Livio Zefiro*
*Dip.Te.Ris, Universita' di Genova, Italy
(Email address:
livio.zefiro@fastwebnet.it)
Notes

Fig.1a) View, normal to its (111) regular hexagonal face, of the Archimedean truncated octahedron (AtO)  Fig.1b) View, normal to its (100) regular octagonal face, of the Archimedean truncated cube (AtC) 
Truncation by a rhombdodecahedron (RD) of the cuboctahedron (CO) and the Archimedean truncated octahedron (AtO) and truncated cube (AtC) 

CO  AtO  AtC  
d_{cube} /d_{octahedron}  √3/2  2/√3  √6√3 
Indices of the Catalan dual  {110}  {210}  {11 √21} 
If d_{RD} =1,
the rhombdodecahedron is tangent to CO, AtO and AtC, respectively, in coincidence with the following values: 

d_{cube }  √2/2  2√2/3  √2/2 
d_{octahedron }  √2/√3  √2/√3  (√2+1)/√6 
Vertextransitive solids 

d_{RD}  (2+√2)/4 = 0.8535  (4+√2)/6 = 0.9024  
Indices of the dual hexakisoctahedra  {√2 1 √21}  {2√2 1 √21}  
In each dual the eight dihedral angles
between contiguous faces sharing a vertex along [100] are equal and measure: 
26.899°  15.722°  
Vertextransitive solids 

d_{RD}  5/6 = 0.8333  (4+√2)/6 = 0.9024  
Indices of the dual hexakisoctahedra  {321}  {3 √2+1 2√21}  
In each dual the six dihedral angles between contiguous faces sharing a vertex along [111] are equal and measure: 
21.787°  11.152°  
Vertextransitive solids with square 

d_{RD}  3/4  5/6 = 0.8333  (2+√2)/4 = 0.8535 
Indices of the dual deltoidicositetrahedra  {211}  {411}  {√2 11} 
In each dual, the three dihedral angles
between contiguous faces sharing a vertex along [111] are equal and measure: 
33.557°  60°  16.842° 
In each dual, the four dihedral angles between contiguous faces sharing a vertex along [100] are equal and measure: 
48.190°  27.266°  60° 
Zone axis of the dual forms  
[uvw] such as hu+kv+lw=0  [1 1 1]  [1 2 2]  [√2 1 1] 
Archimedean truncated cuboctahedron (tCO), rhombcuboctahedron (RCO) and solids including regular faces, obtained by varying the central distance of the intersecting rhombdodecahedron 

RCO  tCO  
d_{cube }/d_{octahedron}  √3/(2√21)  (3√2)/√3 
d_{RD }/d_{octahedron}  √3/(2√21)  (2√21)/√3 
If d_{RD} =1:  
d_{cube}  1  (5√21)/7 
d_{octahedron}  (2√21)/√3  √3/(2√21) 
Indices of the Catalan duals  {√2+1 1 1} 
{3√2 1 √21} 
In each Catalan solid, the dihedral angles between each couple of contiguous faces are equal and measure:  41.882°  24.918° 
Solids including regular octagonal faces of cube 

d_{RD}  (3√22)/2  1 
Indices of the dual hexakisoctahedra  {√2+1 √2 2√2}  {3√2 1 √21} 
In each dual, the eight dihedral angles between contiguous faces sharing a vertex along [100] are equal and measure:

23.650°  24.918° 
Solids including regular hexagonal faces of octahedron 

d_{RD}  (√2+2)/3 
1 
Indices of the dual hexakisoctahedra  {3(√2+1) 3+√2 3√2} 
{3√2 1 √21} 
In each dual, the six dihedral angles between contiguous faces sharing a vertex along [111] are equal and measure:

26.804°  24.918° 
Solids including both square faces of cube and equilateral triangular faces of octahedron 

d_{RD}  1  (11+√2)/14 
Indices of the dual deltoidicositetrahedra  {√2+1 1 1}  {3√22 1 1} 
In each dual, the three dihedral angles between contiguous faces sharing a vertex along [111] are equal and measure:

41.882°  38.709° 
In each dual, the four dihedral angles between contiguous faces sharing a vertex along [100] are equal and measure:

41.882°  44.317° 
NonArchimedean truncated octahedra deriving from RCO and tCO by increasing the value of d_{RD} 

d_{RD}  (4√2)/2  3/(4√2) 
Indices of the dual tetrakishexahedra  {√2+1 2 0}  {3√2 √2 0} 
In each dual, the four dihedral angles between contiguous faces sharing a vertex along [100] are equal and measure:

53.629°  56.151° 
Zone axis of the dual forms  
[uvw] such as hu+kv+lw=0  [2(√21) 1 1]  [√2 √23 √23] 
Fig.2 shows the views, along the [111] direction, of the solids including regular hexagonal faces (as the Archimedean truncated octahedron already shown in Fig.1, on the left), which can be obtained by the intersection of cube, octahedron and rhombdodecahedron.
View along the [111] direction of cubic forms including regular hexagonal faces derived from the intersection of Archimedean polyhedra with a rhombdodecahedron, compared with the Archimedean truncated cuboctahedron 

Archimedean truncated cuboctahedron (tCO) 
Polyhedron with regular hexagonal faces derived from the rhombcuboctahedron (RCO) by varying the central distance of the rhombdodecahedron but letting unchanged the ratio d_{cube}/d_{octahedron} 
Polyhedron including regular hexagonal faces derived from the truncation of the Archimedean cuboctahedron (CO) by a rhombdodecahedron 
Polyhedron including regular hexagonal faces derived from the truncation of the Archimedean truncated cube (AtC) by a rhombdodecahedron 
View along the [100] direction of cubic forms, derived from the intersection of Archimedean solids with a rhombdodecahedron and including regular octagonal faces of cube, compared with the Archimedean truncated cuboctahedron 

Archimedean truncated cuboctahedron (tCO) 
Polyhedron including regular octagonal faces derived from the rhombcuboctahedron (RCO) by varying the central distance of the rhombdodecahedron but letting unchanged the ratio d_{cube}/d_{octahedron} 
Polyhedron including regular octagonal faces, derived from the truncation of the cuboctahedron (CO) by a rhombdodecahedron 
Polyhedron including regular octagonal faces, derived from the truncation of the Archimedean truncated octahedron (AtO) by a rhombdodecahedron 
Truncated cuboctahedron (tCO) compared with the four similar polyhedra, including regular hexagonal or octagonal faces, obtained by intersecting the rhombcuboctahedron (RCO) and the cuboctahedron (CO) with a rhombdodecahedron 

Polyhedron derived from RCO, including regular hexagonal faces
of octahedron 
Polyhedron derived from RCO, including regular octagonal faces
of cube

Being an Archimedean solid, the truncated cuboctahedron (tCO) includes only regular faces: six octagonal faces of cube, eight hexagonal faces of octahedron and twelve square faces of rhombdodecahedron 

Polyhedron derived from CO, including regular octagonal faces
of cube

Polyhedron derived from CO, including regular hexagonal faces
of octahedron 
Comparison between the Catalan hexakisoctahedron, dual of the Archimedean truncated cuboctahedron (tCO), and four very similar hexakisoctahedra, dual of polyhedra derived from CO and RCO which include faces having an octagonal or hexagonal regular shape 

Hexakisoctahedron
{3(√2+1) 3+√2 3√2}
dual of the solid, derived from RCO, including regular hexagonal faces. 
Hexakisoctahedron
{√2+1 √2 2√2}
dual of the solid, derived from RCO, including regular octagonal faces. 


Hexakisoctahedron {3√2 1 √21}, dual of the Archimedean truncated cuboctahedron (tCO): as it is a Catalan solid, all the dihedral angles between each couple of edgesharing faces are equal and measure = 24.918° 

Hexakisoctahedron
{√2
1 √21} dual of the solid, derived from CO,
including regular octagonal faces.

Hexakisoctahedron {321} dual of the solid, derived from CO, including regular hexagonal faces.

Animated sequence highlighting the differences between the truncated cuboctahedron and the solids, including hexagonal or octagonal regular faces, derived from the cuboctahedron (CO) and the rhombcuboctahedron (RCO). 
Animated sequence highlighting the small differences between the Catalan hexakisoctahedron, dual of tCO, and the other hexakisoctahedra, which are the duals of solids, including hexagonal or octagonal regular faces, derived from CO and RCO. 
Comparison between the intersection of tCO with the dual Catalan hexakisoctahedron and the intersection with the relative duals of the polyhedra, derived from the Archimedean CO and RCO, which include hexagonal or octagonal regular faces 

Solid resulting from the intersection with its dual of the polyhedron, derived from RCO, including regular hexagonal faces of octahedron. 
Solid resulting from the intersection with its dual of the polyhedron, derived from RCO, including regular octagonal faces of cube. 


Solid resulting from the intersection between the Archimedean truncated cuboctahedron (tCO) and the dual Catalan hexakisoctahedron, including square, regular hexagonal and regular octagonal faces. 



Solid resulting from the intersection with its dual of the polyhedron, derived from CO, including regular octagonal faces of cube. 
Solid resulting from the intersection with its dual of the polyhedron, derived from CO, including regular hexagonal faces of octahedron. 
Duals of the polyhedra obtained from the intersection between solids including regular faces derived from Archimedean polyhedra and the relative duals 

Solid derived from RCO, in which all the six dihedral angles between each couple of contiguous faces sharing a vertex along [111] measure 23.817° 
Solid derived from RCO, in which all the eight dihedral angles between each couple of contiguous faces sharing a vertex along [100] measure 22.306° 


Solid dual of the intersection between the Archimedean
truncated octahedron and the dual Catalan hexakisoctahedron, made of the
deltoidicositetrahedron
{3√22
1 1},
the triakisoctahedron
{3+√2 3+√2
√2}
and the
tetrakiscube
{3√2
1 0} In it: 

Solid derived from CO, in which all the eight
dihedral angles between each couple of contiguous faces sharing a vertex along [100] measure 25.529° 
Solid derived from CO, in which all the six dihedral angles between each couple of contiguous faces sharing a
vertex along [111] measure 19.188° 
Animated sequences highlighting the differences which characterize the solids reported in Fig.7 and Fig.8 
Comparison between the truncated cuboctahedron and the polyhedra, including regular faces, derived from the intersection with a rhombdodecahedron (RD) of the Archimedean truncated cube (AtC) and truncated octahedron (AtO); in addition, their duals and other related composite forms are shown. 

Polyhedron with regular hexagonal faces derived from the truncation by RD of the Archimedean truncated cube (AtC)  Archimedean truncated cuboctahedron  Polyhedron with regular octagonal faces derived from the truncation by RD of the Archimedean truncated octahedron (AtO) 
Hexakisoctahedron
{3√2+1 2√21}
dual of the solid with regular hexagonal faces derived from the Archimedean truncated cube (AtC) 
Hexakisoctahedron {3√2
1
√21} It is the Catalan dual of the Archimedean truncated cuboctahedron

Hexakisoctahedron
{2√2
1 √21} dual of the solid with regular octagonal faces derived from
the Archimedean truncated octahedron (AtO) 
Solid including regular hexagonal faces,
obtained from the intersection of the polyhedron derived from AtC with its dual 
Solid resulting from the intersection with its Catalan dual of the Archimedean truncated cuboctahedron 
Solid including regular octagonal faces,
obtained from the intersection of the polyhedron derived from AtO with its dual 
Solid derived from AtC, in which all the dihedral angles between the six couples of contiguous faces sharing a vertex along [111]
measure 9.70° 
Solid dual of the intersection between the Archimedean truncated cuboctahedron and the dual Catalan hexakisoctahedron
(the values of the dihedral angles have already been reported in Fig.8) 
Solid derived from AtO, in which all the dihedral
angles between the eight couples of contiguous faces sharing a vertex along [100] measure 14.657° 
The sequences of solids including regularshaped faces, obtained by the intersection with RD of non Archimedean truncated octahedra or truncated cubes belonging to the different intervals just defined, are visualized in Fig.11.
Solids which include faces having the shape of regular polygons, obtained by the intersection with the rhombdodecahedron of generic nonArchimedean truncated octahedra and truncated cubes, compared with the Archimedean truncated cuboctahedron 

Truncated octahedra and truncated cubes 
Vertextransitive solids which include regular faces having hexagonal, octagonal or square shape 
Vertextransitive solids including square faces together with triangular equilateral faces 
2√3/3 < d_{cube} /d_{octahedron}
< √3


d_{RD } = 1 
d_{RD} = 0.951: octagonal regular faces  d_{RD} = 0.917 
√3/(2√21) < d_{cube
}/d_{octahedron}
<
2√3/3 

d_{RD} = 1 
d_{RD}
= 0.933: hexagonal regular faces d_{RD} = 0.883: octagonal regular faces 
d_{RD} = 0.8 
(3√2)√3/3
< d_{cube} /d_{octahedron}
< √3/(2√21) 

d_{RD} = 1 
d_{RD} = 0.871 : hexagonal regular faces d_{RD} = 0.865 : octagonal regular faces d_{RD} = 0.818 : square faces 
d_{RD} = 0.769 
d_{cube} /d_{octahedron } = (3√2)/√3 

d_{RD} = 3/(4√2)
=1.160 implies tangency of the rhombdodecahedron {110} to the truncated octahedron in which:
d_{oct} = √3/(2√21), d_{cube} = (5√21)/7 
An Archimedean solid, the truncated cuboctahedron (tCO), results if also:
d_{octahedron} /d_{RD}
= √3/(2√21)
It follows that, when d_{RD} = 1:
d_{oct} =
√3/(2√21),
d_{cube} = (5√21)/7

When d_{RD}
= (√2+11)/14 = 0.887,
each vertex is shared by four faces if:
d_{oct} =
√3/(2√21),
d_{cube} = (5√21)/7

Alternatively,
d_{RD} = 1 implies tangency to the truncated octahedron in which: d_{octahedron}= √2/√3 and d_{cube}= √2(3√2)/3 
The Archimedean tCO can be obtained when d_{RD} =
(4√2)/3
= 0.862 if:
d_{octahedron}= √2/√3 and d_{cube}= √2(3√2)/3 
When d_{RD}=
(6√2)/6= 0.764, each vertex is shared by four faces if:
d_{octahedron}=
√2/√3
and d_{cube}=
√2(3√2)/3

√3/2 < d_{cube
} /d_{octahedron }
< (3√2)/√3


d_{RD} = 1 
d_{RD
}= 0.931 : square faces d_{RD} = 0.858 : octagonal regular faces d_{RD} = 0.848 : hexagonal regular faces 
d_{RD} = 0.757 
√3/(√2+1)
< d_{cube} /d_{octahedron}
< √3/2


d_{RD} = 1 
d_{RD}
= 0.920 : octagonal regular faces d_{RD} = 0.865 : hexagonal regular faces 
d_{RD} = 0.797 
√3/3 < d_{cube} /d_{octahedron}
< √3/(√2+1) 

d_{RD} = 1 
d_{RD} = 0.946 : hexagonal regular faces 
d_{RD} = 0.919 
Relations holding between the central distance of the truncating rhombdodecahedron and the central distances of cube and octahedron in case of each series of solids including regular faces 
Vertextransitive solids including regular hexagonal faces of octahedron 
d_{RD} = (d_{octahedron} +√3 d_{cube} )/√6 when the ratio d_{cube}/d_{octahedron} varies in the range √3/3 < d_{cube} /d_{octahedron} ≤ 2√3/3 
Vertextransitive solids including regular octagonal faces of cube 
d_{RD} = (√3 d_{octahedron} + (√21) d_{cube})/2 when the ratio d_{cube}/d_{octahedron} varies in the range (√21)√3 ≤ d_{cube} /d_{octahedron}<√3 
Vertextransitive solids including square faces of rhombdodecahedron 
d_{RD} = (√2+1) (√3d_{octahedron} √2 d_{cube}) when the ratio d_{cube}/d_{octahedron} varies in the range √3/2 < d_{cube} /d_{octahedron} <√3/(2√21) 
Vertextransitive solids including square faces of cube and triangular equilateral faces of octahedron 
d_{RD} = √2 (√3d_{octahedron} + d_{cube} )/4 when the ratio d_{cube}/d_{octahedron} varies in the range √3/3 < d_{cube} /d_{octahedron} <√3 
Set of vertextransitive solids including regular hexagonal faces of octahedron 

Orthographic view of the sequence of solids, going from the Archimedean truncated octahedron to the cube and including also the Archimedean truncated cuboctahedron, characterized by regular hexagonal faces of octahedron 
View along the [111] 3fold axis, normal to an hexagonal face of octahedron, of the sequence on the left 
Set of vertextransitive solids including regular octagonal faces of cube 

Orthographic view of the sequence of solids, going from the octahedron to the Archimedean truncated cube and including also the Archimederan truncated cuboctahedron, characterized by regular octagonal faces of cube 
View along the [100] 4fold axis, normal to an octagonal face of cube, of the sequence on the left 
Set of vertextransitive solids including square faces of rhombdodecahedron 

Orthographic view of the sequence of solids, going from the cuboctahedron to the rhombcuboctahedron and including also the truncated cuboctahedron, characterized by square faces of rhombdodecahedron 
View along the [110] 2fold axis, normal to a square face of rhombdodecahedron, of the sequence on the left 
Set of vertextransitive solids including square faces of cube and equilateral triangular faces of octahedron 

Orthographic view of the sequence of solids, going from the cube to the octahedron and including also the Archimedean rhombcuboctahedron, characterized by square faces of cube and equilateral triangular face of octahedron. 
View of the same sequence along the [100] 4fold axis, normal to a square face of each solid of the series. 
View of the same sequence along the [110] 2fold axis, normal to
an usually rectangular face of each solid of the series. 
View of the same sequence along the [111] 3fold axis, normal to an equilateral triangular face of
each solid of the series. 
Solids which include faces having the shape of regular polygons, obtained from the intersection of each truncated tetrahedron with a cube 

Pseudo octahedron resulting from the
intersection of two tetrahedra having an equal central distance:
d_{{111}} =
d_{{111}
}= √3/3 
Pseudo Archimedean truncated octahedron
resulting from the intersection of the pseudo octahedron (on the left) with a cube when:
d_{cube} = 2/3

Pseudo Archimedean cuboctahedron resulting from the intersection of the pseudo octahedron with a cube when:
d_{cube}
= 1/2 
Truncated tetrahedron 
Solids with {111} or {111} faces having the shape
of regular hexagons, as result of the intersection of the truncated tetrahedron (on the left) with a cube,
whose faces are set sequentially at the following distances from the barycenter:
d_{cube} = 5/6 and
d_{cube} = 13/18 
Solid (dual of a deltoiddodecahedron), including {111} and {111} equilateral triangular faces,
obtained when the central distance of the cube intersecting the truncated tetrahedron is:
d_{cube} = 7/12

Archimedean truncated tetrahedron, including regular hexagonal faces of the tetrahedron
{111}
d_{{111}} = 5√3/9;
d_{{111}}
= √3/3 
Solid with regular hexagonal faces of the {111} tetrahedron, derived from the intersection of the Archimedean truncated tetrahedron
with a cube when:
d_{cube} = 7/9 
Solid (dual of a deltoiddodecahedron), including {111} and {111} equilateral triangular faces,
obtained when the central distance of the cube which intersects the Archimedean truncated tetrahedron is:
d_{cube} = 2/3

Truncated tetrahedron 
Solid with regular hexagonal faces of the {111} tetrahedron,
resulting from the intersection of the truncated tetrahedron shown on the left with a cube when:
d_{cube} = 5/6 
Solid (dual of a deltoiddodecahedron), including {111} and {111} equilateral triangular faces,
obtained when the central distance of the cube which intersects the truncated tetrahedron is:
d_{cube} = 3/4

Duals of solids, including regular faces, obtained by intersecting an Archimedean truncated tetrahedron and a pseudo octahedron with a cube 

Triakistetrahedron {311}: it is the dual of the Archimedean truncated tetrahedron in which the faces of the tetrahedron
{111} are nearer to the center of the solid
(d_{{111} }/d_{{111}}=
5/3) 
Cube: it is the dual of the pseudooctahedron obtained from the intersection of the couple of tetrahedra {111}
and {111} when:
d_{{111} }/d_{{111}}=
1

Triakistetrahedron {311}:
it is the dual of the Archimedean truncated tetrahedron in which the faces of the tetrahedron
{111} are nearer to the center of the solid
(d_{{111} }/d_{{111}}=
3/5) 
Hexakistetrahedron {753}: dual of the solid, including regular hexagonal faces, derived from the intersection of the Archimedean truncated tetrahedron (in which d_{{111}}/d_{{111}}= 5/3) with a cube, when d_{cube} = 7/9 
Tetrakishexahedron {210}: dual of the pseudo Archimedean truncated octahedron resulting from the intersection of a pseudooctahedron (obtained from two tetrahedra if: d_{{111}}/d_{{111}}= 1) with a cube, when d_{cube} = 2/3 
Hexakistetrahedron {753}: dual of the solid including regular hexagonal faces derived from the intersection of the Archimedean truncated tetrahedron (in which d_{{111}}/d_{{111}}= 3/5) with a cube, when d_{cube} = 7/9 
Deltoiddodecahedron {221}: dual of the solid, including equilateral triangular and square faces,
derived from the intersection with a cube (when d_{cube } = 2/3) of
the Archimedean truncated tetrahedron (in which:
d_{{111} }/d_{{111}}= 5/3)
. 
Rhombdodecahedron {110}: dual of the pseudo cuboctahedron resulting from the intersection of a pseudooctahedron (obtained if:
d_{{111}}/d_{{111}}= 1)
with a cube, when d_{cube} = 1/2.

Deltoiddodecahedron {221}: dual of
the solid including regular triangular
and square faces, derived from the intersection with a cube (when d_{cube} = 2/3) of the Archimedean truncated tetrahedron (in which:
d_{{111} }/d_{{111}}= 3/5).

Relations holding between the central distance of the truncating cube and the central distances of the two tetrahedra in case of each series of solids including regular faces 
Vertextransitive solids including regular hexagonal faces of the tetrahedron {111} 
d_{cube} = √3(d_{{111}}+ 3d_{{111}})/6 when the ratio d_{{111}}/d_{{111}} varies in the range 3/5 ≤ d_{{111}}/d_{{111}} < 3 
Vertextransitive solids including regular hexagonal faces of the tetrahedron {111} 
d_{cube} = √3(d_{{111}}+ 3d_{{111}})/6 when the ratio d_{{111}}/d_{{111}} varies in the range 1/3 ≤ d_{{111}}/d_{{111}} < 5/3 
Vertextransitive solids including triangular equilateral faces of both tetrahedra 
d_{cube} = √3(d_{{111}}+ d_{{111}})/4 when the ratio d_{{111}}/d_{{111}} varies in the range 1/3 < d_{{111}}/d_{{111}} < 3 
Animated sequence relative to the solids characterized by regular hexagonal faces of the tetrahedron {111} (on the left) or {111} (on the right), deriving from the intersection with a cube of two series of truncated tetrahedra (obtained as the ratio d_{{111}}/d_{{111}} varies within proper ranges). In each sequence, the only frame in which the faces of both tetrahedra have a regular hexagonal shape concerns the pseudo Archimedean truncated octahedron, obtained when the central distances of the two tetrahedra are equal and d_{tetrahedra} /d_{cube} = √3/2. 