Notes

Fig. 1a) Animated sequence of views of the icosidodecahedron.
An orthographic view is followed by the views 
Fig. 1b)
Animated sequence of views of the rhombtriacontahedron, Catalan dual of
the icosidodecahedron, consisting of thirty rhombic face: an orthographic view is followed by
the views along the [τ01] 
Fig. 2a) Animated sequence of views of the Archimedean truncated
icosahedron AtI, including an orthographic view followed by the views:
The dual of AtI is the Catalan pentakisdodecahedron {3 0 1/τ} 
Fig. 2b)
Animated sequence of views of the Archimedean truncated
dodecahedron AtD, including an orthographic view followed by the views:
The dual of AtD is the Catalan triakisicosahedron {τ+1/τ 1/τ^{2} 0} 
Truncation by a rhombtriacontahedron (RT) of the icosidodecahedron (ID) and of two other Archimedean polyhedra: the truncated icosahedron (AtI) and the truncated dodecahedron (AtD) 

ID  AtI  AtD  
d_{dod} /d_{icos}  
Indices of the Catalan dual  {100}  {3 0 1/τ}  {τ+1/τ 1/τ^{2} 0} 
If d_{RT} =1,
the rhombtriacontahedron is tangent to ID, AtI and AtD, respectively, in coincidence with the following values of d_{dod} and d_{icos}: 

d_{dod}  
d_{icos}  
Vertextransitive solids 

d_{RT}  (τ+4)/6  4(τ^{2}+1)/15  
Indices of the dual hexakisicosahedra  {τ 1 1/3τ}  {2 1 1/3τ^{2}}  
In each dual the six dihedral angles between contiguous faces sharing a vertex along [111] are equal and measure: 
12.363°  6.518°  
Vertextransitive solids 

d_{RT}  (4τ+3)/10  4(τ^{2}+1)/15  
Indices of the dual hexakisicosahedra  {τ 1 1}  {4τ1 2 5τ}  
In each dual the ten dihedral angles between contiguous faces sharing a vertex along [τ01] are equal and measure: 
16.535°  10.527°  
Vertextransitive solids with pentagonal 

d_{RT}  (τ^{2}+1)/4  (τ+4)/6  (4τ+3)/10 
Indices of the dual deltoidhexecontahedra  {τ10}  {τ+1/τ 2 0}  {210} 
In each dual, the three dihedral angles between contiguous faces sharing a vertex along [111] are equal (due to the 3fold axis) and measure: 
18.699°  36°  9.799° 
In each dual, the five dihedral angles between contiguous faces sharing a vertex along [τ01] are equal (due to the 5fold axis) and measure: 
30.480°  19.188°  36° 
Fig. 3a)
Animated sequence of views of the Archimedean rhombicosidodecahedron
RID, including an orthographic view followed by the views:
The dual of RID is the Catalan deltoidhexecontahedron {1+1/τ^{2} 1 0} 
Fig. 3b)
Animated sequence of views of the Archimedean truncated icosidodecahedron tID, including an orthographic view followed by the views:
The dual of tID is the Catalan hexakisicosahedron {2 3/τ 1} 
Archimedean truncated icosidodecahedron (tID), rhombicosidodecahedron (RID) and solids including regular faces, obtained by varying the central distance of the intersecting rhombtriacontahedron 

RID  tID  
d_{dod }/d_{icos}  
d_{icos }/d_{RT}  
If d_{RT} =1:  
d_{icos}  
d_{dod}  
Indices of the Catalan duals  {1+1/τ^{2} 1 0} 
{2 3/τ 1} 
In each Catalan solid, the dihedral angles between all the couples of contiguous faces are equal and measure: 
25.879°  15.112° 
Vertextransitive solids including regular hexagonal faces
of icosahedron 

d_{RT} 
1 

Indices of the dual hexakisicosahedra  {3(1+1/τ^{2}) 3 2/τ^{2}} 
{2 3/τ 1} 
In each dual, the six dihedral angles between contiguous faces sharing a vertex along [111] are equal and measure:

16.981°  15.112° 
Vertextransitive solids including regular decagonal faces of dodecahedron 

d_{RT}  1  
Indices of the dual hexakisicosahedra  {5 5/(1+1/τ^{2}) 2/τ^{2}}  {2 3/τ 1} 
In each dual, the ten dihedral angles between contiguous faces sharing a vertex along [τ01] are equal and measure:

14.112°  15.112° 
Vertextransitive solids including regular pentagonal faces of dodecahedron and equilateral triangular faces of icosahedron
(in RID also the faces of rhombtriacontahedron are square) 

d_{RT}  1  
Indices of the dual deltoidhexecontahedra  {1+1/τ^{2} 1 0}  {4τ5 1 0} 
In each dual, the three dihedral angles between contiguous faces sharing a vertex along [111] are equal
(due to the 3fold axis) and measure:

25.879°  22.954° 
In each dual, the five dihedral angles between contiguous faces sharing a vertex along
[τ01] are equal (due to the 5fold axis) and measure:

25.879°  27.769° 
NonArchimedean truncated icosahedra deriving from RID and tID by increasing the value of d_{RT} 

d_{RT}  
Indices of the dual pentakisdodecahedra  {τ^{5}+1 0 1}  {3 0 1/τ^{4}} 
In each dual, the five dihedral angles between contiguous faces sharing a vertex
along [τ01] are equal (due to the 5fold axis) and measure:

30.942°  33.042° 
Forms including faces having the shape of regular pentagons and equilateral triangles, derived from the intersection between a rhombtriacontahedron (RT) and each of four Archimedean polyhedra with icosahedral symmetry 

Vertex transitive solid deriving from the
intersection of RT with the Archimedean truncated icosahedron (in
which d_{dod} =
(1+1/3τ^{2} )/√1+1/τ^{2} and
d_{icos} = τ/√3)
when d_{RT} = (τ+4)/6.
Its dual is the deltoidhexecontahedron {τ+1/τ 2 0} 
Vertex transitive solid deriving from the Archimedean
truncated icosidodecahedron (tID) when the value of d_{RT} decreases from
1 to 
Vertex transitive solid deriving from the intersection of RT with the icosidodecahedron ID (in
which d_{dod} = 1 /√1+1/τ^{2} and
d_{icos} =
τ/√3)
when Its dual is the deltoidhexecontahedron {τ10} 
Vertex transitive solid deriving from the
intersection of RT with the Archimedean truncated
dodecahedron AtD (in
which d_{dod} = 1/√1+1/τ^{2}
and d_{icos} = (1+1/(τ^{5}+τ^{3}))τ/√3 when Its dual is the deltoidhexecontahedron {210} 
Fig. 5 shows the views along the [111] direction of the solids, obtained by the intersection of dodecahedron, icosahedron and rhombtriacontahedron, which include regular hexagonal faces (as the Archimedean truncated icosahedron, already shown in Fig. 2a).
View along the [111] direction of forms, having icosahedral symmetry, derived from the intersection of Archimedean polyhedra with a rhombtriacontahedron and including regular hexagonal faces, compared with the Archimedean truncated icosidodecahedron 

Archimedean truncated icosidodecahedron 
Polyhedron with regular hexagonal faces derived from the rhombicosidodecahedron (RID) by increasing the central distance of the faces of rhombtriacontahedron, but letting unchanged the ratio d_{dod} /d_{icos} 
Polyhedron including regular hexagonal faces derived from the truncation by a rhombtriacontahedron of the Archimedean icosidodecahedron 
Polyhedron including regular hexagonal faces derived from the truncation by a rhombtriacontahedron of the Archimedean truncated dodecahedron 
View along the [τ01] direction of forms, having icosahedral symmetry, derived from the intersection of Archimedean polyhedra with a rhombtriacontahedron and including regular hexagonal faces, compared with the Archimedean truncated icosidodecahedron 

Archimedean truncated icosidodecahedron (tID) 
Polyhedron including regular decagonal faces derived from the rhombicosidodecahedron (RID) by increasing the central distance of the faces of rhombtriacontahedron, but letting unchanged the ratio d_{dod} /d_{icos} 
Polyhedron including regular decagonal faces, derived from the truncation of the icosidodecahedron by a rhombtriacontahedron 
Polyhedron including regular decagonal faces, derived from the truncation by a rhombtriacontahedron of the Archimedean truncated icosahedron 
Truncated icosidodecahedron (tID) compared with the four similar polyhedra including regular hexagonal or decagonal faces, obtained by intersecting the rhombicosidodecahedron (RID) and the icosidodecahedron (ID) with a rhombtriacontahedron (RT) 

Polyhedron including regular hexagonal faces of icosahedron, derived from RID
by a proper increase of the central distance d_{RT} 
Polyhedron including regular decagonal faces of dodecahedron, derived from RID
by a proper increase of the central distance d_{RT}



Being an Archimedean solid, the truncated icosidodecahedron (tID) includes only regular faces: twelve decagonal faces of dodecahedron, twenty hexagonal faces of icosahedron and thirty square faces of rhombtriacontahedron 

Polyhedron including regular decagonal faces of dodecahedron, derived from ID by a proper
decrease of the central distance d_{RT}

Polyhedron including regular hexagonal faces
of icosahedron, derived from ID by a proper decrease of the central distance d_{RT} 
Comparison between the Catalan hexakisicosahedron, dual of the Archimedean truncated icosidodecahedron (tID), and four very similar hexakisicosahedra, dual of polyhedra, derived from ID and RID, which include regular hexagonal or decagonal faces 

Hexakisicosahedron {3(1+1/τ^{2}) 3 2/τ^{2}},
dual of the solid, derived from RID, which include regular hexagonal faces. 
Hexakisicosahedron {5 5/(1+1/τ^{2})
2/τ^{2}},
dual of the solid, derived from RID, which include regular decagonal faces. 


Hexakisicosahedron {2 3/τ 1}, dual of the Archimedean truncated icosahedron (tID): as it is a Catalan solid, all the dihedral angles between each couple of edgesharing faces are equal and measure = 15.112° 

Hexakisicosahedron {τ11}, dual of the solid derived from ID which includes regular decagonal faces.
The ten dihedral angles between each couple of contiguous faces, sharing a vertex along the [τ01] axis, measure 16.535° 
Hexakisicosahedron {τ1τ^{3}},
dual of the solid derived from ID which includes regular hexagonal faces.
The six dihedral angles between each couple of contiguous faces, sharing a vertex along the [111] axis, measure 12.363° 
Animated sequence highlighting the differences between the truncated icosidodecahedron (tID) and the solids, including hexagonal or decagonal regular faces, derived from the rhombicosidodecahedron (RID) and the icosidodecahedron (ID) 
Animated sequence highlighting the small differences between the Catalan hexakisicosahedron, dual of tID, and the other hexakisicosahedra, which are the duals of solids, including hexagonal or decagonal regular faces, derived from ID and RID. 
Comparison between the intersection of tID with the dual Catalan hexakisicosahedron and the intersection, with the relative duals, of the polyhedra, derived from the Archimedean ID and RID, which include hexagonal or decagonal regular faces 

Solid resulting from the intersection with its dual of the polyhedron, derived from RID, which includes regular hexagonal faces of icosahedron 
Solid resulting from the intersection with its dual of the polyhedron, derived from RID, which includes regular decagonal faces of dodecahedron 


Solid resulting from the intersection between the Archimedean truncated icosidodecahedron (tID) and the dual Catalan hexakisicosahedron, including square, regular hexagonal and regular decagonal faces 



Solid resulting from the intersection with its dual of the polyhedron, derived from ID, which includes regular decagonal faces of dodecahedron 
Solid resulting from the intersection with its dual of the polyhedron, derived from ID, which includes regular hexagonal faces of icosahedron 
Duals of the polyhedra obtained from the intersection between solids including regular faces derived from Archimedean polyhedra and the relative duals 

Solid derived from RID, in which all the six dihedral angles between each couple of contiguous faces sharing a vertex along [111] measure 14.856° 
Solid derived from RID, in which all the ten dihedral angles between each couple of contiguous faces sharing a vertex along [τ01] measure 13.521° 


Solid, consisting of the
deltoidhexecontahedron {4τ5 1 0} and of the couple of symmetric forms:
triakisicosahedron {4τ+1 1 0} and
pentakisdodecahedron {4τ+1 0 1}, which is the dual of the polyhedron
obtained by the intersection of the Archimedean truncated icosahedron with its dual, the Catalan hexakisicosahedron.
In such solid:


Solid derived from ID, in which all the ten dihedral angles between each couple of contiguous faces
sharing a vertex along [τ01] measure 15.887° 
Solid derived from ID, in which all the six dihedral angles between each couple of contiguous faces sharing a vertex
along [111] measure 10.764° 
Animated sequences highlighting the differences which characterize the solids reported in Fig. 10 and Fig. 11 
Archimedean truncated icosidodecahedron (tID) compared with the polyhedra, including regular faces, which derive from the truncation by a rhombtriacontahedron of the Archimedean truncated dodecahedron (AtD) and truncated icosahedron (AtI), followed by their duals and other related forms 

Polyhedron with regular hexagonal faces derived from the Archimedean truncated dodecahedron
(AtD) 
Archimedean truncated icosidodecahedron (tID) consisting of regular hexagonal, regular decagonal and square faces 
Polyhedron with regular decagonal faces derived from the Archimedean truncated icosahedron
(AtI) 
Hexakisicosahedron {2 1 1/(3τ^{2})}, dual of the solid with
regular hexagonal faces deriving from AtD 
Catalan hexakisicosahedron {2 3/τ 1}, dual of tID 
Hexakisicosahedron {4τ1 5τ 2}, dual of the solid with regular
decagonal faces deriving from AtI 
Solid including regular hexagonal faces, obtained from the intersection of the polyhedron deriving from AtD with its dual 
Solid resulting from the intersection of tID with its dual 
Solid including regular decagonal faces, obtained from the intersection of the polyhedron deriving from AtI with its dual

Solid, dual of the intersection of the polyhedron deriving from AtD with its dual, in which the dihedral angles between the six couples of contiguous faces sharing a vertex along [111]
measure 5.653° 
Solid dual of the intersection between tID
and the dual Catalan hexakisicosahedron (the values of the dihedral angles have already been reported in Fig.
11) 
Solid, dual of the intersection of the polyhedron deriving from AtI with its dual, in which the dihedral angles between the ten couples of contiguous faces sharing a vertex
along [τ01] measure 10.054° 
A series of solids including regularshaped faces, obtained sequentially by the intersection with RT of
selected non Archimedean truncated icosahedra tI or truncated dodecahedra tD belonging to the six intervals just defined,
is visualized in Fig. 14.
Solids which include faces having the shape of regular polygons, obtained by the intersection with the rhombtriacontahedron of generic nonArchimedean truncated icosahedra and truncated dodecahedra, compared with the Archimedean truncated icosidodecahedron 

Truncated icosahedra 
Vertextransitive solids which include regular faces having hexagonal, decagonal or square shape 
Vertextransitive solids, dual of deltoidhexecontahedra, which include
regular faces having the shape of pentagons and triangles 


d_{RT }= 1 
d_{RT} = 0.9736: decagonal regular faces  d_{RT} = 0.9523 


d_{RT} = 1 
d_{RT}
= 0.9841: hexagonal regular faces d_{RT} = 0.9604: decagonal regular faces 
d_{RT} = 0.9284 


d_{RT} = 1 
d_{RT} = 0.9554 : hexagonal regular faces d_{RT} = 0.9525 : decagonal regular faces d_{RT} = 0.9382 : square faces 
d_{RT} = 0.9141 


b)
An increase of the value of d_{RT} from 1 to 3/(3τ2) =
1+1/(τ(τ^{5}+1))
= 1.051 implies that the rhombtriacontahedron is tangent to thirty edges of a truncated icosahedron in which the ratio
d_{dod }/d_{icos}
derive from the same d_{dod} and
d_{icos} values relative to tID

a)
If, in addition to such value of the ratio d_{dod} /d_{icos},
the ratio d_{icos} /d_{RT}
assumes the value √3/(52τ),
the result of the truncation by RT is the Archimedean truncated icosidodecahedron (tID).
If one assumes d_{RT} = 1, it follows that: d_{icos} = √3/(52τ), d_{dod} = 5/((3τ2)√1+τ^{2}) 
c)
Letting unchanged the values of d_{dod}
and d_{icos} relative to tID,
a decrease of the value of d_{RT}
from 1 to (3τ+4)/(2(τ+3) =

a)
Alternatively, when d_{RT} = 1 the faces of the rhombtriacontahedron
are tangent to thirty edges of the truncated icosahedron tI in which the same ratio
d_{dod }/d_{icos} derive from: d_{dod}= 5/(3√1+τ^{2}) and d_{icos}= τ/√3 
b)
Letting unchanged the values of d_{dod} and d_{icos} relative
to the truncated icosahedron tI,
a decrease of the value of d_{RT} from
1 to (3τ2)/3 = 1 1/(3τ^{4}) = 0.951 leads to tID

c)
Letting unchanged the values of d_{dod} and d_{icos}
relative to the truncated icosahedron tI,
a decrease of the value of d_{RT}
from 1 to 1 (74τ)/6 = 0.912
leads to the dual of the deltoidhexecontahedron 


d_{RT} = 1 
d_{RT
}= 0.9691 : square faces d_{RT} = 0.9499 : decagonal regular faces d_{RT} = 0.9459 : hexagonal regular faces 
d_{RT} = 0.9093 


d_{RT} = 1 
d_{RT}
= 0.9781 : decagonal regular faces d_{RT} = 0.9530 : hexagonal regular faces 
d_{RT} = 0.9295 


d_{RT} = 1 
d_{RT} = 0.9780 : hexagonal regular faces 
d_{RT} = 0.9670 
Relations holding between the central distance of a truncating rhombtriacontahedron and the central distances of dodecahedron and icosahedron belonging to the solid to be truncated, in case of each series of vertextransitive solids including a certain kind of regular faces 
The regular faces of the series of solids are pentagonal faces of dodecahedron and triangular equilateral faces of icosahedron if:

The regular faces of the series of solids are hexagonal faces of icosahedron if:

The regular faces of the series of solids are decagonal faces of dodecahedron if:

The regular faces of the series of solids are square faces of rhombtriacontahedron if:

Set of vertextransitive solids including pentagonal faces of dodecahedron and equilateral triangular faces of icosahedron
obtained when the ratio d_{dod}/d_{icos} varies in the range: 

Orthographic view of the sequence of vertex transitive solids, characterized by pentagonal faces of dodecahedron and equilateral triangular face of icosahedron, going from the dodecahedron to the icosahedron and including also the Archimedean rhombicosidodecahedron RID. 
View of the same sequence along the [100] 2fold axis, normal to an usually rectangular face of each intermediate solid of the series. 
Set of vertextransitive solids including regular hexagonal faces of icosahedron obtained
when the ratio d_{dod}/d_{icos} varies in the range:


Orthographic view of the sequence of solids, characterized by regular hexagonal faces of icosahedron, going from the Archimedean truncated icosahedron AtI to the dodecahedron and including also the Archimedean truncated icosidodecahedron tID. 
View of the same sequence along the [111] 3fold axis, normal to a face of icosahedron having a regular hexagonal shape. 
Set of vertextransitive solids including regular decagonal faces of dodecahedron
obtained when the ratio d_{dod}/d_{icos} varies in the range: 

Orthographic view of the sequence of solids, characterized by regular decagonal faces of dodecahedron, going from the Archimedean truncated dodecahedron AtD to the icosahedron and including also the Archimedean truncated icosidodecahedron tID. 
View of the same sequence along the [τ01] 5fold axis, normal to a face of dodecahedron having a regular decagonal shape. 
Set of vertextransitive solids including square faces of rhombtriacontahedron obtained
when the ratio d_{dod}/d_{icos} varies in the range: 

Orthographic view of the sequence of solids characterized by square faces of rhombtriacontahedron RT, going from the icosidodecahedron ID to the rhombicosidodecahedron RID and including also the truncated icosidodecahedron tID. 
View of the same sequence along the [100] 2fold axis, normal to a square face of rhombtriacontahedron. 