Review of the alternative choices concerning face colouring of all the regular convex polyhedra and a pair of Catalan polyhedra, the rhombic dodecahedron and the rhombic triacontahedron

Livio Zefiro, DIP.TE.RIS., Universita' di Genova, Italy

TABLE of CONTENTS

  • Introduction
  • Tetrahedron
  • Cube
  • Octahedron
  • Icosahedron
  • Dodecahedron
  • Rhombic dodecahedron
  • Rhombic triacontahedron
  • Acknowledgements
  • Links
  • Suggested lectures
  • 4-coloured Tetrahedron 3-coloured Cube 2-coloured Octahedron 4-coloured Dodecahedron 5-coloured Icosahedron 4-coloured rhombic Dodecahedron 5-coloured rhombic Triacontahedron
    The names of the polyhedra appear positioning  the pointer of the mouse above each image
    INTRODUCTION

        
    Similarly to what happens in the realization of maps, where it is necessary to represent contiguous regions or states with different colours in order to distinguish them easily, also in the drawing of polyhedra it can be useful to colour adjacent faces differently, in particular when the polyhedra are rather complex.
    The most obvious colouring procedure assigns different colours to faces sharing an edge; a more restrictive condition can also prevent the sharing of vertices by equally coloured faces.

    The previous criteria are essentially aesthetical; conversely, if face colouring is used to represent some features of the current crystallographic symmetry of a polyhedron, its colouring will be different and sharing of edges may occur between faces having an equal colour and belonging to the same simple form.
    In the sequel, colouring of the five convex regular (or Platonic) polyhedra and of two interesting Catalan polyhedra, the rhombic dodecahedron and rhombic triacontahedron, will be discussed.

    Note: when the mouse hovers on underlined words or some images, the symbol of a pointer appears: if the web browser is set up to visualize VRML (Virtual Reality Modeling Language) files, by clicking with the left button of the mouse it should be possible to visualize in a new window a dynamic image, with which one can interact by rotating it, shifting it, changing its dimensions...
    A good Web3D visualizer can be found at the following address:
    http://www.parallelgraphics.com/products/cortona/

    The tetrahedron, consisting of four equilateral triangular faces, six edges and four vertices, is the only regular polyhedron with an equal number of faces and vertices: consequently, it is also the only autodual polyhedron (remember that duality implies the exchange of faces and vertices between two polyhedra, concerning both their number and their position).
    The images of these tetrahedra and most of the followings have been obtained by Shape 7.0, a program for drawing the morphology of crystals provided by Shape Software.

    The symmetry elements characterizing the tetrahedron are:

    • four 3-fold rotation axes, each one perpendicular to a face and passing through the opposite vertex
    • three 4-fold rotoinversion axes, each one passing through the middle points of a pair of opposite edges
    • six mirror planes, each one passing through both the edge shared by two faces and the middle point of the opposite edge shared by the other two faces split into 2 equal parts by the plane itself.
    Consequently, from a crystallographic point of view it belongs to the 43m point group (in the symbol of the point group, 4 indicates a 4-fold rotoinversion axis, 3 a 3-fold rotation axis and m a mirror plane).

    The cube, characterized by 6 square faces, 8 vertices and 12 edges, is the dual of the octahedron, in turn characterized by 8 triangular equilateral faces, 6 vertices and 12 edges (according to the well-known rule stating that in every polyhedron the sum of the number of faces and vertices corresponds to the number of edges increased by two).

    The two polyhedra share the same symmetry elements, characteristic of m3m crystallographic point group:

    • three 4-fold rotation axes, each one perpendicular to a pair of parallel faces of the cube and passing through two opposite vertices of the octahedron
    • three mirror planes perpendicular to these 4-fold rotation axes, named principal planes
    • four 3-fold rotation axes each one perpendicular to a pair of parallel faces of the octahedron and passing through two opposite vertices of the cube
    • six 2-fold axes passing through the middle points of opposite edges in both polyhedra
    • three mirror planes perpendicular to these 2-fold rotation axes, named  secondary planes
    • an inversion centre, as a consequence of the association of even rotation axes (both 2-fold and 4-fold) with planes placed orthogonally to them

    The polyhedron formed by twelve rhombic faces, perpendicular to the directions corresponding to the six two-fold axes previously mentioned, is named rhombic dodecahedron and belongs to the family of Catalan polyhedra, duals of semiregular Archimedean polyhedra. In particular, the rhombic dodecahedron is the dual of the cuboctahedron, a polyhedron with 14 faces (6 square and 8 equilateral triangular ones) given by a particular combination of two forms, a cube and an octahedron. Actually in the cuboctaedron every vertex is shared by two pairs of squares and equilateral triangles as a consequence of a proper ratio between the distances, from the centre of the cuboctahedron, of the faces belonging to cube and octahedron.

    Concerning the two remaining regular polyhedra, the icosahedron, characterized by 20 triangular faces, 12 vertices and 30 edges, is the dual of the dodecahedron, in turn characterized by 12 pentagonal faces, 20 vertices and 30 edges.

    These two polyhedra share the same symmetry elements, relative to the m35 icosahedral point group:
    • ten 3-fold axes, orthogonal to every couple of parallel faces in the icosahedron and passing through pairs of opposite vertices in the dodecahedron
    • six 5-fold axes, orthogonal to every couple of parallel faces in the dodecahedron and passing through pairs of opposite vertices in the icosahedron
    • fifteen 2-fold axes (which, together with the fifteen mirror planes orthogonal to them, imply the presence of an inversion centre) passing through the middle points of opposite edges in both polyhedra .
    (Note: in the representation of polyhedra with icosahedral symmetry,  the planes of symmetry haven't been drawn in order to avoid clutter)

    The polyhedron formed by 30 rhombic faces, placed in pairs perpendicular to each of the fifteen 2-fold axes present in both the m35 and the 235 icosahedral point groups, is named rhombic triacontahedron. Like the rhombic dodecahedron, it belongs to the family of Catalan polyhedra and is the dual of the icosi-dodecahedron, a semiregular Archimedean polyhedron having 32 faces, in turn given by a particular combination of two forms, an icosahedron and a dodecahedron. As a matter of fact, in the icosidodecahedron every vertex is shared by two pairs of pentagons and triangles, both regular polygons, as a consequence of a proper ratio between the distances, from the centre of the icosidodecahedron, of the faces belonging to the icosahedron and the dodecahedron.

    Incidentally, one can note that, in case of polyhedra made of different forms, where each face does not share any edge with other faces belonging to the same form (as it happens for example with the icosidodecahedron and the cuboctahedron, classified as quasi-regular polyhedra for this reason), in order to highlight the single faces it is sufficient to assign a different colour to each form of the polyhedron.

    Whereas in the rhombic dodecahedron the ratio between the lengths of the diagonals of each rhombic face is equal to √2, in the rhombic triacontahedron such ratio is equal to (√5+1)/2=1.61803..., the irrational number corresponding to the golden ratio.

    The following review of the schemes for colouring different polyhedra, starting from the notions contained in George Hart's page:
    http://www.georgehart.com/virtual-polyhedra/colorings.html, goes into further depth concerning the crystallographic facet of polyhedra colouring.

    TETRAHEDRON

    The easiest way to differentiate the 4 faces of a tetrahedron consists in assigning a different colour to each face.
    The maximum crystallographic symmetry of the tetrahedron, which coincides with the geometrical symmetry, corresponds to the 43m point group. On the other hand, any kind of symmetry would be absent if one take into account the different colouring of the faces.
    Introducing the n-fold chromatic symmetry operators, represented by a prime mark (n'), which relate equivalent faces ignoring their colours, the symbol of the chromatic point group describing the 4-coloured tetrahedron becomes: 4'3'm'.

    CUBE
    If the same colour is assigned to each pair of opposite parallel faces, one obtains a 3-coloured cube.
    In consequence of the symmetry decrease due to face colouring, the crystallographic point group from m3m becomes mmm, so that the 3-coloured cube can be described as a collection of three pinacoids; the symbol of the related chromatic point group is m3'm' since, unlike the chromatic secondary mirror planes m', the principal mirror planes m put in correspondence pairs of faces having an equal colour.

    OCTAHEDRON
    Analogously to the procedure applied to the cube, one can apply the same colour to each of the four pairs of opposite parallel faces of the octahedron, obtaining in this way a 4-coloured octahedron; also in this case  faces with an equal colour do not share any vertex. Since the point group of the 4-coloured octahedron is 1, the decrease of crystallographic symmetry due to face colouring is more drastic in the octahedron than in the cube: the octahedron can be described as a collection of four pinacoids. In turn, the symbol of the chromatic point group is m'3'm'.

    A second way of colouring the faces of the octahedron consists in assigning the same colour to each one of the two autodual tetrahedra in which the octahedron can be decomposed, thus obtaining a 2-coloured octahedron.
    The symmetry decrease consequent to this alternative colouring turns out to be rather limited, since the point group, firstly m3m, becomes 43m. On the other hand, the chromatic point group turns out to be m'3m since, whereas the chromatic principal  mirror planes m' reflect faces differently coloured, both 3-fold axes and secondary mirror planes m relate faces with an equal colour.

    ICOSAHEDRON

    In the icosahedron, one can identify twelve groups of five equilateral triangular faces that share one of the twelve vertices where a 5-fold axis passes. The most obvious procedure of face colouring consists in assigning a different colour to each face of a group.
    Of course the colours of the faces in the different groups are not independent, since each of the twenty faces belongs simultaneously to three of the twelve groups, as one can deduce from the relationship giving the total number of the faces:
         (5x12)/3=20
    All the faces having an equal colour are positioned like the faces of a tetrahedron, so that one can assume that the 5-coloured icosahedron is generated from the intersection of five different monochromatic tetrahedra.

    The indicization of the faces belonging to every tetrahedron is not straightforward.
    First of all, the indices of the 20 faces of the icosahedron derive from all possible permutations (8 and 12, respectively) of the two different kinds of indices, both rational and irrational:
         (  1 1 1 )  and (τ 1/τ 0)
    where τ denotes the golden ratio.

    The tetrahedron whose faces (in red colour) are characterized by the indices:
        (  1 1 1 )   (111)   (11 1)   (1 1 1)
    can be chosen as first tetrahedron A

    (clicking on the images below it is possible to visualize the corresponding VRML files)

    From this choice it follows that it is not possible to identify as second tetrahedron the one with indices:
         (  1 1 1 )   (  1 1 1 )   (  1 1 1 )    (  1 1 1 )
    since these faces, together with the faces of the first tetrahedron, would give rise to a { 1 1 1 } octahedron  and, in such a case, the remaining twelve faces of the icosahedron could not be distributed among three other tetrahedra. Nevertheless just starting from the single faces of this "forbidden" tetrahedron it is possible to arrive at the identification of the other four tetrahedra constituting the icosahedron.

    As starting point for positioning the tetrahedron B (yellow), one can choose for example the (  1 1 1 )  face; consequently to the application of a clockwise rotation of 44.48░ around the [  1 1 1 ]   direction, the three other faces of this tetrahedron, with the indices (111), (111) and (111), undergo a rotation of 41.81░, assuming the same position of the three faces of the icosahedron with the following indices : (τ -1/τ 0), (0 τ -1/τ ) and (-1/τ 0 τ ).
    Such set of three faces, together with the initial (  1 1 1 )  face, identifies a regular tetrahedron: actually the three faces form the same angle of  109.46░, characteristic of  the regular tetrahedron, both each other and with the initial (  1  1  1 )  face.
    (clicking on the images below it is possible to visualize the corresponding VRML files)

    For tetrahedron C (green), the rotation takes place around the direction perpendicular to the (111) face; in this case the indices of the set of three faces from:
       1 1 1 )  (111) (111)
    become:
         (τ -1/τ 0) (0 τ 1/τ ) (1/τ 0 τ )

    (clicking on the images below it is possible to visualize the corresponding VRML files)

    Analogously, for tetrahedron D (blue) the rotation  takes place around the direction perpendicular to the (111) face, and the indices of the set of three faces from:
        ( 1 1 1 ) (111) (  1 1 1 )
    become:
         (τ 1/τ 0) (0 τ 1/τ ) (-1/τ 0 τ )
    (clicking on the images below it is possible to visualize the corresponding VRML files)

    Finally, for tetrahedron E (violet) the rotation takes place around the direction perpendicular to (111) face, and in this last case the indices of the set of three faces from:
         (111) ( 1 1 1 ) (111)
    become:
         ( τ 1/τ 0) (0 τ  -1/τ ) (1/τ 0 τ)
    (clicking on the images below it is possible to visualize the corresponding VRML files)

    To summarize, the indices of the faces of the five tetrahedra whose intersection generates a 5-coloured icosahedron are reported in the following table.
    Colouring of the faces of icosahedron

    Colours Indices of the faces
     A (red) (1 1 1) (1 1 1) (1 1 1) (1 1 1)
     B (yellow) (1 1 1) ( τ  -1/τ  0 ) (0 τ -1/τ ) (-1/τ  0  τ )
     C (green) (1 1 1) τ  -1/τ  0 ) (0 τ 1/τ ) (1/τ  0 τ )
     D (blue) (1 1 1) ( τ  1/τ  0 ) (0 τ 1/τ ) (-1/τ  0  τ )
     E (violet) (1 1 1) ( τ  1/τ  0 ) (0 τ -1/τ ) (1/τ  0  τ )

    After any addition of a tetrahedron to the previous ones, it can be interesting to examine carefully the polyhedra that originate in correspondence of the intermediate steps leading from the initial tetrahedron to the final icosahedron.
    In the following set of images one can see the five polyhedra (all in the same orientation) sequentially formed by the intersection of the differently coloured tetrahedra.

    In order to recognize the single forms making up the different polyhedra originated at each step and to identify the respective point group, it can be useful to orientate properly the different polyhedra, assigning an equal colour to all the faces belonging to each crystallographic form.

    (clicking on the images below it is possible to visualize the corresponding VRML files )

    The three forms intermediate between tetrahedron (43m point group) and icosahedron (m35 point group), are made up of:
    • a trigonal trapezohedron (6 brown faces) and a pinacoid (2 fuchsia faces): 32 point group
    • a trigonal trapezohedron (6 brown faces) and a rhombohedron (6 fuchsia faces): 32 point group
    • a pentagon-dodecahedron (12 brown faces) and a tetrahedron (4 fuchsia faces): 23 point group

    Concerning the colouring of the icosahedron by 5 colours, one must point out that the previous solution is not unique: there is a second alternative, leading to a colour distribution enantiomorphic (or chiral) to the first one. This means that for the icosahedron there are two non-congruent distributions of the 5 colours: in practice, equal colours are applied to the centro-symmetrical faces belonging to the couple of chromatically chiral icosahedra.
    Also in this second case it is possible to follow a procedure for face colouring analogous to the previous one.

    The initial tetrahedron A' (red), dual of the red tetrahedron A belonging to the previous 5-coloured icosahedron (let us remember that the tetrahedron is autodual), this time will be the one whose faces have indices:
       (  1 1 1 )   ( 1 1 1 )   (  1 1 1 )    ( 1 1 1 )

    (clicking on the images below it is possible to visualize the corresponding VRML files)

    Each one of the other four tetrahedra will include only one of the four faces:
         (111) (111) (111) (111
    belonging to the tetrahedron A.
    The 44.48░ rotation (this time counterclockwise), around the direction normal to one of the four faces of tetrahedron A, implies a 41.81░ rotation of the three other faces of the tetrahedron A, leading to a tetrahedron whose faces have, also in this case, the same orientation of  three faces of the icosahedron: the same color is attributed to the faces of this new tetrahedron.
    Repeating this procedure on each other face of tetrahedron A, one obtains the following colour distribution, chiral of the previous one.

    Face colouring of the chiral icosahedron

    Colours Indices of the faces
     A' (red) (1 1 1) (1 1 1) (1 1 1) (1 1 1)
     B'(yellow) (1 1 1) (τ 1/τ 0) (0 τ 1/τ ) (1/τ 0 τ )
     C'(green) (1 1 1) (τ 1/τ 0) (0 τ -1/τ ) (-1/τ 0 τ )
     D' (blue) (1 1 1) (τ -1/τ 0) (0 τ -1/τ ) (1/τ 0 τ )
     E' (violet) (1 1 1) (τ -1/τ 0) (0 τ 1/τ ) (-1/τ 0 τ )

    As already pointed out, from the two previous tables one can note that each face of whatever colour, having the indices (hkl) and belonging to an icosahedron, corresponds to a face with the same colour and indices (h k l) belonging to its chromatically chiral icosahedron.
    In the following figure both the 5-coloured chiral icosahedron and the five tetrahedra whose intersection generates such icosahedron are shown in the same orientation.

    Rotating properly the image of the second icosahedron of the chromatically chiral pair, one can emphasize that the face colourings of the two icosahedra are symmetrical with respect to a vertical mirror placed between the two polyhedra put side by side.

    The difference between the two chiral colourings can be pointed out also by the nets of the chiral couple: in the following images one can ascertain that the face colouring is symmetrical with respect to a reflection line placed horizontally between the two nets (among the many possible alternative nets, the choice made concerns a kind of net showing that the icosahedron comes from a pentagonal euclidean antiprism, where each pentagonal base has been substituted by a pentagonal pyramid formed by five equilateral triangular faces).

    The couple of chiral colourings of the 5-coloured icosahedron can be further highlighted remembering that, based on the duality icosahedron-dodecahedron, the vertices of the dodecahedron correspond to the faces of the icosahedron, and so the respective colourings. This can be pointed out by the net of the dodecahedron, consisting of two groups, made of six pentagonal faces, sharing a side: in each group the central face shares each side with another pentagonal face.
    The coloured arrows show the colour of every vertex (shared by a set of three pentagonal faces of the dodecahedron), corresponding to the colour of the dual face of the icosahedron.

    The following is the net of the alternative chiral colouring, symmetrical of the previous one with respect to a vertical reflection line placed between the two nets put side by side.

    Taking into account the face colouring, the symmetry of the icosahedron, which belongs to the m35 point group, results completely absent in the 5-coloured icosahedron. On the other hand, the symbol of the relative chromatic point group becomes m'3'5'.

    DODECAHEDRON
    Concerning the dodecahedron, to make sure that faces having an equal colour do not share any edge or vertex, one can use for example 4 or 6 different colours.
    The use of six colours implies that a different colour characterizes each of the six pinacoids made up of pairs of parallel pentagonal faces: the crystallographic symmetry of the 6-coloured dodecahedron comes down to the existence of the only inversion centre.
    Regarding the 4-coloured dodecahedron,  the colouring procedure can be the following.
    Let us consider one of the ten 3-fold axes of the dodecahedron, for example the one along the [111] direction. The angle between this direction and each perpendicular to the three nearest faces measures 37.38░. Such faces, with pentagonal shape, have in common a vertex placed along the 3-fold axis and share an edge in pairs: to optimize the visualization of these faces one needs to differentiate them, assigning to each face a different colour.

    Continuing to examine the dodecahedron along the same direction, one can notice the presence of other three faces, equally inclined with respect to the [111] direction, whose perpendiculars form a larger angle (79.19░) with such direction. The indices of these faces are the following:
         ( τ 0 1 ) ( 0 1 τ ) ( 1 τ 0 )
    Since these three faces do not share any edge or vertex, one can assign to them the same colour A = red.

    The next step consists in considering the other three directions that, together with [111], constitute the perpendiculars to the faces of a tetrahedron and then repeating the previous procedure for each direction: every time the same new colour is assigned to the three faces equally inclined with respect to the 3-fold axis,  with which the perpendiculars form an angle of 79.19░.

    Colour B = blue  can be assigned to the three faces with indices:
         ( τ 0 1 ) ( 0 1 τ ) ( 1 τ 0 ),
    whose perpendiculars form the above mentioned angle with the 3-fold axis aligned along the [ 1 1 1 ]  direction; these blue faces are added to the red ones.

    In turn colour C = yellow  is assigned to the three faces with indices:
         ( τ 0 1 ) ( 0 1 τ ) ( 1 τ 0 ),
    symmetrically placed around [1 1 1], which are added to the red and blue faces.
    Finally colour D = green  is assigned to the three faces with indices:
        ( τ 0 1 ) ( 0 1 τ ) ( 1 τ 0 ),
    symmetrically placed around [111], which complete the colouring of the dodecahedron.

    The data relative to the 4-coloured dodecahedron are summarized in the following table.

    4-coloured dodecahedron
    Colours of faces Indices of  faces directions of
    3-fold axis
     A (red) (τ 0 1 ) ( 0 1 τ ) ( 1 τ 0 ) [1 1 1]
     B (blue) ( τ 0 1 ) ( 0 1 τ ) ( 1 τ 0 ) [1 1 1]
     C (yellow) ( τ 0 1) ( 0 1 τ ) ( 1 τ 0 ) [1 1 1]
     D (green) ( τ 0 1) ( 0 1 τ ) ( 1 τ 0 ) [1 1 1]

    Also in the case of the 4-colour dodecahedron there is a second chiral distribution of the colours, to which one arrives following a procedure analogous to the previous one, and whose result is summarized in the relative table.

    4-coloured chiral dodecahedron
    Colours of faces Indices of faces directions of
    3-fold axis
     A' (red) ( τ 0 1) (0 1 τ ) (1 τ 0) [1 1 1]
     B' (blue) ( τ 0 1) (0 1 τ ) (1 τ 0) [1 1 1]
     C' (yellow) ( τ 0 1) (0 1 τ ) (1 τ 0) [1 1 1]
     D' (green) ( τ 0 1) (0 1 τ ) (1 τ 0) [1 1 1]

    Examining the two 4-coloured chiral dodecahedra along the four 3-fold axes previously identified, one can note that they differ in the fact that, for each choice of the colour assigned to the three "exterior" faces whose perpendiculars form an angle of 79.19░ with a 3-fold axis, there is a permutation of the colours assigned to the "interior" faces sharing a vertex placed along the 3-fold axis, with which the perpendiculars to such faces form an angle of 37.38░.

    4-coloured dodecahedron

    4-coloured chiral dodecahedron


    The use of three colours lets one visualize the faces of the dodecahedron parallel to each Cartesian axis. Each colour characterizes a rhombic prism: four of the 3-fold axes present in the dodecahedron, namely the ones along directions with indices of the kind [111], make the connection between the three rhombic prisms by permutation of the colours. In every prism two pairs of faces share an edge, parallel to the zone axis of the prism.
    Whereas the previous colouring of the dodecahedron by 4 colours was completely asymmetric, the crystallographic symmetry of the 3-coloured dodecahedron corresponds to the mmm point group and the chromatic point group becomes m3'5'.

    RHOMBIC DODECAHEDRON
    Also in the case of the rhombic dodecahedron, using for example 4 or 6 different colours, one can make sure that faces having the same colour do not share any edge or vertex.

    .
    The use of six colours implies that a different colour characterizes each of the six pinacoids made up of the pairs of parallel rhombic faces: the crystallographic symmetry of the 6-coloured rhombic dodecahedron comes down to the presence of the only inversion centre.


    The 4-coloured rhombic dodecahedron is definitely more interesting.
    Also in this case it is worthwhile to refer to the directions corresponding to the 3-fold axes.
    Six of the twelve rhombic faces of the rhombic dodecahedron belong to the zone of each 3-fold axis; in turn, each face belongs to two such zones, as it follows from the relationship: (6x4)/2=12. The six faces are equally spaced, since the angle between the perpendiculars of adjacent faces measures 60░.

     The six faces belonging to the zone of the [111] 3-fold axis can be grouped according two trigonal prisms and for example the colour  A = green can be attributed to the trigonal prism that includes the faces:
        (110) (101) (011)
    Concerning the assignment of the colour B = red, one considers the zone of the [ 1 1 1] 3-fold axis, that includes also the (110) face, to which previously the green colour had been assigned. The presence of this face makes unambiguous the choice (among the six faces belonging to the zone) of the faces forming the trigonal prism:
        (110) (101) (  0 1 1 )
    to which the red colour can be assigned.
    The zone of the further [111] 3-fold axis includes both the (101) green face, and the (  0 1 1 ) red face: the other faces of such zone that belong to the third trigonal prism to which the colour C = blue can be assigned, are as follows:
        1 1 0 )  (011) (101)

    Finally, since the zone of the [11 1] 3-fold axis already includes the (011) green face, the (101) red face and the ( 1 1 0 ) blue face, the fourth trigonal prism turns out to be automatically identified: its faces, to which the colour D = yellow results to be assigned, have the indices:
         (110) (011) (101)

    Summarizing, each colour is relative to the faces of a trigonal prism in zone with a 3-fold axis, which, like all the other symmetry operators present, is a chromatic simmetry operator, since it makes the connection between faces that are differently coloured. Therefore the 4-coloured rhombic dodecahedron turns out to be without any operator of crystallographic symmetry.

    4-coloured rhombic dodecahedron
    Colours of faces Indices of faces directions of
    3-fold axis
     A (green) (1 1 0) (1 0 1) (0 1 1) [1 1 1]
     B (red) (1 1 0) (1 0 1) (0 1 1) [1 1 1]
     C (blue) (1 1 0) (0 1 1) (1 0 1) [1 1 1]
     D (yellow) (1 1 0) (0 1 1) (1 0 1) [1 1 1]

    Assigning colour A' to the faces of the other trigonal prism belonging to the same zone of trigonal prism A and following the previous procedure to assign the other colours, a chiral colouring of the rhombic dodecahedron results.

    4-coloured chiral rhombic dodecahedron
    Colours of faces Indices of faces directions of
    3-fold axis
     A' (green) (1 1 0) (1 0 1) (0 1 1) [1 1 1]
     B' (red) (1 1 0) (1 0 1) (0 1 1) [1 1 1]
     C' (blue) (1 1 0) (0 1 1) (1 0 1) [1 1 1]
     D' (yellow) (1 1 0) (0 1 1) (1 0 1) [1 1 1]

    Analogously to the dodecahedron, the use of three colours lets one visualize the faces of the rhombic dodecahedron making up a tetragonal prism parallel to each cartesian axis.
     The chromatic 3-fold axes present in the rhombic dodecahedron make the connection between the three tetragonal prisms by permutation of the colours.
    Since in this case there is only sharing of vertices (and not of edges) between equally coloured faces of each tetragonal prism, the result of prism colouring in the 3-coloured rhombic dodecahedron  is aesthetically better than the already described prism colouring in the 3-coloured dodecahedron. The crystallographic symmetry of the 3-coloured rhombic dodecahedron lowers from m3m to mmm; on the other hand, the corresponding chromatic point group is m3'm', since pairs of faces having equal colour are reflected by the principal mirror planes m.

    RHOMBIC TRIACONTAHEDRON

    As regards the rhombic triacontahedron, the most interesting cases among the possible colouring schemes of faces are based on the use of 3 and 5 colours.

    The use of five colours is related to the fact that, as already seen for the triangular faces of icosahedron, also in the rhombic tracontahedron it is possible to identify twelve groups of five faces  (this time rhombic) sharing a vertex placed along a 5-fold axis, with which the perpendiculars to the faces form a 31.72░ angle. Therefore, five different colours will be assigned to the faces of every group.
    Since each rhombic face is shared between two groups of five faces, the overall number of faces in the triacontahedron, given by the relationship:
         (12x5)/2=30 ,
    is greater than the number of faces in the icosahedron where, as already seen, the overall number of faces is (12x5)/3=20, being each triangular face shared among three groups of five faces.
    Indices of two kinds (rational and irrational) lie with the faces of the triacontahedron; they result from all the cyclical permutations, change of sign included, of  (100) e (τ 1 1/τ) indices, giving a total of 6+24=30 faces.
    Whereas it is immediate to realize that the 6 faces with indices of kind {100} have the same position of the faces of a cube, it is not as easy to realize that also among the remaining 24 faces one can identify 4 groups of 6 faces positioned like the faces of a cube.
    Therefore the triacontahedron can be decomposed into five cubes, each one characterized by a different colour.

    5-coloured rhombic triacontahedron
    Colour
    of each cube
    Indices of the faces
     A (violet) (100) (010) (001) (100) (010) (001)
     B (red) (τ 1 1/τ) (1/τ τ 1) (1 -1/τ τ) (τ 1 -1/τ) (-1/τ τ 1) (1 1/τ τ)
     C (blue) (1 1/τ τ) 1 -1/τ) (1/τ τ 1) (1 -1/τ τ) (τ 1 1/τ) (-1/τ τ 1)
     D (green) (1/τ τ 1) (τ -1 1/τ) (1 1/τ τ) (-1/τ τ 1) (τ 1 -1/τ) (1 -1/τ τ)
     E (yellow) (τ 1 -1/τ) (1 -1/τ τ) (-1/τ τ 1) (τ 1 1/τ) (1 1/τ τ) (1/τ τ 1)

    In the following (clickable) figure, both the five cubes and a 5-coloured triacontahedron generated from their intersection are shown in the same orientation.

    Analogously to the previous case of the 5-coloured icosahedron, also as regards the 5-coloured triacontahedron  it is interesting to examine carefully the polyhedra that originate in correspondence to the intermediate steps, consisting in the addition of a cube to the previous ones and leading from the initial cube to the final  triacontahedron.
    In the following set of images one can see the five polyhedra (all in the same orientation) sequentially formed by the intersection of the differently coloured cubes.

    In order to recognize the single forms making up the different polyhedra originated at each step and to identify the respective point group, also in this case it can be useful to orientate properly the different polyhedra, assigning the same colour to all the faces that belong to each crystallographic form.

    (clicking on the images below it is possible to visualize the corresponding VRML files)

    The three forms, intermediate between cube (m3m point group) and  rhombic triacontahedron  (m35 point group), are made up of:

    • a  ditrigonal scalenohedron (12 brown faces): 3m point group
    • a rhombohedron (6 brown faces), a rhombohedron (6 fuchsia faces) and a hexagonal prism (6 grey faces): 3m point group
    • a diakisdodecahedron or diploid (24 brown faces): m3 point group

    As one can see, only the polyhedron with eighteen faces consist of three forms: two different rhombohedra and a hexagonal prism.

    Operatively, to realize the colouring of the triacontahedron faces by 5 colours it is sufficient, after assignment of a specific colour to a pair of parallel faces, one picks the other two pairs of faces, both perpendicular to the first pair, with which they form a cube; alternatively one can follow the more elaborate procedure that will be described and illustrated in the four figures a-d.

    Fig.a)  Examining a triacontahedron along the [τ01] direction, coinciding with a 5-fold axis where five mirror planes are hinged, one can assume that the five faces, whose perpendiculars form a 31.72░ angle with the 5-fold axis, belong to a pentagonal deltohedron (also named antibipyramid, being the dual of a pentagonal antiprism), 10-face polyhedron which symmetry is the one of point group 5m: a different colour has to be assigned to each of the five vertex-sharing faces.
    Also the perpendiculars to the other five faces of the deltohedron form a 31.72░ angle with the opposite [τ01] direction of the same 5-fold axis: an equal colour will be assigned to the pairs of centrosymmetrical faces of the two sets of five faces. Consequently, the same succession of colours, even if in opposite clockwise and counterclockwise directions, characterizes the two sets of five faces belonging to the deltohedron.
    Fig.b)  Again, regarding the same [τ01] direction, one can assume that the other two sets of five rhombic faces, whose perpendiculars form with the 5-fold axis a 58.28░ angle (therefore greater than the previous one), belong to a second deltohedron.
    In order to prevent equally coloured faces from sharing edges or vertices, the colour assigned to each face of the second deltohedron will be the same of the face, belonging to the adjacent set of five faces of the first deltohedron and lying on the same mirror plane, that is the most distant.
    Fig.c)  The same scheme of face colouring is applied to the other set of five faces belonging to the second deltohedron.
    Fig.d)  At this point there is nothing left to do but to assign a colour to the last ten faces that, being in zone with the 5-fold axis parallel to [τ01] direction, form a decagonal prism. The condition that faces equally coloured cannot share edges and vertices identifies univocally the colour of every face of the decagonal prism.

    (clicking on the underlying images it is possible to visualize the corresponding VRML files)
    First row: views, along a 5-fold axis, of the colouring of :
     a)
    five faces of the first deltohedron
     b) ten of the faces belonging to both deltohedra
    Second row: views, perpendicular to the previous 5-fold axis, of the colouring of:
     c) the twenty faces belonging to the two deltohedra
     d) the remaining faces belonging to decagonal prism
    At this point the colouring of all the 30 faces of the triacontahedron is complete

    Unlike the 5-coloured icosahedron and the 4-coloured dodecahedron, for the 5-coloured triacontahedron there is not any alternative of a chiral colouring, since such colouring is intrinsecally centrosymmetrical.

    The possibility of colouring a triacontahedron by only three colours, without sharing of edges between equally coloured faces, was pointed out by John Dalbec to George Hart, who described it in his already cited page:
    http://www.georgehart.com/virtual-polyhedra/colorings.html
    Each colour (red, green and  blue) characterizes a pair of chains made of five rhombic faces connected through vertices. Adding in succession the green and blue chains to the red ones, one obtains the complete colouring of the triacontahedron by only three colours as shown in the following  clickable images.

    The central faces of every pair of isochromatic chains are parallel, therefore constitute a pinacoid: being the three pinacoids mutually perpendicular, they are the three pairs of opposite faces of a trichromatic cube.
    Since three perpendicular 2-fold axes pass through the centres of the pairs of central face of the chains, in the absence of any further symmetry operator for this arrangement of coloured faces, 222 will be the point group (subgroup of the m35 icosahedral point group) to which the 3-coloured triacontahedron belongs. In addition, the lack both of mirror planes and inversion centre implies the existence of two alternatives for the face colouring.
    Also in this case, by an appropriate rotation of the image of the second 3-coloured triacontahedron it is possible to highlight the fact that colourings of the faces that belong to the two chiral triacontahedra are symmetric with respect to a vertical plane placed between the two polyhedra put side by side.

    The 2-fold axis perpendicular to the central face of each chain relates to each other the  two lateral faces, both connected through a vertex to the central face. The action of the other 2-fold axes perpendicular to the first 2-fold axis relates the two lateral faces of the first chain to the two lateral faces of the second chain having the same colour: the collection of these four faces makes up a rhombic bisphenoid.
    Analogously, the action of the set of three 2-fold axes over the faces placed at the end of the same chain generates a rhombic bisphenoid oriented differently, but identical to the previous one; consequently, in the 3-coloured triacontahedron two rhombic bisphenoids and a pinacoid are present for each pair of equally coloured chains: hence there are a total of six rhombic bisphenoids and three pinacoids.
    Therefore nine colours  would be necessary to differentiate completely  the crystallographic forms altogether present.
    The following image shows, all in the same orientation, the single forms (three pinacoids, making up a 3-coloured cubes, and six rhombic bisphenoids) whose intersection corresponds to the central 9-colour rhombic triacontahedron.

    Finally, in the following summarizing table the different colourings of polyhedra are listed, together with the indication of a possible chiral distribution of the colours.
    The last three columns report respectively:

  • point group of polyhedra
  • point group of n-colored polyhedra
  • chromatic point group of n-coloured polyhedra

  • ALTERNATIVE CHOICES for POLYHEDRA COLOURING

    n-coloured polyhedron

    n colours Chirality of
    n-coloured polyhedron
    Point group of polyhedron Point group of
    n-coloured polyhedron
    Chromatic point group of
    n-coloured polyhedron
     Tetrahedron 4 yes 43m 1 4'3'm'
     Cube 3 no m3m mmm m3'm'
     Octahedron 4 no m3m 1 m'3'm'
     Octahedron 2 no m3m 43m m'3m
     Icosahedron 5 yes m35 1 m'3'5'
     Dodecahedron 6 no m35 1 m'3'5'
     Dodecahedron 4 yes m35 1 m'3'5'
     Dodecahedron 3 no m35 mmm m3'5'
     Rhombic dodecahedron 6 no m3m 1 m'3'm'
     Rhombic dodecahedron 4 yes m3m 1 m'3'm'
     Rhombic dodecahedron 3 no m3m mmm m3'm'
     Rhombic triacontahedron 5 no m35 1 m'3'5'
     Rhombic triacontahedron 3 yes m35 222 m'3'5'
    Clicking with the left button of the mouse over each value of the column "n colours", one can visualize a VRML file reproducing the polyhedron whose faces have been differentiated using the indicated number of colours.

    ACKNOWLEDGEMENTS

    Many thanks are due to Maria Rosa Ardigo' for the fruitful discussions and to Fabio Somenzi for his help concerning the translation of the text.

    LINKS

    SUGGESTED LECTURES

    • O. Wittke et J. Garrido (1959) - SymÚtrie des polyŔdres polychromatiques, Bull. Soc. franš. MinÚr. Crist., LXXXII, 223-30
    • A.V. Shubnikov and V.A. Koptsik (1974), Symmetry in Scienze and Art, Plenum Press, New York & London
    • D. Harker (1976) - A Table of the Colored Crystallographic and Icosahedral Point Group, Including Their Chirality and Diamorphism, Acta Cryst, A32, 133
    • L. Loreto, R. Farinato, F. Pappalardo (1990) - Icosahedral Symmetry, Icosahedral Polyhedra and Indexing methods, in "Topics on contemporary crystallography and quasicrystals", special Issue of "Periodico di Mineralogia", Vol. LIX,  L. Loreto and M. Ronchetti Editors, Roma.