Couples of convex and compound polyhedra obtained from the intersection of deltoidicositetrahedra with their duals, in turn derived from the truncation by rhombdodecahedron of Archimedean solids with m3m cubic symmetry
Livio Zefiro
DISTAV, Universita' di Genova, Italy
(Email address: livio.zefiro@fastwebnet.it)
Notes

The results, step by step, of the truncation through a rhombdodecahedron of Archimedean solids, having m3m cubic symmetry, were already described in two previous works [1,2].
All the vertices of the polyhedra obtained sequentially in the early stages of each truncation process are shared by a face of cube, a face of octahedron and a face of rhombdodecahedron:
indeed such polyhedra are vertextransitive [3] and their duals consist in a continuous series of facetransitive hexakisoctahedra.
On the other hand, all the vertices of the last vertextransitive polyhedron resulting from each truncation process are shared by a square face of cube,
an equilateral triangular face of octahedron and two rectangular faces of rhombdodecahedron; the archetype of these vertex transitive polyhedra,
whose duals are facetransitive deltoidicositetrahedra, is the Archimedean rhombicuboctahedron, in which the lengths of all the edges are equal,
since also the faces of rhombdodecahedron are square.
It is the dual of the Catalan deltoidicositetrahedron [4],
in turn archetype of all the deltoidicositetrahedra.
This work describes firstly a series of solids generated from the intersection of
these vertextransitive polyhedra with their dual deltoidicositetrahedra
and then the duals of such series of solids. In addition, relatively to the solids derived from the rhombicuboctahedron,
the intersection of the last couple of dual polyhedra and also its further dual are reported.
The animated sequences in Fig.1 summarize the convex polyhedra resulting from these processes,
whereas the corresponding nonconvex compound polyhedra are reported in Fig.21.
Fig.1a) Animated sequence of the five Archimedean polyhedra characterized by m3m cubic symmetry,
ordered according to decreasing values of the ratio d_{cube}/d_{octahedron}.

Fig.1b) Animated sequence of five vertextransitive polyhedra, each one dual of a deltoidicositetrahedron. 
Fig.1c) Animated sequence including five polyhedra obtained by the intersection of each solid shown in Fig.1b with its dual deltoidicositetrahedron. 
Fig.1d) Animated sequence including the five duals of the polyhedra shown in Fig.1c. 
Some significant data concerning the three Archimedean polyhedra with m3m cubic symmetry
(or 4/m 3 2/m in the extended notation [5]),
derived from the intersection of a cube and an octahedron, are listed in Table 1,
together with other data relative to the vertex transitive polyhedra, dual of deltoidicositetrahedra, resulting from the truncation of each Archimedean polyhedron
by a rhombdodecahedron (RD) :
Central distances of cube an octahedron leading to cuboctahedron (CO), Archimedean truncated octahedron (AtO) and Archimedean truncated cube (AtC) 

CO  AtO  AtC  
d_{cube}/d_{octahedron}  √3/2  2/√3  √3/(√2+1) 
If d_{RD}=1, the faces of rhombdodecahedron are tangent to the vertices of CO or to the edges of AtO and AtC when the central distances d_{cube } and d_{octahedron} assume the following values: 

d_{cube}  √2/2  2√2/3  √2/2 
d_{octahedron}  √2/√3  √2/√3  (√2+1)/√6 
Features of the deltoidicositetrahedra dual of the vertextransitive polyhedra, including square faces of cube, equilateral triangular faces of octahedron and rectangular faces of rhombdodecahedron, obtained from the truncation of CO, AtO and AtC by a rhombdodecahedron 

from CO  from AtO  from AtC  
If d_{cube} and d_{octahedron} assume the values previously reported, the truncation by a rhombdodecahedron of the Archimedean polyhedra CO, AtO and AtC can give rise to a vertextransitive polyhedron, dual of a deltoidicositrahedron, when the value of d_{RD} becomes, respectively, equal to: 
3/4  5/6  (2+√2)/4 
Generalized Miller indices of the three deltoidicositetrahedra dual of the vertextransitive polyhedra derived from the truncation by RD of CO, AtO and AtC 
{211}  {411}  {√2 11} 
The three dihedral angles between contiguous faces sharing a vertex along [111] in each deltoidicositetrahedron measure: 
33.557°  60°  16.842° 
The four dihedral angles between contiguous faces sharing a vertex along [100]in each deltoidicositetrahedron measure: 
48.190°  27.266°  60° 
The corresponding data relative to rhombicuboctahedron and truncated cuboctahedron, namely the Archimedean polyhedra
with m3m cubic symmetry generated from the intersection of cube, octahedron and rhombdodecahedron,
are listed in Table 2.
Central distances of the single forms constituting the rhombicuboctahedron (RCO) and the Archimedean truncated cuboctahedron (AtCO) 

RCO  AtCO  
d_{octahedron}/d_{cube}  (2√21)/√3  √3/(3√2) 
d_{RD}/d_{cube}  1  (5√2+1)/7 
d_{RD}/d_{octahedron}  √3/(2√21)  (2√21)/√3 
Consequently, if d_{RD}= 1, the values of d_{cube} and d_{octahedron} become:  
d_{cube}  1  (5√21)/7 
d_{octahedron}  (2√21)/√3  √3/(2√21) 
Features of the deltoidicositetrahedra dual of either RCO or the vertextransitive polyhedron obtained by decreasing the central distance of the faces of rhombdodecahedron in AtCO 

RCO  from AtCO  
If d_{cube} and d_{octahedron} assume the values previously reported, the values of d_{RD} relative to RCO and to the vertextransitive polyhedron, dual of a deltoidicositrahedron, obtained by decreasing the central distance of the faces of rhombdodecahedron in AtCO, are: 
1  1[(3√2)/14] 
Generalized Miller indices of the Catalan deltoidicositetrahedra dual of either RCO or the deltoidicositetrahedron, dual of the vertextransitive polyhedron obtained by decreasing the central distance of RD in AtCO 
{√2+1 1 1}  {3√22 1 1} 
The three dihedral angles between contiguous faces sharing a vertex along [111] in each deltoidicositetrahedron measure:

41.882°  38.709° 
The four dihedral angles between contiguous faces sharing a vertex along [100] in each deltoidicositetrahedron measure:

41.882°  44.317° 
Central distances of the nonArchimedean truncated octahedra obtained from RCO and AtCO by increasing properly the value of d_{RD} 

Starting from RCO or AtCO, one obtains the two related nonArchimedean truncated octahedra if, left unchanged the respective values of d_{octahedron} and d_{cube}, d_{RD} increases from 1 to a value: 
≥ (2√2)/2  ≥ (3√22)/14 
Rescaling of the central distances, in order to get a closer comparison with AtO and CO of the nonArchimedean truncated octahedra derived from RCO or AtCO 

Starting from the respective values of the ratio d_{octahedron}/d_{cube}: 
(2√21)/√3  √3/(3√2) 
if one attributes to RCO and AtCO the same value of d_{octahedron}:  √2/√3  √2/√3 
the corresponding values of d_{cube } are:  √2/(2√21)  (3√22)/3 
Solids derived from cuboctahedron
At this point, one can take into consideration the truncation of the vertextransitive polyhedron by its dual, the deltoidicositetrahedron {211}.
The sequence reporting the most significant steps of the complete truncation process is reported in the Appendix (Fig.22a), together with the sequence of the respective duals
(Fig.22b).
In particular, the step leading to a solid with triangular faces of octahedron is described in the animated sequence of Fig.4b;
the convex polyhedron resulting from the truncation and the relative compound polyhedron are shown in Fig.4a and Fig.4c, respectively.
Fig.2a) Archimedean cuboctahedron, made of six square and eight triangular faces, resulting when the value of the ratio d_{cube}/d_{octahedron} is equal to √3/2 
Fig.2b) Animated image showing the correlation between cuboctahedron and the corresponding compound polyhedron consisting in the combination of cube and octahedron, when d_{cube}/d_{octahedron}= √3/2 
Fig.2c) Compound polyhedron, made of a cube and an octahedron, resulting when d_{cube}/d_{octahedron}= √3/2 
Fig.3a) Vertextransitive polyhedron, dual of the deltoid icositetrahedron {211} obtained truncating a cuboctahedron (in which d_{cube}= √2/2 and d_{octahedron}= √2/√3) by its dual rhombdodecahedron, whose faces are at a central distance d_{RD}= 3/4 
Fig.3b) Animated sequence showing the derivation of: 
Fig.3c) Compound polyhedron consisting in the combination of a cuboctahedron with a rhombdodecahedron, whose faces are at a central distance d_{RD}= 3/4 
Fig.4a) Convex polyhedron with triangular faces of octahedron resulting from the intersection of the vertextransitive polyhedron reported in Fig.3a with its dual, the deltoidicositetrahedron {211} 
Fig.4b) Animated sequence correlating the convex and compound polyhedra which result, respectively, from the intersection and combination of the solid reported in Fig.3a with its dual, the deltoidicositetrahedron {211} 
Fig.4c) Compound polyhedron resulting from the combination of the solid reported in Fig.3a with its dual, the deltoidicositetrahedron {211} 
Fig.5a) Dual of the solid reported in Fig.4a: it consists in the intersection of the hexakisoctahedron {421} with the triakisoctahedron {332} 
Fig.5b) Animated sequence correlating the convex polyhedron shown in Fig.5a and the compound polyhedron shown in Fig.5c and including also the two single forms hexakisoctahedron {421} and triakisoctahedron {332} 
Fig.5c) Compound polyhedron made of the combination of the single forms hexakisoctahedron {421} and triakisoctahedron {332} 
Solids derived from the Archimedean truncated octahedron
Fig.6a) Convex vertextransitive polyhedron derived from the intersection of an Archimedean truncated octahedron AtO, in which
d_{cube}= 2√2/3
and d_{octahedron}= √2/√3,
with a rhombdodecahedron, whose faces are at a central distance d_{RD}= 5/6
Fig.6b) Animated sequence showing the vertextransitive convex polyhedron
shown in Fig.6a and the compound polyhedron shown in Fig.6c
Fig.6c) Compound polyhedron consisting in the combination of an Archimedean truncated octahedron AtO, in which
d_{cube}= 2√2/3
and d_{octahedron}= √2/√3,
with a rhombdodecahedron whose faces are at a central distance d_{RD}= 5/6
Fig.7a)
Convex polyhedron; with square faces of cube resulting from the intersection of the polyhedron
reported in Fig.6a
with its dual, the deltoidicositetrahedron {411}
Fig.7b)
Animated sequence correlating the convex and compound polyhedra which result, respectively, from the intersection and combination of the solid reported in
Fig.6a with its dual, the deltoidicositetrahedron {411}
Fig.7c)
Compound polyhedron resulting from the combination of the solid reported in Fig.6a with its dual,
the deltoidicositetrahedron {411}
Fig.8a)
Convex polyhedron dual of the solid reported in Fig.7a,
consisting in the intersection of two single forms, the hexakisoctahedron {11 4 3} and the tetrakishexahedron {410}
Fig.8b)
Animated sequence correlating the convex polyhedron shown in Fig.8a
and the compound polyhedron shown in Fig.8c,
and including also the single forms:
hexakisoctahedron {11 4 3} and tetrakishexahedron {410}
Fig.8c)
Compound polyhedron made of the combination of two single forms, the hexakisoctahedron {11 4 3} and the tetrakishexahedron {410}
When the value of the ratio d_{cube}/d_{octahedron} is included in the range
√3/2< d_{cube}/d_{octahedron}<√3,
the intersection of cube and octahedron gives rise to a truncated octahedron, consisting of square faces of cube and hexagonal faces of octahedron,
mostly nonregular polygons.
Relatively to whichever truncated octahedron in which the central distance of the face of octahedron is
d_{octahedron}= √2/√3, a central distance d_{RD}= 1 of the
faces of a rhombdodecahedron implies, irrespective of the value of d_{cube},
the tangency of all the faces of rhombdodecahedron to the twelve edges shared
by as many couples of faces of octahedron.
By inserting the value
d_{octahedron}= √2/√3
in the formula
√2d_{cube}+√6d_{octahedron}=
4d_{RD},
relative to the distances of the single forms in each vertextransitive polyhedron dual of a deltoidicositetrahedron, one
can infer that the value of d_{RD} must decrease from 1
to (√2d_{cube}+2)/4
(with the central distance of the cube varying in the range:
When the ratio d_{cube}/d_{octahedron}
increases from √3/2, value characterizing the cuboctahedron,
to its reciprocal 2/√3, the intersection of cube and octahedron generates another semiregular polyhedron made of six square and eight regular hexagonal faces, the
Archimedean truncated octahedron AtO.
It is shown in the animated sequence of Fig.6b together with a) a rhombdodecahedron,
b) a compound polyhedron made just of the combination of AtO and a rhombdodecahedron, and
c) the corresponding vertextransitive convex polyhedron made of square, triangular and rectangular faces,
dual of the deltoidicositetrahedron with indices {411}, obtained intersecting AtO with a rhombdodecahedron: it follows that, assigned the values
d_{octahedron}= √2/√3
and consequently d_{cube}= 2√2/3 to the central distances of octahedron and cube
in AtO, the faces of rhombdodecahedron in the dual of the deltoidicositetrahedron {411} are at a central distance d_{RD}
= 5/6; furhermore, relatively to this polyhedron, √2/3
is the ratio between the lengths of the sides that the square and triangular faces share with a rectangular face of rhombdodecahedron.
The vertextransitive convex polyhedron dual of {411} and the related compound polyhedron are shown in Fig.6a and Fig.6c, respectively.
The intersection of the deltoidicositetrahedron {411} with its dual can lead to a convex polyhedron (shown in Fig.7a)
in which the faces of cube are square; the corresponding compound polyhedron is shown in Fig.7c.
The couple of dual polyhedra and the results of their intersection and combination are reported in the animated sequence of Fig.7b.
Other polyhedra obtained at different steps of the intersection process between {411} and its dual are reported, together with their duals,
in Fig.23a and Fig.23b of the Appendix.
The dual of the solid reported in
Fig.7a consists in the intersection of the hexakisoctahedron {11 4 3} with the tetrakishexahedron {410};
it is shown in Fig.8a, the corresponding compound polyhedron in Fig.8c
and the animated sequence correlating the convex and compound polyhedra in Fig.8b.
Solids derived from the Archimedean truncated cube
Fig.9a)
Vertextransitive polyhedron obtained from the intersection of an Archimedean truncated cube AtC, in which the central distances of the faces are
d_{cube}= √2/2
and d_{octahedron}= √2+1)/√6,
with a rhombdodecahedron, whose faces are at a central distance d_{RD}= (2+√2)/4
Fig.9b)
Animated sequence correlating the convex and compound polyhedra derived from the intersection and combination of AtC
with a rhombdodecahedron whose faces are at a central distance
Fig.9c)
Compound polyhedron consisting in the combination with AtC of a rhombdodecahedron whose faces are at a central distance
Fig.10a)
Convex polyhedron with triangular faces of octahedron resulting from the intersection of the polyhedron reported in
Fig.9a with its dual,
the deltoidicositetrahedron
Fig.10b)
Animated sequence correlating the convex and compound polyhedra which result, respectively, from the intersection and combination of the solid reported in
Fig.9a with its dual, the deltoidicositetrahedron
Fig.10c)
Compound polyhedron resulting from the combination of the polyhedron reported in Fig.9a
with its dual, the deltoidicositetrahedron {√2 1 1}
Fig.11a)
Convex polyhedron dual of the solid reported in Fig.10a
and consisting in the intersection of two single forms,
the hexakisoctahedron {2√2 2√21 2}
and the triakisoctahedron {√2+1 √2+1 2}
Fig.11b)
Animated sequence correlating the convex polyhedron shown in Fig.11a
and the compound polyhedron shown in Fig.11c,
and including also the two single forms:
hexakisoctahedron {2√2 2√21 2}
and triakisoctahedron {√2+1 √2+1 2}
Fig.11c)
Compound polyhedron made of the combination of two single forms,
the hexakisoctahedron {2√2 2√21 2}
and the triakisoctahedron
When the value of the ratio d_{cube}/d_{octahedron}
is included in the range
√3/3 < d_{cube}/d_{octahedron}< √3/2,
the intersection of cube and octahedron gives rise to a truncated cube,
consisting of equilateral triangular faces of octahedron and octagonal faces of cube, mostly nonregular polygons.
Relatively to whichever truncated cube in which the central distance of the faces of cube is
d_{cube}= √2/2,
a central distance d_{RD }= 1 of the faces of rhombdodecahedron implies,
irrespective of the value of d_{octahedron},
the tangency of the faces of rhombdodecahedron to all the twelve edges shared by as many couples of faces of cube.
By inserting the value d_{cube}= √2/2
in the formula correlating the central distances in vertextransitive polyhedra dual of deltoidicositetrahedra,
one can infer that the value of d_{RD}
must decrease from 1 to
When the value of the ratio d_{cube}/d_{octahedron}
becomes equal to √3/(√2+1),
the intersection of cube and octahedron generates another semiregular solid made of six regular octagons and eight equilateral triangular faces,
the Archimedean truncated cube AtC.
It is shown in the animated sequence of Fig.9b together with a) a rhombdodecahedron, b)
a compound polyhedron, consisting in the combination of AtC with the truncating rhombdodecahedron,
and c) the corresponding vertextransitive convex polyhedron made of square, triangular and rectangular faces,
dual of the deltoidicositetrahedron with indices {√211},
obtained intersecting the Archimedean truncated cube with a rhombdodecahedron.
In this vertextransitive polyhedron (where 2+√2 is the ratio of the lengths of the sides
of the square and triangular faces) the faces of rhombdodecahedron are at a central distance
d_{RD}= (2+√2)/4,
accordingly to the assignment of the value d_{cube}= √2/2,
and consequently d_{octahedron}= (√2+1)/√6,
to the central distances of cube and octahedron in the Archimedean truncated cube from which it derives.
The vertextransitive convex polyhedron dual of {√2 1 1}
and the related compound polyhedron are shown separately in Fig.9a and Fig.9c, respectively.
The intersection of the deltoidicositetrahedron {√2 1 1} with its dual
can lead to a convex polyhedron (shown in Fig.10a) in which the faces of octahedron are triangular
(other polyhedra obtained at different steps of the intersection process are reported, together with their duals, in Fig.24a and Fig.24b of the Appendix)
and the corresponding compound polyhedron is shown in Fig.10c.
The couple of dual polyhedra and the results of their combination are reported in the animated sequence of Fig.10b.
The dual of the solid reported in Fig.10a consists in the intersection of the hexakisoctahedron
{2√2 2√21 2}
with the triakisoctahedron {√2+1 √2+1 2}:
it is shown in Fig.11a, the corresponding compound polyhedron in Fig.11c,
and the animated sequence correlating the convex and compound polyhedra in Fig.11b.
Solids derived from the Archimedean truncated cuboctahedron
As stated in the introduction, the intermediate steps of the truncation
process by a rhombdodecahedron of whichever truncated octahedron lead to a continuous series of vertextransitive polyhedra, dual of hexakisoctahedra:
in correspondence to each ratio d_{octahedron}/d_{cube},
the resulting series of duals of hexakisoctahedra
is characterized by ratios d_{RD}/d_{cube} included in the following range:
(√2+√6d_{octahedron}/d_{cube})/4
< d_{RD}/d_{cube}
< (√3/√2)(d_{octahedron}/d_{cube}).
When the value of the ratio
d_{octahedron}/d_{cube}
is set to √3/(3√2),
the range becomes:
The dual of AtCO is the Catalan hexakisoctahedron
{2√2+1 √2+1 1}, archetype of all the hexakisoctahedra
dual of vertextransitive solids obtained from the intersection with a rhombdodecahedron of whichever truncated octahedron or truncated cube.
The animated sequence reported in Fig.12b includes AtCO and two related compound polyhedra,
obtained starting either from the single forms, cube and octahedron, or from their intersection, shown in Fig.12a
together with the corresponding compound polyhedron, generated from the combination of cube and octahedron.
A further reduction of the ratio d_{RD}/d_{cube}
from the value relative to AtCO, (5√2+1)/7,
to (8√2+3)/14 leads to a vertextransitive convex polyhedron, dual of the deltoidicositetrahedron
If one inserts in d_{RD}/d_{cube}=
(8√2+3)/14 the value
d_{cube}= (5√21)/7
characterizing also AtCO when d_{RD}=1,
the resulting value of d_{RD} in the dual of the deltoidicositetrahedron
The animated sequence reported in Fig.12c includes, together with this vertextransitive convex polyhedron,
two compound polyhedra derived from the combination of the rhombdodecahedron with each of the two solids shown in Fig.12a.
One can realize that, relatively to the compound polyhedron resulting from the combination of the three single forms,
its exterior shape does not include the rhombdodecahedron, whose faces result to be internally tangent to the compound polyhedron, made of the only cube and octahedron;
therefore it is identical to the compound polyhedron shown in Fig.12a.
The intersection of the deltoidicositetrahedron {3√22 1 1}
with its dual vertextransitive polyhedron can lead to a convex polyhedron (shown in Fig.13a), in which the faces of octahedron are triangular,
whereas the corresponding compound polyhedron is shown in Fig.13c
(also in this case, other polyhedra obtained at different steps of the intersection process between the deltoidicositetrahedron
{3√22 1 1} and its dual are reported, together with their duals, in Fig.25a and Fig.25b of the Appendix).
The couple of dual polyhedra and the results of their combination are reported in the animated sequence of Fig.13b.
The dual of the solid reported in Fig.13a consists in the intersection of the hexakisoctahedron
{10√2+12 4√2+6 4√23}
with the triakisoctahedron {3√21 3√21 2}:
it is shown in Fig.14a, the corresponding compound polyhedron is shown in Fig.14c,
whereas the animated sequence correlating the convex and compound polyhedra is shown in Fig.14b.
Fig.12a)
Compound polyhedron and relative nonArchimedean convex polyhedron resulting from the intersection of a cube and an octahedron
when the value of the ratio d_{octahedron}/d_{cube}
is equal to √3/(3√2),
as in Archimedean truncated cuboctahedron
Fig.12b)
Animated sequence showing the Archimedean truncated cuboctahedron and two related compound polyhedra resulting from the combination
with a rhombdodecahedron of either the two single forms cube and octahedron or their intersection shown in Fig.12a,
when, fixed the ratio
d_{octahedron}/d_{cube}= √3/(3√2),
d_{RD}/d_{cube} becomes equal to (5√2+1)/7 and consequently
d_{RD}/d_{octahedron} =
(2√21)/√3
Fig.12c)
Animated sequence showing, together with the vertex transitive convex polyhedron, dual of the deltoidicositetrahedron
{3√22 1 1}, obtained by decreasing the value of d_{RD} in AtCO,
two related compound polyhedra resulting from the combination with a rhombdodecahedron, of either the single forms cube and octahedron or their intersection
shown in Fig.12a
Fig.13a)
Convex polyhedron, including triangular faces of octahedron, which results from the intersection
with the deltoidicositetrahedron having indices {3√22 1 1}
of its dual, the vertextransitive polyhedron obtained by decreasing the central distance of the rhombdodecahedron in AtCO
Fig.13b)
Animated sequence correlating the solids reported in Fig.13a and Fig.13c,
resulting from the intersection and combination of the deltoidicositetrahedron
{3√22 1 1} with its vertextransitive dual
Fig.13c)
Compound polyhedron resulting from the combination of the deltoidicositetrahedron {3√22 1 1}
with its dual, the vertextransitive polyhedron obtained by decreasing the central distance of the rhombdodecahedron in AtCO
Fig.14a)
Convex polyhedron dual of the solid reported in Fig.13a:
it consists in the intersection of two single forms, the hexakisoctahedron
{10√2+12 4√2+6 4√23}
and the triakisoctahedron {3√21 3√21 2}
Fig.14b)
Animated sequence correlating the convex polyhedron shown in Fig.14a
and the compound polyhedron shown in Fig.14c,
and including also two single forms, the hexakisoctahedron
{10√2+12 4√2+6 4√23}
and the triakisoctahedron {3√21 3√21 2}
Fig.14c)
Compound polyhedron made of the combination of two forms, the hexakisoctahedron
{10√2+12 4√2+6 4√23}
and the triakisoctahedron {3√21 3√21 2}
Solids derived from rhombicuboctahedron
Another noteworthy combination of a cube with an octahedron is obtained when the value of the ratio
d_{cube}/d_{octahedron}
is equal to √3/(2√21):
the resulting compound polyhedron is shown in Fig.15a together with the corresponding convex polyhedron,
a truncated octahedron slightly different from the one, shown in Fig.12a, derived from AtCO.
After the usual series of vertextransitive polyhedra, dual of hexakisoctahedra,
obtained from the intersection of this truncated octahedron with a rhombdodecahedron,
a further decrease of d_{RD} leads to a peculiar polyhedron when its value
coincides with the value of d_{cube } and
consequently also d_{RD}/d_{octahedron}= √3/(2√21);
such polyhedron, dual of the Catalan deltoidicositetrahedron {√2+1
1 1}, is the Archimedean rhombicuboctahedron (RCO), made of equilateral triangular faces of octahedron and square faces of both cube and rhombdodecahedron
(it is alternatively called small rhombicuboctahedron, in analogy with the name great rhombicuboctahedron given to AtCO).
It is interesting to note that the values of the ratio
d_{RD}/d_{octahedron}
in RCO and AtCO are reciprocal: √3/(2√21)
and (2√21)/√3, respectively.
The animated sequence of Fig.15b describes the derivation of both RCO and a related compound polyhedron from the intersection and combination
with a rhombdodecahedron of the truncated octahedron shown in Fig.15a.
In Fig.15c another animated sequence shows RCO and a further related compound polyhedron which derives from the combination
with a rhombdodecahedron of the compound polyhedron shown in Fig.15a.
One can verify that, relatively to the compound polyhedron resulting from the combination of the three single forms: cube, octahedron and rhombdodecahedron,
also in this case its exterior shape does not include the rhombdodecahedron, whose faces result to be internally tangent to the compound polyhedron
made of the only cube and octahedron, therefore identical to the compound polyhedron shown in Fig.15a.
In general this feature characterizes each compound polyhedron, related to the dual of a deltoidicositetrahedron, resulting from the combination of the three single forms.
The intersection of RCO with the dual Catalan deltoidicositetrahedron {√2+1
1 1} can lead to a convex polyhedron with trapezoidal faces of deltoidicositetrahedron, in which the faces of cube and rhombdodecahedron are still square,
and the faces of octahedron are still triangular but all are differently oriented in respect to RCO:
it is shown in Fig.16a and the corresponding compound polyhedron is shown in Fig.16c.
The couple of dual polyhedra and the results of their combination are reported in the animated sequence of Fig.16b.
(Also in case of the intersection between RCO and its dual deltoidicositetrahedron, other polyhedra obtained at different steps
of the intersection process are reported, together with their duals, in Fig.26a and Fig.26b of the Appendix).
The dual of the solid reported in Fig.16a consists in the intersection of the triakisoctahedron
{2√2 1 1}
with the tetrakishexahedron {√2+1 1 0},
having the same central distance; it is shown in Fig.17a and the corresponding compound polyhedron is shown in Fig.17c.
The animated sequence correlating the convex and compound polyhedra is reported in Fig.17b.
The convex polyhedron shown in Fig.18a can be obtained from the intersection of the dual couple of polyhedra reported in Fig.16a and Fig.17a:
the related compound polyhedron is shown in Fig.18c and the animated sequence correlating the convex and compound polyhedra in Fig.18b.
The dual of the polyhedron shown in Fig.18a consists in the intersection of three hexakisoctahedra,
having the same central distance, and a deltoidcositetrahedron.
Fig.19a shows an orthographic view of this polyhedron, whereas the view along the vertical direction [001] and the corresponding stereonet projection
of the faces are reported in Fig.19c and Fig.19b, respectively.
In detail, the animated sequence of Fig.20a includes the following single forms:
Fig.15a) Compound polyhedron, made of a cube and an octahedron, and the related convex polyhedron resulting from their intersection when the value of the ratio d_{cube}/d_{octahedron} is √3/(2√21) as in the Archimedean rhombicuboctahedron RCO 
Fig.15b)
Animated sequence showing the derivation of RCO and the related compound polyhedron from the intersection and combination
with a rhombdodecahedron, whose faces are at the central distance d_{RD}= 1,
of the convex polyhedron shown in Fig.15a, when
d_{octahedron}= (2√21)/√3
and 
Fig.15c) Animated sequence showing RCO and the derivation of the related compound polyhedron, made of the single forms, from the combination with a rhombdodecahedron of the compound polyhedron shown in Fig.15a 
Fig.16a) Convex polyhedron (inscribable in a sphere as the duals of all the deltoidicositetrahedra) resulting from the intersection of the Archimedean rhombicuboctahedron with its dual, the Catalan deltoidicositetrahedron {√2+1 1 1} 
Fig.16b) Animated sequence showing the combination and intersection of the Archimedean rhombicuboctahedron with its dual, the Catalan deltoidicositetrahedron {√2+1 1 1}, resulting in the convex polyhedron shown in Fig.16a and the related compound polyhedron shown in Fig.16c 
Fig.16c) Compound polyhedron resulting from the combination of the Archimedean rhombicuboctahedron with its dual, the Catalan deltoidicositetrahedron {√2+1 1 1} 
Fig.17a)
Dual of the convex polyhedron reported in Fig.16a and consisting in the intersection
of two single forms, the triakisoctahedron 
Fig.17b)
Animated sequence correlating the convex polyhedron shown in
Fig.17a and the compound polyhedron shown in
Fig.17c; also the single forms
triakisoctahedron 
Fig.17c) Compound polyhedron made of the combination of the triakisoctahedron {2√2 1 1} and the tetrakishexahedron {√2+1 1 0} 
Fig.18a) Convex polyhedron deriving from the intersection of the solid shown in Fig.16a and its dual shown in Fig.17a 
Fig.18b) Animated sequence correlating the convex polyhedron shown in Fig.18a and the compound polyhedron shown in Fig.18c 
Fig.18c) Compound polyhedron corresponding to the convex polyhedron shown in Fig.18a 


Fig.19a) Convex polyhedron dual of the solid reported in Fig.18a: it includes three hexakisoctahedra and a deltoidicositetrahedron. 
Fig.19b) Stereonet projection of the 168 faces belonging to the four forms of the polyhedron shown in the side views. 
Fig.19c) View along the vertical direction [001] of the polyhedron reported in Fig.19a 
Fig.20a) Animated sequence including the three hexakisoctahedra and the deltoidicositetrahedron, whose intersection generates the solid shown in Fig.19a 
Fig.20b) Sequential steps of the intersection processes relative to the three hexakisoctahedra and the deltoidicositetrahedron, leading to the solid shown in Fig.19a 
Fig.21a) Animated sequence consisting of the compound polyhedra related to the five Archimedean cubic polyhedra ordered as in Fig.1a 
Fig.21b) Double animated sequence of the compound polyhedra related to the five vertextransitive duals of deltoidicositetrahedra: the second one in each sequence derives from RCO, whereas the others derive from the truncation of the remaining Archimedean polyhedra with m3m symmetry. 
Fig.21c) Animated sequence consisting of the compound polyhedra related to the polyhedra shown in Fig.1c, derived from the intersection of each deltoidicositetrahedron with its vertextransitive dual. 
Fig.21d) Animated sequence consisting of the compound polyhedra related to the polyhedra shown in Fig.1d 
Significant steps of the progressive truncation by
the deltoidicositetrahedron {211} of its vertextransitive dual, in turn
derived from the intersection of a cuboctahedron with a rhombdodecahedron
A result of the truncation by the deltoidicositetrahedron {211} of its vertextransitive dual,
in turn derived from the intersection of a cuboctahedron with its dual rhombdodecahedron, is the polyhedron, shown in Fig.4a, in which the faces of octahedron
have the shape of equilateral triangles.
Fig.22a consists in an animated sequence including also the polyhedra obtained
at other significant steps of the entire truncation process by the deltoidicositetrahedron {211};
the corresponding duals are reported in Fig.22b.
Starting from the vertextransitive dual of the deltoidicositetrahedron {211}, the subsequent steps include:
Fig.22a) Animated sequence consisting of the significant steps of the progressive truncation, by the deltoidicositetrahedron {211}, of its vertextransitive dual, in turn obtained from the truncation of a cuboctahedron by a rhombdodecahedron 
Fig.22b) Animated sequence including the duals of each polyhedron belonging to the sequence reported in Fig.22a 
Significant steps of the progressive truncation by the deltoidicositetrahedron {411} of its vertextransitive dual, in turn
derived from the intersection of AtO with a rhombdodecahedron
A result of the truncation by the deltoidicositetrahedron {411} of its vertextransitive dual, derived from the intersection of the Archimedean truncated octahedron AtO
with a rhombdodecahedron, is the polyhedron shown in Fig.7a, in which the faces of cube are square.
Fig.23a consists in an animated sequence including also the polyhedra obtained at other significant steps of the entire truncation process by the deltoidicositetrahedron {411};
the corresponding duals are reported in Fig.23b.
Starting from the vertextransitive dual of the deltoidicositetrahedron {411}, the subsequent steps include:
Fig.23a) Animated sequence consisting of significant steps of the progressive truncation, by the {411} deltoidicositetrahedron, of its vertextransitive dual, in turn resulting from the intersection of the Archimedean truncated octahedron with a rhombdodecahedron 
Fig.23b) Animated sequence including the duals of each polyhedron belonging to the sequence reported in Fig.23a 
Significant steps of the progressive truncation by the deltoidicositetrahedron {√211}
of its vertextransitive dual, in turn derived from the intersection of AtC with a rhombdodecahedron
A result of the truncation by the deltoidicositetrahedron {√211}
of its vertextransitive dual, derived from the intersection of the Archimedean truncated cube AtC with a rhombdodecahedron,
is the polyhedron shown in Fig.10a, in which the face of octahedron have the shape of equilateral triangles.
Fig.24a consists in an animated sequence including also the polyhedra obtained at other significant steps of the entire truncation process by the deltoidicositetrahedron
{√211}; the corresponding duals are reported in Fig.24b.
Starting from the vertextransitive dual of the deltoidicositetrahedron {√211}, the subsequent steps include:
Fig.24a) Animated sequence consisting of significant steps of the progressive truncation, by the deltoidicositetrahedron {√2 1 1}, of its vertextransitive dual, in turn resulting from the intersection of the Archimedean truncated cube with a rhombdodecahedron 
Fig.24b) Animated sequence including the duals of each polyhedron belonging to the sequence reported in Fig.24a 
Significant steps of the progressive truncation, by the deltoidicositetrahedron {3√22 1 1},
of its vertextransitive dual, in turn obtained by decreasing the central distance of the faces of rhombdodecahedron in the Archimedean truncated cuboctahedron
A result of the truncation by the deltoidicositetrahedron {3√22 1 1} of its vertextransitive dual,
obtained by decreasing the central distance of the faces of rhombdodecahedron in the Archimedean truncated cuboctahedron, is the polyhedron shown in Fig.13a,
in which the face of octahedron have the shape of equilateral triangles.
Fig.25a consists in an animated sequence including also the polyhedra obtained at other significant steps of the entire truncation process by the deltoidicositetrahedron
Starting from the vertextransitive dual of the deltoidicositetrahedron {3√22 1 1}, the subsequent steps include:
Fig.25a) Animated sequence consisting of significant steps of the progressive truncation by the deltoidicositetrahedron {3√22 1 1} of its vertextransitive dual, in turn obtained decreasing the central distance of the rhombdodecahedron in the Archimedean truncated cuboctahedron 
Fig.25b) Animated sequence including the duals of each polyhedron belonging to the sequence reported in Fig.25a 
Significant steps of the progressive truncation of the Archimedean rhombicuboctahedron by the Catalan deltoidicositetrahedron
A result of the truncation of the Archimedean
rhombicuboctahedron RCO by its dual, the Catalan deltoidicositetrahedron {√2+1 1 1},
is the polyhedron reported in Fig.16a, in which, in addition to the faces of deltoidicositetrahedron having a trapezoidal shape,
the face of octahedron have the shape of equilateral triangles, whereas the faces of cube and rhombdodecahedron are both square.
Fig.26a consists in an animated sequence including the polyhedra obtained at other significant steps of the entire truncation process
of RCO by the deltoidicositetrahedron {√2+1 1 1}; the corresponding duals are reported in Fig.26b.
Starting from RCO, the subsequent steps include:
Fig.26a) Animated sequence consisting of significant steps of the progressive truncation of the Archimedean rhombicuboctahedron by its dual, the Catalan deltoidicositetrahedron {√2+1 1 1} 
Fig.26b) Animated sequence including the duals of the solid belonging to the sequence reported in Fig.26a 
Fig.27a) Animated sequence consisting of the solids obtained from the intersection with its dual of each polyhedron which derives from the progressive truncation of RCO by the Catalan deltoidicositetrahedron 
Fig.27b) Animated sequence including the duals of the solid belonging to the sequence reported in Fig.27a 
{411} and dual coming from AtO  Catalan {√2+1 1 1} and RCO  {3√22 1 1} and dual coming from AtCO  {211} and dual coming from CO  {√2 11} and dual coming from AtC 
regular octagonal faces of cube  regular octagonal faces of cube  regular hexagonal faces of octahedron  regular hexagonal faces of octahedron  regular hexagonal faces of octahedron 
square faces of cube  regular hexagonal faces of octahedron  regular octagonal faces of cube  equilateral triangular faces of octahedron  equilateral triangular faces of octahedron 
vanishing of the faces of cube  equilateral triangular faces of octahedron  equilateral triangular face of octahedron  regular octagonal faces of cube  vanishing of the faces of octahedron 
and square faces of both cube and rhombdodecahedron 

regular hexagonal faces of octahedron 
vanishing of the faces of octahedron  square faces of cube  vanishing of the faces of octahedron  regular octagonal faces of cube 
equilateral triangular faces of octahedron 
vanishing of the faces of cube and rhombdodecahedron 
vanishing of the faces of octahedron  square faces of cube  square faces of cube 
vanishing of the faces of rhombdodecahedron 
vanishing of the faces of rhombdodecahedron 
vanishing of the faces of rhombdodecahedron 
vanishing of the faces of rhombdodecahedron 

vanishing of the faces of octahedron 
vanishing of the faces of cube  vanishing of the faces of cube  vanishing of the faces of cube 
Fig.28a) Series of dual polyhedra with 4/m 3 2/m cubic symmetry and relative intersections, all derived from the Archimedean rhombicuboctahedron 
Fig.28b) Series of duals with 2/m 3 5 icosahedral symmetry and relative intersections, all derived from the Archimedean rhombicosidodecahedron: the analogies with the corresponding images of the series in Fig.27a, taking into account the different symmetry, are evident. 
LINKS and REFERENCES