Regular-shaped faces in solids with cubic symmetry derived from truncation processes by rhomb-dodecahedron or cube

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


Notes
  • Tested with Internet Explorer 8.0, Mozilla Firefox 6.0, Opera 11.50, Google Chrome 11.0 and Safari 5 at 1024x768 and 1280x1024 pixels
  • All the images were created by SHAPE 7.0, a software program for drawing the morphology of crystals, distributed by Shape Software

INTRODUCTION
In a previous work, describing the truncation of Platonic and Archimedean polyhedra with cubic or icosahedral symmetry [1], it has been pointed out that, during each truncation process, different faces of the resulting solid can sequentially acquire the shape of regular polygons.
Among such peculiar solids, in the present work the ones with cubic symmetry will be described, together with the relative duals, the solids resulting from the intersection of each couple of dual polyhedra and, finally, the duals of these last solids.
In addition, the conditions leading to solids which include different kinds of regular-shaped faces, starting from generic non Archimedean polyhedra (truncated cubes, truncated octahedra or truncated tetrahedra), will be pointed out.
 
SOLIDS INCLUDING REGULAR-SHAPED FACES DERIVED FROM THE INTERSECTION OF CUBE, OCTAHEDRON AND RHOMB-DODECAHEDRON
If the positions of their barycenter coincide, the intersection of a cube and an octahedron gives rise to a composite polyhedron when the ratio between the distances from the center of the two solids of the faces belonging to cube and octahedron ranges in the interval:
3/3 < dcube /doctahedron < √3
In correspondence of the ratio dcube /doctahedron = √3/2 one obtains an Archimedean solid, the cuboctahedron, in which the faces consist of six squares and eight equilateral triangles.
The dual of the cuboctahedron is the Catalan rhomb-dodecahedron (RD): when the value of the central distance dRD of its twelve rhombic faces becomes equal to 1, such faces are tangent to the vertices of a cuboctahedron in which the central distances of the faces of cube and octahedron are dcube = √2/2 and doctahedron = √2/√3 respectively.
When dRD = 1, the faces of the rhomb-dodecahedron are also tangent to twelve edges of each truncated cube in which:
dcube = √2/2  and  √2/√3 < doctahedron<√3/√2
and to twelve edges of each truncated octahedron in which:
2/2 < dcube <√2  and doctahedron = √2/√3
In particular, as shown in Fig.1:
  • dcube = 2√2/3 and  doctahedron = √2/√3  in case of the Archimedean truncated octahedron (AtO), including regular hexagonal faces
  • dcube = √2/2 and  doctahedron = (√2+1)/√6  in case of the Archimedean truncated cube (AtC), including regular octagonal faces

  • Fig.1a) View, normal to its (111) regular hexagonal face, of the Archimedean truncated octahedron (AtO) Fig.1b) View, normal to its (100) regular octagonal face, of the Archimedean truncated cube (AtC)
    Fig.1

    If the distance of the truncating rhomb-dodecahedron decreases from the value dRD = 1, one can obtain:
    From the cuboctahedron
  • a polyhedron with regular octagonal faces of cube when dRD = (√2+2)/4
  • a polyhedron with regular hexagonal faces of octahedron when dRD = 5/6
  • a polyhedron with equilateral triangular faces of octahedron and square faces of cube when dRD = 3/4
    From the Archimedean truncated octahedron
  • a polyhedron with regular octagonal faces of cube when dRD = (√2+4)/6
  • a polyhedron with equilateral triangular faces of octahedron and square faces of cube when dRD = 5/6
    From the Archimedean truncated cube
  • a polyhedron with regular hexagonal faces of octahedron when dRD = (√2+4)/6
  • a polyhedron with equilateral triangular faces of octahedron and square faces of cube when dRD = (√2+2)/4

  • The previous data are resumed in the following Table A.
     

    Truncation by a rhomb-dodecahedron (RD) of the cuboctahedron (CO) and the Archimedean truncated octahedron (AtO) and truncated cube (AtC)

      CO AtO AtC
    dcube /doctahedron 3/2 2/√3 6-√3
    Indices of the Catalan dual {110} {210} {11 √2-1}
    If dRD =1, the rhomb-dodecahedron is tangent
    to CO, AtO and AtC, respectively,
    in coincidence with the following values:
     
    dcube 2/2 2√2/3 2/2
    doctahedron 2/√3 2/√3 (√2+1)/√6
     

    Vertex-transitive solids
    with regular octagonal faces

     
    dRD (2+√2)/4 = 0.8535 (4+√2)/6 = 0.9024  
    Indices of the dual hexakis-octahedra {√2  1 √2-1} {2√2  1 √2-1}
    In each dual the eight dihedral angles
    between contiguous faces sharing a vertex
    along [100] are equal and measure:
    26.899 15.722
     

    Vertex-transitive solids
    with regular hexagonal faces

     
    dRD 5/6 = 0.8333   (4+√2)/6 = 0.9024
    Indices of the dual hexakis-octahedra {321} {3 √2+1 2√2-1}
    In each dual the six dihedral angles
    between contiguous faces sharing a vertex
    along [111] are equal and measure:
    21.787 11.152
     

    Vertex-transitive solids with square
    faces and equilateral triangular faces

     
    dRD 3/4 5/6 = 0.8333 (2+√2)/4 = 0.8535
    Indices of the dual deltoid-icositetrahedra {211} {411} {√2 11}
    In each dual, the three dihedral angles
    between contiguous faces sharing a vertex
    along [111] are equal and measure:
    33.557 60 16.842
    In each dual, the four dihedral angles
    between contiguous faces sharing a vertex
    along [100] are equal and measure:
    48.190 27.266 60
     
    Zone axis of the dual forms  
    [uvw] such as hu+kv+lw=0 [1 1 1] [1 2 2] [√2 1 1]
    Table A

    As evidenced also by the background color of the cells in the previous table, one obtains vertex-transitive polyhedra [2] including faces with different regular shapes in correspondence of equal values of the central distance dRD of the rhomb-dodecahedron truncating couples of Archimedean solids.
     
    The intersection of cube, octahedron and rhomb-dodecahedron can generate the Archimedean rhomb-cuboctahedron (RCO), in which each vertex is shared by an equilateral triangular face of octahedron and three equidimensional square faces, one coming from the cube and two from the rhomb-dodecahedron, when the values of the ratio of the central distances are:
    dcube /dRD = 1 and  dcube /doctahedron = dRD /doctahedron = √3/(2√2 -1)
    Assigning to the central distance of the rhomb-dodecahedron the value dRD = 1, it follows that:
     dcube = 1 and  doctahedron = (2√2 -1)/√3 = 1.056
    Letting unchanged the ratio dcube /doctahedron and increasing the value of dRD:
  • when dRD = (3√2-2)/2 = 1.121, the solid obtained includes regular octagonal faces of cube
  • when dRD = (√2+2)/3 = 1.138, the solid obtained includes regular hexagonal faces of octahedron
  • when dRD = (4-√2)/2= 1.293, one obtains a (non Archimedean) truncated octahedron, with the faces of the rhomb-dodecahedron tangent to twelve of its edges.

  • The other Archimedean solid which can be obtained by the intersection of cube, octahedron and rhomb-dodecahedron is the Archimedean truncated cuboctahedron (tCO): it includes regular octagonal faces of cube, regular hexagonal faces of octahedron and square faces of rhomb-dodecahedron.
    In this case the values of the ratio of the central distances are:
    dcube /doctahedron = (3-√2)/√3 = 0.916 and  dRD /doctahedron = (2√2 -1)/√3 = 1.056
    (As one can see, in RCO and tCO the ratio dRD /doctahedron assumes reciprocal values: √3/(2√2 -1) and (2√2 -1)/√3 respectively)
    Assigning also in this case the value dRD = 1 to the central distance of the rhomb-dodecahedron, it follows that:
    doctahedron = √3/(2√2 -1) = 0.947 and  dcube = (3-√2)/(2√2 -1) = (5√2 -1)/7 = 0.867
    Letting unchanged the ratio dcube /doctahedron and varying only dRD :
  • an increase of the value of dRD from 1 to 3/(4-√2) = 1.160 leads to a (non-Archimedean) truncated octahedron, with the faces of the rhomb-dodecahedron tangent to twelve of its edges
  • a decrease of the value of dRD from 1 to (11+√2)/14 = 0.887 leads to a vertex-transitive polyhedron in which each vertex is shared by a square face of cube, an equilateral triangular face of octahedron and two rectangular faces of rhomb-dodecahedron: its dual is the deltoid-icositetrahedron {3√2-2 1 1} (whereas the rather similar Catalan deltoid-icositetrahedron {√2+1 1 1} is the dual of RCO)

  • The data relative to the solids deriving from RCO and tCO are resumed in Table B.

    Archimedean truncated cuboctahedron (tCO), rhomb-cuboctahedron (RCO) and solids including regular faces, obtained by varying the central distance of the intersecting rhomb-dodecahedron

      RCO tCO
    dcube /doctahedron 3/(2√2-1) (3-√2)/√3
    dRD /doctahedron 3/(2√2-1) (2√2-1)/√3
    If dRD =1:  
    dcube 1  (5√2-1)/7
    doctahedron (2√2-1)/√3 3/(2√2-1)
    Indices of the Catalan duals {√2+1 1 1}

    {3-√2  1 √2-1}

    In each Catalan solid, the dihedral angles between each couple of contiguous faces are equal and measure: 41.882 24.918
     

    Solids including regular octagonal faces of cube

     
    dRD (3√2-2)/2 1
    Indices of the dual hexakis-octahedra {√2+1 √2  2-√2} {3-√2  1 √2-1}
    In each dual, the eight dihedral angles between contiguous faces sharing a vertex along [100] are equal and measure:
    23.650 24.918
     

    Solids including regular hexagonal faces of octahedron

     
    dRD (√2+2)/3

    1

    Indices of the dual hexakis-octahedra {3(√2+1)  3+√2  3-√2}

    {3-√2  1 √2-1}

    In each dual, the six dihedral angles between contiguous faces sharing a vertex along [111] are equal and measure:
    26.804 24.918
     

    Solids including both square faces of cube and equilateral triangular faces of octahedron

     
    dRD 1 (11+√2)/14
    Indices of the dual deltoid-icositetrahedra {√2+1 1 1} {3√2-2 1 1}
    In each dual, the three dihedral angles between contiguous faces sharing a vertex along [111] are equal and measure:
    41.882 38.709
    In each dual, the four dihedral angles between contiguous faces sharing a vertex along [100] are equal and measure:
    41.882 44.317
     

    Non-Archimedean truncated octahedra deriving from RCO and tCO by increasing the value of dRD

     
    dRD (4-√2)/2 3/(4-√2)
    Indices of the dual tetrakis-hexahedra {√2+1 2 0} {3-√22 0}
    In each dual, the four dihedral angles between contiguous faces sharing a vertex along [100] are equal and measure:
    53.629 56.151
     
    Zone axis of the dual forms  
    [uvw] such as hu+kv+lw=0 [2(√2-1) 1 1] [√22-3 √2-3]
    Table B

    Fig.2 shows the views, along the [111] direction, of the solids including regular hexagonal faces (as the Archimedean truncated octahedron already shown in Fig.1, on the left), which can be obtained by the intersection of cube, octahedron and rhomb-dodecahedron.


    View along the [111] direction of cubic forms including regular hexagonal faces derived from the intersection of Archimedean polyhedra with a rhomb-dodecahedron, compared with the Archimedean truncated cuboctahedron

    Archimedean truncated cuboctahedron (tCO)

    Polyhedron with regular hexagonal faces derived from the rhomb-cuboctahedron (RCO) by varying the central distance of the rhomb-dodecahedron but letting unchanged the ratio dcube/doctahedron

    Polyhedron including regular hexagonal faces derived from the truncation of the Archimedean cuboctahedron (CO) by a rhomb-dodecahedron

    Polyhedron including regular hexagonal faces derived from the truncation of the Archimedean truncated cube (AtC) by a rhomb-dodecahedron

    Fig.2

    Fig.3 shows the views, along the [100] direction, of the solids including regular octagonal faces (as the Archimedean truncated cube already shown in Fig.1, on the right), which can be obtained by the intersection of cube, octahedron and rhomb-dodecahedron.

    View along the [100] direction of cubic forms, derived from the intersection of Archimedean solids with a rhomb-dodecahedron and including regular octagonal faces of cube, compared with the Archimedean truncated cuboctahedron

    Archimedean truncated cuboctahedron (tCO)

    Polyhedron including regular octagonal faces derived from the rhomb-cuboctahedron (RCO) by varying the central distance of the rhomb-dodecahedron but letting unchanged the ratio dcube/doctahedron

    Polyhedron including regular octagonal faces, derived from the truncation of the cuboctahedron (CO) by a rhomb-dodecahedron

    Polyhedron including regular octagonal faces, derived from the truncation of the Archimedean truncated octahedron (AtO) by a rhomb-dodecahedron

    Fig.3

    In Fig.4 the Archimedean truncated cuboctahedron (tCO) is compared to the rather similar solids which include regular faces having the shape of hexagons or octagons, obtained starting from two Archimedean solids, the cuboctahedron (CO) and the rhomb-cuboctahedron (RCO), by a proper variation of the central distance dRD of the rhomb-dodecahedron.
     

    Truncated cuboctahedron (tCO) compared with the four similar polyhedra, including regular hexagonal or octagonal faces, obtained by intersecting the rhomb-cuboctahedron (RCO) and the cuboctahedron (CO) with a rhomb-dodecahedron

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

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

    Polyhedron derived from CO, including regular octagonal faces of cube
    Polyhedron derived from CO, including regular hexagonal faces of octahedron
    Fig.4

    The hexakis-octahedra dual of the solids reported in Fig.4 are shown in the following Fig.5.

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

    Hexakis-octahedron {3(√2+1) 3+√2 3-√2} dual of the solid, derived from RCO, including regular hexagonal faces.
    The six dihedral angles between each couple of contiguous faces, sharing a vertex along the [111] axis, measure 26.804

    Hexakis-octahedron {√2+1 √2 2-√2} dual of the solid, derived from RCO, including regular octagonal faces.
    The eight dihedral angles between each couple of contiguous faces, sharing a vertex along the [100] axis, measure 23.650

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

    Hexakis-octahedron {√2 1 √2-1} dual of the solid, derived from CO, including regular octagonal faces.
    The eight dihedral angles between each couple of contiguous faces, sharing a vertex along the [100] axis, measure 26.899

    Hexakis-octahedron {321} dual of the solid, derived from CO, including regular hexagonal faces.
    The six dihedral angles between each couple of contiguous faces, sharing a vertex along the [111] axis, measure 21.787

    Fig.5
     
    In order to highlight the small differences among the images of the solids shown in Fig. 4 and Fig. 5, the corresponding animated sequences have been reported in Fig.6.

    Animated sequence highlighting the differences between the truncated cuboctahedron and the solids, including hexagonal or octagonal regular faces, derived from the cuboctahedron (CO) and the rhomb-cuboctahedron (RCO).

    Animated sequence highlighting the small differences between the Catalan hexakis-octahedron, dual of tCO, and the other hexakis-octahedra, which are the duals of solids, including hexagonal or octagonal regular faces, derived from CO and RCO.

    Fig.6
    As already shown in a previous paper [3], the intersection of the Archimedean truncated octahedron with its dual, the Catalan hexakis-octahedron {3-√2 1 √2-1}, can generate an interesting solid, in turn including regular octagonal faces of cube, regular hexagonal faces of octahedron and square faces of rhomb-dodecahedron as the Archimedean tCO, whereas the faces coming from the dual hexakis-octahedron have the shape of scalene triangles.
    In Fig.7 such solid is compared with four solids resulting from the intersection of the couples of dual polyhedra, derived from CO and RCO, which have been reported in Fig.4 and Fig.5: also these solids include faces with hexagonal or octagonal regular shape.

    Comparison between the intersection of tCO with the dual Catalan hexakis-octahedron and the intersection with the relative duals of the polyhedra, derived from the Archimedean CO and RCO, which include hexagonal or octagonal regular faces

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

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

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

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

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

    Fig.7

    The solid resulting from the intersection of tCO with its dual, the Catalan hexakis-octahedron, is clearly not vertex-transitive, but its vertices are all equidistant from the barycenter.
    Therefore a further dual of such solid includes faces which are different but equidistant from the barycenter; it is interesting to notice that, in it, the following dihedral angles are egual:
  • the 6 dihedral angles between contiguous faces sharing a vertex along [111]
  • the 8 dihedral angles between contiguous faces sharing a vertex along [100]
  • the 4 dihedral angles between contiguous faces sharing a vertex along [110]
  • In Fig.8 it is compared with four similar solids, derived from CO and RCO, which are the duals of the solids reported in Fig.7.
     

    Duals of the polyhedra obtained from the intersection between solids including regular faces derived from Archimedean polyhedra and the relative duals

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

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

    Solid dual of the intersection between the Archimedean truncated octahedron and the dual Catalan hexakis-octahedron, made of the deltoid-icositetrahedron {3√2-2 1 1}, the triakis-octahedron {3+√2  3+√2  √2} and the tetrakis-cube {3-√2  1 0} In it:
  • the 6 dihedral angles between contiguous faces sharing a vertex along [111] measure 22.062
  • the 8 dihedral angles between contiguous faces sharing a vertex along [100] measure 23.556
  • the 4 dihedral angles between contiguous faces sharing a vertex along [110] measure 17.976
  • Solid derived from CO, in which all the eight dihedral angles between each couple of contiguous faces sharing a vertex along [100] measure 25.529
    Solid derived from CO, in which all the six dihedral angles between each couple of contiguous faces sharing a vertex along [111] measure 19.188
    Fig.8

    In order to highlight the differences among the images of the solids shown in Fig.7 and Fig.8, the corresponding animated sequences have been reported in Fig.9.

    Animated sequences highlighting the differences which characterize the solids reported in Fig.7 and Fig.8

    Fig.9

    The first row of Fig.10 shows, in the center, the Archimedean truncated cuboctahedron (tCO) compared with the solids including regular hexagonal or octagonal faces, derived from the intersection with a rhomb-dodecahedron of the Archimedean truncated cube (AtC) and truncated octahedron (AtO), respectively.
    The relative duals are reported in the second row, whereas the third row shows the polyhedra, including regular faces, which result from the intersections of each dual couple of solids reported in the first two rows; finally the duals of the solids belonging to the third row are shown in the fourth row.

    Comparison between the truncated cuboctahedron and the polyhedra, including regular faces, derived from the intersection with a rhomb-dodecahedron (RD) of the Archimedean truncated cube (AtC) and truncated octahedron (AtO); in addition, their duals and other related composite forms are shown.

    Polyhedron with regular hexagonal faces derived from the truncation by RD of the Archimedean truncated cube (AtC) Archimedean truncated cuboctahedron Polyhedron with regular octagonal faces derived from the truncation by RD of the Archimedean truncated octahedron (AtO)
    Hexakis-octahedron {3√2+1 2√2-1} dual of the solid with regular hexagonal faces derived from the Archimedean truncated cube (AtC)
    Hexakis-octahedron {3-√2 1 √2-1}
    It is the Catalan dual of the Archimedean truncated cuboctahedron
    Hexakis-octahedron {2√2  1 √2-1} dual of the solid with regular octagonal faces derived from the Archimedean truncated octahedron (AtO)
    Solid including regular hexagonal faces, obtained from the intersection of the polyhedron derived from AtC with its dual
    Solid resulting from the intersection with its Catalan dual of the Archimedean truncated cuboctahedron
    Solid including regular octagonal faces, obtained from the intersection of the polyhedron derived from AtO with its dual
    Solid derived from AtC, in which all the dihedral angles between the six couples of contiguous faces sharing a vertex along [111] measure 9.70
    Solid dual of the intersection between the Archimedean truncated cuboctahedron and the dual Catalan hexakis-octahedron (the values of the dihedral angles have already been reported in Fig.8)
    Solid derived from AtO, in which all the dihedral angles between the eight couples of contiguous faces sharing a vertex along [100] measure 14.657
    Fig.10

    Solids including regular-shaped faces can be obtained also by the intersection with a rhomb-dodecahedron (RD) of whichever non-Archimedean truncated octahedron (tO) or truncated cube (tC). Starting from any situation in which RD is tangent to twelve edges of tO or tC, by decreasing its central distance dRD one obtains always, in the early stages, vertex-transitive polyhedra, whose duals are hexakis-octahedra.
    Afterwards, when dRD decreases to a proper value, the truncation of each tO and tC by RD leads to a different vertex-transitive polyhedron (whose dual is a deltoid-icositetrahedron), in which each vertex is shared by a square, an equilateral triangle and, usually, two rectangles which become squares only in case of RCO.
    Concerning the possibility to get, at intermediate steps of the truncation process by RD, other solids including regular faces, one must distinguish among different ranges of the ratio dcube /doctahedron:
  • when 2√3/3 ≤ dcube/doctahedron< √3, the intersection of each truncated octahedron with RD can generate a solid including regular octagonal faces (the ratio dcube /doctahedron= 2√3/3 is relative to the Archimedean truncated octahedron, which in turn includes regular hexagonal faces of octahedron)
  • when √3/(2√2-1) < dcube /doctahedron< 2√3/3, the intersection of each truncated octahedron with RD generates, in sequence, two solids including hexagonal or octagonal regular faces
  • when (3-√2)√3/3 < dcube /doctahedron ≤ √3/(2√2-1), three are the solids including regular faces which can be sequentially generated from the intersection of each truncated octahedron with RD: the two solids including regular hexagonal faces of octahedron or regular octagonal faces of cube are followed by a solid including square faces of rhomb-dodecahedron (if dcube/doctahedron= √3/(2√2-1), such solid is the Archimedean rhomb-cuboctahedron which, together with square faces of rhomb-dodecahedron, includes equilateral triangular faces of octahedron and square faces of cube too)
  • when dcube /doctahedron= (3-√2)√3/3, an unique value dRD, relative to the central distance of the rhomb-dodecahedron intersecting the truncated octahedron, leads to the Archimedean truncated cuboctahedron, which include simultaneously regular faces of three kinds, having the shape of hexagons, octagons and squares
  • when √3/2 < dcube /doctahedron < (3-√2)√3/3, the three solids including regular faces are obtained in an inverse order, since the solid including square faces of rhomb-dodecahedron comes before the solid with regular octagonal faces of cube, followed by the solid with regular hexagonal faces of octahedron
  • when √3/(√2+1) < dcube /doctahedron ≤ √3/2, from the intersection with RD of each truncated cube (becoming a cuboctahedron, if dcube /doctahedron = √3/2) at first one obtains a solid with regular octagonal faces of cube, and then a solid with regular hexagonal faces of octahedron
  • when √3/3 < dcube /doctahedron ≤ √3/(√2+1), the intersection of each truncated cube with RD can generate a solid including regular hexagonal faces of octahedron (the ratio dcube /doctahedron = √3/(√2+1) is relative to the Archimedean truncated cube, which in turn includes regular octagonal faces of cube)
  • The sequences of solids including regular-shaped faces, obtained by the intersection with RD of non Archimedean truncated octahedra or truncated cubes belonging to the different intervals just defined, are visualized in Fig.11.

    Solids which include faces having the shape of regular polygons, obtained by the intersection with the rhomb-dodecahedron of generic non-Archimedean truncated octahedra and truncated cubes, compared with the Archimedean truncated cuboctahedron

    Truncated octahedra and truncated cubes
    Vertex-transitive solids which include regular faces having hexagonal, octagonal or square shape
    Vertex-transitive solids including square faces together with triangular equilateral faces

    2√3/3 < dcube /doctahedron < √3

    If  dcube = 5√2/6  and  doctahedron = √2/√3  then:

    dRD = 1

    dRD = 0.951: octagonal regular faces dRD = 0.917
     

    3/(2√2-1) < dcube /doctahedron < 2√3/3

    If  dcube = 3√2/5  and  doctahedron = √2/√3  then:

    dRD = 1

    dRD = 0.933: hexagonal regular faces
    dRD = 0.883: octagonal regular faces

    dRD = 0.8

     

    (3-√2)√3/3 < dcube /doctahedron < √3/(2√2-1)

    If dcube = (12√2-1)/21  and  doctahedron = √2/√3  then:

     dRD = 1

    dRD = 0.871 : hexagonal regular faces
    dRD = 0.865 : octagonal regular faces
    dRD = 0.818 : square faces

    dRD = 0.769

     

    dcube /doctahedron = (3-√2)/√3

    dRD = 3/(4-√2) =1.160 implies tangency of the rhomb-dodecahedron {110} to the truncated octahedron in which:
    doct = √3/(2√2-1),  dcube = (5√2-1)/7
    An Archimedean solid, the truncated cuboctahedron (tCO), results if also:
    doctahedron /dRD = √3/(2√2-1)
    It follows that, when dRD = 1:
    doct = √3/(2√2-1),  dcube = (5√2-1)/7
    When dRD = (√2+11)/14 = 0.887, each vertex is shared by four faces if:
    doct = √3/(2√2-1),  dcube = (5√2-1)/7
    Alternatively, dRD = 1 implies tangency to the truncated octahedron in which:
    doctahedron= √2/√3 and  dcube= √2(3-√2)/3
    The Archimedean tCO can be obtained when dRD = (4-√2)/3 = 0.862 if:
    doctahedron= √2/√3 and  dcube= √2(3-√2)/3
    When dRD= (6-√2)/6= 0.764, each vertex is shared by four faces if:
    doctahedron= √2/√3 and  dcube= √2(3-√2)/3
     

    3/2 < dcube /doctahedron < (3-√2)/√3

    If  dcube = (9√2-4)/12 and  doctahedron = √2/√3  then:

    dRD = 1

    dRD = 0.931 : square faces
    dRD = 0.858 : octagonal regular faces
    dRD = 0.848 : hexagonal regular faces

    dRD = 0.757

     

    3/(√2+1) < dcube /doctahedron < √3/2

    If  dcube = √2/2  and  doctahedron = (2/(2√2-1))√2/√3  then:

    dRD = 1

    dRD = 0.920 : octagonal regular faces
    dRD = 0.865 : hexagonal regular faces

    dRD = 0.797

     

    3/3 < dcube /doctahedron < √3/(√2+1)

    If  dcube = √2/2   and  doctahedron = √3/(3-√2)  then:

    dRD = 1

    dRD = 0.946 : hexagonal regular faces

    dRD = 0.919

    Fig.11

    In case of the solids including one or more regular faces, obtained from the intersection of the rhomb-dodecahedron with each polyhedron which derives from the reciprocal truncation of cube and octahedron, the relations holding between dRD and the couple of central distances dcube and doctahedron (as the ratio dcube/doctahedron varies in well definited ranges) are reported in Table C.

    Relations holding between the central distance of the truncating rhomb-dodecahedron and the central distances of cube and octahedron in case of each series of solids including regular faces

    Vertex-transitive solids including regular hexagonal faces of octahedron

    dRD = (doctahedron +√3 dcube )/√6  when the ratio dcube/doctahedron varies in the range √3/3 < dcube /doctahedron ≤ 2√3/3

    Vertex-transitive solids including regular octagonal faces of cube

    dRD = (√3 doctahedron + (√2-1) dcube)/2  when the ratio dcube/doctahedron varies in the range (√2-1)√3dcube /doctahedron<√3

    Vertex-transitive solids including square faces of rhomb-dodecahedron

    dRD = (√2+1) (√3doctahedron -√2 dcube)  when the ratio dcube/doctahedron varies in the range √3/2 < dcube /doctahedron <√3/(2√2-1)

    Vertex-transitive solids including square faces of cube and triangular equilateral faces of octahedron

    dRD = √2 (√3doctahedron + dcube )/4  when the ratio dcube/doctahedron varies in the range √3/3 < dcube /doctahedron <√3
    Table C

    Animated sequences, relative to the solids including the different regular faces, are shown in the following images, from Fig.12 to Fig.15.

    Set of vertex-transitive solids including regular hexagonal faces of octahedron

    Orthographic view of the sequence of solids, going from the Archimedean truncated octahedron to the cube and including also the Archimedean truncated cuboctahedron, characterized by regular hexagonal faces of octahedron

    View along the [111] 3-fold axis, normal to an hexagonal face of octahedron, of the sequence on the left

    Fig.12

    Set of vertex-transitive solids including regular octagonal faces of cube

    Orthographic view of the sequence of solids, going from the octahedron to the Archimedean truncated cube and including also the Archimederan truncated cuboctahedron, characterized by regular octagonal faces of cube

    View along the [100] 4-fold axis, normal to an octagonal face of cube, of the sequence on the left

    Fig.13

    Set of vertex-transitive solids including square faces of rhomb-dodecahedron

    Orthographic view of the sequence of solids, going from the cuboctahedron to the rhomb-cuboctahedron and including also the truncated cuboctahedron, characterized by square faces of rhomb-dodecahedron

    View along the [110] 2-fold axis, normal to a square face of rhomb-dodecahedron, of the sequence on the left

    Fig.14

    Set of vertex-transitive solids including square faces of cube and equilateral triangular faces of octahedron

    Orthographic view of the sequence of solids, going from the cube to the octahedron and including also the Archimedean rhomb-cuboctahedron, characterized by square faces of cube and equilateral triangular face of octahedron.

    View of the same sequence along the [100] 4-fold axis, normal to a square face of each solid of the series.

    View of the same sequence along the [110] 2-fold axis, normal to an usually rectangular face of each solid of the series.
    It is only in case of the Archimedean rhomb-cuboctahedron that all the rectangular faces becomes square.

    View of the same sequence along the [111] 3-fold axis, normal to an equilateral triangular face of each solid of the series.
    Fig.15

    SOLIDS INCLUDING REGULAR FACES GENERATED BY THE INTERSECTION WITH A CUBE OF EACH TRUNCATED TETRAHEDRON

    The intersection of two tetrahedra {111} and {111}, 90 rotated around a direction passing through the midpoints of two opposite edges, gives rise to a composite polyhedron when the ratio between the distances of the faces of the two tetrahedra from the barycenter ranges in the interval: 1/3 < d{111} /d{111} < 3.
    In correspondence of the ratio d{111} /d{111} = 1, from the intersection of the two tetrahedra one obtains a solid, identical to an octahedron only geometrically, since the octahedron is not compatible with the 43m symmetry (or Td according to the notation adopted by Schoenflies) which characterizes the tetrahedron.
    When 1/3 < d{111} /d{111} < 1, the intersection of the two tetrahedra originates solids consisting in the tetrahedron {111} truncated by the tetrahedron {111}, whereas solids consisting in the tetrahedron {111} truncated by the tetrahedron {111} are obtained when:
    1 < d{111} /d{111} < 3
    Since the two families of solids so obtained are equivalent by symmetry, it is sufficient to examine in detail only one of the two intervals.
    A cube, whose central distance has the value dcube = 1, is tangent to the vertices of the pseudo-octahedron, made of the two tetrahedra having faces equidistant from the center, when:  d{111} = d{111} = √3/3.
    Such cube is tangent also to six edges of each truncated tetrahedron in which:  d{111} = √3/3  and  √3/3 < d{111} <√3
    By decreasing the central distance of the cube from the value dcube = 1, in the early stages one obtains always vertex-transitive polyhedra, whose duals are hexakis-tetrahedra.
    By decreasing furtherly dcube to a value depending from the ratio d{111} /d{111}, the intersection of each truncated tetrahedron with the cube leads to a different vertex-transitive polyhedron (whose dual is a deltoid-dodecahedron), in which every vertex is shared by a rectangle and two equilateral triangles; the rectangles become squares and the sizes of all the equilateral triangles become equal in case of the pseudo cuboctahedron (whose dual is the rhomb-dodecahedron), which can be obtained, by the intersection with a cube, starting from the pseudo octahedron.
    Concerning the possibility to get other solids including regular faces at intermediate steps of the truncation process by cube, one must consider the different ranges of the ratio d{111} /d{111} between the central distances of the two tetrahedra.
     In particular:
  • when d{111} /d{111} = 1, by decreasing the value of dcube from 1 to 2/3, the pseudo octahedron becomes a pseudo Archimedean truncated octahedron, consisting of square and regular hexagonal faces (a further decrease to 1/2 leads to a pseudo cuboctahedron, made of square and triangular equilateral faces)
  • when 1< d{111} /d{111} < 5/3, two are the solids, including regular hexagonal faces of the {111} or {111} tetrahedron, obtained sequentially by proper decreases of the value of dcube
  • when d{111} /d{111} = 5/3, the intersection of the two tetrahedra originates an Archimedean truncated tetrahedron, which itself includes regular hexagonal faces belonging to the tetrahedron {111}; by decreasing the value of dcube from 1 to 7/9, the intersection of the Archimedean truncated tetrahedron with a cube generates a solid, in which the regular hexagonal faces belong to the tetrahedron {111}
  • when 5/3 < d{111} /d{111} < 3, a proper decrease of the value of dcube leads to a solid including hexagonal faces of the tetrahedron {111}
  •  
    The sequences of solids including faces which consist in regular polygons, obtained by the intersection with a cube of truncated tetrahedra belonging to the different intervals just defined, are visualized in Fig.16.

    Solids which include faces having the shape of regular polygons, obtained from the intersection of each truncated tetrahedron with a cube

    Pseudo octahedron resulting from the intersection of two tetrahedra having an equal central distance:
    d{111} = d{111} = √3/3
    Pseudo Archimedean truncated octahedron resulting from the intersection of the pseudo octahedron (on the left) with a cube when:
    dcube = 2/3
    Pseudo Archimedean cuboctahedron resulting from the intersection of the pseudo octahedron with a cube when:
    dcube = 1/2
     

    Truncated tetrahedron

    d{111} = 4√3/9;  d{111} = √3/3

    Solids with {111} or {111} faces having the shape of regular hexagons, as result of the intersection of the truncated tetrahedron (on the left) with a cube, whose faces are set sequentially at the following distances from the barycenter:
    dcube = 5/6  and   dcube = 13/18
    Solid (dual of a deltoid-dodecahedron), including {111} and {111} equilateral triangular faces, obtained when the central distance of the cube intersecting the truncated tetrahedron is:
    dcube = 7/12
     
    Archimedean truncated tetrahedron, including regular hexagonal faces of the tetrahedron {111}

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

    Truncated tetrahedron

    d{111} = 2√3/3;  d{111} = √3/3

    Solid with regular hexagonal faces of the {111} tetrahedron, resulting from the intersection of the truncated tetrahedron shown on the left with a cube when:
    dcube = 5/6
    Solid (dual of a deltoid-dodecahedron), including {111} and {111} equilateral triangular faces, obtained when the central distance of the cube which intersects the truncated tetrahedron is:
    dcube = 3/4
    Fig.16

    Concerning the following Fig.17, in the first row:
  • the left image shows the triakis-tetrahedron {311}, which is the Catalan dual of the Archimedean truncated tetrahedron (reported in the left image of the third row of Fig.16)
  • the right image shows the triakis-tetrahedron {311}, obtained by a 90 rotation around the [001] direction of the triakis-tetrahedron {311}
  • the central image shows a cube, dual of the pseudo octahedron (reported in the left image of the first row of Fig.16) resulting from the intersection of the two tetrahedra {111} and {111} when: d{111} /d{111}= 1.
    In the second row:
  • the left image shows the hexakis-tetrahedron {753}, which is the dual of the solid (reported in the central image of the third row of Fig.16) resulting from the intersection of the pseudo-octahedron with a cube when: dcube = 7/9
  • the right image shows the hexakis-tetrahedron {753}, obtained by a 90 rotation around the [001] direction of the hexakis-tetrahedron {753}
  • the central image shows the Catalan tetrakis-hexahedron {210}, dual of the pseudo Archimedean truncated octahedron (reported in the central image of the first row of Fig.16) resulting from the intersection of the pseudo-octahedron with a cube when: dcube = 2/3.
    In the third row:
  • the left image shows the deltoid-dodecahedron {221}, which is the dual of the solid (reported in the right image of the third row of Fig.16) resulting from the intersection of the pseudo-octahedron with a cube when: dcube = 2/3
  • the right image shows the deltoid-dodecahedron {221}, obtained by a 90 rotation around the [001] direction of the deltoid-dodecahedron {221}
  • the central image shows the rhomb-dodecahedron {110}, dual of the pseudo cuboctahedron (reported in the right image of the first row of Fig.16) resulting from the intersection of the pseudo-octahedron with a cube when: dcube = 1/2

  • From the comparison of the duals forms in Fig.17, it is rather evident that the cube may be considered the solid intermediate (with the halving of the number of faces) between the two 90 rotated triakis-tetrahedra, whereas the tetrakis-hexahedron {210} and the rhomb-dodecahedron {110} may be considered the solids intermediate (with an equal number of faces) between the two 90 rotated hexakis-tetrahedra and deltoid-dodecahedra, respectively.

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

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

    Hexakis-tetrahedron {753}: dual of the solid, including regular hexagonal faces, derived from the intersection of the Archimedean truncated tetrahedron (in which d{111}/d{111}= 5/3) with a cube, when dcube = 7/9

    Tetrakis-hexahedron {210}: dual of the pseudo Archimedean truncated octahedron resulting from the intersection of a pseudo-octahedron (obtained from two tetrahedra if: d{111}/d{111}= 1) with a cube, when dcube = 2/3

    Hexakis-tetrahedron {753}: dual of the solid including regular hexagonal faces derived from the intersection of the Archimedean truncated tetrahedron (in which d{111}/d{111}= 3/5) with a cube, when dcube = 7/9

     
    Deltoid-dodecahedron {221}: dual of the solid, including equilateral triangular and square faces, derived from the intersection with a cube (when dcube = 2/3) of the Archimedean truncated tetrahedron (in which: d{111} /d{111}= 5/3) .
    Rhomb-dodecahedron {110}: dual of the pseudo cuboctahedron resulting from the intersection of a pseudo-octahedron (obtained if: d{111}/d{111}= 1) with a cube, when dcube = 1/2.
    Deltoid-dodecahedron {221}: dual of the solid including regular triangular and square faces, derived from the intersection with a cube (when dcube = 2/3) of the Archimedean truncated tetrahedron (in which: d{111} /d{111}= 3/5).
    Fig.17

    Also in case of the solids including one or more regular faces, which derive from the intersection of two tetrahedra with a cube, the relations holding between dcube and the couple of central distances d{111} and d{111} of the two tetrahedra (as the ratio d{111}/d{111} varies in well definited ranges) have been found and reported in Table D.

    Relations holding between the central distance of the truncating cube and the central distances of the two tetrahedra in case of each series of solids including regular faces

    Vertex-transitive solids including regular hexagonal faces of the tetrahedron {111}

    dcube = √3(d{111}+ 3d{111})/6 when the ratio d{111}/d{111} varies in the range 3/5 ≤ d{111}/d{111} < 3

    Vertex-transitive solids including regular hexagonal faces of the tetrahedron {111}

    dcube = √3(d{111}+ 3d{111})/6 when the ratio d{111}/d{111} varies in the range 1/3 ≤ d{111}/d{111} < 5/3

    Vertex-transitive solids including triangular equilateral faces of both tetrahedra

    dcube = √3(d{111}+ d{111})/4 when the ratio d{111}/d{111} varies in the range 1/3 < d{111}/d{111} < 3
    Table D

    The left animated image of Fig.18 show all the truncated tetrahedra obtained when the ratio of the central distances d{111}/d{111} varies in the range: 1/3 < d{111}/d{111}< 3; the right image shows the corresponding vertex-transitive solids, including triangular equilateral faces of both tetrahedra, obtained from the intersection of each truncated tetrahedron with a cube, when dcube = √3(d{111}+ d{111})/4

    Fig.18

    The two series of vertex-transitive solids which include regular hexagonal faces of a tetrahedron, obtained by a proper intersection of each truncated tetrahedron with a cube, are shown in Fig.19. In particular:
  • (left image) when dcube = √3(d{111}+ 3d{111})/6, the intersection between the cube and each truncated tetrahedron, obtained as the ratio d{111}/d{111} varies in the range 1/3 < d{111}/d{111} ≤ 5/3, leads to a series of solids in which the faces having the shape of regular hexagons belong to the tetrahedron {111}
  • (right image) when dcube = √3(d{111}+ 3d{111})/6, the intersection between the cube and each truncated tetrahedron, obtained as the ratio d{111}/d{111} varies in the range 3/5 ≤ d{111}/d{111} < 3, leads to a series of solids in which the faces having the shape of regular hexagons belong to the tetrahedron {111}

  • Animated sequence relative to the solids characterized by regular hexagonal faces of the tetrahedron {111} (on the left) or {111} (on the right), deriving from the intersection with a cube of two series of truncated tetrahedra (obtained as the ratio d{111}/d{111} varies within proper ranges). In each sequence, the only frame in which the faces of both tetrahedra have a regular hexagonal shape concerns the pseudo Archimedean truncated octahedron, obtained when the central distances of the two tetrahedra are equal and dtetrahedra /dcube = √3/2.

    Fig.19

    LINKS and REFERENCES
    1. Zefiro L.
      Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra
      VisMath, volume 13, No. 2, 2011
    2. Cromwell P.
      Polyhedra
      Cambridge University Press, 1997, ISBN 9-521-55432-2
    3. Zefiro L., Ardigo' M.R.
      Platonic and Catalan polyhedra as archetypes of forms belonging to the cubic and icosahedral systems
      VisMath, volume 11, No. 2, 2009