PLATONIC AND CATALAN POLYHEDRA AS ARCHETYPES OF FORMS
BELONGING TO THE CUBIC AND ICOSAHEDRAL SYSTEMS
Livio Zefiro* and Maria Rosa Ardigo'
*Dip.Te.Ris., Universita' di Genova, Italy
Sequential intersections of the seven forms, corresponding to Platonic and Catalan solids,
which characterize both m3m cubic
and m 3 5 icosahedral point groups. 
Notes
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Chrome at 1024x768 and 1280x1024 pixels
 all the images have been created by SHAPE 7.0, a software program for drawing the morphology of crystals,
distributed by Shape Software
 when the symbol of a pointer appears hovering with the mouse on the images,
if your web browser is set up to visualize VRML (Virtual Reality Modeling Language) 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...
A good Web3D visualizer can be found at this
address, even if, at present, it does not work with Google Chrome and the last release of Firefox.

The attractiveness of crystals must be credited mainly to their regular
morphology, which is due to the presence of symmetry operators. This holds true, in particular, of crystals belonging to the cubic system
(Figure 1),
and, to an even higher degree, of the icosahedral quasicrystals (Figure 2).
PYRITE
m3 point group

SPHALERITE
43m
point group


Figure 1
 Pleasant examples of cubic natural crystals deriving from the intersection
of many single forms, whose name and Miller's indices are displayed hovering with the mouse on every face of the three
crystals. 
(clicking on the following images, one can visualize the corresponding VRML files)
Figure 2
 Composite icosahedral forms belonging
to 235 (left) and m 3 5
(right) point groups. The central form is consistent with both point groups. 
Unlike the {hkl} general form, which is different
in the m 3 5
and 235 icosahedral point groups since only the former has mirror planes and inversion centre,
each of the six special forms is shared by the two icosahedral point groups [1].
On the whole, the forms belonging to each icosahedral point group are seven, as many as the forms in each of the five
cubic point groups, which, in general, are more familiar than the icosahedral
ones.
Focusing on the holohedral symmetries, both m3m cubic and
m 3 5 icosahedral point groups include,
in addition to two forms equal to Platonic solids
[2],
each dual of the other (cube and octahedron in
m3m, dodecahedron and
icosahedron in m 3 5),
another form with fixed Miller’s indices, equal to a Catalan solid:
in m3m point group it is the rhombdodecahedron {110},
dual of the quasiregular cuboctahedron,
whereas in m 3 5 it is the rhombtriacontahedron
{100}, dual of another quasiregular polyhedron, the icosidodecahedron (Figure 3).




cube 
dodecahedron 
rhombdodecahedron 
rhombtriacontahedron 




octahedron 
icosahedron 
cuboctahedron 
icosidodecahedron 
Figure 3  Couples of dual polyhedra made of two
Platonic polyhedra ( 1st and 2nd column), or a Catalan and a quasiregular polyhedra
(3rd and 4th column). It must be pointed out that a convex
quasiregular polyhedron is an Archimedean polyhedron in which
each edge is shared by two different faces consisting in regular polygons with
m
and nsides.
In these images, the faces are coloured in such a way that the two faces
belonging to each couple of edgesharing faces have different colours. 
The other four forms belonging to each point group are:

triakisicosahedron, deltoidal hexecontahedron, pentakisdodecahedron, hexakisicosahedron in case of
the m 3 5 point group

triakisoctahedron,
deltoidal icositetrahedron, tetrakishexahedron, hexakisoctahedron in case of
the m3m point group
All the forms characterizing the
m 3 5 and m3m
point groups can be compared in Table 1: the correlated icosahedral and cubic forms are shown on the same row.
(It must be pointed out that
m 3 5
and m3m are the
HermannMauguin point group symbols adopted by the International Tables
for Crystallography [1], in the short version; the more explanatory full symbols are
2/m 3 5 and
4/m 3 2/m, respectively).
Correlated cubic and icosahedral forms characterizing
the m 3 5
and m3m point groups 
Forms of the m 3 5
icosahedral point group (golden ratio τ = 1.61803...) 
Forms of the m3m cubic point group
(h>k>l) 
{1τ 0}
dodecahedron 
{100} cube 
{111} icosahedron 
{111} octahedron 
{100} rhombtriacontahedron 
{110} rhombdodecahedron 
{hk0} triakisicosahedron
(where: 0 < k/h <1/τ^{2}) 
{hhk} triakisoctahedron 
{hk0} deltoidal hexecontahedron
(where: 1/τ < h/k <τ^{2}) 
{kkh} deltoidal icositetrahedron 
{hk0} pentakisdodecahedron
(where: 0 < h/k <1/τ) 
{hk0} tetrakishexahedron 
{hkl} hexakisicosahedron 
{hkl} hexakisoctahedron 
Table 1 
Miller's indices and names of
the forms belonging to the m 3 5
and m3m
point groups. 
The forms having variable indices, reported on the last four rows of Table 1, can be led to coincide
with the corresponding Catalan solids by the attribution of appropriate values to their indices.
Therefore each of these Catalan polyhedra,
belonging to the m 3 5 icosahedral
or m3m
cubic point groups (Figure 4), is the archetype of a whole family of isomorphic forms.




triakisicosahedron
{τ^{2}+1 1/τ 0} 
deltoidal hexecontahedron
{τ+1/τ τ 0} 
pentakisdodecahedron
{1 3τ 0} 
hexakisicosahedron
{2 3/τ 1} 




triakisoctahedron
{1 1 √21} 
deltoidal icositetrahedron
{1 1 √2+1} 
tetrakishexahedron
{210} 
hexakisoctahedron
{2√2+1 √2+1
1} 
Figure 4  Further Catalan polyhedra belonging to the m 3 5
icosahedral and m3m cubic point groups
(first and second row, respectively) 
These Catalan polyhedra are the dual of the following semiregular Archimedean solids:

truncated dodecahedron, rhombicosidodecahedron, truncated icosahedron, truncated icosidodecahedron belonging to the
m 3 5 point group

truncated cube, rhombcuboctahedron, truncated octahedron, truncated cuboctahedron
belonging to the m3m point group
which are reported in Figure 5, in the same order of their duals shown in Fig.4.
Figure 5  Semiregular Archimedean solids, belonging to
the m 3 5
(first row) or m3m
point group (second row), duals of the Catalan solids shown in Figure 4.

As one can see in Table 2, analogously to what happens in case of the forms having icosahedral symmetry,
also the Miller's indices of three cubic Catalan forms do not take on integer values.
Sets of {hkl} indices relative to the forms,
belonging to m 3 5 icosahedral
and m3m cubic point groups,
which correspond to archetypal 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 
{τ+1/τ
τ 0}
{τ^{2}
1/τ
1/τ}
{2 1 τ} 
Deltoidal icositetrahedron 
{1
1 √2+1} 
Hexakisicosahedron 
{τ+2/τ
1/τ^{2}
1/τ^{2}}
{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 2 
Set of indices of the archetypal Platonic and Catalan forms belonging to m 3 5
and m3m point groups.
The indices of all the faces of the icosahedral polyhedra can be obtained from the cyclic permutation of a unique set of
indices only in case of the dodecahedron, whereas the sets of indices are two, three or five in case of the other six forms.
(Each set of indices corresponds to one of the forms belonging to the m3
point group, subgroup of m 3 5,
in which the icosahedron and the Catalan polyhedra can be decomposed) 
Whichever form deriving from single forms that belong, as an alternative, to m3m
or m 3 5 point groups, can be classified in one of the following three sets:

A first set of composite forms derives from every possible
intersection of three forms, all with fixed Miller's indices, which consist
of a Catalan solid (rhombdodecahedron or rhombtriacontahedron)
and two mutually dual Platonic solids (cube and octahedron or dodecahedron and icosahedron).
The three forms intersect concurrently only when the ratios between their distances from the centre of the
resulting polyhedron
vary within appropriate intervals; conversely, values of the ratios going out of range lead, at first,
to the coexistence of only two of the three forms and then to a unique form.

A second set includes single forms and intersections of any number
of forms whose archetypes, in m3m
or m 3 5 point groups,
are the other four Catalan solids: in this case the variability concerns both
the central distance and the indices of the forms.

A last set includes every combination of forms belonging coincidently to both previous sets.
More in detail, the composite forms belonging to the first set can be distributed in two subsets including respectively:

Isogonal (or vertex transitive) polyhedra, in which,
according to the definition given by Peter R. Cromwell [2]: "...
any vertex can be carried to any other by a symmetry operation":
their duals are isohedral (or face transitive)
polyhedra which belong to the second set.
When the isogonal polyhedra derive from the intersection of all the
three single forms, generally each vertex is shared by a rectangular face and two
symmetric but nonregular ones: a hexagonal and a decagonal face in
the m 3 5 point group,
a hexagonal and an octagonal face in the m3m point group (Figure 6).
The isohedral duals of such polyhedra are hexakisicosahedra or hexakisoctahedra, depending on the point group which they belong to.
Figure 6  Forms
of the first subset, given by the intersecton of three single forms
and belonging to the m 3 5
(upper row) or m3m point groups
(lower row), whose vertices are all equivalent by symmetry; their
isohedral duals are generic hexakisicosahedra or hexakisoctahedra, respectively. 
On the other hand, for particular values of the ratios between the central distances of the faces belonging to the three single forms
(see the
link),
each vertex of the composite form is shared by four faces, consisting
in two rectangles and two regular polygons: they are a triangle and a pentagon
in the m 3 5 point
group (Figure 7a), a triangle and a square in the m 3 5
point group (Figure 8a).
Their dual isohedral polyhedra are deltoidal hexecontahedra or deltoidal icositetrahedra, respectively.
Figure 7 
Isogonal forms of the first subset, belonging to the icosahedral
system, obtained in case of particular values of the ratio between the central distances of the single forms:
a) Polyhedron, given by the intersection of a dodecahedron, an icosahedron
and a rhombtriacontahedron, in which each vertex is shared by four faces: two regular faces, a triangular and a pentagonal one,
and two rectangular faces; its dual is a deltoidal hexecontahedron.
b) Truncation of a dodecahedron by an icosahedron: each vertex is
shared by a triangular and two nonregular decagonal faces; its dual is a triakisicosahedron
c) Truncation of an icosahedron by a dodecahedron: each vertex is
shared by a pentagonal and two nonregular hexagonal faces; its dual is a pentakisdodecahedron. 
Figure 8 
Isogonal forms of the first subset, belonging to the cubic system,
obtained in case of particular values of the ratio between the central distances of the single forms:
a) Polyhedron, given by the intersection of a cube, an octahedron and a rhombdodecahedron,
in which each vertex is shared by four faces: two regular faces, a triangular and a square one, and two rectangular faces;
its dual is a deltoidal icositetrahedron.
b) Truncation of a cube by an octahedron: each vertex is shared by a triangular
and two nonregular octagonal faces; its dual is a triakisoctahedron
c) Truncation of an octahedron by a cube: each vertex is shared by a square
and two nonregular hexagonal faces; its dual is a tetrakishexahedron. 
In the absence of the quasiregular polyhedron (the rhombtriacontahedron or the rhombdodecahedron,
according to the point group), the intersection of the two remaining Platonic solids gives as result,
depending on the ratio between their central distances:
1) in the m 3 5 point group: truncated dodecahedra (Figure 7b),
truncated icosahedra (Figure 7c) and, for the particular ratio
d_{dodecahedron} / d_{icosahedron}
= √3/√τ^{2}+1,
the icosidodecahedron
2) in the m3m point group: truncated cubes (Figure 8b), truncated octahedra (Figure 8c) and,
for the particular ratio d_{cube}
/ d_{octahedron} = √3/2,
the cuboctahedron.
Their respective duals are the following polyhedra belonging to the second set:
1) triakisicosahedra, pentakisdodecahedra and the rhombtriacontahedron in the m 3 5
point group
2) triakisoctahedra, tetrakishexahedra and the rhombdodecahedron in the m3m point group.
It is important to point out that all the isohedral duals belonging to the second set are isomorphic to the archetypal Catalan solids which,
together with the two Platonic polyhedra, characterize both
m 3 5 and m3m point groups.
b. Nonisogonal polyhedra, whose vertices are not all equivalent by symmetry operations (Figure 9).
When all the three forms are simultaneously present:
1) In the m 3 5 point group,
the pentagonal faces of the dodecahedron and the triangular faces of the icosahedron do not share either sides or vertices,
since they are separated by the nonregular octagonal faces of the rhombtriacontahedron.
Then each vertex is shared by two faces of the rhombtriacontahedron
and a third face, which can be a triangular (in sixty cases) or a pentagonal face (in the other sixty cases).
2) Analogously, in the m3m point group,
the square faces of the cube and the triangular faces of the octahedron do not share either sides or vertices,
since they are separated by the nonregular octagonal faces of the rhombdodecahedron.
Then each vertex is shared by two faces of the rhombdodecahedron and a third face,
which can be a triangular (in twentyfour cases) or a square face (in the other twentyfour cases).
Figure 9 
Examples of nonisogonal composite forms, belonging to the 2nd subset of the 1st set;
the lengths of the sides of the regular polygons are equal only in the
polyhedra of the central column.
In m 3 5 point group (upper row),
there are two kinds of vertices shared among three faces: two faces of rhombtriacontahedron, having a
symmetric but nonregular octagonal shape, and, alternatively, a pentagonal face of dodecahedron or a triangular face of icosahedron.
Analogously, also in the m3m
point group (lower row) there are two kinds of vertices,
shared among three faces: two faces of rhombdodecahedron, having a symmetric but nonregular octagonal shape, and, alternatively,
a square face of cube or a triangular face of octahedron.

Two are the different kinds of vertices even in the presence of only two forms,
namely a rhombtriacontahedron together with an icosahedron or a
dodecahedron in the m 3 5
point group, a rhombdodecahedron together with an octahedron or a cube
in the m3m point group.
The polyhedra resulting from the intersection of the previous two forms are shown in
the side columns of Figure 10, compared, in the central column,
with the polyhedron resulting from the intersection of the
rhombtriacontahedron or the rhombdodecahedron with both Platonic solids.
Figure 10
 Nonisogonal polyhedra resulting from the intersection of two forms
(in the lateral columns) or three forms (in the central column); their duals are shown in Figure 11. 
Consequently, their duals (shown in Figure 11) always consist of two forms, which can be of the same kind (two deltoidal hexecontahedra
in the m3 5 point group,
two deltoidal icositetrahedra in the m3m point group)
or different (a deltoidal hexecontahedron with alternatively an icosahedron or a dodecahedron
in the m 3 5 point group,
a deltoidal icositetrahedron with alternatively an octahedron or a
cube in the m3m point group).
(clicking on the following images, one can visualize the corresponding VRML files)
Figure 11  Duals of all kinds of nonisogonal composite forms
that belong to the 2nd subset of the first set; depending on the point group which they belong to, they are made of:
a deltoidal hexecontahedron and an icosahedron (left), two deltoidal hexecontahedra
(centre), a deltoidal hexecontahedron and a dodecahedron (right)
in m 3 5 point group (upper row).
a deltoidal icositetrahedron and an octahedron (left), two deltoidal
icositetrahedra (centre), a deltoidal icositetrahedron and a cube (right) in
m3m point group (lower row).
One can remark that the forms shown in each row can be also
derived from the partial or total augmentation [2] of an icosidodecahedron or a cuboctahedron, respectively
(the augmentation consists in the erection of a flat pyramid on the pentagonal, square or triangular faces,
so that the resulting polyhedron remains convex).

Examples of composite forms belonging to the second set are shown in
Figure 12 and Figure 13.
The first row of both figures reports three composite forms, deriving from the intersection
of the same three single forms: a tetrakishexahedron, a deltoidal icositetrahedron and a triakisoctahedron in
m3m, a pentakisdodecahedron, a deltoidal hexecontahedron and
a triakisicosahedron in m3 5.
Each image refers to different ratios between the central distances
of the single forms (the faces of the different forms are all equidistant only in the central image).
The archetypes of such three single forms, drawn in
the second row, are their isomorphic Catalan solids, in turn shown in the third row; the fourth row reports the composite forms, resulting from the
intersection of these Catalan solids, to be compared with the composite forms of the first row.
Even if, at a first look, the differences between the corresponding
single forms in the second and third rows are hardly noticeable, the composite forms in the fourth
row, which result from the intersection of the Catalan solids, change
drastically with respect to the composite forms in the first row.
Figure 12
(First row) Three composite forms, deriving from the intersection
of the same three single forms (a tetrakishexahedron, a deltoidal
icositetrahedron and a triakisoctahedron), obtained varying the
relative central distances of the single forms; the faces are all
equidistant only in case of the central image
(Second row) The three single forms whose intersection originates the previous composite forms
(Third row) The corresponding Catalan solids, archetypes of the previous isomorphic single forms
(Fourth row) Three composite forms deriving from the intersections of the three
Catalan solids: also here the faces are all equidistant only in case
of the central image.

Figure 13
(First row) Three composite forms, deriving from the
intersection of the same three single forms (a
pentakisdodecahedron, a deltoidal hexecontahedron and a
triakisicosahedron), obtained varying the relative central distances
of the single forms; the faces are all equidistant only in case of
the central image
(Second row) The three single forms whose intersection originates the previous composite forms
(Third row)
The corresponding Catalan solids, archetypes of the previous isomorphic single forms
(Fourth row)
Three composite forms, deriving from the intersections of the three
Catalan solids: also here the faces are all equidistant only in case
of the central image.

Some examples of generic composite forms belonging to the third set are displayed in
Figure 14.
(clicking on the following images, one can visualize the corresponding VRML files)
Figure 14
Generic composite polyhedra of the third set, resulting from
the intersection of forms belonging to m3m cubic point group
or m 3 5 icosahedral point group.
The single forms generating the composite forms are:
In m3m point group
(upper row)a cube, a rhombdodecahedron, an octahedron, a deltoidal icositetrahedron and two tetrakishexahedron
(on the left)
a cube, a rhombdodecahedron, a deltoidal icositetrahedron, two triakisoctahedra and a hexakisoctahedron
(on the right).
In m 3 5 point group (lower row)
an icosahedron, a rhombtriacontahedron, a triakisicosahedron and a deltoidal hexecontahedron
(on the left)
a dodecahedron, an icosahedron, a rhombtriacontahedron,
a triakisicosahedron, a pentakisdodecahedron and a hexakisicosahedron
(on the right).

Some very particular
composite forms, belonging to the third set, are reported in the next images (from Figure 15 to
Figure 18).
The figures show, in each row, polyhedra whose geometry is anyway correlated,
even if they are characterized by a different symmetry.
In fact, each couple of cubic (left) or icosahedral (right) forms, shown in the upper rows of the figures,
is made of faces having similar shapes (triangles, hexagons, rhombuses, kites...),
with the only exceptions of squares versus pentagons and octagons versus decagons.
Such composite forms derive from the intersection of an Archimedean polyhedron and its Catalan
dual, whose faces are at such a central distance that each vertex is shared by two couples of faces: a first couple consisting of
equal faces coming from the Catalan solid and a second one made of equal or different faces coming from the Archimedean solid.
It is important to point out that, just like the vertices of the isogonal Archimedean polyhedra, also
the vertices of each composite form are equidistant from the centre of the solid.
Consequently also all the faces of the respective duals, shown
in the lower row of the figures, have the same central distance, even if they belong to different forms.
The values of the indices of the single forms which constitute these duals are reported in the caption of the figures:
they are often function of √2 or τ,
depending on their cubic or icosahedral symmetry, respectively. In general, these single forms are not
Catalan solids (except the rhombdodecahedron and the rhombtriacontahedron):
they are only isomorphic to Catalan solids.
(clicking on the following images, one can visualize the corresponding VRML files)
Figure 15
(Upper row) Composite polyhedra resulting from
the intersection of an Archimedean solid and its Catalan dual: the truncated cube and the triakisoctahedron
{1 1 √21},
on the left the truncated dodecahedron and the triakisicosahedron
{τ^{2}+1
1/τ 0}, on the right The vertices are
all equidistant from the centre, in case of appropriate ratios between the
central distances of the faces.
(Lower row) Duals of the composite polyhedra reported in the upper row, consisting of:
the rhombdodecahedron {110} and the deltoidal icositetrahedron {1 1 √2},
on the left the rhombtriacontahedron {100} and the deltoidal hexecontahedron
{210}, on the right. In each dual, all the faces have the same central
distance. 
(clicking on the following images, one can visualize the corresponding VRML files)
Figure 16
(Upper row) Composite polyhedra resulting from
the intersection of an Archimedean solid and its Catalan dual:
the truncated octahedron and the tetrakishexahedron
{210},
on the left
the truncated icosahedron and the pentakisdodecahedron
{1 3τ 0},
on the right.
(Lower row) Duals of the composite polyhedra drawn in the upper row, consisting of:
the rhombdodecahedron
{110} and the deltoidal icositetrahedron
{114},
on the left
the rhombtriacontahedron
{100} and the deltoidal hexecontahedron
{τ+1/τ 2 0},
on the right.

(clicking on the following images, one can visualize the corresponding VRML files)
Figure 17
(Upper row) Composite polyhedra resulting from
the intersection of an Archimedean solid and its Catalan dual: the rhombcuboctahedron and the deltoidal
icositetrahedron
{1
1 √2+1},
on the left the rhombicosidodecahedron and the deltoidal hexecontahedron
{1+1/τ^{2} 1
0}, on the right.
(Lower row) Duals of the composite polyhedra drawn in the upper row, consisting of:
the tetrakishexahedron
{√2+1
1 0} and the triakisoctahedron
{1
1 2√2},
on the left
the pentakisdodecahedron
{1 τ^{3} 0}
and the triakisicosahedron
{τ^{3} 1 0},
on
the right. Therefore both these duals belong to the second set previously defined.

(clicking on the following images, one can visualize the corresponding VRML files)
Figure 18
(Upper row)
Composite polyhedra resulting from
the intersection of an Archimedean solid and its Catalan dual: the truncated cuboctahedron and the hexakisoctahedron
{2√2+1 √2+1
1}, on the left
the truncated icosidodecahedron and the hexakisicosahedron
{2 3/τ 1},
on the right.
(Lower row) Duals of the composite polyhedra drawn in the upper row, consisting of:
the triakisoctahedron {1+3/√2
1+3/√2 1}, the deltoidal icositetrahedron
{1 1 3√22}
and the tetrakishexahedron {3√2 1 0},
on the left the triakisicosahedron {4τ+1 1 0},
the deltoidal hexecontahedron {4τ5 1 0} and the pentakisdodecahedron
{1 4τ+1 0}, on the right.
Therefore also these duals belong to the second set.

A different situation concerns the two couples
consisting of an Archimedean quasiregular solid and its Catalan dual: the cuboctahedron and the rhombdodecahedron in
m3m point group
or the icosidodecahedron and the rhombtriacontahedron in
m 3 5 point group.
As it was already pointed out in a previous
work
relative to the icosahedral polyhedra, neither the Archimedean truncated icosidodecahedron
nor the rhombicosidodecahedron can be obtained just truncating an icosidodecahedron by means of a rhombic triacontahedron,
but they derive
from the intersection of an icosahedron, a dodecahedron and a rhombic triacontahedron
in case of appropriate values of the ratios between their central distances.
Analogously, neither the Archimedean truncated cuboctahedron nor the
rhombcuboctahedron can be obtained truncating a cuboctahedron
by means of a rhombdodecahedron, but they too derive from the intersection of an octahedron,
a cube and a rhombdodecahedron in another welldefined case (incidentally, both names "truncated cuboctahedron" and "truncated
icosidodecahedron" are therefore misleading, so that, as an alternative,
they are named "great rhombcuboctahedron" and "great rhombicosidodecahedron", respectively).
On the other hand, letting unchanged the ratio:
d_{cube} /d_{octahedron} = √3/2
or d_{icosahedron} /d_{dodecahedron} = √(τ^{2}+1)/3
which characterizes the cuboctahedron or the icosidodecahedron,
when the central distance of the faces belonging to the truncating Catalan solids is such that:
d_{rhombdodecahedron} /d_{octahedron}
= (3√3/2)/4 or
d_{rhombtriacontahedron} /d_{icosahedron} = √15/4,
the truncation can generate the particular isogonal forms shown in the upper row of Figure 19.
It must be mentioned that such forms are special cases of the forms reported in Fig.7a and Fig. 8a: one can see
that each vertex is shared by four faces including, in both cubic and icosahedral polyhedra, a triangular and two rectangular faces,
whereas the fourth one is alternatively a square or a pentagonal face
Consequently, each of their duals (shown in the lower row of the same figure) is an isohedral form, belonging to the second set,
which derives from the relative Catalan archetype: the deltoidal icositetrahedron or the deltoidal hexecontahedron.
(clicking on the following images, one can visualize the corresponding VRML files)
Figure 19
Upper row: isogonal polyhedra resulting from the
truncation of an Archimedean quasiregular solid by its Catalan dual (cuboctahedron and rhombdodecahedron
on the left, icosidodecahedron and rhombtriacontahedron on the right);
appropriate ratios between the central distances of the faces lead, in both
cases, to vertices equidistant from the centre of the solid.
Lower row: {112} deltoidal icositetrahedron
(on the left) and {τ10} deltoidal hexecontahedron (on the right);
they are the isohedral duals of the isogonal polyhedra shown in the upper row. 
Altogether, ten out of the thirteen isohedral Catalan solids are equally distributed between
m 3 5 and
m3m point groups.
As regards the other three Catalan solids, shown in
Figure 20:

the triakistetrahedron belongs to the 43m
cubic point group (as well as its Archimedean dual, the truncated tetrahedron,
and the tetrahedron, that is the only selfdual Platonic solid)

the pentagonal icositetrahedron and its Archimedean dual, the snub cube,
belong to the 432 cubic point group

the pentagonal hexecontahedron and its Archimedean dual, the snub dodecahedron,
belong to the 235 icosahedral point group.
Figure 20  Selfdual Platonic
tetrahedra (1st column) and, in the other three columns, the remaining Catalan and Archimedean
dual polyhedra: triakistetrahedron and truncated tetrahedron
(2nd column), pentagonal
icositetrahedron and snub cube (3rd column), pentagonal hexecontahedron and snub dodecahedron
(4th column). 
Figure 21 shows, on the left, the intersection of the Archimedean truncated
tetrahedron with its Catalan dual, the triakistetrahedron {113}, whose faces
are placed at an appropriate central distance so that, in correspondence to each vertex of the resulting polyhedron, two faces
of the triakistetrahedron are shared with two different faces of the
truncated tetrahedron (hexagonal the smaller face, triangular the larger one)
or, alternatively, with the only hexagonal face.
Since all the vertices have an equal central distance, also the faces of its dual, made of two
forms: the deltoiddodecahedron {221} and the triakistetrahedron
{11√3},
are all at the same central distance.
(clicking on the following images, one can visualize the corresponding VRML files)
Figure 21
On the left:
composite polyhedron resulting from the intersection of the Archimedean
truncated tetrahedron with its Catalan dual, the triakistetrahedron
{113}, in case of appropriate ratios between the central distances of
their faces leading to vertices equidistant from the centre.
On the right: its dual
made of the deltoiddodecahedron {221}
and the
triakistetrahedron {11√3}, both at the same central distance. 
The composite polyhedra, having all the vertices at an equal central distance, which result from the
intersection of the last two dual couples made of an Archimedean and a Catalan solid
(the snub cube and the pentagonal icositetrahedron, the snub dodecahedron and the pentagonal hexecontahedron),
are shown in the upper row of Figure 22, whereas in the lower row one can see their respective duals.
(clicking on the following images, one can visualize the corresponding VRML files)
Figure 22
Upper row: composite polyhedra resulting from the intersection of an Archimedean solid
and its Catalan dual (the snub cube and the pentagonal icositetrahedron
on the left, the snub dodecahedron
and the pentagonal hexecontahedron on the right),
in case of appropriate ratios between the central distances of their faces,
leading to vertices shared by four faces: two faces coming from the Archimedean polyhedron and two faces from the Catalan polyhedron.
Lower row: duals of the composite polyhedra drawn in the upper row;
they consist of a rhombdodecahedron and two pentagonal icositetrahedra
(on the left),
a rhombtriacontahedron and two pentagonal hexecontahedra
(on the right),
whose faces have all the same central distance.

In Table 3 the features of all the Platonic and Catalan solids, archetypes of
the cubic and icosahedral single forms, are listed and compared with the
features of the dual Platonic and Archimedean solids.
Table 3  Faces, vertices and edges in Platonic,
Catalan and Archimedean polyhedra. 
From the comparison of Table 4 and Table 5 one can ascertain
that the faces of the cubic and icosahedral forms corresponding to Catalan polyhedra have, in pairs,
analogous shapes (the only unpaired Catalan polyhedron is the triakistetrahedron):
they differ only in the values of the geometrical parameters.
Shape and geometrical
parameters of the faces belonging to the cubic Catalan polyhedra 
Catalan polyhedra 
Shape of the faces 
Geometry of the faces 
Angles
between the sides of each polygon 
Triakistetrahedron 
isosceles triangle 
L_{1}/ L_{2} = 0.532
Height of the triangular pyramids = 0.163L_{2 } 
2x 33.56° 1x 112.88°

Rhombdodecahedron 
rhombus 
Ratio of the
diagonals =
√2
=
1.414 
2x 109.47°
2x 70.53° 
Tetrakishexahedron 
isosceles triangle 
L_{1 }/ L_{2}
= 3/4
Height of the tetragonal pyramids = L_{2 }/ 4 
2x 48.19° 1x
83.62° 
Triakisoctahedron 
isosceles triangle 
L_{1}/ L_{2} = 0.586
Height of the triangular pyramids = 0.099L_{2 } 
2x 31.40° 1x 117.20°

Deltoidal icositetrahedron 
deltoid (or kite) 
L_{1}/ L_{2} = 1.293 
3x 81.58° 1x 115.26°

Hexakisoctahedron 
scalene triangle 
L_{1}/ L_{2} = 1.219
L_{1}/ L_{3} = 1.631 
1x 87.20°
1x 55.03°
1x 37.78° 
Pentagonal icositetrahedron

nonregular symmetric pentagon
with 2 equal long sides and 3 equal short sides 
L_{1}/ L_{2} = 1.420 
4x 114.81° 1x 80.75° 
Table 4 
List of the cubic Catalan polyhedra,
shape and geometrical parameters of their faces, including the angles between the sides
L_{i} of each polygon.

Shape and geometrical
parameters of the faces belonging to the icosahedral Catalan polyhedra

Catalan polyhedra 
Shape of the faces 
Geometry of the faces 
Angles
between the sides of each polygon 
Rhombtriacontahedron 
rhombus 
Ratio of the
diagonals = τ = 1.618 
2x 116.565°
2x 63.435° 
Pentakisdodecahedron 
isosceles triangle 
L_{1}/L_{2} = 0.887
Height of the pentagonal pyramids = 0.251L_{2} 
2x 55.69° 1x 68.62° 
Triakisicosahedron 
isosceles triangle 
L_{1}/L_{2} = 0.580
Height of the triangular pyramids = 0.057L_{2} 
2x 30.48° 1x 19.04° 
Deltoidal hexecontahedron 
deltoid (or kite) 
L_{1}/L_{2} = 1+ τ/3 =
1.539 
2x 86.97° 1x 118.27°
1x 67.78° 
Hexakisicosahedron 
scalene triangle 
L_{1}/L_{2}
= (2/3)(52τ)
= 1.176
L_{2}/L_{3}
= (3/5)τ^{2}
= 1.571
L_{1}/L_{3}
= (2/5)(τ+3) = 1.847 
1x 88.99°
1x 58.24°
1x 32.77° 
Pentagonal hexecontahedron

nonregular symmetric pentagon
with 2 equal long sides and 3 equal short sides 
L_{1}/L_{2} = 1.750 
4x 118.14° 1x 67.44° 
Table 5  List of the icosahedral Catalan polyhedra, shape and geometrical parameters
of their faces, including the angles between the sides L_{i} of each polygon. 
The Platonic and Catalan polyhedra, archetypes of the forms that belong to both
m3m and 432 cubic
point groups, are shown in Figure 23, together with the composite form
resulting from the intersection of all the single forms relative to each point group.
In detail:
 The first row includes the simplest forms belonging to the cubic system, namely the two dual Platonic
polyhedra (cube and octahedron) and the rhombdodecahedron, whereas the third
row includes the other special forms corresponding to Catalan solids.
 The second and the third image from left in the second row show the two general forms corresponding to further Catalan polyhedra,
the hexakisoctahedron and the pentagonal icositetrahedron,
relative to the m3m and 432 cubic point groups,
respectively. The two general forms are different, consequently to the lack, in the 432 point group, of the
other symmetry operators present in the m3m point group, namely the
nine mirror planes and the inversion centre.
 The other two images in the second row, namely the first and the fourth one from left, represent the composite forms,
relative to the m3m and
432 cubic point groups, given by the intersection of the seven single
forms, all with an equal central distance, including the general form which characterizes each point group.
Figure 23 
Composite forms (symmetry elements included) resulting from the
intersection, in m3m and 432 cubic point groups, of the
equidistant faces belonging to the two Platonic and to the five Catalan polyhedra, archetypes of the
relative isomorphic forms. 
Analogously, the Platonic and Catalan polyhedra, archetypes of all the forms that belong to
m 3 5
and
235 icosahedral point groups, are shown in
Figure 24,
together with the composite form resulting from the intersection of all the single forms relative to each point group.
In detail:
 The first row includes the simplest forms belonging to the icosahedral system,
namely the two dual Platonic polyhedra (dodecahedron and icosahedron) and the rhombtriacontahedron,
whereas the third row includes the other special forms corresponding to Catalan solids.

The second and the third image from left in the second row show the two general forms corresponding to further Catalan polyhedra,
the hexakisicosahedron and the pentagonal hexecontahedron,
relative to the m 3 5
and 235 icosahedral point groups, respectively.
The two general forms are different, consequently to the lack, in the 235 point group,
of the other symmetry operators present
in the m 3 5
point group, namely the fifteeen mirror planes and the inversion centre.

The other two images in the second row, namely the first and the fourth one from left, represent the composite forms,
relative to the m 3 5
and 235 icosahedral point groups, given by the intersection of the seven single forms, all with an equal central distance,
including the general form which characterizes each point group.
Figure 24  Composite forms (symmetry elements included) resulting from the intersection, in m 3 5 and
235 icosahedral point groups,
of the equidistant faces belonging to the two Platonic and to the five Catalan polyhedra, archetypes of the
relative isomorphic forms. 
REFERENCES