Icosahedral Polyhedra

Description of the Forms Belonging to the 235 and m35 Icosahedral Point Groups Starting from the Pairs of Dual Polyhedra: Icosahedron-Dodecahedron and Archimedean Polyhedra-Catalan Polyhedra

It is well known that the regular convex polyhedra (or Platonic solids) are only five, namely:
tetrahedron
cube
octahedron
dodecahedron
icosahedron

TETRAHEDRON	CUBE	OCTAHEDRON	DODECAHEDRON	ICOSAHEDRON

FIG.1 - The five regular convex polyhedra (or Platonic solids), coloured in order to evidence the single faces.

The last two polyhedra are less common than the others, because of their higher complexity, and because no specimen of crystals whose habit includes such forms has ever been found in Nature. The reason is the incompatibility of the 5-fold rotation axes, orthogonal to each face of the dodecahedron and passing through each vertex of the icosahedron, with any crystal lattice [1].
The study of the coexistence rules of polyhedral forms, belonging to 235 and m35 icosahedral point groups (the second one named accordingly to [2], whereas elsewhere it is alternatively named m35), remained for a long time a mere theoretical (and aesthetic) exercise. A new motivation to deepen such studies [3-6] was given by the discovery of the so-called quasicrystals: they are essentially metal alloys that, as a consequence of a quick cooling, revealed at morphological and diffractometric examinations the surprising presence of a 5-fold rotational symmetry; it probably results from the presence in the atomic structure of compact packings, made of a limited number of atoms, with a symmetry of this kind.

In addition to the aforementioned dodecahedron and icosahedron, a further polyhedron with a limited multiplicity (i.e., with a low number of faces) is the rhombic triacontahedron, characterized by the presence of thirty rhombic faces. Being the dual of the icosi-dodecahedron, namely a semiregular Archimedean polyhedron, it belongs to the family of Catalan polyhedra (recall that duality implies the exchange of faces and vertices between two polyhedra, both in number and position, whereas the number of edges is unchanged).

The following Archimedean polyhedra, whose faces consist of different pairs of regular polygons:

truncated icosahedron (20 hexagonal and 12 pentagonal faces)

truncated dodecahedron (20 triangular and 12 decagonal faces)

icosi-dodecahedron (20 triangular and 12 pentagonal faces)

are generated by the intersection between a dodecahedron and an icosahedron, when the ratio of the distances of the respective faces from the centre of the resulting polyhedron gets the value appropriate for each pair of polygons.

FIG.2 - The icosi-dodecahedron (at the centre of the sequence) is a semiregular Archimedean polyhedron that can be obtained by the reciprocal truncation of an icosahedron and a dodecahedron, both regular Platonic polyhedra.
At intermediate steps of the truncation process, one can obtain two other Archimedean polyhedra: a truncated icosahedron, starting from the icosahedron, and a truncated dodecahedron, starting from the dodecahedron.
From left to right:
icosahedron
truncated icosahedron
icosi-dodecahedron
truncated dodecahedron
dodecahedron.

Two further Archimedean polyhedra, whose faces belong to different sets of three regular polygons:

truncated icosi-dodecahedron (12 decagonal, 20 hexagonal and 30 square faces)

rhombicosi-dodecahedron (12 pentagonal, 20 triangular and 30 square faces)

are instead generated from the intersection of three polyhedra: a dodecahedron, an icosahedron and a rhombic triacontahedron, also in these cases for appropriate values of the ratios of the distances of the different faces from the centre of the resulting polyhedron (named central distances in the sequel).

FIG.3 - The truncated icosido-decahedron (on the left) and the rhombic icosi-dodecahedron (on the right) are a couple of Archimedean polyhedra generated from the intersection of three polyhedra: an icosahedron, a dodecahedron and a rhombic triacontahedron. When the ratios between the distances of their faces from the centre of the resulting polyhedron get appropriate values, the faces are regular polygons: together with squares, in the first case there are decagons and hexagons that, in the second case, become pentagons and triangles, respectively.

The duals of all the previous Archimedean polyhedra are the following Catalan polyhedra:

pentakis-dodecahedron (60 faces with the shape of an isosceles triangle) is the dual of the truncated icosahedron

triakis-icosahedron (60 faces with the shape of an isosceles triangle) is the dual of the truncated dodecahedron

rhomb-triacontahedron (30 rhombic faces) is the dual of the icosi-dodecahedron

hexakis-icosahedron (120 faces with the shape of a scalene triangle) is the dual of the truncated icosi-dodecahedron

deltoid-hexecontahedron (60 faces with the shape of a deltoid or kite) is the dual of the rhombicosi-dodecahedron

Catalan polyhedra are isohedral (or face-transitive), since in each polyhedron the faces have the shape of a unique, non-regular polygon, even though generally symmetric (except the hexakis-icosahedron). On the other hand, whereas in Catalan polyhedra the angoloids aren't all equal, the dual Archimedean polyhedra are isogonal (or vertex transitive), since their faces meet in identical vertices.


FIG.4 - Catalan polyhedra dual of the Archimedean polyhedra previously listed From left to right: pentakis-dodecahedron, dual of the truncated icosahedron triakis-icosahedron, dual of the truncated dodecahedron rhomb-triacontahedron (or rhomb-triacontahedron), dual of the icosi-dodecahedron hexakis-icosahedron, dual of the truncated icosi-dodecahedron deltoid-hexecontahedron (or deltoid-hexecontahedron), dual of the rhombicosi-dodecahedron.

The hexakis-icosahedron is the sole with 120 asymmetric faces; unlike all the other forms it is compatible only with the m35 point group, whereas another Catalan polyhedron, the pentagonal hexecontahedron, made of 60 pentagonal faces (each symmetric with respect to a mirror plane), belongs only to 235, the other icosahedral point group.

The dual of the pentagonal hexecontahedron is the snub dodecahedron, Archimedean polyhedron with a rather singular look, resulting from the intersection of the three following forms:

a dodecahedron

an icosaedron

a pentagonal hexacontahedron
Even though the sixty faces of this pentagonal hexecontahedron are completely asymmetric (unlike the faces of the pentagonal hexecontahedron dual of the snub dodecahedron), due to the intersection with the dodecahedral and icosahedral faces, they assume the shape of equilateral triangles, having the same dimensions of the twenty triangular faces coming from the icosahedron, for appropriate values of both:

the Miller indices of the faces of the pentagonal hexecontahedron
and

the ratios between the central distances of the three forms.
Therefore all the eighty triangular faces, together with the twelve pentagonal faces of the dodecahedron, originate the isogonal snub dodecahedron made of regular polygons, that is an Archimedean polyhedron.

The absence of mirror planes and centre of symmetry implies that, because of enantiomorphism, in this case there are two chiral couples of dual polyhedra: each couple is made of a snub dodecahedron and a pentagonal hexecontahedron, both right or left.

FIG.5 - Couple of dual polyhedra: Catalan pentagonal hexecontahedron (on the left) and Archimedean snub dodecahedron (on the right).

In all the icosahedral polyhedra made of different forms, the positions, with respect to the reference axes, of dodecahedral, icosahedral and rhomb-triacontahedral faces are always fixed, being perpendicular to the axes characterizing the icosahedral symmetry: 5-fold, 3-fold and 2-fold rotation axes, respectively.
On the other hand, each other single generic form has usually an orientation different from the one of the corresponding Catalan polyhedron, and it is subject only to the condition:

to be parallel to the 2-fold axes (in case of triakis-icosahedra, deltoidal hexecontahedra and pentakis-dodecahedra)
or, as an alternative

to be in a general position as regards every symmetry operator (in case of hexakis-icosahedra or, depending on the point group, of pentagonal hexecontahedra).

In general, all the possible ways to obtain polyhedra having an icosahedral symmetry come from the coexistence of any number of forms previously listed, taking into account that the central distance of the faces of each form can assume any value.
As an example, Fig.6 shows in the upper row a pair of polyhedra, deriving from the intersection of the four forms reported in the lower row and characterized by the simplest indices: (100) for the triacontahedron, (111) for the icosahedron, (110) for a generic deltoidal hexecontahedron and (112) for a generic hexakis-icosahedron.
In the former polyhedron the central distances of the faces belonging to the four forms are different, whereas in the latter they are equal: consequently, the relative extent of the faces varies significantly in the pair of polyhedra, in particular as regards the rhomb-triacontahedral and the deltoid-hexecontahedral faces.



Rhombic triacontahedron	Icosahedron	Deltoid-hexecontahedron	Hexakis-icosahedron
FIG.6 - Pair of polyhedra deriving from the intersection of the same four forms: a rhomb-triacontahedron and an icosahedron, whose faces are orthogonal to 2-fold and 3-fold axes, respectively a deltoidal hexecontahedron and a hexakis-icosahedron, both generic polyhedra where the indices of the faces are different from the ones of the corresponding Catalan polyhedra. The relative extent of the four forms changes in the two polyhedra, as a consequence of the variation of each central distance.

ASSOCIATIONS OF SYMMETRY ELEMENTS

It is useful to examine in detail the associations of symmetry operators characterizing the icosahedral point groups, also with the intent to evidence their relationships with the classic crystallographic points groups.
On the whole, the symmetry operators present in 235 point group (or crystal class) consist of:

fifteen 2-fold axes

ten 3-fold axes

six 5-fold axes

whereas in m35 point group also the centre of symmetry and fifteen mirror planes perpendicular to the 2-fold axes group must be added to the previous rotation axes.
As described in [2], Eulero's formula:

(where α, β, γ indicate the angles of rotation, around an appropriate direction, by which all the crystallographic elements, and the whole crystal form, superimpose) lets one obtains the possible angles ω between each couple of such directions.
In particular, concerning 5-fold axes, one obtains that the least angles between:

couples of 5-fold axes are equal to 63.43�

5-fold axes and 3-fold axes are equal to 37.38� and 79.18�

5-fold axes and 2-fold axes are equal to 31.72�, 58.28� and 90�.

According to these values, the icosahedral point groups can be represented by the following four stereographic projections of their symmetry operators.
In addition to the position of the rotation axes, all and sundry reported in the stereographic projections, in the first three projections also the zones of the fifteen 2-fold axes, the ten 3-fold axes and the six 5-fold axes are sequentially reported (a zone includes all the faces parallel to a certain crystallographic direction); in the fourth projection, together with the zones of all the rotation axes, the fifteen mirror planes, present in m35 point group and coinciding with the zones of the 2-fold axes, are represented (drawn with solid lines).

FIG.7 - Free-hand drawing of the stereographic projections of the icosahedral symmetry operators.
In addition to the position of 2-fold, 3-fold and 5-fold rotation axes:
the fifteen zones of the 2-fold axes (dashed in black)
the ten zones of the 3-fold axes (dashed in blue)
the six zones of the 5-fold axes (dashed in red)
the zones of all the rotation axes, together with the fifteen mirror planes (drawn with solid lines) coinciding with the zones of the 2-fold axes
are sequentially reported in the four images.

Analogously to what happens for the crystallographic 23 point group, subgroup of 235 point group, three mutually orthogonal 2-fold axes (chosen among five set of three) play the role of reference axes.
The values of all the angles formed between each pair of rotation axes are reported in the following table.

TABLE 1- Angles between pairs of rotation axes

2 3 5

2 36� 20.91� 31.72�

2 60� 54.74� 58.28�

2 72� 69.10� 90�

2 90� 90�

3 41.81� 37.38�

3 70.53� 79.19�

5 63.44�

As one can see, the different least values of the angles not concerning 5-fold axes are the following:

36�, 60�, 72�, 90� between pairs of 2-fold axes

20.91�, 54.74�, 69.10�, 90� between a 2-fold and a 3-fold axis

41.81� and 70.53� between pairs of 3-fold axes.

The detailed examination, also by the stereographic projection, of the reciprocal orientation of the rotation axes reveals that:

the fifteen 2-fold axes are placed along five sets of three orthogonal directions
the orientations of four of the ten 3-fold axes coincide with the [111] direction and their equivalent ones, namely they are placed along the diagonal of a cube; the other six 3-fold axes are, in pairs, orthogonal to each of the three 2-fold axes chosen as reference axes.
also the six 5-fold axes are, in pairs, orthogonal to each of the three 2-fold axes chosen as reference axes.
orthogonally to each 3-fold axis one can count three 2-fold axes, spaced 60� from each other
orthogonally to each 5-fold axis one can count five 2-fold axes, spaced 36� from each other
in correspondence of each couple of 2-fold axes belonging to the five sets of three orthogonal 2-fold axes, there is a couple of 3-fold axes: each 3-fold axis forms angles of 20.90� and 69.10� with the 2-fold axes of the couple, and it is orthogonal to the third 2-fold axis.
in correspondence of each couple of 2-fold axes belonging to the five orthogonal set of three 2-fold axes, there is also a couple of 5-fold axes: each 5-fold axis forms angles of 58.28� and 31.72� with the 2-fold axes of the couple, and it is orthogonal to the third 2-fold axis.
in other words, whereas the 3-fold and 5-fold axes are placed at the intersection of the zones of three and five 2-fold axes, respectively, the 2-fold axes are placed at the intersection of six zones, relative to three pairs of 2-fold, 3-fold and 5-fold axes.

DUAL FORMS CHARACTERIZING THE ICOSAHEDRAL POINT GROUPS

Each of the six forms placed in a particular orientation (parallelism and sometimes also orthogonality), as regards the symmetry operators, is compatible with both 235 and m35 point groups; on the contrary, the seventh form, that is in general position, results different in the two point groups, also concerning multiplicity, because of the presence in the m35 point group of both the fifteen mirror planes and the centre of symmetry, all absent in the 235 point group.

DODECAHEDRON AND ICOSAHEDRON

   Both dodecahedron and icosahedron are convex regular polyhedra (or Platonic solids): their faces are made of twelve regular pentagons orthogonal to the 5-fold axes, and twenty equilateral triangles orthogonal to the 3-fold axes, respectively; in both polyhedra 2-fold axes pass in the middle point of every edge.
    An important feature of the pair dodecahedron-icosahedron is that they are duals: duality implies that in every dual pair the two polyhedra interchange faces and vertices (including their number and respective positions), whereas they have the same number of edges, as follows from the known rule: F + V = E + 2.
    In this specific case, twelve faces and twenty vertices of the dodecahedron correspond to twenty faces and twelve vertices of the icosahedron: therefore the edges are thirty both in the dodecahedron and in the icosahedron.

FIG.8

On the left: orthographic view of the dodecahedron {1τ0}, approximately along [621] direction, reporting also all the rotation axes present in the 235 point group

On the right: the relative stereographic projection seen along the crystallographic axis c.

    An important difference between icosahedral and crystallographic point groups consists in the possibility that the indices usually assume the value of irrational numbers, instead of being integers.
    The dodecahedron, compatible with both icosahedral point groups, is the unique form whose faces can be obtained from the twelve possible cyclic permutations, changes of sign included, of just one set of three indices, {1τ0}, where τ = (√5+1)/2 = 1.61803... denotes the golden ratio. In contrast, the indices of the faces belonging to all the other single icosahedral forms assume values coming from two, three or five set of indices, depending on each case.
    For example, eight of the twenty sets of three indices naming the faces of the icosahedron are derived from {111}, taking into account all the possible changes of sign, and twelve from the cyclic permutations, changes of sign included, of the other set of indices {τ 1/τ 0}; such two sets of indices correspond to forms belonging to the m3 cubic point group, subgroup of the m35 point group. They are the octahedron, consisting of faces with the shape of equilateral triangles, and a particular pentagon-dodecahedron, in which every face (with irrational indices) has the shape of a non-regular pentagon, becoming an equilateral triangle as a consequence of its intersections with two adjacent triangular faces of the octahedron.


FIG. 9 On the left: orthographic view of the icosahedron, whose faces have indices deriving from {111} and {τ 1/τ 0}, drawn together with all the rotation axes present in the 235 point group On the right: its stereographic projection, seen along the crystallographic axis c.

The icosahedron can also be considered as the result of the intersection of five tetrahedra:

{111}, {τ 1/τ 0}, {τ -1/τ 0}, {111}, {1/τ 0 τ}

related by the six 5-fold axes.

In the following image the five tetrahedra, whose intersection generates the icosahedron, are shown in the appropriate orientation together with the 5-coloured icosahedron.

FIG.10 - Five-coloured icosahedron generated by the intersection of as many tetrahedra, whose faces are orthogonal to all the 3-fold axes present in the icosahedron.

RHOMBIC TRIACONTAHEDRON AND ICOSIDODECAHEDRON

The action of the set of symmetry elements over each face placed orthogonally to a 2-fold axis generates the rhombic triacontahedron (or rhomb-triacontahedron), made up of thirty rhombic faces. The ratio between the lengths of the diagonals of the faces corresponds to the golden ratio τ. (Analogously one can note the existence, in the cubic system, of a somehow similar form, the rhombic dodecahedron {110}, made up of twelve rhombic faces. The ratio between the lengths of the diagonals of the faces in this case corresponds to the value √2).
In addition to the 2-fold axes orthogonal to the faces, there are also 3-fold and 5-fold axes passing through the vertices shared by three or five faces, respectively; besides, pairs of orthogonal planes, parallel to the diagonal of the rhombic faces, cross one another in correspondence of the 2-fold axes.


FIG.11 - Orthographic view of the rhombic triacontahedron (left) and its stereographic projection (right), seen along the crystallographic axis c.

Analogously to the icosahedron, also the rhombic triacontahedron is made up of faces whose indices derive from two different sets of indices: {100} and {τ 1 1/τ}; as a consequence of their cyclic permutations, changes of sign included, one obtains the overall number of 6+24 = 30 faces.
With reference to the five sets of three orthogonal 2-fold axes previously described, the rhombic triacontahedron may be regarded as deriving from the intersection of five cubes; the faces of each cube are orthogonal to one of the five sets of three 2-fold axes. The five forms, related one another by the 5-fold axes and shown in different colours in Fig.12, together with the 5-coloured rhombic triacontahedron generated by their intersection, can be named:

{100}, {τ 11/τ}, {τ 11/τ}, {11/τ τ}, {1-1/τ τ}

FIG.12 - Five-coloured rhombic triacontahedron and the five cubes, properly orientated, which intersection generates the rhombic triacontahedron.

The rhombic triacontahedron belongs to the family of Catalan polyhedra (so named after the Belgian mathematician Eug�ne Charles Catalan) made up by equal faces, having generally the shape of symmetric but non-regular polygons, and different regular angoloids. The duals of the isohedral Catalan polyhedra are the isogonal semiregular Archimedean polyhedra, in turn characterized by faces made up by two or three different regular polygons (whose sides and angles are therefore all equal) and by equal but non-regular angoloids.

The dual of the rhombic triacontahedron is the icosi-dodecahedron, Archimedean polyhedron consisting of 20 triangular and 12 pentagonal faces, orthogonal to 3-fold and 5-fold axes, respectively, whereas 2-fold axes pass through the 30 vertices. Each vertex is shared by four polygons, namely a pair of equilateral triangles and a pair of regular pentagons, oriented in an opposite direction with respect to triangles and pentagons constituting the faces of icosahedron and dodecahedron. In order to generate the icosi-dodecahedron by the intersection of an icosahedron and a dodecahedron, the ratio between the distances of their faces from the centre of the polyhedron must assume the following value:

It must be pointed out that, just as it happens with the other dual pairs consisting of an Archimedean polyhedron and a Catalan polyhedron, the ratio between the central distances of the triangular and pentagonal faces of the icosi-dodecahedron is the reciprocal of the ratio between the central distances of the two different vertices of the dual rhombic triacontahedron, lying along the 3-fold and 5-fold axes, respectively.

FIG.13 - Orthographic view of the icosi-dodecahedron (left) and its stereographic projection (right), seen along the crystallographic axis c.

In particular, when the central distances of the faces of an icosi-dodecahedron have just the following values:

τ /√τ²+1 = 0.8507 for dodecahedral faces
and

τ /√3 = 0.9342 for icosahedral faces,
the faces, placed at a unit central distance, that belong to the rhombic triacontahedron, dual of the icosi-dodecahedron, result to be exactly tangent to the vertices of the icosi-dodecahedron.

Since, in the instance of the dual pair consisting of triacontahedron and icosi-dodecahedron, the total number of faces and vertices corresponds to 30 + 32 = 62, the edges are sixty in both polyhedra, according to the rule: F+V=E+2.

There are two other semiregular polyhedra, both deriving from the intersections between icosahedron and dodecahedron, of which the icosi-dodecahedron just described can be considered an intermediate form: they are the Archimedean truncated icosahedron and the Archimedean truncated dodecahedron.

TRUNCATED ICOSAHEDRON AND PENTAKIS-DODECAHEDRON

In a generic truncated icosahedron, the icosahedral faces truncated by the pentagonal faces of dodecahedron assume firstly the shape of non-regular hexagons (Fig.14): nevertheless they are symmetric relatively to three mirror lines, forming each other angles of 60�, which intersect in the centre of the faces.

ICOSAHEDRON (d_icosahedron= τ /√3) TRUNCATED BY DODECAHEDRON

d_dodecahedron = 1.10 d_dodecahedron = 1.05

d_dodecahedron = 1 d_dodecahedron = 0.921

d_dodecahedron = 0.9 d_dodecahedron = 0.87

FIG.14 - Series of polyhedra, obtained truncating an icosahedron by a dodecahedron, included between the icosahedron (left upper corner of the figure), and the icosi-dodecahedron (right lower corner). They are characterized by:
pentagonal-shaped faces of dodecahedron with increasing dimensions
icosahedral faces generally with the shape of non-regular (but symmetric) hexagons, except for the Archimedean truncated icosahedron (central image) and the icosi-dodecahedron (last image), where such faces become regular hexagons and equilateral triangles, respectively.
Whereas the central distance of the icosahedral faces is constant in every polyhedron and equal to τ /√3, the central distance of the dodecahedral faces decreases from √1+1/τ² (value at which the faces of dodecahedron are tangent to the vertices of icosahedron), to 1/√1+1/τ², in case of icosi-dodecahedron; for such range of values of the central distances of the dodecahedral and icosahedral faces, the faces of a rhombic triacontahedron having a unit central distance would be tangent:
to the edges between each pair of icosahedral faces, in the icosahedron and in every truncated icosahedron
to every vertices, in the icosi-dodecahedron.

When the ratio between the central distances of dodecahedral and icosahedral faces assumes the following value:

the faces deriving from the icosahedron become regular hexagons and together with the regular pentagonal faces of the dodecahedron originate the Archimedean truncated icosahedron, whose shape is that of a soccer ball.
Each of its vertices is shared by three regular polygons, namely two hexagons and a pentagon. Three- and five-fold axes are orthogonal to hexagonal and pentagonal faces, respectively, whereas two-fold axes pass through the middle point of all the edges common to two hexagonal faces.


FIG.15 - Orthographic view of the Archimedean truncated icosahedron (left) and its stereographic projection (right), seen along the crystallographic axis c.

After the Archimedean truncated icosahedron, as a consequence of the further truncation of the icosahedron by the dodecahedron, one obtains, in addition to dodecahedral faces with the shape of regular pentagons, icosahedral faces having again the shape of non-regular but symmetric hexagons, as shown in Fig.14; the length of the three shorter sides of hexagon decreases progressively, till it vanishes when the hexagonal faces turn into the equilateral triangular faces of the icosidodecahedron.

The dual of the Archimedean truncated icosahedron is the Catalan pentakis-dodecahedron, whose faces are named by indices deriving from the cyclic permutation, changes of sign included, of three sets of indices:

{1 3τ 0}, {τ³ τ 2}, {τ²+1 2τ 1}

Consequently it consists of 12+24+24 = 60 faces with the shape of an isosceles triangle, such that:

a 5-fold axis passes through the vertex between the two equal sides

a 3-fold axis passes through each of the other two vertices

a 2-fold axis passes in the middle of the side different from the others.

The Catalan pentakis-dodecahedron can be supposed to derive from a dodecahedron, substituting each face with a pentagonal pyramid having the height H given by the relation:

H = L/[√1+1/τ² (3 + 1/τ²)] = 0.251L where L is the length of the side of the pentagonal base of the pyramid, namely the dodecahedral face. The triangular faces of the pyramid (in which the length of the two equal sides is 0.887L and the two equal angles measure 55.69�) are inclined 20.08� relatively to the pentagonal base of the pyramid.


FIG.16 - Orthographic view of the Catalan pentakis-dodecahedron (left) dual of the semiregular truncated icosahedron, and its stereographic projection (right), seen along the crystallographic axis c.

In addition to the Catalan pentakis-dodecahedron dual of the Archimedean truncated icosahedron, there are very many other generic pentakis-dodecahedra (duals of the truncated icosahedra different from the Archimedean one) all including faces {hk0} characterized by a ratio between the indices h and k such that:

0 < h/k < 1/τ

The heights of all the pyramids in the different pentakis-dodecahedra range between:

H_min= 0, when the pentagonal pyramid flattens out, assuming the shape of the pentagonal face of the dodecahedron from which the pentakis-dodecahedron derives
H_max= L/2√1+1/τ² = 0.425L, value such that each couple of adjacent faces belonging to different pentagonal pyramids collapses, originating a unique rhombic face of triacontahedron, inclined of 31.72� with respect to the bases of both pentagonal pyramids.

If the height of the pyramids exceeds the value of H_max, the polyhedron wouldn't be convex anymore.

FIG.17 - Forms whose faces have indices {hk0} such that the ratio h/k ranges between 1/τ and 0 a) dodecahedron {1τ 0}, obtained in correspondence of the value H_min= 0, meaning that the pyramids have flattened out completely
b) pentakis-dodecahedron {1 2τ 0}
c) Catalan pentakis-dodecahedron {1 3τ 0}, dual of the Archimedean truncated icosahedron, obtained when the height of the pyramids is H = 0.251L
d) pentakis-dodecahedron {1 8τ 0}
e) rhombic triacontahedron {010}, obtained when the height of the pyramids is H_max= 0.425L.

In relation to the dual pair consisting of truncated icosahedron and pentakis-dodecahedron, the total number of faces and vertices corresponds to:

F + V = 32 + 60 = 92 Then the edges are ninety in both polyhedra.

TRUNCATED DODECAHEDRON AND TRIAKIS-ICOSAHEDRON

As a consequence of the beginning of the truncation by the triangular faces of an icosahedron, the pentagonal faces of a dodecahedron become non-regular decagons (even if symmetric with respect to five mirror lines intersecting in the centre of each decagon).

DODECAHEDRON (d_dodecahedron= 1/√1+1/τ²) TRUNCATED BY ICOSAHEDRON

DODECAHEDRON
d_icosahedron = √3/τ = 1.0705
d_icosahedron = 1.05 d_icosahedron = 1.03

d_icosahedron = 1
ARCHIMEDEAN
TRUNCATED DODECAHEDRON
d_icosahedron = (τ /√3)(7τ-6)/5 = 0.9951
d_icosahedron = 0.97

d_icosahedron = 0.96 d_icosahedron = 0.95
ICOSIDODECAHEDRON
d_icosahedron = τ /√3 = 0.93417

FIG.18- Series of polyhedra, obtained truncating a dodecahedron by an icosahedron, included between the dodecahedron (left upper corner of the figure), and the icosi-dodecahedron (right lower corner). They are characterized by triangular-shaped faces of the icosahedron with increasing dimensions and faces of dodecahedron having generally the shape of non-regular (but symmetric) decagons, except for the Archimedean truncated dodecahedron (central image) and the icosi-dodecahedron (last image), where such faces are regular decagons and regular pentagons, respectively.
Whereas the central distance of the dodecahedral faces is constant in every polyhedron and equal to 1/√1+1/τ², the central distance of the icosahedral faces decreases from √3/τ, (at which value the faces of icosahedron are tangent to the vertices of dodecahedron), to τ/√3, in case of the icosi-dodecahedron; for such range of values, the faces of a dual rhombic triacontahedron placed at a unit central distance would be tangent:
to the edges between each pair of dodecahedral faces, in dodecahedron and in every truncated dodecahedron
to every vertices, in icosi-dodecahedron.

Only when the ratio between the central distances of the two forms becomes equal to:

d_icosahedron / d_dodecahedron = √(τ²+1)/3 (7τ-6)/5 = 1.1698

the faces of the dodecahedron assume the shape of regular decagons, thus obtaining the Archimedean truncated dodecahedron, in which each vertex is shared by three regular polygons: two decagons and a triangle.
Three-fold and five-fold axes are orthogonal to triangular and decagonal faces respectively, whereas two-fold axes pass through the middle of every edge common to two decagonal faces.

FIG.19 - Orthographic view of the Archimedean truncated dodecahedron (left) and its stereographic projection (right), seen along the crystallographic axis c.

After the Archimedean truncated dodecahedron, as a consequence of the further truncation of the dodecahedron by the icosahedron, one obtains, in addition to icosahedral faces with the shape of equilateral triangles, dodecahedral faces having again the shape of non-regular but symmetric decagons; the length of the five shorter sides of decagon decreases progressively, till it vanishes when the decagonal faces turn into the regular pentagonal faces of the icosi-dodecahedron.

The dual of the Archimedean truncated dodecahedron is the Catalan triakis-icosahedron (or trisicosahedron), whose faces are named by indices deriving from the cyclic permutation, changes of sign included, of three sets of indices:

{τ²+1 1/τ 0}, {τ² 2 τ}, {2τ τ 1/τ}

Consequently it consists of 12+24+24 = 60 faces having the shape of an isosceles triangle, such that:

a 3-fold axis passes through the vertex between the two equal sides

a 5-fold axis passes through each of the other two vertices

a 2-fold axis passes in the middle of the side different from the others.

The Catalan triakis-icosahedron can be supposed to derive from an icosahedron, substituting each face with a trigonal pyramid having the following height H:

H = L / [√3(5τ + 2)] = 0.057L

In this relation L is the length of the side of the triangular base of the pyramid, namely the icosahedral face.
The triangular faces of the pyramid (in which the length of the two equal sides is 0.58L and the two equal angles measure 30.48�) are 11.21� inclined with respect to the base of the pyramid.

FIG.20 - Orthographic view of the Catalan triakis-icosahedron (left), dual of the Archimedean truncated dodecahedron, and its stereographic projection (right), seen along the crystallographic axis c.

In addition to the Catalan triakis-icosahedron, dual of the Archimedean truncated dodecahedron, there are vey many other generic triakis-icosahedra (duals of the truncated dododecahedra different from the Archimedean one) all including faces {hk0} characterized by a ratio between the indices h and k such that:

0 < k/h < 1/τ²

The heights of the trigonal pyramids, relative to all the different triakis-icosahedra, range between:

H_min = 0, when the trigonal pyramid flattens out, assuming the shape of the triangular face of the icosahedron from which the triakis-icosahedron derives
H_max= √3L/6τ² = 0.11L, value such that each couple of adjacent faces, belonging to two trigonal pyramids, collapses originating a unique rhombic face of triacontahedron, inclined of 20.91� with respect to the bases of both trigonal pyramids.

Also in this case, if the height of the pyramids exceeds the value of H_max, the polyhedron wouldn't be convex anymore.

FIG.21 - Forms whose faces have indices {hk0} such that the ratio k/h ranges between 0 and 1/τ² a) rhombic triacontahedron {100}, obtained when the height of the pyramids is H_max= 0.11L
b) triakis-icosahedron {3τ² 1/τ 0}
c) Catalan triakis-icosahedron {τ²+1 1/τ 0}, dual of the Archimedean truncated dodecahedron, obtained when the height of the pyramids is H = 0.057L
d) triakis-icosahedron {τ² 1/τ 0}
e) icosahedron {τ 1/τ 0}, obtained in correspondence of the value H_min= 0, meaning that the trigonal pyramids have flattened out completely.

Since the total number of faces and vertices corresponds to:

F + V = 32 + 60 = 92,the edges are ninety in both polyhedra also in case of the dual pair made of truncated dodecahedron and triakis-icosahedron.

TRUNCATING ICOSI-DODECAHEDRON BY RHOMBIC TRIACONTAHEDRON

The truncation, by means of a rhombic triacontahedron, of all the vertices of an icosi-dodecahedron (polyhedron a in Fig.22), at the beginning transforms the triangular and pentagonal faces of the icosi-dodecahedron into hexagonal and decagonal polygons, respectively, both non-regular even though symmetric, whereas the faces of the triacontahedron assume a rectangular shape (ref: polyhedra b-d-f).
Only in two circumstances, for particular values of the central distance of the triacontahedrical faces, it happens that the decagons (rif: polyhedron c) or the hexagons (ref: polyhedron e) from non-regular polygons become regular ones.

Decreasing the central distance of the triacontahedral faces, one obtains the polyhedron g, whose faces, besides the rectangular ones, consist of equilateral triangles and regular pentagons as well as the initial faces of the icosi-dodecahedron, but orientated in an opposite direction.
Proceeding with the truncation, before the complete elimination of the icosahedral faces by the triacontahedral ones, in the resulting polyhedron there is the simultaneous presence of triangular and pentagonal faces, both regular, separated by octagonal faces of triacontahedron, which are non-regular even though symmetric (ref: polyhedron h).

    After the elimination of the icosahedral triangular faces, the remaining faces are the dodecahedral pentagonal faces and the triacontahedral faces which from octagonal become hexagonal, always non-regular but symmetric: the six sides of the hexagonal faces consist of four short sides and two long sides (ref: polyhedron i).
   The length of the pair of longer sides decreases progressively as the further truncation proceeds, arriving at the polyhedron j, in which the lengths of all the sides of the hexagonal faces are equal (the hexagons are not regular polygons, since the six angles between the sides of each hexagon assume two different values).
    Successively the hexagonal faces deform again, lengthening progressively along a direction orthogonal to the direction along which they were stretched previously (ref: polyhedron k, in which the hexagonal faces consist of four short sides and two long sides), until the truncation eliminates also the pentagonal faces: at this point the resulting polyhedron includes only the rhomb-triacontahedral faces (ref: polyhedron l).

Obviously one obtains the same result by keeping constant the central distance of the triacontahedral faces and by increasing alternatively the central distances of both dodecahedral and icosahedral faces, provided that the ratio of the distances: d_dodecahedron / d_icosahedron = 0.9106, characterizing the icosi-dodecahedron, remains always constant.
Therefore no Archimedean polyhedron can be obtained just truncating an icosi-dodecahedron by means of a rhombic triacontahedron.




FIG.22 - Progressive truncation of an icosi-dodecahedron, realized diminishing the central distance of a rhombic triacontahedron (a-h); after the elimination of the icosahedral triangular faces, the further decrease (i-l) of the central distance of the triacontahedral faces causes the reduction and subsequent disappearence also of the pentagonal faces of the dodecahedron.

TWO FURTHER SEMIREGULAR POLYHEDRA: TRUNCATED ICOSI-DODECAHEDRON AND RHOMBICOSI-DODECAHEDRON

The Archimedean truncated icosi-dodecahedron and the rhombicosi-dodecahedron are the two further semiregular polyhedra which, as just described, cannot be obtained truncating directly an icosi-dodecahedron by means of a rhombic triacontahedron, but, more generally, derive from the intersection of the three forms: icosahedron, dodecahedron and rhombic triacontahedron, taking into account possible intermediate steps as described in Fig.23.

Regarding the Archimedean truncated icosi-dodecahedron (name that, for the aforementioned reasons, is not entirely appropriate and consequently it is sometimes replaced by "great rhombicosidodecahedron"), the icosahedral, dodecahedral and triacontahedral faces become regular hexagonal, regular decagonal and square ones, respectively, when the ratios between each pair of central distances assume the following values:

d_dodecahedron / d_icosahedron= (1+1/τ²)√1+1/τ² /√3= 0.9380

d_icosahedron / d_{triacontahedron} = √3 /(1+ 2 /τ²) = 0.9819

d_dodecahedron / d_{triacontahedron} = √1+1/τ ²(1+1/τ²) /(1+2/ τ²) = 0.9210

In the instance of rhombicosi-dodecahedron (named also "small rhombicosidodecahedron"), the icosahedral, dodecahedral and riacontahedral faces take the shape of equilateral triangles, regular decagons and squares, respectively, as a consequence of the following values of the ratio between each pair of central distances:

d_dodecahedron / d_icosahedron = (3√3 / [(τ²+2)√1+1/ τ²] = 0.9571

d_icosahedron / d_{triacontahedron} = (1+ 2 /τ²) /√3 = 1.0184

d_dodecahedron / d_{triacontahedron} = 3/ (τ² √1+1/τ²) = 0.9748

The first three rows of Fig.23 report the different combinations of each pair of polyhedra, from which both the Archimedean truncated icosi-dodecahedron and the rhombicosi-dodecahedron (on the left and on the right of the fourth row, respectively) can be generated, as a consequence of the intersection with the remaining third polyhedron (when the ratios between the respective central distances assume just the aforementioned values.

Generation of an Archimedean truncated icosi-dodecahedron (left) and a rhombicosi-dodecahedron (right) by the combination of a dodecahedron, an icosahedron and a rhombic triacontahedron

FIG.23a - Intermediate step given by the intersection of a dodecahedron and an icosahedron

d_dod. /d_icos. = (1+1/τ²) √1+1/τ² /√3 = 0.9380

d_dod./d_icos. = 3√3 / [(5-1/τ²) √1+1/ τ²] = 0.9571

FIG.23b - Intermediate step given by the intersection of an icosahedron and a triacontahedron

d_icos./d_triacont. = √3/(1+2/τ²)= 0.9819

d_icos./d_triacont. = (1+2/τ²)/√3 = 1.0184

FIG.23c - Intermediate step given by the intersection of a dodecahedron and a triacontahedron

d_dod./d_triacont. = √1+1/τ²(1+1/τ²)/(1+2/τ²) = 0.9210

d_dod./d_triacont. = 3/(τ²√1+1/τ²) = 0.9748

Fig.23d - Archimedean truncated icosi-dodecahedron and rhombicosi-dodecahedron generated by the intersection of each previous pair of forms with the respective third form

d_dodecahedron/ d_icosahedron= 0.9380
d_icosahedron/ d_{triacontahedron}= 0.9819
d_dodecahedron/ d_{triacontahedron}= 0.9210 d_dodecahedron/ d_icosahedron = 0.9571
d_icosahedron/ d_{triacontahedron} = 1.0184
d_dodecahedron/ d_{triacontahedron} = 0.9748

On the basis of the previous considerations, one does not have to draw the conclusion that the Archimedean truncated icosi-dodecahedron and the rhombicosi-dodecahedron are totally unrelated to the icosi-dodecahedron.
In fact, given a unit value to the central distance of the rhomb-triacontahedral faces, if one denotes by:

d_dodecahedron and d_icosahedron

d_dodecahedron and d_icosahedron

d_dodecahedron and d_icosahedron

the pairs of values of the central distances, relative to the dodecahedral and icosahedral faces, concerning respectively:

icosidodecahedron

Archimedean truncated icosi-dodecahedron

rhombicosidodecahedron

it is possible to ascertain that:

It follows that in a plot, which reports the value of d_dodecahedronon the y-axis and the value of d_icosahedron on the x-axis, the points representing every pair of values of such central distances, relatively to the three Archimedean polyhedra, are aligned along a line whose equation is:

d_dodecahedron = d_dodecahedron + √3 /(1+1/τ²) (d_icosahedron - d _icosahedron)

After substitution, in the equation, of d_dodecahedron and d_icosahedron with the values of their central distances in icosi-dodecahedron, one obtains the relation:

d_dodecahedron = (1/√1+1/τ²) (√3 d_icosahedron - 1/τ )

valid not only for the three quoted Archimedean polyhedra, but also for all the solids, originating from the intersection of a dodecahedron with an icosahedron and a triacontahedron, characterized by pairs of values of d_dodecahedron and d_icosahedron included in the intervals:

1/√1+1/τ² < d_dodecahedron < (3/τ²)/√1+1/τ²)

τ /√3 < d_icosahedron < (1 +2/τ²) /√3

As one can see in Fig.24, all these polyhedra include square triacontahedral faces that have increasing dimensions, starting from the icosi-dodecahedron, where the square faces are absent, since the triacontahedron results exactly tangent to the vertices of the icosi-dodecahedron, up to the rhombicosi-dodecahedron, where the square faces reach the maximum dimension: the length of their side (equal to 2/τ³ if d_{triacontahedron} = 1) becomes equal to the length of the sides of the triangular and pentagonal faces.

ICOSIDODECAHEDRON
d_icosahedron = τ /√3 = 0.9342
d_icosahedron = 0.94 d_icosahedron = 0.95

d_icosahedron = 0.97 ARCHIMEDEAN TRUNCATED ICOSIDODECAHEDRON
d_icos = √3 /(1+ 2 /τ²) = 0.9819 d_icosahedron = 0.99

d_icosahedron = 1 d_icosahedron = 1.01
RHOMBICOSIDODECAHEDRON
d_icos = (1 +2/τ²) / √3 = 1.0184

Fig.24 - Series of polyhedra, all characterized by square faces of triacontahedron, included between two Archimedean semiregular polyhedra: the icosi-dodecahedron (left upper corner), where the square faces are absent, being the triacontahedron tangent to the vertices of the icosi-dodecahedron, and the rhombicosi-dodecahedron (right lower corner). In particular, hexagonal and decagonal faces are both regular in the Archimedean truncated icosi-dodecahedron (in the centre of the figure).
Whereas the central distance of the square faces of triacontahedron has always a unit value, the central distance of the dodecahedral faces can be obtained from the central distance of the icosahedral faces by the relation:
d_dodecahedron = (1/√1+1/τ²) (√3 d_icosahedron - 1/τ )
previously reported in the text.

In this series, the faces of the polyhedra different from Archimedean polyhedra are respectively:

non-regular decagons, symmetric with respect to five mirror lines intersecating in correspondence of the 5-fold rotation axis, in the centre of every decagon
non-regular hexagons, symmetric with respect to three mirror lines intersecating in correspondence of the 3-fold rotation axis, in the centre of every hexagon.

RHOMBICOSIDODECAHEDRON AND DELTOIDAL HEXECONTAHEDRON

In the rhombicosi-dodecahedron, Archimedean polyhedron consisting of:

twelve pentagonal faces coming from the dodecahedron

twenty triangular faces from the icosahedron

thirty square faces from the triacontahedron

each vertex turns out to be shared by four polygons: a regular pentagon, an equilateral triangle and two squares; as regards the axes of rotation, 5-fold, 3-fold and 2-fold axes are orthogonal to pentagonal, triangular and square faces, respectively.


FIG.25 - Orthographic view of the rhombicosi-dodecahedron (left) and its stereographic projection (right), seen along the crystallographic axis c.

The polyhedron dual of the Archimedean rhombicosi-dodecahedron is the Catalan deltoidal hexecontahedron, (or deltoid-hexecontahedron), whose faces are named by indices deriving from the cyclic permutations, changes of sign included, of the 3 sets of indices:

{τ+1/τ τ 0}, {τ² 1/τ 1/τ}, {2 1 τ}

Therefore it consists of 12+24+24 = 60 faces having the shape of a deltoid (or kite), namely a quadrangle having two pairs of adjacent sides that are congruent.

In each deltoidal faces:

the ratio between the length of the sides is equal to 1+τ/3 = 1.5393

the two angles between each couple consisting of a short and a long side are equal and measure 86.97�

the angle between the couple of short sides measures 118.27�

the angle between the couple of long sides measures 67.78�

A 3-fold axis and a 5-fold axis pass through the two vertices shared by the couple of short sides and long sides, respectively, whereas a 2-fold axis passes through each of the two other vertices of the face.

FIG.26 - Orthographic view of the Catalan deltoidal hexecontahedron (left), dual of the rhombicosi-dodecahedron, and its stereographic projection (right), seen along the crystallographic axis c.

The (1+1/τ² 1 0) face of the Catalan deltoidal hexecontahedron makes an angle of 14.98� with the (τ 1/τ 0) face of the icosahedron and an angle of 22.39� with the (1τ 0) face of the dodecahedron.
The corresponding face in every other generic deltoidal hexecontahedron has indices (hk0) such that 1/τ < h/k < τ²; the possible orientations of all these faces are included in an overall angular interval 37.37� wide.

FIG.27 - Forms whose faces have indices {hk0} such that the ratio h/k ranges between τ² and 1/τ
a) icosahedron {τ 1/τ 0}
b) deltoidal hexecontahedron {210}
c) Catalan deltoidal hexecontahedron {τ+1/τ τ 0} dual of the Archimedean rhombicosi-dodecahedron
d) deltoidal hexecontahedron {110}
e) dodecahedron {1τ 0}

These generic deltoidal hexacontahedra are the duals of particular polyhedra in which, as a consequence of the values assumed by the central distances of their faces, each vertex is shared by a pentagonal face of dodecahedron, a triangular face of icosahedron and two rectangular faces, derived from a triacontahedron, becoming square only in case of the Archimedean rhombicosi-dodecahedron.
Fig.28 shows a series of polyhedra of this kind, describing the transformation of dodecahedron into icosahedron passing through the rhombicosi-dodecahedron; such transformation can be obtained, given a unit value to the central distance of the triacontahedral faces, by an increase of d_dodecahedron from 1/√1+1/τ² to √1+1/τ², accompanied by a decrease of d_icosahedron from √3/τ to τ/√3, according to the following relation:
√1+1/τ² d_dodecahedron + √3τ d_icosahedron = 4

DODECAHEDRON d_dod. = 1/√1+1/τ² d_icos.= √3/τ = (1+1/τ⁴) τ /√3	d_icosahedron = (1+4/τ⁷) τ /√3	d_icosahedron = (1+1/2τ³) τ /√3

d_icosahedron = (1+2/τ⁶) (τ /√3)	RHOMBICOSI-DODECAHEDRON d_icos. = (1+ 2 / τ²)/√3 = (1+1/τ⁵) τ /√3 d_dod. = 3/τ²√1+1/τ² = (1+1/τ⁴)/√1+1/τ²	d_icosahedron = (1+2/τ⁷) τ /√3

d_icosahedron = (1+1/τ⁶) τ /√3	d_icosahedron = (1+1/τ⁷) τ /√3	ICOSAHEDRON d_icosahedron = τ /√3 d_dodecahedron = √1+1/τ²

FIG.28 - Series of polyhedra, dual both of Catalan and generic deltoidal hexacontahedra, describing the transformation of dodecahedron into icosahedron passing through an Archimedean polyhedron, the rhombicosi-dodecahedron; in every intermediate polyhedron all the vertices are shared by four polygons: a pentagon, a triangle and two rectangles becoming squares only in the rhombicosi-dodecahedron. For every intermediate polyhedron, given a unit value to the central distance of the triacontahedral faces, the distance of the dodecahedral faces can be obtained, starting from the reported distance of the icosahedral faces, by the relation: d_dodecahedron = (1/√1+1/τ²) (4 -√3τ d_icosahedron)

Concerning the dual pair formed by the Catalan deltoidal hexecontahedron and the Archimedean rhombicosi-dodecahedron (and more in general the pairs made of a generic deltoidal hexecontahedron and its dual), the total number of faces and vertices corresponds to:

F + V = 62 + 60 = 122

Consequently the edges are 120 in each polyhedron of the pair.

TRUNCATED ICOSI-DODECAHEDRON AND HEXAKIS-ICOSAHEDRON

In the Archimedean truncated icosi-dodecahedron, consisting of twelve decagonal faces of dodecahedron, twenty hexagonal faces of icosahedron and thirty square faces of triacontahedron, each vertex is shared by three polygons: a regular decagon, a regular hexagon and a square. As regards the axes of rotation, 5-fold, 3-fold and 2-fold axes are orthogonal to decagonal, hexagonal and square faces, respectively.

FIG.29 - Orthographic view of the Archimedean truncated icosi-dodecahedron (left) and its stereographic projection (right), seen along the crystallographic axis c.

The polyhedron dual of the Archimedean truncated icosi-dodecahedron is the hexakis-icosahedron (named also dis-dyakis-triacontahedron and decakis-dodecahedron), Catalan polyhedron whose faces are named by indices deriving, in this instance, from the cyclic permutations, changes of sign included, of the following five sets of indices:

{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}

Consequently the hexakis-icosahedron consists of 5x24 = 120 faces, having the shape of a scalene triangle (rather similar to a right-angled triangle), whose angles measure 88.99�, 58.24�, 32.77�; differently from all the aforementioned icosahedral polyhedra, it does not include any face with an index equal to zero, namely parallel to a 2-fold axis, being in a general position with respect to every symmetry operator.

The ratios between the lengths of the sides of every triangular face have the following values:

L₁ /L₂ = 2(5-2τ)/3 = 1.1759 L₂ /L₃ = 3τ²/5 = 1.5708 L₁ /L₃ = 2(τ+3)/5 = 1.8472

FIG.30
On the left: orthographic view of the Catalan hexakis-icosahedron, dual of the Archimedean truncated icosi-dodecahedron, visualized together with all the symmetry operators, fifteen mirror planes included
On the right: its stereographic projection, seen along the crystallographic axis c.

In case of the pair of dual polyhedra formed by the Catalan hexakis-icosahedron and the Archimedean truncated icosidodecahedron, the total number of faces and vertices is:

F + V = 120 + 62 = 182

Consequently the edges are 180 in both polyhedra.

PENTAGONAL HEXECONTAHEDRON AND SNUB DODECAHEDRON

    The general forms relative to 235 and m35 icosahedral point groups are different, since in the only m35 point group there is the presence of fifteen mirror planes, each one orthogonal to a 2-fold axis, and the symmetry centre.
    Whereas the general form of the m35 point group is, as already described, the hexakis-icosahedron, whose number of faces (multiplicity) is 120, the general form of the 235 point group is the pentagonal hexecontahedron (or pentagon-hexecontahedron), whose multiplicity is 60; it is the only single form with icosahedral symmetry lacking of both the fifteen mirror planes and the symmetry centre: therefore it can show itself in two chiral forms, named laevo (left) and dextro (right), due to enantiomorphism.

    In a generic pentagonal hexecontahedron, the pentagonal faces, though are asymmetric, own two pairs of equal sides; a 3-fold axis and a 5-fold axis pass through the vertices placed between the two pairs of equal contiguous sides, whereas a 2-fold axis passes in the middle of the fifth side, different from the other sides.

FIG.31 - Comparison between an asymmetric pentagonal face and the symmetric one belonging to the pentagonal hexecontahedron, dual of the snub dodecahedron.
As described in the text, an asymmetric pentagonal face belongs also to the particular pentagonal hexecontahedron whose intersection with icosahedron and dodecahedron gives rise to a snub dodecahedron when the ratio between:
the equal central distance of the triangular faces deriving from the icosahedron or the pentagonal hexecontahedron
and
the central distance of the faces deriving from the dodecahedron
assumes the value of 1.0486

Even though the polyhedron is chiral, therefore lacking mirror planes on the whole, for a particular orientation of the faces it happens that each face of the pentagonal hexecontahedron is locally symmetric with respect to a mirror plane.
As a consequence of the fact that in such a case also the fifth side has the same length of the pair of short sides, four of the angles of the pentagonal faces are equal (118.14�), whereas the fifth angle, the one between the couple of longest sides, is smaller than the others and measures 67.44�.

FIG.32- Orthographic view of the pentagonal hexecontahedron (left), dual of the snub-dodecahedron, and its stereographic projection (right), seen along the crystallographic axis c.

The Archimedean polyhedron dual of this particular pentagonal hexecontahedron is the snub dodecahedron; it consists in the intersection of three forms :

a dodecahedron

an icosahedron

a peculiar pentagonal hexecontahedron

The orientation, with respect to the crystallographic reference axes, of the faces (having the shape of an irregular pentagon) that belong to this pentagonal

hexecontahedron, is different if compared with the orientation of the faces of the pentagonal hexecontahedron dual of the snub dodecahedron (that, in turn, have the shape of a non-regular even though symmetric pentagon): therefore the faces of the two pentagonal hexecontahedra are characterized by different Miller indices.

FIG.33 - View, along the crystallographic axis a, of:
the snub dodecahedron (on the left)
the asymmetric pentagonal hexecontahedron, whose faces belong to the snub dodecahedron (on the right)
an intermediate polyhedron (at the centre), which can be described as a snub dodecahedron deprived of the icosahedron, or, conversely, as the result of the intersection of the asymmetric pentagonal hexecontahedron with the dodecahedron; the faces deriving from the asymmetric pentagonal hexecontahedron in this case become quadrilateral ed assume the shape of a deltoid (or kite).

In the snub dodecahedron the faces deriving from the icosahedron and the pentagonal hexecontahedron have the same central distance: the value of its ratio with respect to the central distance of the dodecahedral faces is 1.0486.
This geometrical configuration generates the 92 faces of the snub dodecahedron, all having the shape of regular polyhedra: in detail, twelve pentagons and eighty triangles. The twelve pentagons and twenty out of the eighty triangles obviously come from the dodecahedron and the icosahedron, respectively, whereas the other sixty triangles come from the pentagonal hexecontrahedron, whose irregular pentagonal faces are drastically modified by the simultaneous intersections with dodecahedral and icosahedral faces.

Since for the dual pair snub dodecahedron-pentagonal hexecontahedron: F+V = 92+60 = 152, both polyhedra have 150 edges.

FIG.34 - Orthographic view of the snub-dodecahedron (left) and its stereographic projection (right), seen along the crystallographic axis c.

The positions of the rotation axes, with respect to the single forms constituting the snub dodecahedron, are the following:

5-fold axes orthogonal to the pentagonal faces of the dodecahedron

3-fold axes orthogonal to the triangular faces of the icosahedron

2-fold axes passing through the middle of the edges shared between pairs of triangular faces which derive from the pentagonal hexecontahedron.

Each pentagonal face of the dodecahedron shares its five vertices with as many triangular faces of the icosahedron, whereas each triangular face of the icosahedron shares its three vertices with as many pentagonal faces of the dodecahedron.
On the whole, each vertex of the snub dodecahedron is shared among five faces, a pentagonal face and four equilateral triangular ones: three of these triangular faces derive from the pentagonal hexecontahedron and one from the icosahedron.

As a valid alternative to snub dodecahedron, the name of snub icosidodecahedron has been proposed for this polyhedron, since both the pentagonal faces of the dodecahedron and the triangular faces of the icosahedron share each side with one side of a triangular face deriving from the pentagonal hexecontahedron, by which therefore they are bordered.
In turn, each triangular face derived from the pentagonal hexecontahedron shares one of its sides with a pentagonal face of the dodecahedron and a second side with a triangular face of the icosahedron, whereas the third side is in common with another triangular face, coming from the pentagonal hexecontahedron, with which it makes a pair. Since in the snub dodecahedron there are in total thirty pairs of these triangular faces, one can describe the snub dodecahedron as a distortion of the rhombicosidodecahedron (Fig.35), assuming that the pair of triangular equilateral faces derived from the pentagonal hexecontahedron replaces each square face of the triacontahedron, with a consequent rotation of the twelve pentagonal faces and the other twenty triangular faces around the 5-fold and 3-fold rotation axes orthogonal to every face.

FIG.35 - Comparison between the rhombicosidodecahedron and the pair of chiral snub-dodecahedra (viewed along the crystallographic axis a).

As alredy pointed out, there are chiral forms for both the pentagonal hexecontahedron and its dual, the snub dodecahedron; in Fig.36 each row shows the chiral pairs, each column the dual pairs.

FIG.36 - Pairs of chiral polyhedra (along each row) and dual polyhedra (along each column)
In detail:
Upper row: chiral snub dodecahedra
Lower row: chiral pentagonal hexacontahedra
Left column: dual pair consisting of the left snub dodecahedron and the left pentagonal hexecontahedron
Right column: dual pair consisting of the right snub dodecahedron and the right pentagonal hexecontahedron.

In Fig.37 one can see some peculiar forms, consisting in the intersection (for appropriate values of the central distances of the faces) of the dual pairs made of an Archimedean polyhedron and the corresponding Catalan polyhedron.

icosidodecahedron
& dual rhombic triacontahedron truncated dodecahedron
& dual triakis-icosahedron truncated icosahedron
& dual pentakis-dodecahedron

rhombicosidodecahedron
& dual deltoidal hexecontahedron

truncated icosidodecahedron
& dual hexakis-icosahedron

snub dodecahedron
& dual pentagonal hexecontahedron

FIG.37 - Rounded icosahedral forms deriving from the intersection of the dual pairs made of an Archimedean semiregular polyhedron and the corresponding Catalan polyhedron: they all have the peculiarity (deriving from the assignation of appropriate values to the central distances of the faces) that each vertex is shared by two faces belonging to the Archimedean solid and two faces of the dual Catalan solid.

In the first of the following summarizing Tables, the central distances of the icosahedral and dodecahedral faces belonging to semiregular Archimedean polyhedra are reported.

In the next Table, one can find the shapes assumed by the faces of Catalan polyhedra, dual of semiregular Archimedean polyhedra, and the relative geometric parameters (L indicates the length of the sides of the polygons).

In the last Table, the sets of {hkl} indices characterizing the Catalan polyhedra are reported together with the corresponding values of the sum: h²+k²+l². Taking into account that the root of such sums is proportional to the reciprocal central distance d_hkl of a (hkl) face, the sum h²+k²+l² must be constant for the set of indices of each Catalan polyhedron, since all its faces have the same central distance.
For instance, the faces of the rhomb-triacontahedron derive from two different kinds of indices, written as:

{100} and {τ/2 1/2 1/2τ} in order to obtain just the same value (equal to 1, in this particular case) for the sum h²+k²+l²; more in general, one can see that such sum is a simple linear function of τ.
Only in the case of the pentagonal hexecontahedron it has not been possible to express the indices as a simple function of the golden ratio τ.

The values, given in the present work, of both the indices of the faces and the central distances of the polyhedra have been obtained by means of a procedure that will be described in a next publication.

COMBINATIONS OF FORMS HAVING ICOSAHEDRAL SIMMETRY

    The icosahedral forms having a particular importance are the ones with minor multiplicity, namely dodecahedron, icosahedron and rhomb-triacontahedron.
As already seen, it is from their intersection (analogously to what happens in the cubic system for the three forms: cube, octahedron and rhomb-dodecahedron), that the Archimedean polyhedra with icosahedral symmetry can be generated (except the snub dodecahedron, since it includes also the pentagonal hexecontahedron), attributing appropriate values to the respective central distances.
    The duals of these isogonal Archimedean polyhedra are the isohedral Catalan polyhedra characterizing the icosahedral symmetry (or the cubic symmetry, regarding the other Catalan polyhedra, dual of Archimedean polyhedra which derive from the intersection of cube, octahedron and rhomb-dodecahedron).

    The rounded polyhedron shown in Fig.38 results from the intersection of all the seven forms belonging to the m35 point group (two Platonic polyhedra, the dodecahedron and the icosahedron, and five Catalan polyhedra), in the particular event that the central distances of all the faces are equal.

FIG.38 - Rounded composite polyhedron originated from all the seven forms characterizing the m35 point group:
1) {1 τ 0} dodecahedron: 12 yellow faces
2) {τ 1/τ 0} icosahedron: 20 red faces
3) {1 0 0} rhomb-triacontahedron: 30 green faces
4) {1 3τ 0} Catalan pentakis-dodecahedron: 60 ochre faces
5) {τ+1/τ τ 0} Catalan deltoidal hexecontahedron: 60 violet faces
6) {τ²+1 1/τ 0} Catalan triakis-icosahedron: 60 fuchsia faces
7) {τ² 1 2/τ²} Catalan hexakis-icosahedron: 120 pale-blue faces
for a total of 362 faces, in this particular case all equidistant from the centre of the composite polyhedron and belonging to Platonic and Catalan solids.
(all the forms are identified, for simplicity, by only one of the sets of indices marking them).

Its stereographic projection, showing clearly the arrangement of the faces with respect to each rotation axis and mirror plane, is reported in Fig.39.

FIG. 39 - View along the crystallographic axis c and stereographic projection of the rounded composite polyhedron made of the seven single forms belonging to the m35 point group and having all the same central distance, which coincide with the Catalan and Platonic solids.

Starting from the truncated icosidodecahedron, in Fig.40 each dodecahedral, icosahedral and triacontahedral face is progressively substituted, through an "augmentation" process, by a decagonal, hexagonal and square pyramid, respectively. Each pyramid is made of pairs of faces, symmetrically orientated with respect to the rotation axes, belonging to two polyhedra out of the set of three: triakis-icosahedron, pentakis-dodecahedron and deltoidal hexecontahedron.

FIG. 40 - Progressive substitution of each face of a truncated icosidodecahedron with pyramids made of pairs of faces, coming from the set of three polyhedra: triakis-icosahedron, pentakis-dodecahedron and deltoidal hexecontahedron, placed in symmetric position with respect to the 2-fold, 3-fold and 5-fold rotation axes.
Each form in the lower row derives from the corresponding form in the upper row by the substitution of the dodecahedral faces with very flat decagonal pyramids.

In general, all the polyhedra having icosahedral symmetry can be considered to derive from the intersection of whichever number of forms (each of them made of faces at a different central distance), belonging to two different sets:

the first set includes dodecahedron, icosahedron and triacontahedron, characterized by faces in fixed position, since they are orthogonal to 2-fold, 3-fold and 5-fold axes, respectively

the second set includes all the other single forms previously described which, on the other hand, usually assume orientations different from the ones corresponding to the Catalan polyhedra, dual of the Archimedean polyhedra.

235 and m35 point groups:
dodecahedron {1τ0}
rhomb-triacontahedron {100}
generic triakis-icosahedron {410}
generic pentakis-dodecahedron {120}
m35 point group:
icosahedron {τ 1/τ 0}
rhomb-triacontahedron {100}
generic deltoidal hexecontahedron {210}
generic hexakis-icosahedron {421}
235 point group:
icosahedron {τ 1/τ 0}
dodecahedron {1τ0}
generic triakis-icosahedron {410}
generic pentagonal hexecontahedron {421}

FIG.41 - Composite forms belonging to icosahedral 235 and m35 point groups, each consisting in the intersection of four different polyhedral forms (identified, for simplicity, by only one of the sets of indices marking them).

LIST OF PLATONIC, ARCHIMEDEAN AND CATALAN POLYHEDRA

The number of faces, edges and vertices relative to all the regular Platonic and semiregular Archimedean polyhedra is reported in Table 5; the number of faces and vertices is obviously interchanged in their duals (Platonic and Catalan polyhedra, respectively), whereas the number of edges holds steady.

TABLE 5 - Platonic and Archimedean polyhedra, and the dual Catalan polyhedra.

In conclusion, it is noteworthy to point out that, by an identical procedure, one can describe the other Platonic and Archimedean polyhedra, as well as their dual Catalan polyhedra, characterized by a cubic symmetry, and consequently all the possible forms deriving from their combination.

ACKNOWLEDGEMENTS

Many thanks are due to Fabio Somenzi for his precious help concerning the translation of the text.

REFERENCES

[1] Donnay J.D.H. & Donnay G. - A graphical derivation of the crystallographic rotation axes
    Canadian Mineralogist, vol.14, 567-570 (1976)
[2] Rigault G. - Sul gruppo puntuale 235
    Estratti dagli "Atti dell'Accademia delle Scienze di Torino", vol. 117(1983)
[3] Loreto L., Farinato R., Pappalardo F. - Icosahedral Symmetry, Icosahedral Polyhedra and Indexing methods
    in "Topics on contemporary crystallography and quasicrystals", special Issue of "Periodico di Mineralogia"
    Vol. LIX (1990) - L. Loreto and M. Ronchetti Editors
[4] Cahn J.W. & Gratias D. - Indexing of Icosahedral Quasiperiodic Crystals
    J. Mater. Res., vol.1, 13-26 (1986)
[5] Hahn T. & Klapper K. - Non crystallographic point group
    in "International Tables for Crystallography", Volume A: Space-Group Symmetry, 796 (2005) - Theo Hahn Editor
[6] Weissbach B. & Martini H. - On the Chiral Archimedean Solids
     Contribution to Algebra and Geometry, vol.43, No.1, 121-133 (2002)

LINKS

TABLE 1- Angles between pairs of rotation axes
	2	3	5
2	36�	20.91�	31.72�
2	60�	54.74�	58.28�
2	72�	69.10�	90�
2	90�	90�
3		41.81�	37.38�
3		70.53�	79.19�
5			63.44�


FIG.13 - Orthographic view of the icosi-dodecahedron (left) and its stereographic projection (right), seen along the crystallographic axis c.

ICOSAHEDRON (d_icosahedron= τ /√3) TRUNCATED BY DODECAHEDRON
	d_dodecahedron = 1.10	d_dodecahedron = 1.05

d_dodecahedron = 0.9	d_dodecahedron = 0.87

FIG.14 - Series of polyhedra, obtained truncating an icosahedron by a dodecahedron, included between the icosahedron (left upper corner of the figure), and the icosi-dodecahedron (right lower corner). They are characterized by: pentagonal-shaped faces of dodecahedron with increasing dimensions icosahedral faces generally with the shape of non-regular (but symmetric) hexagons, except for the Archimedean truncated icosahedron (central image) and the icosi-dodecahedron (last image), where such faces become regular hexagons and equilateral triangles, respectively. Whereas the central distance of the icosahedral faces is constant in every polyhedron and equal to τ /√3, the central distance of the dodecahedral faces decreases from √1+1/τ² (value at which the faces of dodecahedron are tangent to the vertices of icosahedron), to 1/√1+1/τ², in case of icosi-dodecahedron; for such range of values of the central distances of the dodecahedral and icosahedral faces, the faces of a rhombic triacontahedron having a unit central distance would be tangent: to the edges between each pair of icosahedral faces, in the icosahedron and in every truncated icosahedron to every vertices, in the icosi-dodecahedron.

DODECAHEDRON (d_dodecahedron= 1/√1+1/τ²) TRUNCATED BY ICOSAHEDRON
DODECAHEDRON d_icosahedron = √3/τ = 1.0705	d_icosahedron = 1.05	d_icosahedron = 1.03

d_icosahedron = 1	ARCHIMEDEAN TRUNCATED DODECAHEDRON d_icosahedron = (τ /√3)(7τ-6)/5 = 0.9951	d_icosahedron = 0.97

d_icosahedron = 0.96	d_icosahedron = 0.95	ICOSIDODECAHEDRON d_icosahedron = τ /√3 = 0.93417

FIG.18- Series of polyhedra, obtained truncating a dodecahedron by an icosahedron, included between the dodecahedron (left upper corner of the figure), and the icosi-dodecahedron (right lower corner). They are characterized by triangular-shaped faces of the icosahedron with increasing dimensions and faces of dodecahedron having generally the shape of non-regular (but symmetric) decagons, except for the Archimedean truncated dodecahedron (central image) and the icosi-dodecahedron (last image), where such faces are regular decagons and regular pentagons, respectively. Whereas the central distance of the dodecahedral faces is constant in every polyhedron and equal to 1/√1+1/τ², the central distance of the icosahedral faces decreases from √3/τ, (at which value the faces of icosahedron are tangent to the vertices of dodecahedron), to τ/√3, in case of the icosi-dodecahedron; for such range of values, the faces of a dual rhombic triacontahedron placed at a unit central distance would be tangent: to the edges between each pair of dodecahedral faces, in dodecahedron and in every truncated dodecahedron to every vertices, in icosi-dodecahedron.


FIG.19 - Orthographic view of the Archimedean truncated dodecahedron (left) and its stereographic projection (right), seen along the crystallographic axis c.

FIG.23a - Intermediate step given by the intersection of a dodecahedron and an icosahedron

d_dod. /d_icos. = (1+1/τ²) √1+1/τ² /√3 = 0.9380	d_dod./d_icos. = 3√3 / [(5-1/τ²) √1+1/ τ²] = 0.9571

FIG.23b - Intermediate step given by the intersection of an icosahedron and a triacontahedron

d_icos./d_triacont. = √3/(1+2/τ²)= 0.9819	d_icos./d_triacont. = (1+2/τ²)/√3 = 1.0184

FIG.23c - Intermediate step given by the intersection of a dodecahedron and a triacontahedron

d_dod./d_triacont. = √1+1/τ²(1+1/τ²)/(1+2/τ²) = 0.9210	d_dod./d_triacont. = 3/(τ²√1+1/τ²) = 0.9748

Fig.23d - Archimedean truncated icosi-dodecahedron and rhombicosi-dodecahedron generated by the intersection of each previous pair of forms with the respective third form

d_dodecahedron/ d_icosahedron= 0.9380 d_icosahedron/ d_{triacontahedron}= 0.9819 d_dodecahedron/ d_{triacontahedron}= 0.9210	d_dodecahedron/ d_icosahedron = 0.9571 d_icosahedron/ d_{triacontahedron} = 1.0184 d_dodecahedron/ d_{triacontahedron} = 0.9748

ICOSIDODECAHEDRON d_icosahedron = τ /√3 = 0.9342	d_icosahedron = 0.94	d_icosahedron = 0.95

d_icosahedron = 0.97	ARCHIMEDEAN TRUNCATED ICOSIDODECAHEDRON d_icos = √3 /(1+ 2 /τ²) = 0.9819	d_icosahedron = 0.99

d_icosahedron = 1	d_icosahedron = 1.01	RHOMBICOSIDODECAHEDRON d_icos = (1 +2/τ²) / √3 = 1.0184

Fig.24 - Series of polyhedra, all characterized by square faces of triacontahedron, included between two Archimedean semiregular polyhedra: the icosi-dodecahedron (left upper corner), where the square faces are absent, being the triacontahedron tangent to the vertices of the icosi-dodecahedron, and the rhombicosi-dodecahedron (right lower corner). In particular, hexagonal and decagonal faces are both regular in the Archimedean truncated icosi-dodecahedron (in the centre of the figure). Whereas the central distance of the square faces of triacontahedron has always a unit value, the central distance of the dodecahedral faces can be obtained from the central distance of the icosahedral faces by the relation: d_dodecahedron = (1/√1+1/τ²) (√3 d_icosahedron - 1/τ ) previously reported in the text.


icosidodecahedron & dual rhombic triacontahedron	truncated dodecahedron & dual triakis-icosahedron	truncated icosahedron & dual pentakis-dodecahedron

rhombicosidodecahedron & dual deltoidal hexecontahedron	truncated icosidodecahedron & dual hexakis-icosahedron	snub dodecahedron & dual pentagonal hexecontahedron
FIG.37 - Rounded icosahedral forms deriving from the intersection of the dual pairs made of an Archimedean semiregular polyhedron and the corresponding Catalan polyhedron: they all have the peculiarity (deriving from the assignation of appropriate values to the central distances of the faces) that each vertex is shared by two faces belonging to the Archimedean solid and two faces of the dual Catalan solid.