers
[1,2], all the icosahedral compound forms, and in particular the ones belonging to the
holohedral m
3
5 point goup, can be
derived from seven single forms, corresponding to two Platonic and five Catalan polyhedra.
The two Platonic polyhedra are the dodecahedron and the icosahedron: both are
characterized by Miller's indices having a fixed value. The same holds true for the rhombtriacontahedron,
a Catalan polyhedron dual of a quasiregular Archimedean polyhedron, the icosidodecahedron.
Conversely, the indices of the other four forms can assume variable values, in the intervals defined in the following table.
Miller's indices of the single forms characterizing the m 3 5 icosahedral point group 
Forms of the m 3 5 icosahedral point group (golden ratio τ = 1.61803...) 
{1τ0} dodecahedron 
{τ 1/τ 0} icosahedron 
{100} rhombtriacontahedron 
{hk0} triakisicosahedron (where: 0 < k/h <1/τ^{2 }) 
{hk0} deltoidal hexecontahedron (where: 1/τ <h/k <τ^{2 }) 
{hk0} pentakisdodecahedron (where: 0 < h/k <1/τ ) 
{hkl} hexakisicosahedron (where: 0 < k/h <1/τ^{2}, l/h <1/τ  τk/h, with h > 0) 
Miller's indices and names of the forms belonging to the
m
3
5 point group. 
(clicking on each following image by the left button of the mouse, one can
visualize the corresponding VRML dynamic image)
The seven single forms
1) dodecahedron
2) icosahedron
3) rhombtriacontahedron
4) triakisicosahedron
5) deltoidhexecontahedron
6) pentakisdodecahedron
7) hexakisicosahedron
Three further compound forms
including the rhombtriacontahedron
Two compound forms made of two forms
2+3
1+3
Compound form
made of three forms
1+2+3
Last compound form including
an icosahedron and a dodecahedron
Compound form
made of two forms
1+2
The compound form on the left derives from the intersection of the four single forms on the right:
a triakisicosahedron, an icosahedron and two different pentakisdodecahedra.  
The compound form on the left derives from the intersection of the four single forms on the right:
a rhombtriacontahedron, an icosahedron and two different deltoidhexecontahedra  
The compound form on the left derives from the intersection of the four single forms on the right:
a dodecahedron, a hexakisicosahedron and two different triakisicosahedra.
