Dissection of the rhomb-icosidodecahedron in elementary regular-faced polyhedra and subsequent reassembly leading to a set of Johnson's polyhedra

Livio Zefiro
Dip.Te.Ris, Universita' di Genova, Italy
(E-mail address: livio.zefiro@fastwebnet.it)

Notes
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  • The almost entirety of the images was created by SHAPE 7.0, a software program for drawing the morphology of crystals, distributed by Shape Software
  • Hovering with the mouse on some images, the symbol of a pointer appears: if your web browser is set up to visualize VRML files, by clicking with the left button of the mouse it should be possible to visualize in a new window the corresponding dynamic image, that can be enlarged, rotated, shifted... Cortona 3D Viewer is a good Web3D visualizer, at present working also with the updated versions of the main browsers, that can be downloaded from here.

 Animation showing the rhomb-icosidodecahedron and the derived Johnson's polyhedra obtained by the rotation and/or removal of the pentagonal cupolas.


TABLE of CONTENTS
(one can jump directly to each item by clicking on it)
  • INTRODUCTION
  • Rhomb-icosidodecahedron
  • J5 Pentagonal cupola
  • J72 Gyrate rhomb-icosidodecahedron
  • J73 Parabigyrate rhomb-icosidodecahedron
  • J74 Metabigyrate rhomb-icosidodecahedron
  • J75 Trigyrate rhomb-icosidodecahedron
  • J76 Diminished rhomb-icosidodecahedron
  • J77 Paragyrate diminished rhomb-icosidodecahedron
  • J78 Metagyrate diminished rhomb-icosidodecahedron
  • J79 Bigyrate diminished rhomb-icosidodecahedron
  • J80 Parabidiminished rhomb-icosidodecahedron
  • J81 Metabidiminished rhomb-icosidodecahedron
  • J82 Gyrate bidiminished rhomb-icosidodecahedron
  • J83 Tridiminished rhomb-icosidodecahedron
  • CONCLUSIONS
  • REFERENCES and LINKS
  • Appendix A
  • Appendix B
  • Appendix C

  • Introduction

       Only four out of the thirteen Archimedean solids can originate regular-faced polyhedra by dissection [1], as shown in Figure1.
    According to the nomenclature introduced by Norman W. Johnson [2], the "elementary" polyhedra so obtained are:
  • a tridiminished rhomb-icosidodecahedron and three pentagonal cupolas, starting from the rhomb-icosidodecahedron
  • two pentagonal rotundas, from the icosidodecahedron
  • two triangular cupolas, from the cuboctahedron
  • an octagonal prism and two square cupolas, from the rhomb-cuboctahedron.
    The elementary polyhedra derived from the dissection can be reassembled after an appropriate relative rotation, leading to isomeric forms of the Archimedean solids. A detailed description of the features of the rhomb-cuboctahedron and its isomer, the elongated square gyrobicupola (or pseudo rhomb-cuboctahedron), has been given in a previous work [3].
    As regards rhomb-icosidodecahedron, it is the only one which numbers four isomeric Johnson's polyhedra, obtained assembling the tridiminished rhomb-icosidodecahedron and the three pentagonal cupolas after the rotation of one, two or three cupolas (there are two alternative choices concerning the couple of cupolas to be rotated).

  • rhomb-icosidodecahedron icosidodecahedron cuboctahedron rhomb-cuboctahedron
    Fig. 1
    UPPER ROW: the four Archimedean solids which can be dissected into regular-faced polyhedra: rhomb-icosidodecahedron, icosidodecahedron, cuboctahedron, rhomb-cuboctahedron.
    LOWER ROW: the related animated views describing both the dissection into elementary polyhedra and the subsequent reassembly leading, after the rotation of the cupolas or the rotundas, to the respective isomers; the rhomb-icosidodecahedron is the only one which numbers four isomers, including the trigyrate rhomb-icosidodecahedron shown in the first image on the left.

    In addition to the four isomers of the rhomb-icosidodecahedron, seven further Johnson's polyhedra can be obtained assembling the tridiminished rhomb-icosidodecahedron with one or two pentagonal cupolas (rotated, if needed). Therefore in total the Johnson's polyhedra derived from the rhomb-icosidodecahedron are thirteen, if one includes the tridiminished rhomb-icosidodecahedron and also the pentagonal cupola itself.
     
    Rhomb-icosidodecahedron
         
    The rhomb-icosidodecahedron is an Archimedean polyhedron obtained by the intersection of a rhomb-triacontahedron, an icosahedron and a dodecahedron, placed at proper distances [4] from the centre of the polyhedron.
    It consists of the following regular-shaped faces:
    • thirty square faces, placed at a unit central distance, coming from the rhomb-triacontahedron
    • twenty triangular faces, at a central distance equal to (1+2/τ3)/√3 = 1.0184, coming from the icosahedron
    • twelve pentagonal faces at a central distance equal to 3/(τ√τ2+1) = 0.9748, coming from the dodecahedron, which is characterized by the Miller indices {1τ 0}, where τ is the golden ratio: 
      τ
      = (√5 +1)/2 = 1.61803...
    Each vertex of the rhomb-icosidodecahedron turns out to be shared among four polygons: two squares, an equilateral triangle and a regular pentagon; as regards the presence of axes of rotation, there are fifteen 2-fold, ten 3-fold and six 5-fold axes orthogonal to the square, triangular and pentagonal faces, respectively.
    Because of the further presence of fifteen mirror planes, orthogonal to the 2-fold axes (and consequently also of a centre of inversion), the rhomb-icosidodecahedron belongs to m 3 5, the holohedral point group of the icosahedral system, even if it can be generated, as an alternative, from the action of the only rotation axes characterizing 235, the other icosahedral point group (Figure 2).
     
    (clicking on the following image one can visualize the corresponding VRML file)
    Fig. 2
    The Archimedean rhomb-icosidodecahedron with the 2-fold, 3-fold and 5-fold rotation axes characterizing the 235 point group.

    The stereographic projections of the rhomb-icosidodecahedron viewed along the 2-fold, 3-fold and 5-fold axes are sequentially reported in Figure 3 (the small circles and the x-shaped characters represent normals to the faces directed upwards or downwards, respectively).

    Fig. 3a - The rhomb-icosidodecahedron and its stereographic projection along a 2-fold rotation axis.
    Fig. 3b - The rhomb-icosidodecahedron and its stereographic projection along a 3-fold rotation axis.
    Fig. 3c - The rhomb-icosidodecahedron and its stereographic projection along a 5-fold rotation axis.

        As already shown in [4], even if it is not possible to obtain the rhomb-icosidodecahedron by the direct truncation of an icosi-dodecahedron, the two polyhedra share the relationship:

    ddodecahedron = (1/√1+1/τ2 ) (√3 dicosahedron - 1/τ )

    between the distances, from the centre of each polyhedron, of their icosahedral and dodecahedral faces, given a unit value to the central distance of the rhomb-triacontahedral faces.
    The relationship holds true, in the intervals:

    1/√1+1/τ2  < ddodecahedron < (3/τ2)/√1+1/τ2   and   τ /√ 3 < dicosahedron < (1+2/τ2) /√3

    also in case of all the polyhedra intermediate between the icosi-dodecahedron and the rhomb-icosidodecahedron, including the Archimedean truncated icosidodecahedron.
        As one can see in Figure 4, in all these polyhedra the triacontahedral faces consist of squares with increasing dimensions, going from the icosidodecahedron to the rhomb-icosidodecahedron, where the square faces reach the maximum dimension: the length of their side (equal to 2/τ3 if dtriacontahedron = 1) becomes equal to the length of the sides of the triangular and pentagonal faces, being the rhomb-icosidodecahedron a regular-faced Archimedean polyhedron.
     
    Fig. 4 - Transition from the icosidodecahedron to the rhomb-icosidodecahedron, through the Archimedean truncated icosi-dodecahedron: the red triacontahedral faces, tangent to the vertices and therefore absent in the  icosidodecahedron,  are square in all the intermediate polyhedra.

    In addition to the square faces, the other faces of each non-Archimedean polyhedron belonging to this series are, as shown in Figure 5:
    • non-regular decagons, symmetrical relatively to five mirror planes intersecting in the centre of every decagon in correspondence to a 5-fold rotation axis
    • non-regular hexagons, symmetrical relatively to three mirror planes intersecting in the centre of every hexagon in correspondence to a 3-fold rotation axis.
    Fig. 5 - Mirror planes and rotation axes which characterize a generic non-Archimedean polyhedron belonging to the series of polyhedra intermediate between the icosi-dodecahedron and the rhomb-icosidodecahedron:
  • couples of mirror planes intersecting along the 2-fold axes normal to the square faces
  • sets of five mirror planes intersecting along the 5-fold axes normal to the non-regular decagonal faces
  • sets of three mirror planes intersecting along the 3-fold axes normal to the non-regular hexagonal faces
  • The polyhedron dual [5] of the Archimedean rhomb-icosidodecahedron is the Catalan deltoidal hexecontahedron shown in Figure 6, whose faces are identified by indices deriving from the cyclic permutations, changes of sign included, of the following Miller indices:

    {τ+1/τ  τ  0}, {τ2  1/τ  1/τ}, {2 1 τ}
    (with the only exception of the dodecahedron, the indices of all the icosahedral forms derive from the cyclic permutation, changes of sign included, of up to five Miller indices; for example, relatively to the other single forms belonging to the rhomb-icosidodecahedron, two are the indices both in case of the icosahedron, {111} and {τ  1/τ  0}, and of the rhomb-triacontahedron, {100} and {τ  1 1/τ}).
    Therefore the Catalan deltoidal hexecontahedron consists of 12+24+24 = 60 deltoidal (or kite-shaped) faces, namely a quadrangle having two disjoint pairs of congruent adjacent sides and two equal angles between the pairs of non-congruent sides.
    (clicking on the following image one can visualize the corresponding VRML file)
    Fig. 6
    Clinographic view of the Catalan deltoidal hexecontahedron, dual of the Archimedean rhomb-icosidodecahedron, showing the positioning of the fifteen 2-fold, the ten 3-fold and the six 5-fold axes.

    The views along the 2-fold, 3-fold and 5-fold axes of the Catalan deltoidal hexecontahedron and its corresponding stereographic projections are sequentially reported in Figure 7.

    Fig. 7a - The deltoidal hexecontahedron, dual of the rhomb-icosidodecahedron, and its stereographic projection viewed along a 2-fold rotation axis (to be compared with Fig. 3a).
    Fig. 7b - Deltoidal hexecontahedron, dual of the rhomb-icosidodecahedron, and its stereographic projection viewed along a 3-fold rotation axis  (to be compared with Fig. 3b).
    Fig. 7c - The deltoidal hexecontahedron, dual of the rhomb-icosidodecahedron, and its stereographic projection viewed along a 5-fold rotation axis (to be compared with Fig. 3c).

    If one describes the Catalan deltoidal hexecontahedron by the simmetry operators relative to the m3 point group, subgroup of m 3 5, the forms associated to the different Miller indices are:
  • {2 1 τ} diploid (or diakisdodecahedron)
  • {τ2  1/τ  1/τ} deltoidal icositetrahedron
  • {τ+1/τ  τ  0} pentagon-dodecahedron

  • In Figure 8 one can see a clinographic view of the Catalan deltoidal hexecontahedron and the three forms belonging to the m3 point group in which it can be decomposed.
     
    Fig. 8
    Decomposition of the Catalan deltoidal hexecontahedron in three single forms belonging to the m3 point group:
    a) {2 1 τ} diploid (or diakisdodecahedron)
    b) {τ2  1/τ  1/τ} deltoidal icositetrahedron
    c) {τ+1/τ  τ  0} pentagon-dodecahedron

    Pentagonal cupola (J5)
       
    The pentagonal cupola is the fifth polyhedron in the Johnson's list of non-Archimedean convex polyhedra with regular faces [2].
    A 5-fold cupola consists of two parallel faces, having pentagonal and decagonal shapes, connected by a ring of other ten polygons, where squares alternate to equilateral triangles.
    All the squares and triangles of the ring share a side with the decagon, whereas the opposite side (in case of the squares) or the opposite vertex (in case of the triangles) are shared with the pentagon.
    Then ten out of the fifteen vertices of the 5-fold cupola are shared among three faces, a decagonal, a triangular and a square one, whereas the other five vertices are shared among four faces:
    one of them is pentagonal, one is triangular, and two are square.
    The projection of a 5-fold cupola along the normal to the pentagonal face and the relative stereographic net are reported in Figure 9.
    (clicking on the  image of the 5-fold cupola, one can visualize the corresponding VRML file)
    Fig. 9 - Projection of a pentagonal cupola along the normal to the pentagonal face and the relative stereographic net.
    The angles between  the normal to the pentagonal face and  each normal to the square and triangular faces  measure 31.72 and 37.38, respectively.

    A rhomb-icosidodecahedron (henceforth, RID) can be sliced in correspondence to each decagonal base of its twelve 5-fold cupolas. Since all the cupolas share three faces (a square and two triangular ones) with each one of the five contiguous cupolas, one can infer that at most three cupolas can be obtained simultaneously from a RID.
    The rotation of one, two or three 5-fold cupolas by an angle of 180 (or, in general, an odd multiple of 36) and the subsequent reattachment to the RID lead to four Johnson's polyhedra which are isomers of the RID (Fig. 10a):
    (clicking on the following twelve images one can visualize the corresponding VRML files)
    FIG. 10a

    Gyrate RID (J72)

    Parabigyrate RID (J73)

    Metabigyrate RID (J74)

    Trigyrate RID (J75)


    The removal (instead of the rotation) from the RID of one, two or three 5-fold cupolas leads to other four Johnson's polyhedra (Fig. 10b):
    FIG. 10b

    Diminished RID (J76)

    Parabidiminished RID (J80)

    Metabidiminished RID (J81)

    Tridiminished RID (J83)


    A last set of four Johnson's polyhedra derives from the simultaneous removal and rotation of two or three 5-fold cupolas (Fig. 10c):
    FIG. 10c

    Paragyrate diminished RID (J77)

    Metagyrate diminished RID (J78)

    Bigyrate diminished RID (J79)

    Gyrate bidiminished RID (J82)

    In addition to the previous twelve Johnson's polyhedra, the thirteenth one deriving from the RID is the pentagonal cupola itself.

    Gyrate rhomb-icosidodecahedron (J72)
     
    Fig. 11 - Animation showing the derivation of the gyrate RID from the RID by a 180 rotation of the highlighted pentagonal cupola. Both polyhedra are viewed along a direction corresponding to a 5-fold axis.

       
    Rotating a pentagonal cupola of the RID by an angle of 180 around the normal to its central pentagonal face (Fig. 11), one obtains the simplest isomer of the RID: it is the 72nd polyhedron in the Johnson's list of non-Archimedean convex polyhedra with regular faces, named gyrate rhomb-icosidodecahedron by Johnson himself.
    Relatively to the maximum symmetry shown by the RID (which includes six 5-fold axes, ten 3-fold axes, fifteen couples consisting of a 2-fold axis and a perpendicular mirror plane, one centre of inversion), the symmetry of the gyrate RID is drastically lowered by the rotation of the pentagonal cupola.
    As a matter of fact, the point group of the gyrate RID is 5m, the same point group of the pentagonal cupola responsible for the lowering of symmetry in the gyrate RID: the only remaining symmetry operators are, as shown in Figure 12, the five mirror planes intersecting along a 5-fold axis perpendicular to the centre of the pentagon belonging to the rotated cupola.

    Fig. 12 - Clinographic view of the gyrate RID, showing the five mirror planes and the 5-fold axis.

    The gyrate RID viewed along the 5-fold axis and the corresponding stereographic net are shown in Figure 13; the indices of the faces of the rotated cupola in J72, compared with the indices of the faces of the corresponding cupola in  the RID, are reported in the Appendix A.

    Fig. 13
    Projection of the gyrate RID (J72) along its 5-fold axis and the relative stereographic net (to be compared with Fig. 3c).
    Only the faces of the rotated pentagonal cupola are exactly superimposed to the corresponding faces of the opposite pentagonal cupola which has not been rotated.

    Fifty of the the sixty-two faces of the gyrate RID can be grouped, according to the 5m point group, into ten sets of five faces equivalent by rotation around the 5-fold axis: therefore the corresponding single forms are ten pentagonal pyramids (five upwards and five downwards), whose intersection generates the gyrate RID after the addition of the other twelve faces, in turn grouped in three form: a decagonal prism, parallel to the 5-fold axis, and two monohedra (or pedions), perpendicular to the 5-fold axis.

    Supposing that the axis of the rotated cupola is aligned along the [τ 01] direction, the view along the [010] direction of the polyhedron dual of the gyrate RID is shown in Figure 14, compared with the equivalent view of the dual of the RID.

    Fig. 14 - The polyhedron dual of the gyrate RID (on the left), projected along the direction [010] and compared with the dual of the RID (on the right).

    The ten dark-blue faces of the dual of the gyrate RID have a four-sided shape, different from the other fifty faces (in pale blue), which are kite-shaped just like all the faces of the Catalan deltoidal hexecontahedron, dual of the RID.
    As one can see in Figure 15 from the views, along the [τ 01] direction, of the duals of both RID and gyrate RID, the changes between the two polyhedra involve fifteen faces: whereas the ten dark-blue faces have a different four-sided shape but an equal orientation in respect to the corresponding kite-shaped faces of the dual of the RID, the five faces nearest to the 5-fold axis, and contiguous to the dark-blue faces, have the same shape (and pale-blue color) of the faces of the dual of the RID but they are rotated of 180 around the 5-fold axis.

    Fig. 15
    Comparison between the dual of the gyrate RID (on the left) and the dual of the RID (on the right), projected along the [τ 01] direction.
    The ten faces in dark-blue, making a ring, have a generic four-sided shape: they differ from the other fifty kite-shaped faces, in pale-blue, of the dual of the gyrate RID,
    equal to the faces of the dual of the RID. Moreover in the gyrate RID the five kite-shaped faces inside the ring of ten dark-blue faces are 180 rotated relatively to the corresponding five faces of the dual of the RID.


    Due to the different symmetry of the duals of RID and gyrate RID (m 3 5 and 5m, respectively), both consisting of sixty faces, in the former polyhedron the faces belong to an only form, namely the Catalan deltoidal hexecontahedron, whereas the latter polyhedron derives from the intersection of eight pyramidal forms: in addition to the dark-blue decagonal pyramid, there are another decagonal pyramid, four pentagonal and two dipentagonal ones, all made of pale-blue faces.
    All these forms may be recognized more easily by their stereographic projections reported in Figure 16; the indices of the five rotated faces in the dual of J72, compared with the indices of the corresponding faces in  the dual of the RID, are reported in the Appendix B.

    Fig. 16
    Comparison between the stereographic projections of the duals of gyrate RID (left) and RID (right) along the [τ 01] pentagonal axis.
    As a consequence of the 180 rotation in J72 of the fifteen vertices of the 5-fold cupola, the faces of the dual of J72 corresponding to such rotated vertices undergo an equal rotation so that, analogously to the faces belonging to the two 10-fold pyramids, also the faces of the rotated 5-fold pyramid are exactly superimposed to the faces of the underlying 5-fold pyramid, which has not been rotated.
     

    Parabigyrate rhomb-icosidodecahedron (J73)

    Fig. 17 - Parabigyrate RID obtained by a 180 rotation of the highlighted pentagonal cupolas of a RID.
       

        The simultaneous rotation of two opposite pentagonal cupolas of the RID by an angle of 180 around the direction perpendicular to the central pentagonal face of both cupolas leads to a second isomer of the RID, the parabigyrate rhomb-icosidodecahedron, shown in Figure 17 (according to Norman Johnson [2]: "Where there are two different ways of adjoining, removing, or turning a pair of pieces, these are distinguished by the further prefix para- if the pieces are opposite each other and meta- if they are not").
    The parabigyrate RID (J73) is characterized by the same symmetry elements present in the gyrate RID, with the addition of five 2-fold axes normal to the five mirror planes: consequently also the centre of inversion is present. Therefore the paragyrate RID belongs to the 5m point group and its complete symmetry is: 5(2/m)(2/m)(2/m)(2/m)(2/m)1.
    Its view along the [010] 2-fold axis, perpendicular to both the 5-fold axis and a mirror plane, is shown in Figure 18.

    Fig. 18 - View of the parabigyrate RID along the [010] direction, showing the five 2-fold axes: each of them is normal to both the 5-fold axis and a mirror plane.

    The projection of the parabigyrate RID along the 5-fold axis and the corresponding stereographic net are shown in Figure 19.

    Fig. 19
    Projection of the parabigyrate RID (J73) along its 5-fold axis and the relative stereographic net: from the comparison with Fig. 3c one can note the 180 rotation of the red and yellow marks, placed on the most internal circumferences, representing the corresponding square and triangular faces of the rotated cupolas.

    According to the 5m point group, the 62 faces of the parabigyrate RID can be grouped, as shown in Fig. 20a, into five different 5-fold deltohedra (or trapezohedra), a 10-fold prism and a pinacoid (or parallelohedron): the single forms with the same colour have also an equal central distance.
    The intermediate steps in the reconstruction of the parabigyrate RID, given by the intersection of the equidistant forms, are shown in Fig. 20b, together with the resulting parabigyrate RID.
    The pale-blue polyhedron consists in a dodecahedron, whereas, in the re-building of the parabigyrate RID, the less usual red and yellow polyhedra play the same role of the rhomb-triacontahedron and the icosahedron generating the RID along with the dodecahedron (Fig.20c).

    Fig. 20a - Decomposition of the parabigyrate RID into 5-fold deltohedra (shown, not to scale, in the first five images), a 10-fold prism and a pinacoid (shown together in a clinographic view reported in the last image).
    Fig. 20b - View along the [010] direction of the intermediate polyhedra obtained by the intersection of the single forms sharing both colour and  central distance: the only recognizable polyhedron is the dodecahedron, deriving from the intersection of the pale-blue deltohedron and the pinacoid. The fourth image concerns the paragyrate RID, obtained by the intersection of  the red, yellow and pale-blue polyhedra.
    Fig. 20c - View along the [010] direction of a rhomb-triacontahedron, an icosahedron and a dodecahedron, together with the RID resulting from their intersection in case of  appropriate ratios between their central distances.
    Rhomb-triacontahedron and icosahedron can be compared with the corresponding red and yellow polyhedra in Fig. 20b.

        The views of the dual of the parabigyrate RID along the directions [010] and [τ 01] are shown in Figure 21.
    Consequently to the rotation of two opposite pentagonal cupolas in the parabigyrate RID, in its dual there are two rings of ten faces (in dark-blue) having a four-sided shape different from the other forty faces (in pale blue) which are kite-shaped just like all the faces of the Catalan deltoidal hexecontahedron, dual of the RID.


    Fig. 21
    The polyhedron dual of the parabigyrate RID, projected along the directions: [010] (on the left) and [τ 01] (on the right).
    The twenty faces in dark-blue, distributed in two rings, have a four-sided shape: they differ from the forty kite-shaped faces, in pale blue, identical to the faces of the dual of the RID.

    According to the 5m point group, in the dual of the parabigyrate RID the sixty faces are grouped in four forms: two different pentagonal deltohedra, a dipentagonal scalenohedron and a decagonal dipyramid (Fig. 22).

    Fig. 22
    View, along the [010] direction, of the four forms (not drawn to scale), whose intersection generates the dual of  the parabigyrate RID. They are, in sequence:
    a) a flat pentagonal deltohedron
    b) a decagonal dipyramid
    c) a dipentagonal scalenohedron
    d) a sharp pentagonal deltohedron

    It must be pointed out that the single forms resulting from the decomposition of both J73 and its dual are equal to the forms in which the RID and its dual could be decomposed if one ascribes also to them the symmetry of the point group 5m, subgroup of m3 5 ; the only difference consists in the positioning, in respect to the other single forms, of the flat deltohedra: two in J73 and one in its dual. Such difference is due to the rotation of the 5-fold cupola involving a change in its dual, too.
    On the other hand, the positioning of the 10-fold dipyramid does not vary (as one can see in Figure 23), even if the shape of the twenty 4-sided dark blue faces constituting the dipyramid in the dual of J73 differs from the deltoidal shape of the corresponding faces in the dual of the RID.

    Fig. 23 - Comparison, along the [τ01] 5-fold axis, between the stereographic nets of the duals of parabigyrate RID (left) and RID (right).
    As a consequence of the 180 rotation of the fifteen vertices of both 5-fold cupolas, in the dual of the parabigyrate RID the faces of the flat 5-fold deltohedron undergo a corresponding rotation, whereas the 4-sided dark-blue faces of the 10-fold dipyramid belonging to the dual of J73 have, on the whole, the same orientation of the deltoidal pale-blue faces of the 10-fold dipyramid belonging to the dual of the RID: only the shape of the faces is different.

    Metabigyrate rhomb-icosidodecahedron (J74)
     
    Fig. 24 - Metabigyrate RID obtained by a 180 rotation of the two highlighted pentagonal cupolas of a RID, aligned along the [τ 01] and [τ 01] directions.

        In addition to the parabigyrate RID, there is a second isomer of the RID obtained by the 180 rotation of two pentagonal cupolas of the RID: it is the metabigyrate rhomb-icosidodecahedron (J74). Whereas in the parabigyrate RID the rotation concerns two centrosymmetric cupolas aligned along the same 5-fold axis, in the metabigyrate RID the axes of the rotated cupolas are placed at an angle φ = 2arctg(τ) = 116.56 (Fig. 24): it implies a drastic lowering of the symmetry of the metabigyrate RID in respect to the RID.
    If the two rotated 5-fold cupolas of the RID are the ones centered along the [τ 01] and [τ 01] directions, in J74 there is a 2-fold rotation axis, normal to the (001) square face, in correspondence to the intersection of two orthogonal mirror planes, (100) and (010): the mirror (100) reflects each other the two gyrate cupolas, whereas the mirror (010) divides each cupola in two specular parts (Fig. 25).
    Therefore mm2 is the symbol of the point group describing the orthorhombic symmetry of J74.

    Fig. 25 - Clinographic view of the metabigyrate RID, showing the two mirror planes and the 2-fold rotation axis placed at their intersection.

    The J74 polyhedron viewed along the [001] 2-fold axis and the corresponding stereographic net are shown in Figure 26.

    Fig. 26
    Projection of the metabigyrate RID along the [001] 2-fold axis and the relative stereographic net: from the comparison with Fig. 3a relative to the RID, one can note that, whereas in the RID all the red and yellow marks representing the square and triangular faces non parallel to [001] are superimposed in pairs, in the metabigyrate RID many of these marks are not superimposed, due to the rotation of the two pentagonal cupolas.

    According to the mm2 point group, the 62 faces of the metabigyrate RID can be grouped in nine rhombic pyramids, a rhombic prism, nine sphenoids (or domes or dihedra), a pinacoid (or parallelohedron) and two pedions (or monohedra).

        In Figure 27 one can see, on the left, the projection along the directions [010] of the dual of the metabigyrate RID and, on the right, a clinographic view showing also the two mirror planes intersecting along the [001] 2-fold axis.
    Analogously to the orientation of the two rotated pentagonal cupolas in J74, even in its dual the two rings of ten faces (in dark-blue), having a four-sided but non deltoidal shape, are centered along the [τ 01] and [τ 01] directions: in particular, each face of a couple of contiguous faces of a ring shares a side with a specular face belonging to the other ring.

    Fig. 27
    Polyhedron dual of the metabigyrate RID: on the left its projection along the [010] direction and, on the right, a clinographic view showing the [001] 2-fold axis and the (100) and (010) mirror planes characterizing the mm2 point group.
    The twenty faces in dark-blue, having a four-sided but non-deltoidal shape, are arranged in two rings centered along the
    [τ01] and [τ01] directions.

    As one can infer from the stereographic net along the [001] 2-fold axis (Fig. 28), in the dual of the parabigyrate RID the sixty faces are arranged, according to the mm2 point group, into the following forms: twelve rhombic pyramids, four dihedra (domes or sphenoids) and a rhombic prism.

    Fig. 28
    Projection, along the [001] 2-fold axis, of the dual of the metabigyrate RID and relative stereographic net; in mm2 point group the sixty faces are arranged into the following forms:  twelve rhombic pyramids, four sphenoids (or domes or dihedra) and a rhombic prism (it should be compared with Fig. 7a).


    Trigyrate rhomb-icosidodecahedron (J75)
     
    Fig. 29 - View along the [111] 3-fold axis, at the intersection of three mirror planes, of the trigyrate RID, obtained by a 180 rotation of three  pentagonal cupolas of a RID aligned along the equivalent [τ 01], [01τ] and [1τ 0] directions.

         The fourth isomer of the RID is the trigyrate RID (J75), obtained from the RID by the rotation of three pentagonal cupolas (Fig. 29): 116.56 is the angle between the axes of each couple of rotated cupolas, whereas 79.19 is the angle between the axes of all the cupolas and the 3-fold axis perpendicular to the triangular face placed in a symmetrical position among the rotated cupolas. The 3-fold axis is positioned at the intersection of three mirror planes: every mirror divides a rotated cupola in two specular parts and reflects the other two rotated cupolas (Figure 30). Therefore 3m is the symbol of the point group describing the symmetry of the trigyrate RID.

    Fig. 30
    View of the trigyrate RID including the three mirror planes which intersect along the 3-fold axis perpendicular to the (111) triangular face.
    In consequence of the rotation of the three pentagonal cupolas, a side is shared by each square face of the rotated cupolas with a square face that has not been rotated.

    The trigyrate RID viewed along the 3-fold axis, normal to the (111) triangular face and the corresponding stereographic net are shown in Figure 31.
    Fig. 31
    Projection of the trigyrate RID along the 3-fold axis normal to the (111) triangular face and the relative stereographic net (to be compared with Fig. 3b).

    From the view along the [111] 3-fold axis and the relative stereographic net, one can infer that, according to the 3m point group, the sixty-two faces of the trigyrate RID are arranged in seventeen single forms: two monohedra or pedions (perpendicular to the 3-fold axis), ten 3-fold pyramids and five ditrigonal pyramids.

    The polyhedron dual of the trigyrate RID is shown in Fig.32, where also the three mirror planes intersecting along the 3-fold axis are reported.
    Fig. 32 - View of the dual of the trigyrate RID including three mirror planes intersecting along the [111] 3-fold axis.

    In consequence of the rotation in J75 of three pentagonal cupolas, its dual includes three rings of ten faces (in blue) having a four-sided but non-deltoidal shape.
    From the stereographic net along the [111] 3-fold axis shown in Figure 33, one can infer that in the dual of the trigyrate RID the sixty faces are arranged into four trigonal and eight ditrigonal pyramids, according to the 3m point group.

    Fig. 33 - Projection of the dual of the trigyrate RID along the [111] 3-fold axis and relative stereographic net showing that its sixty faces are arranged into four trigonal and eight ditrigonal pyramids (to be compared with Fig. 7b).


    Diminished rhomb-icosidodecahedron (J76)
     
    Fig. 34 - The removal of a pentagonal cupola from the RID leads to the diminished RID, a regular-faced polyhedron in which a regular decagon substitutes the pentagonal cupola.

    Whereas the rotation of pentagonal cupolas leads to Johnson's polyhedra with 62 faces which are isomers of the RID, the removal of one, two or three 5-fold cupolas, letting a decagonal face in the place of each substituted cupola, leads to Johnson's polyhedra with 52, 42 or 32 faces, respectively.
    The simplest case consist in the removal from the RID of an only 5-fold cupola (Fig. 34), leading to the diminished RID (J76): in Figure 35 one can see, on the left, its view along the [001] direction making an angle of 58.28 with the [τ 0 1] axis of the removed cupola and, on the right, the view along the [010] direction, perpendicular to the axis of the removed cupola.

    Fig.35 - Views of the diminished RID along the [001] (left) and [010] (right) directions making an angle of 58.28 and 90, respectively, with the normal to the [τ 0 1] axis of the removed cupola.

    The diminished RID, viewed along the 5-fold axis normal to the decagonal face, and the corresponding stereographic net are shown in Figure 36.

    Fig. 36 - Projection of the diminished RID along the 5-fold axis normal to the decagonal face (on the left) and relative stereographic net (on the right); it should be compared with Fig. 3c.

    In the diminished RID, the symmetry decrease relative to the RID, due to the removal of a 5-fold cupola, is equal to the symmetry decrease found in the gyrate RID, due to the rotation of a 5-fold cupola: in fact 5m is the point group relative to both Johnson's polyhedra. Consequently, the fifty-two faces of the diminished RID consist of eight pentagonal pyramids, a decagonal prism and two monohedra (or pedions).

    Fig. 37 - Clinographic view of the dual of the diminished RID, showing the ten triangular face forming a 10-fold pyramid.

        The dual of J76, shown in Figure 37 in the same orientation as J76 in Figure 34, is rather different from the dual of the RID and also from the duals of its isomers: the lack in J76 of the five vertices of the pentagon belonging to the removed cupola implies the lack of five faces in the dual; consequently the ten faces contiguous to the missing ones extend their dimension. They share a vertex placed along the 5-fold axis normal to the removed cupola and their shape from deltoidal  becomes triangular. A comparison between the dual of J76 and the dual of the RID, viewed along the [010] axis, can be seen in Figure 38.

    Fig. 38 - View along the direction [010] of the dual of the diminished RID compared with the dual of the RID.

    From the stereographic net along the [τ 0 1] 5-fold axis, shown in Figure 39, one can infer that, in the dual of the diminished RID, the fifty-five faces are arranged into three pentagonal, two dipentagonal and two decagonal pyramids, according to the 5m point group.

    Fig. 39 - Projection of the dual of the diminished RID along the [τ01] 5-fold axis (on the left) and relative stereographic net (on the right). It is not difficult to recognize the presence of five mirror planes which characterize the 5m point group, together with the 5-fold axis placed at their intersection.


    Paragyrate diminished rhomb-icosidodecahedron (J77)
     
    Fig. 40 -  The removal of a pentagonal cupola and the simultaneous rotation of the opposite pentagonal cupola transform the RID into the paragyrate diminished RID, which includes a decagonal face in substitution of the removed pentagonal cupola.

        The removal of a pentagonal cupola from the RID can be accompanied by the rotation of another 5-fold cupola; when the rotation concerns the cupola opposite to the removed one (Fig. 40), the resulting polyhedron is the paragyrate diminished RID (J77): its view along the [010] direction, perpendicular to the [τ 0 1] axis of both cupolas, the removed and the rotated one, is shown in Figure 41.

    Fig. 41 - View of the paragyrate diminished RID along the [010] direction, perpendicular to a square face and also to the axis of both removed and rotated 5-fold cupolas.

    In Figure 42 one can see the view of J77, along the 5-fold axis of the rotated and removed cupolas, together with the corresponding stereographic net.
     
    Fig. 42 - Projection of the paragyrate diminished RID along the 5-fold axis normal to the pentagonal face of the rotated 5-fold cupola (on the left) and relative stereographic net (on the right);  it should be compared with Fig. 3a.

    The simultaneous rotation and diminution do not introduce in J77 variations of symmetry in respect to the gyrate RID or the diminished RID: therefore also the paragyrate diminished RID can be described in the 5m point group and its fifty-two faces can be grouped, analogously to J76, in eleven forms: eight pentagonal pyramids, a decagonal prism and two monohedra (or pedions).

    Figure 43 shows (on the left) the dual of J77 and (on the right) the dual of the RID, viewed along the [010] direction.

    Fig.43 -  The dual of the paragyrate diminished RID (on the left), compared with the dual of the RID (on the right), shows the two 10-fold pyramidal forms, consisting of ten triangular faces in blue and ten four-sided, but non-deltoidal, faces  in dark-blue, which shape is different from the shape of the deltoidal faces in pale-blue.

        From the comparison with the dual of the RID, one can realize that the dual of J77 includes two 10-fold pyramidal forms: the first, consisting of elongated triangular faces, which is present even in the dual of J76 and the second, consisting in a ring of ten four-sided (but non-deltoidal) faces, which is present even in the dual of J72. Thus the dual of J77 includes faces with three different shapes, whose geometrical features will be analyzed and compared in the Appendix C.
    On the whole, also the fifty-five faces of the dual of J77 are arranged (according to the 5m point group) into three pentagonal, two dipentagonal and two decagonal pyramids, as shown in Figure 44 by the view along the [τ 0 1] 5-fold axis and the corresponding stereographic net.

    Fig. 44 - Projection of the dual of the paragyrate diminished RID along the [τ01] 5-fold axis (on the left) and relative stereographic net (on the right). It is not difficult to recognize the presence of the five mirror plane characterizing, together with the 5-fold axis, the 5m point group.


    Metagyrate diminished rhomb-icosidodecahedron (J78)
     
    Fig. 45 - The removal of a 5-fold cupola and the simultaneous rotation of another one, aligned along a direction making an angle of 116.56 with the axis of the previous cupola, transform the RID into the metagyrate diminished RID, which includes a decagonal face in substitution of the removed pentagonal cupola.

         The removal from the RID of a pentagonal cupola, accompanied by the simultaneous rotation of another one, can be accomplished also when the axes of the two cupolas, instead of being aligned as in J77,  form an angle φ = 2arctg(τ) = 116.56 (equal to the angle between the two rotated cupolas in J74): the resulting polyhedron is the metagyrate diminished RID (J78), reported in Figure 45.
    Its view along the [010] direction, perpendicular to both axes of the cupolas, is shown in Figure 46.

    Fig. 46 - View of the metagyrate diminished RID along the [010] direction, perpendicular to the axes of both rotated and removed 5-fold cupolas, which make an angle of 116.56, being aligned along the [τ 01] and [τ 01] directions.

    The view along the [001] direction, shown in Figure 47, reveals the presence in J78 of an only (010) mirror plane, placed vertically. Among the Johnson's polyhedra derived from the RID, the metagyrate diminished RID shows, together with J79 and J82, the lowest symmetry: as a matter of fact, it belongs to the m point group and its fifty-two faces are grouped in twenty domes (or dihedra), ten pedions (or monohedra) and a pinacoid (or parallelohedron).

    Fig. 47 - Projection of the metagyrate diminished RID along the [001] direction (on the left) and relative stereographic net (on the right): only a vertical (010) mirror plane is present. It should be compared with Fig. 3a.

    In Figure 48 one can see the variations occurring in the dual of J78 relatively to the dual of the RID, in consequence of the simultaneous rotation and removal of two 5-fold cupolas whose axes are at an angle of 116.56.

    Fig. 48 - View along the [010] direction of the dual of the metagyrate diminished RID compared with the dual of the RID.

    According to the presence in the m point group of an only mirror, the fifty-five faces of the dual of J78 are arranged into many single forms: twenty-six domes (or dihedra) and three pedions (or monohedra), as shown in Figure 49, reporting the view along the [001] direction and the corresponding stereographic net.

    Fig. 49 - Projection of the dual of the metagyrate diminished RID along the [001] direction (on the left) and relative stereographic net (on the right). The presence of a (010) mirror plane placed vertically is easily recognizable.


    Bigyrate diminished rhomb-icosidodecahedron (J79)
     
    Fig. 50 - Bigyrate diminished RID derived from the RID when two of a set of three 5-fold cupolas (each other at 116.56) are rotated and the third one is removed: a decagonal face replaces the removed pentagonal cupola. 

    The simultaneous rotation of two 5-fold cupolas, aligned along the axes [1 τ 0] and [1 τ 0], and the removal of a third one, aligned along the [τ 0 1] axis, lead to the bigyrate diminished RID (J79), shown in Figure 50.
    The view of J79 along the [10τ] direction, normal to the axis of the removed 5-fold cupola, highlights the symmetry decrease relatively to the RID (Fig. 51): among all the forty-eight symmetry elements characterizing the RID (identity included), only the vertical (010) mirror plane m survives in J79.
     
    Fig. 51 - Comparison between the bigyrate diminished RID (on the left) and the RID (on the right) viewed along the [1 0 τ]  direction, normal to the axis of the removed 5-fold cupola; the positions of the two rotated cupolas are symmetrical in respect to the vertical (010) mirror plane, the only symmetry operator present in J79.

    The view of J79 along the [001] direction, normal to both axes of the rotated 5-fold cupolas and forming an angle of 58.28 with the axis of the decagonal face, is shown in Figure 52, together with the relative stereographic net.
    Even in J79 the low degree of symmetry implies the grouping of the fifty-two faces in a large number of forms: twenty-one domes (or dihedra) and ten pedions (or monohedra).

    Fig. 52 - View along the [001] direction of J79 (left) and relative stereographic net (right); the two rotated cupolas are  symmetrical in respect to to the vertical (010) mirror plane.

        In the dual of J79, viewed along the [10τ] direction (as J79 in Fig.51) on the left of Figure 53, five pale-blue deltoidal faces, 180 rotated relatively to the corresponding faces of the dual of the RID (shown on the right), and a ring of ten 4-sided dark blue faces derive from each rotated cupola; ten blue triangolar faces are instead the consequence of the removal of the third 5-fold cupola.

    Fig. 53 - Views, along the [10τ] direction, of the dual of the bigyrate diminished RID (on the left) and of the dual of the RID,  shown on the right by comparison.

    According to the m point group, the fifty-five faces of the dual of J79 can be grouped in twenty-six domes (or dihedra) and three pedions (or monohedra), as one can infer from the view along the [001] direction and the relative stereographic projection reported in Figure 54.

    Fig. 54 - View, along the [001] direction, of the dual of J79 (left) and relative stereographic net (right). Also in this case, the presence of a (010) mirror plane placed vertically is easily recognizable.

    Concerning both rings of ten dark-blue faces, since the [1 τ 0] and [1 τ 0] directions of the axes of the two rotated cupolas lie in the plane of the stereographic net, the small circles and the x-shaped characters, representing the perpendiculars to the faces directed upwards or downwards, respectively, are superimposed in pairs; this circumstance does not occur with the ten blue faces, because the axis of the removed cupola makes an angle of 58.28 with the [001] direction normal to the stereographic net.


    Parabidiminished rhomb-icosidodecahedron (J80)
     
    Fig. 55 - The simultaneous removal of two opposite pentagonal cupolas of the RID leads to the parabidiminished RID, which includes two decagonal faces in substitution of the removed pentagonal cupolas.

    The axes of two 5-fold cupolas simultaneously removed from the RID can coincide or make an angle of 116.56, leading to different Johnson's polyhedra having forty-two faces.
    When the axes of the two removed 5-fold cupolas coincide, the resulting polyhedron is the parabidiminished RID (J80) shown in Figure 55. It belongs to the 5m point group, analogously to the parabigyrate RID derived from the rotation of the two 5-fold cupolas: their comparison can be seen in Figure 56.

    Fig. 56 - Comparison between the parabidiminished RID (on the left), and the parabigyrate RID (on the right): their projections along the [010] direction show the removal or the rotation of two opposite 5-fold cupolas centered along the [τ 0 1] and [τ 0 1] opposite directions.
    As one can infer from the presence of the 5-fold axis and the five couples made of a 2-fold axis and an orthogonal mirror, both Johnson's polyhedra belong to the 5m point group.

    In Figure57 the view of J80 along the [τ 0 1] direction, together with the corresponding stereographic net, shows the presence of five mirror planes, normal to the 2-fold axes, intersecting along a 5-fold rotation axis passing through the centres of the two superimposed decagonal faces; the further presence of a centre of inversion implies that the forty-two faces can be grouped in three deltohedra, a decagonal prism and a pinacoid (or parallelohedron).

    Fig. 57 - View along the [τ 0 1]  direction of the parabidiminished RID (left) and relative stereographic net (right).
    In addition to the evident five mirror planes intersecting along
    the 5-fold rotation axis directed along [τ 0 1], also five 2-fold axes normal to each mirror and a centre of inversion are present.

    Due to the lack of two 5-fold cupolas in J80, its dual is characterized by the presence of a decagonal dipyramid consisting of twenty blue triangular faces. In  Figure 58 its view along [010] is compared with the corresponding view of the dual of J73.

    Fig. 58 - Comparison between the dual of the parabidiminished RID (on the left) and the dual of the parabigyrate RID (on the right): in the former, a decagonal dipyramid, consisting of twenty triangular blue faces, replaces the decagonal dipyramid, consisting of dark blue 4-sided (but non-deltoidal) faces, and the flat pale-blue deltohedron, both present in the latter.

    According to the 5m point group, the fifty faces of the dual of the parabidiminished RID can be grouped in three forms: in addition to the 10-fold dipyramid, there are a dipentagonal scalenohedron and a sharp pentagonal deltohedron, as one can infer also from the stereographic net reported in Figure 59.

    Fig. 59 - View along the [τ 0 1]  direction, corresponding to the 5-fold rotation axis, of the dual of the parabidiminished RID (left) and relative stereographic net (right).

    The three single form, which intersection generates the parabidiminished RID, are viewed along the [010] direction in Figure 60 (not to scale).

    Fig. 60 - From left: the 10-fold dipyramid, the dipentagonal scalenohedron and the sharp pentagonal deltohedron whose intersection, in case of faces having the same central distance, generates the dual of the parabidiminished RID. This figure can be compared with Fig.22, where one can see that, among  the single forms deriving from the decomposition of the dual of  the parabigyrate RID, also a flat pale-blue deltohedron was present.


    Metabidiminished rhomb-icosidodecahedron (J81)
     
    Fig. 61 - The simultaneous removal from the RID of two pentagonal cupolas, aligned along two directions making an angle of 116.56,  leads to the metabidiminished RID.

    When the axes of the two cupolas removed from the RID make an angle of 116.56, the resulting polyhedron is the metabidiminished RID (J81), shown in Figure 61; two decagonal faces substitute the removed pentagonal cupolas.
    In Figure 62 one can see two views of J81 along the orthogonal directions [010] (a) and [001] (b).

    Fig. 62 - The views of the metabidiminished RID along two orthogonal directions: [010] (on the left) and [001] (on the right), highlight the presence of two orthogonal mirror planes intersecting along the [001] 2-fold axis.

    The view along [010] shows the result of the removal from the RID of two 5-fold cupolas, whose normals are aligned along the [τ 0 1] and [τ 0 1] directions, both normal to [010].
    From the view along [001] one can detect the presence in J81 of two orthogonal mirror planes intersecting along a 2-fold axis directed along [001]: therefore the metabidiminished RID belongs to the mm2 point group, analogously to the metabigyrate RID which derive from the rotation of two 5-fold cupolas.
    As one can infer from the corresponding stereographic nets reported in Figure 63, relative to the projections of J81 along the directions [010] (a) and [001] (b), the forty-two faces of J81 can be grouped in five rhombic pyramids, a rhombic prism, seven dihedra (or domes or sphenoids), a parallelohedron (or pinacoid) and two monohedra (or pedions).

    Fig. 63 - Stereographic net of J81 along the directions [010] (a) and [001] (b), showing the presence of two orthogonal mirror planes., intersecting along the [001] 2-fold axis.

    Due to the lack in J81 of the two 5-fold cupolas aligned along the [τ 0 1] and [τ 0 1] directions, its dual polyhedron is characterized, as shown in Figure 64, by the presence of two sets of ten triangular faces which derive, by a duality relation, from the vertices belonging to the decagonal base of each cupola. 

    Fig. 64 - Views of the dual of the metabidiminished RID along the orthogonal directions [010] (a) and [001] (b), highlighting the presence of two orthogonal mirror planes intersecting along the [001] 2-fold axis.

    Each set of ten triangular faces would constitute a 10-fold pyramid, being the dihedral angle between all the couples of contiguous faces equal to 25.88, but their concomitant presence lowers the overall symmetry of the dual of J81: in fact it belongs, as J81, to the mm2 point group.
    From the corresponding stereographic projections, reported in Figure 65, of the dual of J81 along the orthogonal directions [010] (a) and [001] (b), one can identify the single forms in which the fifty faces of the dual of J81 can be decomposed according to the mm2 point group: ten rhombic pyramids, a rhombic prism and three dihedra (or domes or sphenoids).

    Fig. 65 - Stereographic projections of the dual of the metabidiminished RID along the orthogonal directions [010] (a) and [001] (b). The stereographic net (a) is symmetric in respect to a vertical (010) mirror plane, whereas in (b) both vertical (010) and horizontal (100) mirror planes, intersecting along the [001] 2-fold axis, are highlighted.

     
    Gyrate bidiminished rhomb-icosidodecahedron (J82)
     
    Fig. 66 - Removal of two cupolas and simultaneous rotation of the third one leading to the gyrate bidiminished RID.

    Given a set of three pentagonal cupolas of the RID (each other at 116.56), the removal of two cupolas and the simultaneous 180 rotation of the third one lead to the gyrate bidiminished RID (J82), which includes two decagonal faces in substitution of the removed pentagonal cupolas (Figure 66).
    If the axes of the two removed 5-fold cupolas are aligned along the directions [1τ 0] and [1 τ 0] and the axis of the rotated cupola along [τ 01], in Figure 67 the  four views of J82, included between the orthogonal directions [001] and [100], reveal the existence of an only (010) mirror plane, reflecting each other the two decagonal faces and bisecting the rotated cupola: the point group of J82  is therefore m, analogously to J78 and J79.

    Fig. 67 - Views of the gyrate bidiminished RID along four directions: in addition to the orthogonal directions, [001] (a) and [100] (d), both normal to a square face, the intermediate directions are: [10 τ ] (b), normal to the axis of the rotated cupola, and [τ 0 1] (c), normal to a pentagonal face.
    All the views highlight the presence of an only mirror plane, orthogonal to the [010] direction; the (a) and (d) views can be compared with the (a) and (b) views in Fig.62, relative to the metabidiminished RID, which is characterized by the the removal of two 5-fold cupola, without any rotation of the third cupola.

    From the stereographic projections along the directions [001], orthogonal to the axes of the two decagonal faces deriving from the 5-fold removed cupolas, and [10τ], orthogonal to the axis of the gyrate 5-fold cupola (Figure 68), one can recognize the presence of twelve pedions (or monohedra) and fifteeen domes (or dihedra), according to the m point group.

    Fig. 68 - Stereographic projections of the gyrate bidiminished RID along the directions [001] (a) and [10 τ ] (b)

    The four views of the dual of J82 in Figure 69 correspond to the view of J82 in Figure 67 and include faces having three different shapes: triangles, kites and 4-sided polygons. The two groups of ten blue triangular faces derive from the removed 5-fold cupolas, the ten 4-sided dark blue faces and the five rotated pale-blue faces derive from the rotated cupola.

    Fig. 69 - Views of the dual of the gyrate bidiminished RID along the same four directions of  Fig.67: in addition to the orthogonal directions, [001] (a) and [100] (d), the intermediate directions are [10 τ ] (b) and [τ 0 1] (c).

    From the stereographic projections of the dual of J82 reported in Figure 70, viewed along the directions [001] and [10τ] (as J82 in Fig. 68), one can recognize the presence of 23 domes (parallelohedra) and 4 pedions (monohedra), according to the m point group.

    Fig. 70 - Stereographic projections of the dual of the gyrate bidiminished RID along the directions [001] (a) and [10 τ ] (b).


    Tridiminished rhomb-icosidodecahedron (J83)
     
    Fig. 71 - The simultaneous removal of three pentagonal cupolas of the RID (each other at 116.56) leads to the tridiminished RID, which includes three decagonal faces in substitution of the removed pentagonal cupolas.

    The last Johnson's polyhedron derived from the RID is shown in Figure 71.
    It has been already pointed out that the 5-fold cupolas which can be simultaneously rotated and/or removed from a RID are three at most; the different possibilities are resumed in Figure 72:
  • the rotation of three cupolas leads to the trigyrate RID (J75)
  • the rotation of two cupolas and the removal of the third one lead to the bigyrate diminished RID (J79)
  • the removal of two cupolas and the rotation of the third one lead to the gyrate bidminished RID (J82)
  •  the removal of three cupolas leads to the tridiminished RID (J83).

  • Among the Johnson's polyhedra deriving from the RID, J83 is the only numbering 32 faces.

     

    Trigyrate RID (J75)

     

    Bigyrate diminished RID (J79)

    RID

    Gyrate bidiminished RID (J82)

     

    Tridiminished RID (J83)

      
    Fig. 72 - The four Johnson polyhedra obtained from the RID by a 180 rotation and/or removal of three 5-fold cupolas.

    Two views of the tridiminished RID (with the respective stereographic nets) are shown in Figure 73:
    • the first one along a 3-fold axis, at the intersection of three mirror plane; 116.56 is the angle between the axes of the couples of decagonal faces substituting the removed cupolas, whereas 79.19 is the angle between the axis of each decagonal face and the 3-fold axis perpendicular to the triangular face at the centre of the image
    • the second one along a direction making an angle of 58.28 with the axis of a decagonal face and orthogonal to the axes of the other two decagonal faces.
    Fig. 73 - Views of the tridiminished RID and corresponding stereographic nets:
    (a) along the 3-fold axis, at the intersection of three mirror plane
    (b) along a direction orthogonal to the axes of the two decagonal faces replacing the 5-fold cupolas.

    Since in J83 three mirror planes intersect along a three fold axis, the relative point group is 3m and the thirty-two faces can be grouped in eight trigonal pyramids, a ditrigonal pyramid and two pedions (or monohedra), as one can infer from Figure 73a.

    The dual of J83 includes three 10-fold triangular faces deriving, by a duality relation, from the vertices of the three decagonal faces belonging to J83.
    Two views of the dual of J83 and the respective stereographic nets are shown in Figure 74.

    Fig. 74
    Views of the tridiminished RID (left) and relative stereographic nets (right):
    (a) along the 3-fold axis, at the intersection of three mirror plane
    (b) along a direction perpendicular to the axes of  two 10-fold pyramids and making an angle of  58.28 with the axis of the third 10-fold pyramid.

    According  to 3m point group, the forty-five faces can be grouped in three trigonal pyramids and six ditrigonal pyramids, as one can infer from Figure 74a.

        The features of the Johnson's polyhedra deriving from the RID and also of their duals, including the number of faces of the single forms in which each polyhedron can be decomposed, are resumed in the next table.
    Despite the different notation, the classification of the Johnson's polyhedra according to the respective point group is analogous to the classification given by Norman Johnson in his fundamental paper [2] concerning the convex polyhedra with regular faces.

    Decomposition in single forms of both the Johnson's polyhedra derived from the RID and their duals

    Johnson's
    polyhedra

    Single forms

    Faces
    in each
    form

    Total
    number
    of faces

    Point
    group
    Duals of
    Johnson's
    polyhedra
    Single forms

    Faces
    in each
    form

    Total
    number
    of faces

    J72

    10 pentagonal pyramids
      1 decagonal prism
      2 monohedra

    5
    10
    1

    62

    5m

    dual J72

     4 pentagonal pyramids
     2 dipentagonal pyramids
     2 decagonal pyramid

    5
    10
    10

    60

    J73

      5 pentagonal deltohedra
      1 decagonal prism
      1 parallelohedron

    10
    10
    2

    62 5m dual J73

     2 pentagonal deltohedra    
     1 dipentagonal scalenohedron 
     1 decagonal dipyramid

    10
    20
    20

    60

    J74

      9 rhombic pyramids
      1 rhombic prism
      9 dihedra
      1 parallelohedron
      2 monohedra

    4
    4
    2
    2
    1

    62   mm2 dual J74

    12 rhombic pyramids
      1 rhombic prism
      4 dihedra

    4
    4
    2

    60

    J75

      5 ditrigonal pyramids
    10 trigonal pyramids
      2 monohedra

    6
    3
    1

    62 3m dual J75

      8 ditrigonal pyramids
      4 trigonal pyramids

    6
    3
    60

    J76

      8 pentagonal pyramids 
      1 decagonal prism
      2 monohedra

    5
    10
    1

    52  5m dual J76

     3 pentagonal pyramids
     2 dipentagonal pyramids
     2 decagonal pyramid

    5
    10
    10
    55

    J77

      8 pentagonal pyramids 
      1 decagonal prism
      2 monohedra

    5
    10
    1

    52  5m dual J72

     3 pentagonal pyramids
     2 dipentagonal pyramids
     2 decagonal pyramid

    5
    10
    10
    55

    J78

    20 dihedra
    10 monohedra
      1 parallelohedron

    2
    1
    2

    52   m dual J78

    26 dihedra
      3 monohedra

    2
    1
    55

    J79

    21 dihedra
    10 monohedra

    2
    1

    52   m dual J79

    26 dihedra
      3 monohedra

    2
    1
    55

    J80

      3 pentagonal deltohedra
      1 decagonal prism
      1 parallelohedron

    10
    10
    2

    42

    5m dual J80  1 decagonal dipyramid 
     1 dipentagonal scalenohedron    
     1 pentagonal deltohedron

    20
    20
    10

    50

    J81

      5 rhombic pyramids
      1 rhombic prism
      7 dihedra
      1 parallelohedron
      2 monohedra

    4
    4
    2
    2
    1

    42

      mm2

    dual J81

    10 rhombic pyramids
      1 rhombic prism
      3 dihedra

    4
    4
    2

    50

    J82

    15 dihedra
    12 monohedra

    2
    1

    42

    m

    dual J82

    23 parallelohedra
     4  monohedra

    2
    1

    50

    J83

     1 ditrigonal pyramid
     8 trigonal pyramids
     2 monohedra

    6
    3
    1

    32

    3m

    dual J83

     6 ditrigonal pyramids
     3 trigonal pyramids

    6
    3

    45

    The duals of the Johnson's polyhedra derived from the RID are listed in Figure 75, in the same sequence of the Johnson's polyhedra reported in Figure 10, namely:

  • duals of isomers of the RID, obtained by rotation of cupolas of the RID (Fig. 75a)
  • duals of polyhedra obtained by the removal of cupolas from the RID (Fig. 75b)
  • duals of polyhedra obtained by the simultaneous removal and rotation of different cupolas of the RID (Fig. 75c).
    (clicking on the following twelve images one can visualize the corresponding VRML files)
    FIG. 75a

    Dual of gyrate RID (J72)

    Dual of parabigyrate RID (J73)

    Dual of metabigyrate RID (J74)

    Dual of trigyrate RID (J75)

    FIG. 75b

    Dual of diminished RID (J76)

    Dual of parabidiminished RID (J80)

    Dual of metabidiminished RID (J81)

    Dual of tridiminished RID (J83)

    FIG. 75c

    Dual of paragyrate diminished RID (J77)

    Dual of metagyrate diminished RID (J78)

    Dual of bigyrate diminished RID (J79)

    Dual of gyrate bidiminished RID (J82)

    Fig. 75 - Duals of all the Johnson's polyhedra (except the pentagonal cupola) deriving from the RID.

    A dynamic comparison of the duals of the Johnson polyedra deriving from the RID, viewed along the same direction, is shown in the next image.

     Animation showing the dual of the RID and the duals of the Johnson's polyhedra derived from the RID by the rotation and/or removal of its pentagonal cupolas.

    Conclusions

        The procedure leading to twelve Johnson's polyhedra, by rotation and/or removal of up to three 5-fold cupolas from a rhomb-icosidodecahedron, has been described in detail.
    When a cupola is removed from the RID, eleven faces (one pentagonal, five triangular and five square faces) are replaced by a decagonal face: consequently, the number of faces decreases from sixty-two to fifty-two, forty-two or thirty-two, depending on the number of cupolas eliminated.
    Both rotation and removal of cupolas from the RID imply a change of the symmetry: it decreases from m 3 5 in the RID to:

  • 5 m in J73-J80
  • 5 m in J72-J76-J77 (and J5)
  • 3 m in J75-J83
  • mm2 in J74-J81
  • m in J78-J79-J82

    The same symmetry of each Johnson's polyhedron characterizes also its dual.
    The faces of the duals of the Johnson's polyhedra may include, in addition to the deltoidal (or kite-shaped) faces characterizing the deltoidal hexecontahedron, dual of the RID:
  • sets of ten triangular vertex-sharing faces, deriving from the elimination of each 5-fold cupola
  • rings of ten 4-sided (but non-deltoidal) faces deriving from the rotation of each 5-fold cupola.
    The number of faces of the duals of the Johnson's polyhedra decreases from sixty (when the cupolas of the corresponding Johnson's polyhedra are not removed but only rotated) to fifty-five, fifty or forty-five, depending, also in this case, on the number of cupolas removed.
    In addition, the single forms in which both the Johnson's polyhedra and the corresponding duals can be decomposed, relatively to the point group which they belong to, are listed.
     

    References and links

    1)  Peter R. Cromwell
    Polyhedra, Cambridge University Press, 1997

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

    3) http://www.mi.sanu.ac.rs/vismath/zefiro2009/___pseudo_RCO_Zefiro-Ardigo'.htm

    4) http://www.mi.sanu.ac.rs/vismath/zefirocorrection/__Zefiro-Ardigo'_icosahedral_polyhedra_updating.htm

    5) http://www.georgehart.com/virtual-polyhedra/archimedean-duals-info.html


    Appendix A

    Labelling of the faces of the5-fold cupolas, aligned along the [τ 01] direction, belonging to the RID (on the left) or to the gyrate RID (on the right)p = pentagonal face, Si = square faces, ti = triangular faces. The different orientation of each face, except p, derives from the 180 rotation of the cupola around its [τ 01] axis.

     Indices of the corresponding pentagonal, square and triangular faces of the RID and J72

    RID J72
       (τ01)  p   (τ01)
     S1  (100)  S1  (102)
     S2  (τ  1  1/τ)  S2  (5τ-7 -3τ+4 1)
     S3  (τ  -1  1/τ)  S3  (5τ-7  3τ-4 1)
     S4  (1  1/τ  τ)  S4  (7τ+4 -5  3-τ)
     S5  (1 -1/τ  τ)  S5  (7τ+4  5  3-τ)
     t1   (1/τ 0 τ)  t1   (5τ+2  0  -1)
     t2    (1 -1 1)  t2   (3  2τ-1 1)
     t3   (1  1 1)  t3   (3 -2τ+1 1)
     t4   (τ  -1/τ  0)  t4   (1  3τ-4 2)
     t5   (τ  1/τ  0)  t5   (1 -3τ+4 2)


    Appendix B

    A duality relation subsists among the five labelled kite-shaped faces belonging to the dual of the RID (on the left) and the five vertices of the pentagonal face, perpendicular to the [τ 01] direction, of the RID . As a consequence of the 180 rotation, occurring in J72, of the fifteen vertices of the 5-fold cupola aligned along  [τ 01] , in the dual of J72 the relative dual faces rotate accordingly (on the right).
    The final result is a 180 rotation of  the five labelled faces, whereas the contiguous ten faces (making a ring) apparently do not rotate, being 180 a multiple of 360 /10. The different shape of the ten faces in the dual of J72 is due solely to the intersection with the 180 rotated five faces.

     Indices of ki = kite-shaped faces in the duals of the RID and (after a 180 rotation) of J72

    DUAL of the RID DUAL of J72
     k1  (τ 0  τ +1/τ)  k1  (5τ-2  0 1)
     k2  (2 1 τ)  k2  (6τ+2 -5  7τ-6)
     k3  (2 -1 τ)  k3  (6τ+2  5  7τ-6)
     k4  2  1/τ  1/τ)  k4  (8τ+1 -5  6τ+7)
     k5  2  -1/τ  1/τ)  k5  (8τ+1  5  6τ+7)

  • Appendix C

    Geometrical features of the faces characterizing the duals of the Johnson's polyhedra derived from the RID

    The geometrical features of the faces characterizing the duals of the Johnson's polyhedra derived from the RID can be determined starting from the dual of the diminished rhomb-icosidodecahedron (J76), having fifty-five faces: in fact it numbers forty-five of the sixty kite-shaped faces present in the dual of the RID, and, in substitution of the other fifteen faces, ten triangular 10-fold faces forming a pyramid.
    The truncation of such pyramid, in correspondence to its apex, by a flatter 5-fold pyramid (having kite-shaped faces) leads to different results depending on the orientation of this 5-fold pyramid.
    If the faces of the 5-fold pyramid are orientated as the corresponding faces in the dual of the RID,  the truncation of the ten triangular faces leads to as many kite-shaped faces: consequently, the sixty congruent kite-shaped faces resulting altogether constitute the deltoidal hexecontahedron, dual of the RID.
    The truncation of the 10-fold pyramid (having triangular faces), with the same 5-fold pyramid, but 180 rotated along its axis, gives as result ten 4-sided faces which, together with the other fifty faces, constitute the dual of J72.

    Animation showing the truncation of the triangular faces of the 10-fold pyramid present in the dual of J76 by the kite-shaped faces of a flatter 5-fold pyramid: it gives rise to the dual of the RID or, after a 180 rotation along its axis, to the dual of J72.


    dual RID

    dual J76

    dual J72

    AB = BC=1

    AD = CD = 0.6496 = 3/(
    τ+3)

     

     

    AB = 1

    AE = 2.1708 = 3(τ+2)/5

    BE = 2.3416 = 2(+1)/5

     

    AB = 1

    FG = 0.6496 = 3/(τ+3)

    AG = 0.8292 = 3/(τ+2)

    BF = 0.8204 = (6τ+4)/(+7)

    The set of four values of the angles in the kite-shaped and in the 4-sided faces are identical, but their sequence is different: in the kite-shaped faces belonging to the dual of the RID the two equal angles (having the intermediate value) are opposite each other, whereas they are contiguous in the 4-sided faces belonging to the dual of J72.
    Two angles of the triangular faces belonging to the dual of J76 are equal to the corresponding angles present in the faces of both duals of J72 and RID, whereas the third angle of the triangular faces has a smaller value.

    A side having the same length of the two long sides of the kites which characterize the dual of the RID is also present both in the 4-sided faces of the dual of J72 and in the triangular faces of the duals of J76; conversely, a side having the same length of the two short sides of the kites is present only in the 4-sided faces.
    In the dual of J72 the other two sides of the 4-sided faces have lengths intermediate between the lenghts of the first two sides, equal to the sidelengths of the kites, whereas in the dual of J76 the length of the other two sides of the triangular faces more than double the length of the first side, having the same length of the long side of the kites.