What Became of the Controversial Fourteenth Archimedean Solid, the Pseudo RhombCuboctahedron?
Livio Zefiro* and Maria Rosa Ardigo'
*Dip.Te.Ris, Universita' di Genova, Italy
Notes
A good Web3D visualizer can be downloaded from here.
It is not sure that it can work with the most recent releases of the browsers.

INTRODUCTION
In his work "De nive sexangula" [1], quoted
in [2], rather surprisingly Johannes Kepler mentioned
fourteen Archimedean solids, unlike the usual thirteen ones. Probably he was
refering to the pseudo rhombcuboctahedron, also known as Miller's solids, since
J.C.P. Miller, as reported in [35], obtained it accidentally while trying
to build a model of an Archimedean rhombcuboctahedron starting from an
incorrect net. Later, such solid was rediscovered many times, i.e. by V.G.
Ashkinuze (so that, consequently, it is also named MillerAshkinuze solid),
and sometimes included in the list of the Archimedean solids.
Due to the lack not only of the particular actractiveness
typical of the Archimedean solids, but mainly of vertextransitivity [3],
the pseudo rhombcuboctahedron does not number among the Archimedean solids.
Analogously, its dual, achievable by means of a 45° rotation of one half of
the deltoidicositetrahedron (dual of the rhombcuboctahedron), is merely a pseudo Catalan solid, not being facetransitive.
THE RHOMBCUBOCTAHEDRON AND ITS ISOMER, THE PSEUDO RHOMBCUBOCTAHEDRON
Fig.1  Animation showing the transformation of the rhombcuboctahedron
into the pseudo rhombcuboctahedron

Table 1  Platonic and Archimedean solids compared with their duals  
Vertices  Faces  Edges  Faces  Vertices  Point Group  
Platonic solids  Dual Platonic solids  
tetrahedron  4  4  6  4  4  tetrahedron  43m 
octahedron  6  8  12  6  8  cube  m3m 
cube  8  6  12  8  6  octahedron  m3m 
icosahedron  12  20  30  12  20  dodecahedron  m 3 5 
dodecahedron  20  12  30  20  12  icosahedron  m 3 5 
Archimedean solids  Dual Catalan solids  
truncated tetrahedron  12  8  18  12  8  triakistetrahedron  43m 
cuboctahedron  12  14  24  12  14  rhombdodecahedron  m3m 
truncated cube  24  14  36  24  14  triakisoctahedron  m3m 
truncated octahedron  24  14  36  24  14  tetrakishexahedron  m3m 
rhombcuboctahedron  24  26  48  24  26  deltoidicositetrahedron  m3m 
truncated cuboctahedron  48  26  72  48  26  hexakisoctahedron  m3m 
snub cube (chiral) 
24  38  60  24  38 
pentagonal icositetrahedron (chiral) 
432 
icosidodecahedron  30  32  60  30  32  rhombtriacontahedron  m 3 5 
truncated dodecahedron  60  32  90  60  32  triakisicosahedron  m 3 5 
truncated icosahedron  60  32  90  60  32  pentakisdodecahedron  m 3 5 
rhombicosidodecahedron  60  62  120  60  62  deltoidhexecontahedron  m 3 5 
truncated icosidodecahedron  120  62  180  120  62  hexakisicosahedron  m 3 5 
snub dodecahedron (chiral)  60  92  150  60  92  pentagonal hexecontahedron (chiral)  235 
Given two of three values among the number of faces, edges and vertices, the third one
can be obtained by the wellknown relation:
The last column of the table reports the crystallographic
point group relative to each pair of dual polyhedra, according to the International notation (or
HermannMauguin notation).
Four out of the thirteen Archimedean solids can be dissected (Fig.2), originating regularfaced polyhedra: according to the nomenclature introduced by Norman W. Johnson [6], the names of the "elementary" polyhedra so obtained are:

The rhombicosidodecahedron has four other
isomers, whereas icosidodecahedron, cuboctahedron and
rhombcuboctahedron have only one other isomeric
form. Particularly noteworthy in case of the rhombcuboctahedron, since all its solids angles are congruent.
Focusing our attention on the rhombcuboctahedron, its dissection into
elementary polyhedra gives an octagonal prism and two square cupolas (Fig.3). A square cupola includes two
parallel polygons, a square and an octagon, connected
by a ring of other eight polygons, where squares alternate to equilateral
triangles. Both squares and triangles of the ring share one side with the
octagon, whereas the opposite side (in case of the squares) or vertex (in case
of the triangles) are shared with the
square parallel to the octagon.
upper square cupola  octagonal prism  lower square cupola 
Fig.3  Square cupolas and octagonal prism obtained dissecting the
rhombcuboctahedron into regularfaced polyhedra 
Fig.4  Dissection of the rhombcuboctahedron in
regularfaced polyhedra, 45° rotation of the lower square cupola and subsequent
reassembly leading to the pseudo rhombcuboctahedron.

In addition to pseudo rhombcuboctahedron and Miller's solid, a further name, deriving from Johnson's nomenclature and showing how the solid can be built starting just from regularfaced elementary polyhedra, is elongated square gyrobicupola, since it includes two square cupolas, reciprocally rotated of 45° (gyro) and separated by an octagonal prism, which makes the resulting form more elongated than a square gyrobicupola (if one applies the same nomenclature to the Archimedean solids, an alternative name of the rhombcuboctahedron could be elongated square bicupola).
Two nets of the rhombcuboctahedron (left) and
pseudo rhombcuboctahedron (right) are shown in Fig.5: in
the rhombcuboctahedron a square of each cupola is connected, on opposite sides,
to the same square of the octagonal prism, whereas in the pseudo
rhombcuboctahedron similar squares, belonging to the two cupolas, are connected (always on opposite
sides) to contiguous squares of the octagonal prism.
Fig.5  Nets of the rhombcuboctahedron (left) and pseudo rhombcuboctahedron (right);
clicking here
one can see alternative coloured nets of the two isomers. 
Fig.6  Rhombcuboctahedron (on the left) and pseudo rhombcuboctahedron (on the right) drawn with their respective symmetry operators,
namely mirrors, rotation and rotoinversion axes.

Fig.7  View along the [001] direction and relative stereographic projection of rhombcuboctahedron (upper row) and pseudo rhombcuboctahedron (lower row). 
In the Schoenflies notation, D_{4v} is the symbol of the symmetry group
relative to the pseudoRCO, whereas in the International notation it is
8m2, indicating that the four mirror planes
present in the pseudoRCO, at 45° from each other, intersect along a line that
is not only a
Orthogonally to the 8fold rotoinversion axis and symmetrically interposed between the mirrors,
there are also four 2fold rotation axes. The absence of the centre of symmetry implies that the mirror planes are not orthogonal
to the evenfold rotation axes of the pseudoRCO.
At this point one can ascertain that in
RCO the action of the symmetry operators makes all the vertices equivalent:
for example, each 3fold rotation axis, orthogonal to a triangle, relates the
three vertices of the triangle itself.
In pseudoRCO, on the contrary, only two vertices of each triangle are related by a mirror,
whereas there is no symmetry operator relating the third vertex (the
one at a corner of the square basis of the cupola) to the others.
Therefore, concerning the isomeric couple consisting of RCO and
pseudoRCO, only RCO is vertextransitive (like all the other
Archimedean solids). It can be useful to recall the definition given by Peter R. Cromwell [3]:
"A polyhedron is vertextransitive (or isogonal) if any vertex can be carried to any other by a symmetry operation".
Consequently, the lack of vertextransitivity is the objective property that,
added to aesthetical reasons, prevents pseudoRCO from being numbered among the Archimedean solids.
In other words, according to Viktor A. Zalgaller [7]:
"Besides two infinite series of prisms and antiprisms, ... there further
exist only 13 semiregular polyhedra (the bodies of Archimedes). If instead
of the compatibility of the vertices under selfcoincidence of the polyhedron
as a whole, we require only the local commonness of the vertices, then here
one more, the fourteenth, polyhedron exist."
DUALS OF THE RHOMBCUBOCTAHEDRON AND THE PSEUDO RHOMBCUBOCTAHEDRON
Fig.8  Animation showing the transition from the Catalan
deltoidicositetrahedron, dual of rhombcuboctahedron, to the dual of the pseudo
rhombcuboctahedron, by a 45° rotation of the lower half of the polyhedron.

Fig.9  Steps leading from the dual of
the rhombcuboctahedron to the dual of the pseudo rhombcuboctahedron, through the 45°
rotation of one half of the polyhedron.

have
an equal distance from the centre of the respective solid and an identical shape (they are all kiteshaped),
whereas their different features derive from the vertex transitivity of the RCO alone.
The dual of the Archimedean RCO is obviously a Catalan solid, named
deltoidicositetrahedron (or trapezohedron), belonging to m3m,
the same point group of RCO: by the action of the relative symmetry operators, one face can be related to all the others.
From a crystallographic point of view, the form, made of 24 kiteshaped (or
deltoidal) faces, can be identified by the generalized Miller's indices
The pseudoRCO and its unnamed dual belong to the same point
group,
8m2, whose symmetry operators
are instead unable to relate one vertex to all the other, in case of
pseudoRCO, and one face to all the others, in case of its dual.
It follows that the deltoidicositetrahedron, like all the Catalan solids, is
facetransitive or isohedral (according again to Peter R. Cromwell [3]: "A polyhedron is said to
be face transitive if for any pair of faces, there is a symmetry of the
polyhedron which carries the first face onto the second"), whereas the
dual of the pseudoRCO is not face transitive: as described in the next chapter, it can be derived from the intersections
of two single forms, having obviously a lower molteplicity.
In Fig.10 the duals of RCO and pseudo RCO are shown with their symmetry operators.
Fig.10  The

Fig.11  View along the [001] direction, and relative stereographic projection,
of the Archimedean deltoidicositetrahedron, dual of rhombcuboctahedron
(upper row), and the dual of pseudo rhombcuboctahedron (lower row). 
Fig.12  Fourcoloured net of deltoidicositetrahedron, dual of
rhombcuboctahedron and twocoloured net of the dual of pseudo rhombcuboctahedron. 
DECOMPOSITION OF RHOMBCUBOCTAHEDRON, PSEUDO RHOMBCUBOCTAHEDRON AND
THEIR DUALS IN SINGLE FORMS
In Fig.13 the dual of the pseudo RCO (central image) is
decomposed into two single forms, an octagonal bipyramid (on the left) and a tetragonal kiteshaped isosceles trapezohedron (or deltohedron)
on the right, both compatible with the
8m2 point group to which also
the pseudo RCO belongs.
Fig.13  Scale views of the

Taking a backward step, since both RCO and pseudoRCO are not face transitive, they can be decomposed (Fig.14) in different single forms that, taking into account the respective m3m and 8m2 point groups, are:
Fig.14  From the decomposition of a rhombcuboctahedron in single
forms (left), one obtains a cube, an octahedron and a rhombdodecahedron,
whereas the pseudo rhombcuboctahedron (right) gives an octagonal prism,
a pinacoid and two different 4fold deltohedra (isosceles trapezohedra with
kiteshaped faces). The indices of the two deltohedra are {101} and {111},
if one assumes a monometric set of three orthogonal reference axes. 
It may be interesting to visualize the single forms originating from
the decomposition of RCO, pseudoRCO and their duals in the different point
groups which are subgroups of m3m
or 8m2 (Fig.15
and Fig.16, respectively).
Decomposition of the rhombcuboctahedron and the deltoidicositetrahedron, its dual, into sets of forms with lower symmetry, belonging to cubic, trigonal, tetragonal and orthorhombic point groups, all subgroups of the m3m point group 
CUBIC POINT GROUPS 
1 cube (6) 
1 cube (6) 
1 cube (6) 
1 cube (6) 2 tetrahedra (4) 1 rhombdodecahedron (12) 
1 cube (6) 2 tetrahedra (4) 1 rhombdodecahedron (12) 
m3m  432  m3  43m  23 
1 deltoidicositetrahedron(24) 
1 deltoidicositetrahedron(24) 
1 deltoidicositetrahedron(24) 
2 triakistetrahedra (12) 
2 triakistetrahedra (12) 
TRIGONAL POINT GROUPS 
3 rhombohedra (6) 
3 rhombohedra (6) 
3 rhombohedra (6) 
6 trigonal pyramids (3) 
6 trigonal pyramids (3) 
3m  3  32  3m  3 
2 rhombohedra (6) 
4 rhombohedra (6) 
4 rhombohedra (6) 
4 trigonal pyramids (3) 
8 trigonal pyramids (3) 
TETRAGONAL POINT
GROUPS 
2 tetragonal bipyramids
(8) 
2 tetragonal bipyramids
(8) 
2 tetragonal bisphenoids
(4) 
2 tetragonal bisphenoids
(4) 
(4/m)mm  422  4m2  42m 
1 bipyramid ditetragonal
(16) 
2 tetragonal trapezohedra (8) 
2 tetragonal bisphenoid
(4) 
2 tetragonal bisphenoid
(4) 
4 pyramid tetragonal (4) 
2 tetragonal bipyramids
(8) 
4 bisphenoids (4) 
4 pyramid tetragonal (4) 
4mm  4/m  4  4 
2 pyramid ditetragonal (8) 2 pyramid tetragonal (4) 
3 bipyramid tetragonal (8) 
6 tetragonal bisphenoid (4) 
6 pyramid tetragonal (4) 
ORTHORHOMBIC POINT GROUPS

1 rhombic bipyramid (8) 
2 rhombic bisphenoids (4) 
2 rhombic pyramids (4) 
mmm  222  mm2 
3 rhombic bipyramid (8) 
6 rhombic bisphenoids (4) 
6 rhombic pyramids (4) 
Fig.17  Symmetry and forms constituting the rhombcuboctahedron and its dual in
all the subgroups of m3m,
except the ones belonging to the monoclinic and triclinic systems. 
The decomposition of the pseudo rhombcuboctahedron and its
dual into single forms belonging to point subgroups of
8m2 is shown in
Fig.18.
Decomposition of the pseudo rhombcuboctahedron and its dual in sets of forms, with lower symmetry, belonging to subgroups of the 8m2 point group 
2 tetragonal deltohedra
(8) 
2 tetragonal deltohedra
(8) 
4 tetragonal pyramids (4) 
8m2  8  4 
1 octagonal bipyramid (16) 1 tetragonal deltohedron (8) 
3 tetragonal deltohedra
(8) 
6 tetragonal pyramids (4) 
2 tetragonal deltohedra
(8) 
4 rhombic bisphenoids (4) 
4 tetragonal pyramids (4) 
4 dihedra (2) 
42*2*  22*2*  4mm  mm2 
2 tetragonal bipyramids
(8) 
4 rhombic bisphenoids (4) 
2 ditetragonal pyramids
(8) 
6 rhombic pyramids (4) 
The asterisks in 42*2* and 22*2* indicates that the horizontal 2fold axes are rotated 22.5° with respect to their usual orientation in 422 and 222 cristallographic point groups, in accordance with their orientation in the 8m2 point group. Then the (100) faces of the ditetragonal prism, by rotation along the nearest 2fold axes, is related to the contiguous (110) and (110) faces, placed at 45° from (100) face. Hence, geometrically, the prism is an octagonal prism.
The only subgroups in common between m3m and 8m2 are:
The list, relative to the eight shared point subgroups, of the single forms (with
their respective multiplicity) deriving from the decomposition of RCO and pseudoRCO is
reported in Table 2; the analogous list relative to the decomposition of
their duals is reported in Table 3.
POINT GROUPS  RHOMBCUBOCTAHEDRON
(26 faces) 
PSEUDO RHOMBCUBOCTAHEDRON (26 faces) 
422  2 tetragonal bipyramids (8)  2 tetragonal deltohedra (8) 
2 tetragonal prisms (4)  1 ditetragonal prism (8)  
1 pinacoid (2)  1 pinacoid (2)  
4mm  4 tetragonal pyramid (4)  4 tetragonal pyramids (4) 
2 tetragonal prisms (4)  2 tetragonal prisms (4)  
2 pedions (1)  2 pedions (1)  
4  4 tetragonal pyramid (4)  4 tetragonal pyramids (4) 
2 tetragonal prisms (4)  2 tetragonal prisms (4)  
2 pedions (1)  2 pedions (1)  
222  2 rhombic bisphenoids (4)  4 rhombic bisphenoids (4) 
3 rhombic prisms (4)  2 rhombic prisms (4)  
3 pinacoids (2)  1 pinacoid (2)  
mm2  2 rhombic pyramids (8)  4 dihedra (2) 
4 dihedra (2)  2 rhombic pyramids (4)  
1 prism (4)  1 rhombic prism (4)  
2 pinacoids (2)  2 pinacoids (2)  
2 pedions (1)  2 pedions (1)  
m  8 dihedra (2)  8 dihedra (2) 
1 pinacoids (2)  1 pinacoids (2)  
8 pedions (1)  8 pedions (1)  
2  8 sphenoid (2)  8 sphenoid (2) 
4 pinacoids (2)  4 pinacoids (2)  
2 pedions (1)  2 pedions (1)  
1  26 pedions (1)  26 pedions (1) 
Table2  Decomposition in single forms of the rhombcuboctahedron and the pseudo rhombcuboctahedron, relatively to the shared subgroups of m3m and 8m2; the multiplicity of each form, namely the number of its faces, is reported in round brackets. 
POINT GROUPS  Dual of the RHOMBCUBOCTAHEDRON (24 faces) 
Dual of the PSEUDO RHOMBCUBOCTAHEDRON (24 faces) 
422  1 tetragonal bipyramid (8)  2 tetragonal bipyramids (8) 
2 tetragonal trapezohedra (8)  1 tetragonal deltohedron (8)  
4mm  2 ditetragonal pyramids (8)  2 ditetragonal pyramids (8) 
2 tetragonal pyramids (4)  2 tetragonal pyramids (4)  
4  6 tetragonal pyramids (4)  6 tetragonal pyramids (4) 
222  6 rhombic bisphenoids (4)  4 rhombic bisphenoids (4) 
2 rhombic prisms (4)  
mm2  6 rhombic pyramids (4)  6 rhombic pyramids (4) 
m  12 dihedra (2)  11 dihedra (2) 
2 pedions(2)  
2  12 sphenoids (2)  12 sphenoids (2) 
1  24 pedions (1)  24 pedions (1) 
Table3  Decomposition in single forms of the duals of
the rhombcuboctahedron and the pseudo rhombcuboctahedron, relatively to the
shared subgroups of m3m
and 8m2. 
Symmetry of the rhombcuboctahedron in 422 point group and visualization of the constituting forms 
Fig.19  Rhombcuboctahedron (centre) with the rotation axes relative to
the 422 point group; on the left
the 
Symmetry of the pseudo
rhombcuboctahedron in 422 point group and visualization of the
constituting
forms 
 
Fig.20
Upper row: pseudo rhombcuboctahedron (left)
with the rotation axes relative to the 422 point group; on the right the Lower row: {111} red tetragonal deltohedron (left) and {101} yellow tetragonal deltohedron (right) Also in this case (and in the following ones) the indices have been calculated assuming a monometric set of three orthogonal reference axes. 
Symmetry of the
dual of the rhombcuboctahedron in 422 point group and visualization
of the single constituting forms 
Fig.21
Upper row: dual of the rhombcuboctahedron (left) with the simmetry axes relative to
the 422 point group;
on the right the
Lower row: {√2+1 1 1} ochre tetragonal deltohedron (left) and 
Symmetry of the dual of the pseudo
rhombcuboctahedron in 422 point group and visualization of the single constituting
forms 
 
Fig.22
Upper row: the dual of pseudo rhombcuboctahedron with the simmetry axes relative to the 422 point group
(left) and the
Lower row: {√2+1 1 1} ochre tetragonal bipyramid (left) and 
In summary, with regard to the isomeric couple RCOpseudo RCO evaluated in 422
point group, in addition to the two
Concerning the duals, also the
REFERENCES and LINKS
1) J. Kepler
De nive sexangula, Prague, 1611
2) C. Hardie
The Sixcornered Snowflake, Oxford University Press, 1966
3)
Peter R. Cromwell
Polyhedra,
Cambridge University Press, 1997
4)
W.W. Rouse Ball, H.S.M. Coxeter
Mathematical Recreations and Essays, Dover Publications,
1987
5)
http://www.georgehart.com/virtualpolyhedra/pseudorhombicuboctahedra.html
6) Norman W. Johnson
Convex Polyhedra with Regular Faces, Canadian Journal of Mathematics,
18, 169200, 1966
7) Viktor A. Zalgaller
Convex Polyhedra with Regular Faces, Seminars in Mathematics  V.A.
Steklov Mathematical Institute, Leningrad  Volume 2, Consultants Bureau, 1969