What Became of the Controversial Fourteenth Archimedean Solid, the Pseudo Rhomb-Cuboctahedron?

Livio Zefiro* and Maria Rosa Ardigo'
*Dip.Te.Ris, Universita' di Genova, Italy


Notes
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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 rhomb-cuboctahedron, also known as Miller's solids, since J.C.P. Miller, as reported in [3-5], obtained it accidentally while trying to build a model of an Archimedean rhomb-cuboctahedron starting from an incorrect net. Later, such solid was re-discovered many times, i.e. by V.G. Ashkinuze (so that, consequently, it is also named Miller-Ashkinuze 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  vertex-transitivity [3], the pseudo rhomb-cuboctahedron does not number among the Archimedean solids.
   Analogously, its dual, achievable by means of a 45 rotation of one half of the deltoid-icositetrahedron (dual of the rhomb-cuboctahedron), is merely a pseudo Catalan solid, not being face-transitive.

THE RHOMB-CUBOCTAHEDRON AND ITS ISOMER, THE PSEUDO RHOMB-CUBOCTAHEDRON

Fig.1 - Animation showing the transformation of the rhomb-cuboctahedron into the pseudo rhomb-cuboctahedron

    Many are the possible classifications of the polyhedra based on geometrical features: among the convex polyhedra whose faces consist in regular polygons, without forgetting the 92 non-uniform regular faced Johnson's solids [6], one must surely mention the regular Platonic solids and  the semiregular Archimedean solids.
   Table1 reports the number of faces, edges and vertices of the Platonic and Archimedean solids compared with their duals (Platonic and Catalan solids, respectively).

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 triakis-tetrahedron 43m
cuboctahedron 12 14 24 12 14 rhomb-dodecahedron m3m
truncated cube 24 14 36 24 14 triakis-octahedron m3m
truncated octahedron 24 14 36 24 14 tetrakis-hexahedron m3m
rhomb-cuboctahedron 24 26 48 24 26 deltoid-icositetrahedron m3m
truncated cuboctahedron 48 26 72 48 26 hexakis-octahedron m3m

snub cube (chiral)

24 38 60 24 38

pentagonal icositetrahedron (chiral)

432
icosidodecahedron 30 32 60 30 32 rhomb-triacontahedron m 3 5
truncated dodecahedron 60 32 90 60 32 triakis-icosahedron m 3 5
truncated icosahedron 60 32 90 60 32 pentakis-dodecahedron m 3 5
rhomb-icosidodecahedron 60 62 120 60 62 deltoid-hexecontahedron m 3 5
truncated icosidodecahedron 120 62 180 120 62 hexakis-icosahedron 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 well-known relation: F + V = E + 2.
The last column of the table reports the crystallographic point group relative to each pair of dual polyhedra, according to the International notation (or Hermann-Mauguin notation).

   Four out of the thirteen Archimedean solids can be dissected (Fig.2), originating regular-faced polyhedra: according to the nomenclature introduced by Norman W. Johnson [6], the names of the "elementary" polyhedra so obtained are:

  • tridiminished rhomb-icosidodecahedron
  • pentagonal rotunda
  • triangular, square and pentagonal cupolas
  • octagonal prism.
    Such elementary polyhedra can be reassembled in different ways, generating isomeric forms of the original Archimedean solid.

    rhomb-icosidodecahedron icosidodecahedron cuboctahedron rhomb-cuboctahedron

    Fig.2 - The four Archimedean solids which can be dissected into regular-faced polyhedra (upper row) and examples of their isomers obtained after reassembly of the elementary polyhedra (lower row).

        The rhomb-icosidodecahedron has four other isomers, whereas icosidodecahedron, cuboctahedron and rhomb-cuboctahedron have only one other isomeric form. Particularly noteworthy in case of the rhomb-cuboctahedron, since all its solids angles are congruent.
       
        Focusing our attention on the rhomb-cuboctahedron, 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 rhomb-cuboctahedron into regular-faced polyhedra

    If, after the rhomb-cuboctahedron dissection, one of the two cupolas is rotated by 45 and then the polyhedron is reassembled, the resulting form is the pseudo rhomb-cuboctahedron: the two forms, made of the same faces, are isomers (Fig.4).

    Fig.4 - Dissection of the rhomb-cuboctahedron in regular-faced polyhedra, 45 rotation of the lower square cupola and subsequent reassembly leading to the pseudo rhomb-cuboctahedron.

        In addition to pseudo rhomb-cuboctahedron and Miller's solid, a further name, deriving from Johnson's nomenclature and showing how the solid can be built starting just from regular-faced 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 rhomb-cuboctahedron could be elongated square bicupola).

        Two nets of the rhomb-cuboctahedron (left) and pseudo rhomb-cuboctahedron (right) are shown in Fig.5: in the rhomb-cuboctahedron a square of each cupola is connected, on opposite sides, to the same square of the octagonal prism, whereas in the pseudo rhomb-cuboctahedron 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 rhomb-cuboctahedron (left) and pseudo rhomb-cuboctahedron (right); clicking here one can see alternative coloured nets of the two isomers.

        The comparison of the rhomb-cuboctahedron with the pseudo rhomb-cuboctahedron (henceforth, RCO and pseudo-RCO) lets to highlight the differences between the two solids.
    • All the vertices of RCO and pseudo-RCO, equidistant from the centre of the respective polyhedron, are shared by four faces (three squares and a triangle): the solid angles are all congruent.
    • The initial and final positions of the RCO coincide if one applies, perpendicularly to each pair of parallel faces, a rotation of:
      • 120 in case of the eight triangular faces
      • 180 in case of the twelve square faces sharing edges with the triangular faces
      •   90 in case of the other six square faces
    • With respect to all its faces, the pseudo-RCO shows the same behaviour only for a 90 rotation along a direction perpendicular to the square bases of the two cupolas resulting from the dissection. Moreover, it is invariant also in case of a rotation along four 2-fold axes perpendicular both to the previous direction and to the edges between the square faces of the octagonal prism.
    • In addition, mirror planes are also present: nine in RCO and four in pseudo-RCO; in turn, the centre of symmetry is present only in RCO.
    Then, in order to detect all the different features, it is opportune to draw RCO and pseudo-RCO showing their symmetry operators and attributing different colours to each single form (Fig.6).

    (clicking on the images one can visualize the corresponding VRML files)
    Fig.6 - Rhomb-cuboctahedron (on the left) and pseudo rhomb-cuboctahedron (on the right) drawn with their respective symmetry operators, namely mirrors, rotation and roto-inversion axes.

    The views along the [001] direction of RCO and pseudo-RCO and the stereographic projections of their faces are shown in Fig.7, emphasizing that the faces belonging to the two cupolas are superimposed only in RCO.

    Fig.7 - View along the [001] direction and relative stereographic projection of rhomb-cuboctahedron (upper row) and pseudo rhomb-cuboctahedron (lower row).


       The symbol of the symmetry group to which RCO belongs is Oh in the Schoenflies notation and m3m in the International notation. Finally, adopting the more explicit extended notation, it becomes 4/m 3 2/m.
        The last notation highlights the presence of mirror planes orthogonal both to 4-fold and 2-fold rotation axes, in addition to 3-fold rotoinversion axes (it must be remembered that the action of an odd rotoinversion axis is equivalent to the disjoint action of the corresponding odd rotation axis and the centre of symmetry). In total there are 24 symmetry elements (if one includes also identity):
  • three pairs made of a 4-fold rotation axis and an orthogonal mirror m
  • six pairs made of a 2-fold rotation axis and an orthogonal mirror m
  • four 3-fold rotation axes
  • centre of symmetry
  • identity

    In the Schoenflies notation, D4v is the symbol of the symmetry group relative to the pseudo-RCO, whereas in the International notation it is 8m2, indicating that the four mirror planes present in the pseudo-RCO, at 45 from each other, intersect along a line that is not only a 4-fold rotation axis, but also a 8-fold rotoinversion axis (equivalent to a 8-fold rotoreflection axis).
    Orthogonally to the 8-fold rotoinversion axis and symmetrically interposed between the mirrors, there are also four 2-fold rotation axes. The absence of the centre of symmetry implies that the mirror planes are not orthogonal to the even-fold rotation axes of the pseudo-RCO.

        At this point one can ascertain that in RCO the action of the symmetry operators makes all the vertices equivalent: for example, each 3-fold rotation axis, orthogonal to a triangle, relates the three vertices of the triangle itself.
    In pseudo-RCO, 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 pseudo-RCO, only RCO is vertex-transitive (like all the other Archimedean solids). It can be useful to recall the definition given by Peter R. Cromwell [3]: "A polyhedron is vertex-transitive (or isogonal) if any vertex can be carried to any other by a symmetry operation".
    Consequently, the lack of vertex-transitivity is the objective property that, added to aesthetical reasons, prevents pseudo-RCO 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 RHOMB-CUBOCTAHEDRON AND THE PSEUDO RHOMB-CUBOCTAHEDRON

    Fig.8 - Animation showing the transition from the Catalan deltoid-icositetrahedron, dual of rhomb-cuboctahedron, to the dual of the pseudo rhomb-cuboctahedron, by a 45 rotation of the lower half of the polyhedron.

    The transition from the dual of RCO to the dual of pseudo-RCO can be accomplished by the following steps (Fig.9):
  • split of the first dual in two halves
  • 45 rotation of the lower half
  • reassembly of the two parts of the solid  
    Fig.9 - Steps leading from the dual of the rhomb-cuboctahedron to the dual of the pseudo rhomb-cuboctahedron, through the 45 rotation of one half of the polyhedron.

        In common, the twenty-four faces of both duals of RCO and pseudo-RCO

     have an equal distance from the centre of the respective solid and an identical shape (they are all kite-shaped), 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 deltoid-icositetrahedron (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 kite-shaped (or deltoidal) faces, can be identified by the generalized Miller's indices {1.1.2+1}.
        The pseudo-RCO 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  pseudo-RCO, and one face to all the others, in case of its dual.
       It follows that the deltoid-icositetrahedron, like all the Catalan solids, is face-transitive 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 pseudo-RCO 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.

    (clicking on the images one can visualize the corresponding VRML files)
    Fig.10 - The {1 1 √2+1} Catalan deltoid-icositetrahedron, dual of the Archimedean rhomb-cuboctahedron (left) and the dual of pseudo rhomb-cuboctahedron (right), drawn with the symmetry operators relative to m3m and 8m2 point groups, respectively.

    The views along the [001] direction of the duals of RCO and pseudo-RCO and the stereographic projections of their faces are shown in Fig.11, emphasizing that, in case of the dual of pseudo-RCO, not all the faces are superimposed.

    Fig.11 - View along the [001] direction, and relative stereographic projection, of the Archimedean deltoid-icositetrahedron, dual of rhomb-cuboctahedron (upper row), and the dual of pseudo rhomb-cuboctahedron (lower row).

    The nets of the duals of RCO and pseudo-RCO are shown in Fig.12.

    Fig.12 - Four-coloured net of deltoid-icositetrahedron, dual of rhomb-cuboctahedron and two-coloured net of the dual of pseudo rhomb-cuboctahedron.


    DECOMPOSITION OF RHOMB-CUBOCTAHEDRON, PSEUDO RHOMB-CUBOCTAHEDRON 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 kite-shaped 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 {√2+1 1 1} octagonal bipyramid (left) and the {1 1 √2+1} tetragonal deltohedron (right), whose intersection generates the dual of the pseudo rhomb-cuboctahedron (centre) when the ratio of their cental distances takes a proper value. The values of the indices have been calculated in relation to a monometric set of three orthogonal reference axes.


    On the other hand, the deltoid-icositetrahedron being face transitive, it is not further decomposable in m3m point group.

        Taking a backward step, since both RCO and pseudo-RCO 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:

  • a cube, an octahedron and a rhomb-dodecahedron for RCO
  • an octagonal prism, a pinacoid and two 4-fold isosceles trapezohedra (or deltohedra) for pseudo-RCO.

    Conversely, the intersection of such single forms, placed at appropriate distances from the centre of the polyhedron, re-build RCO and pseudo-RCO.

     Fig.14 - From the decomposition of a rhomb-cuboctahedron in single forms (left), one obtains a cube, an octahedron and a rhomb-dodecahedron, whereas the pseudo rhomb-cuboctahedron (right) gives an octagonal prism, a pinacoid and two different 4-fold deltohedra (isosceles trapezohedra with kite-shaped 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, pseudo-RCO and their duals in the different point groups which are subgroups of m3m or 8m2 (Fig.15 and Fig.16, respectively).

    Fig.15 - Flow chart showing the subgroups of cubic m3m point group.

    Fig.16 - Flow chart showing the subgroups of 8m2 point group.

        The subgroups belonging to monoclinic and triclinic systems, the ones with the lowest symmetry, have not been taken into consideration concerning the visualization of the forms relative to such subgroups.
    The decomposition of the rhomb-cuboctahedron and the deltoid-icositetrahedron, its dual, into single forms belonging to point subgroups of  m3m is shown in Fig.17.

    Decomposition of the rhomb-cuboctahedron and the deltoid-icositetrahedron, 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 octahedron (8)
    1 rhomb-dodecahedron (12)

    1 cube (6)
    1 octahedron (8)
    1 rhomb-dodecahedron (12)

    1 cube (6)
    1 octahedron (8)
    1 rhomb-dodecahedron (12)

    1 cube (6)
    2 tetrahedra (4)
    1 rhomb-dodecahedron (12)
    1 cube (6)
    2 tetrahedra (4)
    1 rhomb-dodecahedron (12)
    m3m 432 m3 43m 23

    1 deltoid-icositetrahedron(24)

    1 deltoid-icositetrahedron(24)

    1 deltoid-icositetrahedron(24)

    2 triakis-tetrahedra (12)

    2 triakis-tetrahedra (12)


    TRIGONAL POINT GROUPS

    3 rhombohedra (6)
    1 hexagonal prism (6)
    1pinacoid (2)

    3 rhombohedra (6)
    1 hexagonal prism (6)
    1pinacoid (2)

    3 rhombohedra (6)
    2 trigonal prisms (3)
    1pinacoid (2)

    6 trigonal pyramids (3)
    1 hexagonal prism (6)
    2 pedions (1)

    6 trigonal pyramids (3)
    2 trigonal prisms (3)
    2 pedions (1)

    3m 3 32 3m 3

    2 rhombohedra (6)
    1 ditrigonal scalenohedron (12)

    4 rhombohedra (6)

    4 rhombohedra (6)

    4  trigonal pyramids (3)
    2 ditrigonal pyramids (6)

    8 trigonal pyramids (3)


    TETRAGONAL POINT GROUPS

    2 tetragonal bipyramids (8)
    2 tetragonal prisms (4)
    1 pinacoid (2)

    2 tetragonal bipyramids (8)
    2 tetragonal prisms (4)
    1 pinacoid (2)

    2 tetragonal bisphenoids (4)
    1 tetragonal bipyramid (8)
    2 tetragonal prisms (4)
    1 pinacoid (2)

    2 tetragonal bisphenoids (4)
    1 tetragonal bipyramid (8)
    2 tetragonal prisms (4)
    1 pinacoid (2)

    (4/m)mm 422 4m2 42m

    1 bipyramid ditetragonal (16)
    1 bipyramid tetragonal (8)

    2 tetragonal trapezohedra (8)
    1 bipyramid tetragonal (8)

    2 tetragonal bisphenoid (4)
    2 tetragonal scalenohedra (8)

    2 tetragonal bisphenoid (4)
    2 tetragonal scalenohedra (8
    )


    4 pyramid tetragonal (4)
    2 tetragonal prisms (4)
    2 pedions (1)

    2 tetragonal bipyramids (8)
    2 tetragonal prisms (4)
    1 pinacoid (2)

    4 bisphenoids (4)
    2 tetragonal prisms (4)
    1 pinacoid (2)

    4 pyramid tetragonal (4)
    2 tetragonal prisms (4)
    2 pedions (1)

    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)
    3 rhombic prisms (4)
    3 pinacoids (2)

    2 rhombic bisphenoids (4)
    3 rhombic prisms (4)
    3 pinacoids (2)

    2 rhombic pyramids (4)
    4 dihedra (2)
    1 rhombic prism (4)
    2 pinacoids (2)
    2 pedions (1)

    mmm 222 mm2

    3 rhombic bipyramid (8)

    6 rhombic bisphenoids (4)

    6 rhombic pyramids (4)

    Fig.17 - Symmetry and forms constituting the rhomb-cuboctahedron and its dual in all the subgroups of m3m, except the ones belonging to the monoclinic and triclinic systems.

    One may note that:

    The decomposition of  the pseudo rhomb-cuboctahedron and  its dual into single forms belonging to point subgroups of  8m2 is shown in Fig.18.

    Decomposition of the pseudo rhomb-cuboctahedron and its dual in sets of forms, with lower symmetry, belonging to subgroups of the 8m2 point group


    2 tetragonal deltohedra (8)
    1 octagonal prism (8)
    1 pinacoid (2)

    2 tetragonal deltohedra (8)
    1 octagonal prism (8)
    1 pinacoid (2)

    4 tetragonal pyramids (4)
    2 tetragonal prisms (4)
    2 pedions (1)

    8m2 8 4
    1 octagonal bipyramid (16)
    1 tetragonal deltohedron (8)
    3 tetragonal deltohedra (8)
    6 tetragonal pyramids (4)

    2 tetragonal deltohedra (8)
    1 ditetragonal prism (8)
    1 pinacoid (2)

    4 rhombic bisphenoids (4)
    2 rhombic prisms (4)
    1 pinacoid (2)

    4 tetragonal pyramids (4)
    2 tetragonal prisms (4)
    2 pedions (1)

    4 dihedra (2)
    2 rhombic pyramids (4)
    1 rhombic prism (4)
    2 pinacoids (2)
    2 pedions (1)

    42*2* 22*2* 4mm mm2

    2 tetragonal bipyramids (8)
    1 tetragonal deltohedron (8)

    4 rhombic bisphenoids (4)
    2 rhombic prisms (4)

    2 ditetragonal pyramids (8)
    2 tetragonal pyramids (4)

    6 rhombic pyramids (4)

    Fig.18 - Symmetry and forms constituting the pseudo rhomb-cuboctahedron and its dual in all the subgroups of 8m2.

        The asterisks in 42*2* and 22*2*  indicates that the horizontal 2-fold 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 2-fold 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:

  • 422, 4mm and 4 (tetragonal system)
  • 222 and mm2 (orthorombic system)
  • 2 and m (monoclinic system)
  • 1 (triclinic system)

    The list, relative to the eight shared point subgroups, of the single forms (with their respective multiplicity) deriving from the decomposition of RCO and pseudo-RCO is reported in Table 2; the analogous list relative to the decomposition of their duals is reported in Table 3.

    POINT GROUPS RHOMB-CUBOCTAHEDRON
    (26 faces)
    PSEUDO RHOMB-CUBOCTAHEDRON
    (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 rhomb-cuboctahedron 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 RHOMB-CUBOCTAHEDRON
     (24 faces)
     Dual of the PSEUDO RHOMB-CUBOCTAHEDRON
     (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 rhomb-cuboctahedron and the pseudo rhombcuboctahedron, relatively to the shared subgroups of m3m and 8m2.

    In particular, concerning the 2 point subgroup, the unique 2-fold axis is directed along [100] or [110] direction if the subgroup derives from m3m, whereas it is directed along [100] or [√2+1 1 0] direction if the subgroup derives from 8m2.

    As an example, relatively to the 422 point subgroup, in the next figures one can see RCO, pseudo RCO and their duals, together with the single forms (all iso-orientated), originating from their decomposition.
     

    Symmetry of the rhomb-cuboctahedron in 422 point group and visualization of the constituting forms

    Fig.19 - Rhomb-cuboctahedron (centre) with the rotation axes relative to the 422 point group; on the left the {111} red and the {101} yellow tetragonal bipyramids, on the right the {110} ochre and the {100} pale-blue tetragonal prisms closed by the {001} dark-blue pinacoid.
    The indices have been calculated assuming a monometric set of three orthogonal reference axes.


    Symmetry of the pseudo rhomb-cuboctahedron in 422 point group and visualization of the constituting forms

     

    Fig.20
    Upper row: pseudo rhomb-cuboctahedron (left) with the rotation axes relative to the 422 point group; on the right the {110} pale-blue ditetragonal prism (geometrically corresponding, as already pointed out, to an octagonal prism).
    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 rhomb-cuboctahedron in 422 point group and visualization of the single constituting forms

    Fig.21
    Upper row: dual of the rhomb-cuboctahedron (left) with the simmetry axes relative to the 422 point group; on the right the {1 1 √2+1} violet tetragonal bipyramid.
    Lower row: {√2+1 1 1} ochre tetragonal deltohedron (left) and {1 √2+1 1} yellow tetragonal deltohedron (right).

    Symmetry of the dual of the pseudo rhomb-cuboctahedron in 422 point group and visualization of the single constituting forms

     

    Fig.22
    Upper row: the dual of pseudo rhomb-cuboctahedron with the simmetry axes relative to the 422 point group (left) and the {1 1 √2+1} violet tetragonal deltohedron (right).
    Lower row: {√2+1 1 1} ochre tetragonal bipyramid (left) and {1 √2+1 1} yellow tetragonal bipyramid (right).

        In summary, with regard to the isomeric couple RCO-pseudo RCO evaluated in 422 point group, in addition to the two {110} and {100} tetragonal prisms transforming into the {100} ditetragonal prism (with octagonal bases), the {111} and the {101} tetragonal bipyramids, originating from the decomposition of RCO, become the corresponding {111} and {101} tetragonal deltohedra (or isosceles trapezohedra) in case of pseudo-RCO.
        Concerning the duals, also the {1 1 √2+1} tetragonal bipyramid, originating from the decomposition of the dual of RCO, becomes the {1 1 √2+1} tetragonal deltohedron in case of the dual of pseudo-RCO, whereas the {√2+1 1 1} and {1 √2+1 1} tetragonal deltohedra, coming from the dual of RCO, become the {√2+1 1 1} and {1 √2+1 1} tetragonal bipyramids in case of the dual of pseudo-RCO.


    ACKNOWLEDGEMENTS

    Many thanks are due to Riccardo Basso for the fruitful discussions and to Fabio Somenzi for his precious help concerning the translation of the text.


    REFERENCES and LINKS

    1)  J. Kepler
    De nive sexangula, Prague, 1611

    2)  C. Hardie
    The Six-cornered 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/virtual-polyhedra/pseudo-rhombicuboctahedra.html

    6)  Norman W. Johnson
    Convex Polyhedra with Regular Faces, Canadian Journal of Mathematics, 18, 169-200, 1966

    7)  Viktor A. Zalgaller
    Convex Polyhedra with Regular Faces, Seminars in Mathematics - V.A. Steklov Mathematical Institute, Leningrad - Volume 2, Consultants Bureau, 1969