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
  • tested with Internet Explorer 6, Mozilla Firefox 2, Opera 9.51 and Google 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 quasi-crystals (Figure 2).

PYRITE
m3 point group

SPHALERITE
43m point group

GARNET
m3m 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 rhomb-dodecahedron {110}, dual of the quasi-regular cuboctahedron, whereas in m 3 5 it is the rhomb-triacontahedron {100}, dual of another quasi-regular polyhedron, the icosi-dodecahedron (Figure 3).

cube dodecahedron rhomb-dodecahedron rhomb-triacontahedron

octahedron icosahedron cuboctahedron icosi-dodecahedron

Figure 3 - Couples of dual polyhedra made of two Platonic polyhedra ( 1st and 2nd column), or a Catalan and a quasi-regular polyhedra (3rd and 4th column). It must be pointed out that a convex quasi-regular polyhedron is an Archimedean  polyhedron in which each edge is shared by two different faces consisting in regular polygons with m- and n-sides.
In these images, the faces are coloured in such a way that the two faces belonging to each couple of edge-sharing faces have different colours.

The other four forms belonging to each point group are:

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 Hermann-Mauguin 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} rhomb-triacontahedron {110} rhomb-dodecahedron
{hk0} triakis-icosahedron (where: 0 < k/h <1/τ2) {hhk} triakis-octahedron
{hk0} deltoidal hexecontahedron (where: 1/τ < h/k <τ2) {kkh} deltoidal icositetrahedron
{hk0} pentakis-dodecahedron (where: 0 < h/k <1/τ) {hk0} tetrakis-hexahedron
{hkl} hexakis-icosahedron {hkl} hexakis-octahedron
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.

triakis-icosahedron
{τ2+1 1/τ  0}		 deltoidal hexecontahedron
{τ+1/τ  τ 0}		 pentakis-dodecahedron
{1 3τ 0}		 hexakis-icosahedron
{2 3/τ 1}

triakis-octahedron
{1 1 √2-1}		 deltoidal icositetrahedron
{1 1 √2+1}		 tetrakis-hexahedron
{210}		  hexakis-octahedron
{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 semi-regular Archimedean solids: which are reported in Figure 5, in the same order of their duals shown in Fig.4.
Figure 5 - Semi-regular 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}
Rhomb-triacontahedron {100}
{τ 1 1/τ}		 Rhomb-dodecahedron {110}
Pentakis-dodecahedron {1/τ 3 0}
{τ2 1 2/τ}
{τ+1/τ 2 1/τ}		 Tetrakis-hexahedron {210}
Triakis-icosahedron {τ+1/τ 1/τ2 0}
{τ 2/τ 1}
{2 1 1/τ2}		 Triakis-octahedron {1 1  √2-1}
Deltoidal hexecontahedron {τ+1/τ  τ 0}
{τ2 1/τ 1/τ}
{2 1 τ} 		Deltoidal icositetrahedron {1 1  √2+1} 
Hexakis-icosahedron {τ+2/τ 1/τ2 1/τ2}
{τ2 1 2/τ2}
{2+1/τ2 τ 1/τ2}
 {2 3/τ 1}
 {τ+1/τ  2/τ  1+1/τ2}		 Hexakis-octahedron
  {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:
  1. 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 (rhomb-dodecahedron or rhomb-triacontahedron) 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.
  2. 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.
  3. 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:
  1. 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 non-regular 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 hexakis-icosahedra or hexakis-octahedra, 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 hexakis-icosahedra or hexakis-octahedra, 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 icosi-tetrahedra, 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 rhomb-triacontahedron, 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 non-regular decagonal faces; its dual is a triakis-icosahedron
c) Truncation of an icosahedron by a dodecahedron: each vertex is shared by a pentagonal and two non-regular hexagonal faces; its dual is a pentakis-dodecahedron.


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 rhomb-dodecahedron, 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 non-regular octagonal faces; its dual is a triakis-octahedron
c) Truncation of an octahedron by a cube: each vertex is shared by a square and two non-regular hexagonal faces; its dual is a tetrakis-hexahedron.

In the absence of the quasi-regular polyhedron (the rhomb-triacontahedron or the rhomb-dodecahedron, 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 ddodecahedron / dicosahedron = √3/√τ2+1, the icosi-dodecahedron
2) in the m3m point group: truncated cubes (Figure 8b), truncated octahedra (Figure 8c) and, for the particular ratio dcube / doctahedron = √3/2,  the cuboctahedron.
Their respective duals are the following polyhedra belonging to the second set:
1) triakis-icosahedra, pentakis-dodecahedra and the rhomb-triacontahedron in the m 3 5 point group
2) triakis-octahedra, tetrakis-hexahedra and the rhomb-dodecahedron 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.   Non-isogonal 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 non-regular octagonal faces of the rhomb-triacontahedron.
Then each vertex is shared by two faces of the rhomb-triacontahedron 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 non-regular octagonal faces of the rhomb-dodecahedron.
Then each vertex is shared by two faces of the rhomb-dodecahedron and a third face, which can be a triangular (in twenty-four cases) or a square face (in the other twenty-four cases).

Figure 9  - Examples of non-isogonal 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 rhomb-triacontahedron, having a symmetric but non-regular 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 rhomb-dodecahedron, having a symmetric but non-regular 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 rhomb-triacontahedron together with an icosahedron or a dodecahedron in the m 3 5 point group, a rhomb-dodecahedron 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 rhomb-triacontahedron or the rhomb-dodecahedron with both Platonic solids.
 

Figure 10 - Non-isogonal 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 icosi-tetrahedra 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 icosi-tetrahedron 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 non-isogonal 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 icosi-tetrahedron and an octahedron (left), two deltoidal icosi-tetrahedra (centre), a deltoidal icosi-tetrahedron 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 icosi-dodecahedron 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 tetrakis-hexahedron, a deltoidal icosi-tetrahedron and a triakis-octahedron in m3m, a pentakis-dodecahedron, a deltoidal hexecontahedron and a triakis-icosahedron 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 tetrakis-hexahedron, a deltoidal icosi-tetrahedron and a triakis-octahedron), 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 pentakis-dodecahedron, a deltoidal hexecontahedron and a triakis-icosahedron), 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 rhomb-dodecahedron, an octahedron, a deltoidal icosi-tetrahedron and two tetrakis-hexahedron (on the left)
  • a cube, a rhomb-dodecahedron, a deltoidal icosi-tetrahedron, two triakis-octahedra and a hexakis-octahedron (on the right).
     In m 3 5  point group (lower row)
  • an icosahedron, a rhomb-triacontahedron, a triakis-icosahedron and a deltoidal hexecontahedron (on the left)
  • a dodecahedron, an icosahedron, a rhomb-triacontahedron, a triakis-icosahedron, a pentakis-dodecahedron and a hexakis-icosahedron (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 rhomb-dodecahedron and the rhomb-triacontahedron): 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 triakis-octahedron {1 1 √2-1}, on the left
  • the truncated dodecahedron and the triakis-icosahedron {τ2+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 rhomb-dodecahedron {110} and the deltoidal icositetrahedron {1 1 √2}, on the left
  • the rhomb-triacontahedron {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 tetrakis-hexahedron {210}, on the left
  • the truncated icosahedron and the pentakis-dodecahedron {1 3τ 0}, on the right.
    (Lower row) Duals of the composite polyhedra drawn in the upper row, consisting of:
  • the rhomb-dodecahedron {110} and the deltoidal icositetrahedron {114}, on the left
  • the rhomb-triacontahedron {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 rhomb-cuboctahedron and the deltoidal icositetrahedron {1 1  √2+1}, on the left
  • the rhomb-icosidodecahedron and the deltoidal hexecontahedron {1+1/τ2 1 0}, on the right.
    (Lower row) Duals of the composite polyhedra drawn in the upper row, consisting of:
  • the tetrakis-hexahedron {√2+1 1 0} and the triakis-octahedron {1 1  2-√2}, on the left
  • the pentakis-dodecahedron {1 τ3 0} and the triakis-icosahedron {τ3 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 hexakis-octahedron {2√2+1 √2+1 1}, on the left
  • the truncated icosi-dodecahedron and the hexakis-icosahedron {2 3/τ 1}, on the right.
    (Lower row) Duals of the composite polyhedra drawn in the upper row, consisting of:
  • the triakis-octahedron {1+3/√2 1+3/√2 1}, the deltoidal icosi-tetrahedron {1 1 3√2-2} and the tetrakis-hexahedron {3-√2 1 0}, on the left
  • the triakis-icosahedron {4τ+1 1 0}, the deltoidal hexecontahedron {4τ-5 1 0} and the pentakis-dodecahedron {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 quasi-regular solid and its Catalan dual: the cuboctahedron and the rhomb-dodecahedron in m3m point group or the icosi-dodecahedron and the rhomb-triacontahedron 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 icosi-dodecahedron nor the rhomb-icosidodecahedron 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 rhomb-cuboctahedron can be obtained truncating a cuboctahedron by means of a rhomb-dodecahedron, but they too derive from the intersection of an octahedron, a cube and a rhomb-dodecahedron in another well-defined case (incidentally, both names "truncated cuboctahedron" and "truncated icosi-dodecahedron" are therefore misleading, so that, as an alternative, they are named "great rhomb-cuboctahedron" and "great rhomb-icosidodecahedron", respectively).
    On the other hand, letting unchanged the ratio:
    dcube /doctahedron = √3/2   or   dicosahedron /ddodecahedron = √(τ2+1)/3
    which characterizes the cuboctahedron or the icosi-dodecahedron, when the central distance of the faces belonging to the truncating Catalan solids is such that:
    drhomb-dodecahedron /doctahedron = (3√3/2)/4   or   drhomb-triacontahedron /dicosahedron = √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 icosi-tetrahedron 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 quasi-regular solid by its Catalan dual (cuboctahedron and rhomb-dodecahedron on the left, icosi-dodecahedron and rhomb-triacontahedron 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 icosi-tetrahedron (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 triakis-tetrahedron belongs to the 43m cubic point group (as well as its Archimedean dual, the truncated tetrahedron, and the tetrahedron, that is the only self-dual 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 - Self-dual Platonic tetrahedra (1st column) and, in the other three columns, the remaining Catalan and Archimedean dual polyhedra: triakis-tetrahedron and truncated tetrahedron (2nd column), pentagonal icosi-tetrahedron 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 triakis-tetrahedron {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 triakis-tetrahedron 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 deltoid-dodecahedron {221} and the triakis-tetrahedron {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 triakis-tetrahedron {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 deltoid-dodecahedron {221} and the triakis-tetrahedron {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 icosi-tetrahedron, 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 icosi-tetrahedron 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 rhomb-dodecahedron and two pentagonal icositetrahedra (on the left), a rhomb-triacontahedron 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 triakis-tetrahedron): 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
    Triakis-tetrahedron isosceles triangle L1/ L2 = 0.532
    Height of the triangular pyramids = 0.163L2     		  2x   33.56°
      1x 112.88°
    Rhomb-dodecahedron rhombus Ratio of the diagonals = √2 = 1.414

       2x 109.47°
       2x   70.53°

    Tetrakis-hexahedron isosceles triangle L1 / L2 = 3/4
    Height of the tetragonal pyramids = L2 / 4     		  2x  48.19°
      1x  83.62°
    Triakis-octahedron isosceles triangle L1/ L2 = 0.586
    Height of the triangular pyramids = 0.099L2     		  2x   31.40°
      1x 117.20°
    Deltoidal icositetrahedron deltoid (or kite) L1/ L2 = 1.293   3x  81.58°
      1x 115.26°
    Hexakis-octahedron scalene triangle L1/ L2 = 1.219
    L1/ L3 = 1.631    		  1x  87.20°
     1x  55.03°
     1x  37.78°
    Pentagonal icositetrahedron non-regular
    symmetric pentagon
    with 2 equal long sides
    and 3 equal short sides     		L1/ L2 = 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 Li 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
    Rhomb-triacontahedron rhombus Ratio of the diagonals = τ = 1.618   2x 116.565° 
      2x   63.435°
    Pentakis-dodecahedron isosceles triangle L1/L2 = 0.887
    Height of the pentagonal pyramids = 0.251L2    		 2x   55.69°
    1x   68.62°
    Triakis-icosahedron isosceles triangle L1/L2 = 0.580
    Height of the triangular pyramids = 0.057L2    		 2x   30.48°
    1x   19.04°
    Deltoidal hexecontahedron deltoid (or kite) L1/L2 = 1+ τ/3 = 1.539 2x    86.97°
    1x  118.27°
    1x    67.78°
    Hexakis-icosahedron scalene triangle L1/L2 = (2/3)(5-2τ) = 1.176
    L2/L3 = (3/5)    τ2 = 1.571
    L1/L3 = (2/5)(τ+3) = 1.847    		 1x    88.99°
    1x    58.24°
    1x   32.77°
    Pentagonal hexecontahedron non-regular
    symmetric pentagon
    with 2 equal long sides
    and 3 equal short sides     		L1/L2 = 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 Li 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 rhomb-dodecahedron, 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 hexakis-octahedron and the pentagonal icosi-tetrahedron, 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 rhomb-triacontahedron, 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 hexakis-icosahedron 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