Square faces of rhombtriacontahedron in
presence of decagonal faces of dodecaheron and hexagonal faces of
icosahedron which are both regular only in case of the truncated
icosidodecahedron and become regular pentagons and triangles, respectively,
in case of the rhombicosidodecahedron.
The values, reported in the captions, of the central distance of the faces of dodecahedron
and icosahedron refer to a unit value of the central distance of the
square faces of rhombtriacontahedron. 


d_{dodecahedron }
= 0.8887 d_{icosahedron}
= 0.96 
Archimedean truncated icosidodecahedron
d_{dodecahedron }
= 0.9210 d_{icosahedron}
= 0.9819 


d_{dodecahedron }
= 0.9476 d_{icosahedron}
= 1.0 
Archimedean rhombicosidodecahedron
d_{dodecahedron }
= 0.9748 d_{icosahedron}
= 1.0184 
A series of composite forms including square faces of rhombtriacontahedron
can be obtained starting from the Archimedean rhombicosidodecahedron, by the substitution of the icosahedron with a triakisicosahedron
and/or the substitution of the dodecahedron with a pentakisdodecahedron.
In the next figure, the icosahedron is always
present in the forms of the left column, whereas it is replaced by
the triakisicosahedron {τ^{3}10} in the
forms of the right column.
The dodecahedron, always present in the forms of the first row, is replaced by the pentakisdodecahedron {τ^{2}01}
in the second row
and by the pentakisdodecahedron {τ^{3}01}
in the third row.
One can note that only in the cases c) and f)
of the previous set of images the square faces of rhombtriacontahedron
derive from the intersection with couples of faces characterized by
permutated values of the indices and having consequently the same central
distance; in detail, they are:
icosahedron {τ^{2}10}
and pentakisdodecahedron {τ^{2} 01} in c)
triakisicosahedron {τ^{3}10}
and pentakisdodecahedron {τ^{3 }01} in f)
In the other four cases the couples of faces intersecting each square face are
placed at different central distances.
Another series of composite forms including square faces of rhombtriacontahedron
can be obtained starting from the Archimedean truncated icosidodecahedron by
the substitution of the icosahedron
and/or the dodecahedron, respectively, with:
a couple of polyhedra made of a triakisicosahedron and a
deltoidhexecontahedron having a symmetrical orientation in respect to the
icosahedron
a couple of polyhedra made of
a pentakisdodecahedron and another deltoidhexecontahedron having a symmetrical
orientation in respect to the dodecahedron.
Square faces of deltoidhexecontahedron can be obtained in composite forms
only in case of particular geometric conditions.
Starting from the (τ10) face of the deltoidhexecontahedron, its
intersection with the (τ 1
1/τ) and (τ 1 1/τ)
faces of the rhombtriacontahedron (as pointed out in the Table 1 of the
Appendix, the rhombtriacontahedron includes faces with both {100} and {τ 1
1/τ} indices) give rise to two edges, both parallel to the [1τ 0] direction.
In order to obtain a square face of deltoidhexecontahedron, two other faces
must intersect each face of the
deltoidhexecontahedron along edges parallel to the [001] direction and
therefore perpendicular to the two edges parallel to the [1τ
0] direction: in the
more trivial case, such faces are the (1τ 0) face of the dodecahedron and the
(τ 1/τ
0) face of the icosahahedron, placed at proper central distances.
Also in this case, other composite forms including square faces of
deltoidhexecontahedron can be obtained starting from the previous polyhedron
by the substitution of the icosahedron and/or the dodecahedron with a
deltoidhexecontahedron.
At last, composite forms including square faces of deltoidhexecontahedron can
be obtained also by the substitution of the rhombtriacontahedron, always
present in the previous polyhedra, with a {hkl}
hexakisicosahedron in which the ratio h/k corresponds to
τ.
For example, in the following animated images the
{τ 1 1/τ} face of the rhombtriacontahedron is substituted by the {τ 1
1/τ^{2}} face of the
hexakisicosahedron.
Animated images
showing the transition from rhombtriacontahedral to hexakisicosahedral faces in composite
forms including square faces of deltoidhexecontahedron 


deltoidhexecontahedron
{τ10}
dodecahedron{1τ0}
icosahedron {τ^{2}10}
rhombtriacontahedron {100}
becoming a
hexakisicosahedron {τ 1
1/τ^{2}} 
deltoidhexecontahedron
{τ10}
deltoidhexecontahedron
{110}
deltoidhexecontahedron
{210}
rhombtriacontahedron {100}
becoming a
hexakisicosahedron {τ 1
1/τ^{2}} 
In addition to the faces of rhombtriacontahedron and deltoidhexecontahedron, also the faces
of triakisicosahedron and pentakisdodecahedron can assume a square shape, as shown in the following animated sequence
(viewing direction along the normal to the (τ^{3}10)
face of the triakisicosahedron, on the left, and (τ^{3}01)
face of the pentakisdodecahedron, on the right, respectively).
In both cases it is required the presence of a rhombtriacontahedron and an
appropriate deltoidhexecontahedron, together with a dodecahedron in case of
the square faces of triakisicosahedron or an icosahedron in case of the the
square faces of pentakisdodecahedron.
Square faces resulting from the
sequential truncation of a triakisicosahedron (on the left) and a pentakisdodecahedron (on
the right), by a deltoidhexecontahedron,
an icosahedron or a dodecahedron, and a rhombtriacontahedron. 


triakisicosahedron {τ^{3}10}
deltoidhexecontahedron
{τ^{3}11}
icosahedron {τ^{2}10}
rhombtriacontahedron {100}

pentakisdodecahedron {τ^{3}01}
deltoidhexecontahedron
{τ^{3}11}
dodecahedron {τ01}
rhombtriacontahedron {100}

Appendix
It may be useful to remember that, as already pointed out in
[2] and
[3],
the indices of all the faces of each icosahedral form can be obtained from the cyclic permutation of an
only set
of indices uniquely in case of dodecahedron, whereas the sets of indices are two, three or five in case of the other six icosahedral forms
(see
Table 1, where the indices of the Platonic
and Catalan icosahedral forms
belonging to the
2/m
3
5 point group are reported and compared with the indices of the corresponding
Platonic and Catalan cubic forms belonging to the
4/m
3 2/m
point group).
Sets of {hkl} indices relative to the forms,
belonging to 2/m 3 5 icosahedral and
4/m 3 2/m cubic point groups,
which correspond to Platonic and Catalan polyhedra 
Icosahedral polyhedra

Indices {hkl} 
Cubic polyhedra 
Indices {hkl} 
Dodecahedron 
{1τ 0} 
Cube 
{100} 
Icosahedron 
{111}
{τ 1/τ 0} 
Octahedron 
{111} 
Rhombtriacontahedron 
{100}
{τ 1 1/τ} 
Rhombdodecahedron 
{110} 
Pentakisdodecahedron 
{1/τ 3 0}
{τ^{2}
1 2/τ}
{τ+1/τ 2 1/τ} 
Tetrakishexahedron 
{210} 
Triakisicosahedron 
{τ+1/τ
1/τ^{2} 0}
{τ 2/τ
1}
{2 1 1/τ^{2}} 
Triakisoctahedron 
{1 1 √21} 
Deltoidal hexecontahedron 
{1+1/τ^{2}
1 0}
{τ^{2}
1/τ
1/τ}
{2 1 τ} 
Deltoidal icositetrahedron 
{1
1 √2+1} 
Hexakisicosahedron 
{2+τ^{2}
1/τ
1/τ}
{τ^{2} 1 2/τ^{2}}
{2+1/τ^{2}
τ 1/τ^{2}}
{2 3/τ 1}
{τ+1/τ
2/τ 1+1/τ^{2}} 
Hexakisoctahedron 
{2√2+1 √2+1 1}

Table 1
Each set of indices relative to the icosahedral forms corresponds to a different form in the 2/m 3 cubic point group, subgroup of the
2/m 3
5
icosahedral point group.
Concerning the Catalan deltoidhexecontahedron (made of 60 faces), the three forms belonging to the 2/m
3 cubic point group in
which it can be decomposed are, as shown in Figure 1:
the pentagon dodecahedron {1+1/τ^{2} 1 0}
(12 faces)
the deltoidicositetrahedron {τ^{2}
1/τ 1/τ} (24 faces)
the diploid {2 1 τ} (24 faces)
In the Catalan deltoidhexecontahedron these pentagondodecahedral, deltoidicositetrahedral and diploidal faces are related by
the 5fold axes.
Decomposition of the Catalan deltoidhexecontahedron
in three forms belonging to the 2/m
3 point group: a
pentagondodecahedron, a deltoidicositetrahedron and a diploid 


pentagondodecahedron
{1+1/τ^{2} 1
0} 
deltoidicositetrahedron
{τ^{2}
1/τ 1/τ} 


diploid
{2 1 τ} 
Catalan deltoidhexecontahedron resulting
from the intersection of the three single forms
belonging to the 2/m 3 point group 
Figure 1
In turn, the five forms (all made of 24 faces), in which the Catalan hexakisicosahedron
(made of 120 faces) can be decomposed, are, as shown in
Figure 2:
the deltoidicositetrahedron {2+τ^{2
}1/τ 1/τ} (or {4τ+1 1 1})
the four diploids: {τ^{2} 1 2/τ^{2}},
{τ+1/τ 2/τ 1+1/τ^{2}}, {2
3/τ 1}, {2+1/τ^{2}
τ 1/τ^{2}}
Obviously also in the Catalan hexakisicosahedron these deltoidicositetrahedral and diploidal faces are related by the 5fold axes.
Figure 2
As shown in Figure 3, square faces
of rhombtriacontahedron can be generated only from its intersection with
a deltoidexecontahedron
{hk0} whose ratio h/k, included in the range:
1/τ < h/k < τ^{2},
assumes the value h/k
= 1+1/τ^{2} characterizing the (1+1/τ^{2} 1
0) face of the Catalan deltoidexecontahedron
(central image of Figure 3). In fact, the
(1+1/τ^{2} 1 0) face is the only (hk0) face that, by the action of the [τ^{ }1/τ 0] 3fold axis,
becomes the (τ^{2 }
1/τ 1/τ) face, that is a (hkk) face: the (100) square face of
rhombtriacontahedron can be generated uniquely by the intersection just with (hkk) faces
(when the relative h/k
ratio is equal or greater than τ^{3}). It follows from the orthogonality
of the [011]
and [0 1
1] directions, parallel
to the edges between the (100) face and the contiguous (hkk) and (hkk)
faces, becoming (τ^{2 }
1/τ 1/τ) and (τ^{2 }
1/τ 1/τ) in case of the Catalan deltoidhexecontahedron.
Figure 3
Whereas the intersection with
{hkk} forms characterized by a ratio
h/k ≥ τ
^{3} always
generates square faces of rhombtriacontahedron, generally square faces cannot be generated when the intersection with the rhombtriacontahedron concerns
faces belonging to
{hkk} forms, usually hexakisicosahedra, characterized by a ratio
h/k <τ
^{3}.
In such a case, the (hkk) "deltoidicositetrahedral" face
of the hexakisicosahedron (see Fig.1), whose h/k ratio is included in the range 1<h/k <τ^{3}, does not intersect the
(100) face of the rhombtriacontahedron and the same thing happens,
when
h/k < 1, with the "triakisoctahedral" face of the hexakisicosahedron.
The faces that, being the nearest to the (100) face of the rhombtriacontahedron, intersect it,
usually are instead four {hkl} "diploidal" faces of the hexakisicosahedron,
derived from the (hkk) face by the action of the symmetry operators.
The indices of one of these {hkl} faces are reported, together with the {hkk}
indices,
in the captions of the sequence of images shown in Figure 4, ordered in columns
according to decreasing value of h/k, starting from the deltoidhexecontahedron {τ^{3}11}, whose
intersection with the rhombtriacontahedron generates square faces.
However, as shown in the fifth row of the sequence, one can obtain square (100) faces,
even if the ratio h/k is less than τ^{3},
starting from both {1+2/τ^{2}
1 1} and {1+1/τ^{2}
τ^{2} τ^{2}} hexakisicosahedra.
In fact, such hexakisicosahedra include, by the action of the symmetry operators,
also the "deltoidicositetrahedral" faces with indices
(1+τ^{2} 1/τ^{2} 1/τ^{2})
and (2+1/τ^{2} 1/τ^{2} 1/τ^{2}), respectively,
where the value of the ratio h/k is greater than
τ^{3}; therefore, in both cases, the result of their intersection with the (100) face
of the rhombtriacontahedron is a face having a square shape.
Then, according to the 2/m
3 point
group, the five forms in which the former of these two hexakisicosahedra can
be decomposed are three diploids and the two deltoidicositetrahedra {1+2/τ^{2}
1 1} and {1+τ^{2} 1/τ^{2} 1/τ^{2}},
whereas the latter can originate three diploids, the
deltoidicositetrahedron {2+1/τ^{2} 1/τ^{2} 1/τ^{2}}
and the triakisoctahedron {1+1/τ^{2}
τ^{2} τ^{2}}. On
the other hand, as
already pointed out in Fig.2, the Catalan hexakisicosahedra
can be decomposed in a
deltoidicositetrahedron and four diploids, and, finally, five diploids
can be derived from the
generic hexakisicosahedra.
Resuming, as one can see in
Fig.4, the
{hkk} forms are all hexakisicosahedra, excluding five forms:
the deltoidhexecontahedra {τ^{3}11} and {011},
shown in the 1^{st} row; whereas four "deltoidicositetrahedral" {τ^{3}11}
faces intersect directly the (100) face generating
a square face, the deltoidhexecontahedron {011} intersects the (100)
face by means of four (τ^{2 }1/τ^{2} 1)
"diploidal" faces: the result is a rhombic face
instead of a square face
the pentakisdodecahedra {τ+1/τ 1 1} and {1/τ^{3}
1 1}, intersecting the (100) face by their "pentagondodecahedral
faces": (τ^{4} 0 1)
and (τ^{4} 0 1)
in the first case, (1+τ^{2} 0 1)
and (1+τ^{2} 0 1) in the second case (3^{rd} row); the
common result is a face of rhombtriacontahedron having the shape of nonregular but symmetric hexagons
the icosahedron {111}, intersecting
the (100) face of the rhombtriacontahedron by the faces (τ 1/τ 0) and
(τ 1/τ 0); also in this case the faces of rhombtriacontahedron have the shape
of nonregular but symmetric hexagons; they are orientated orthogonally to the
previous hexagonal faces of rhombtriacontahedron, generated by the
intersection with pentakisdodecahedra.
The couples of composite forms reported in the six intermediate rows of Fig.4
are similar, even though they correspond to {hkk} forms characterized by
values of the ratio h/k greater (left column) or less (right column) than 1.
The faces of the couple of rhombtriacontahedra in the 6^{th} row, generated by the intersection with the hexakisicosahedra
{τ 1 1} and {1/τ 1 1}, have a rhombic shape, characterized by a ratio of the
diagonals equal to τ.
Intersections of the rhombtriacontahedron with {hkk} forms
in which the ratio h/k assumes decreasing values in the interval:
τ^{3} ≥ h/k ≥ 0 


a) (100) face of the rhombtriacontahedron intersected by the (τ^{3}11) face of the
deltoidhexecontahedron with the same indices 
o) (100) face of the rhombtriacontahedron intersected
by the (τ^{2 }1/τ^{2} 1)
face of the deltoidhexecontahedron {011} 


b) (100) face of the rhombtriacontahedron
intersected by the (3τ+1 1/τ^{2} 1)
face of the hexakisicosahedron {τ^{2}11} 
n) (100) face of the rhombtriacontahedron
intersected by
the (5τ1 1/τ^{2}
τ+1/τ) face of the hexakisicosahedron
{1/τ^{4}
11} 


c) (100) face of the rhombtriacontahedron
intersected by the (τ^{4} 01)
face of the pentakisdodecahedron {τ+1/τ 1 1} 
m) (100) face of the rhombtriacontahedron
intersected by
the (1+τ^{2 }
0 1)
face of the pentakisdodecahedron {1/τ^{3}
1 1} 


d) (100) face of the rhombtriacontahedron
intersected by the (3τ^{2} 1/τ^{2}
1)
face of the hexakisicosahedron {211} 
l) (100) face of the rhombtriacontahedron intersected
by the (2+τ^{2} 1/τ^{2}
1)
face of the hexakisicosahedron {1/τ^{2}
11} 


e) (100) square face of rhombtriacontahedron
resulting from the intersection with the
(1+τ^{2} 1/τ^{2} 1/τ^{2})
face of the hexakisicosahedron
{1+2/τ^{2}
1 1} 
k) (100) square face of rhombtriacontahedron
resulting from the intersection with the
(2+1/τ^{2} 1/τ^{2} 1/τ^{2})
face of the hexakisicosahedron
{1+1/τ^{2}
τ^{2} τ^{2}}



f) (100) face of the rhombtriacontahedron
intersected by the (τ^{3}
1/τ 1/τ^{2})
face of the hexakisicosahedron {τ 1 1}

j) (100) face of the rhombtriacontahedron intersected
by the (3
1/τ 1/τ^{2})
face of the hexakisicosahedron {1/τ 1 1}



g) (100) face of the rhombtriacontahedron intersected by the (7τ+5 τ+2 1)
face of the hexakisicosahedron {1+1/τ^{2 }1 1}

i) (100) face of the rhombtriacontahedron intersected by the
(τ^{5} 3τ2 1)
face of the hexakisicosahedron {1 1+1/τ^{2} 1+1/τ^{2}}


h) (100) face of the rhombtriacontahedron
intersected by the (τ 1/τ 0)
face of the icosahedron {111} 
Figure 4
The animated sequence shown in Figure 5
reports the views along the [001] direction and the corresponding stereographic projections relative to twentyfive composite forms
obtained from the intersection of the rhombtriacontahedron with forms that include {hkk} faces, indicated by a red circle in the
stereographic net, in which the ratio h/k varies in the interval 4τ+1 ≥ h/k ≥ 0
(the ratio h/k = 4τ+1 in a {hkk} form corresponds to the Catalan hexakisicosahedron).
REFERENCES and LINKS

International Union of Crystallography
International Tables for Crystallography, Vol. A, Theo Hahn Editor,
Kluwer Academic Publisher, 1989
 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:
IcosahedronDodecahedron and Archimedean PolyhedraCatalan Polyhedra
VisMath, volume 9, No. 4, 2007
 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