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 
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.
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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 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 wellknown 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 polyhedron formed by twelve rhombic faces, perpendicular to the directions corresponding to the six twofold 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.
The polyhedron formed by 30 rhombic faces, placed in pairs perpendicular to each
of the fifteen 2fold axes
present in both the
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 quasiregular 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
The following review of the schemes for colouring
different polyhedra, starting from the notions contained in George Hart's page:
http://www.georgehart.com/virtualpolyhedra/colorings.html,
goes into further depth concerning the crystallographic facet of polyhedra colouring.
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
For tetrahedron C
(green), the
rotation takes place around the direction perpendicular
to the
( 1 1 1 )
(111)
(111)
become:
(τ
1/τ 0)
(0 τ 1/τ )
(1/τ 0 τ )
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 )
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 noncongruent distributions of the 5 colours:
in practice, equal colours are applied to the centrosymmetrical 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 5coloured
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 )
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 ) 
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 5coloured icosahedron can be further highlighted remembering that, based on the duality icosahedrondodecahedron, 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 5coloured icosahedron. On the other hand, the symbol of the relative chromatic point group becomes m'3'5'.
The next step consists in considering the other three directions
that, together with
The data relative to the 4coloured dodecahedron are summarized in the following table.
4coloured dodecahedron  
Colours of faces  Indices of faces  directions of 3fold 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 4colour 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.
4coloured chiral dodecahedron  
Colours of faces  Indices of faces  directions of 3fold 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 4coloured chiral dodecahedra along the four 3fold 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 3fold axis, there is a permutation of the colours
assigned to the "interior" faces sharing a vertex placed along the
3fold axis, with which the perpendiculars to such faces form an angle of 37.38°.
4coloured dodecahedron 
4coloured chiral dodecahedron 
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.
. 
Summarizing, each colour is relative to the faces of a trigonal prism in zone with a 3fold 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 4coloured rhombic dodecahedron turns out to be without any operator of crystallographic symmetry.
4coloured rhombic dodecahedron  
Colours of faces  Indices of faces  directions of 3fold 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.
4coloured chiral rhombic dodecahedron  
Colours of faces  Indices of faces  directions of 3fold 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] 
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)  (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 5coloured
triacontahedron generated from their intersection are shown in the same orientation. 
Analogously to the previous case of the 5coloured icosahedron, also as regards the
5coloured 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.
The three forms, intermediate between cube (m3m point group) and rhombic triacontahedron (m35 point group), are made up of:
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 ad.
Fig.a) Examining a triacontahedron
along the [τ01] direction, coinciding with a 5fold axis where five
mirror planes are hinged, one can assume that the five faces, whose
perpendiculars form a 31.72° angle with the 5fold axis, belong to a
pentagonal deltohedron (also named antibipyramid, being the dual of a
pentagonal antiprism), 10face polyhedron which symmetry is the one of point group
5m: a different colour has to be
assigned to each of the five vertexsharing 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 5fold 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 5fold 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 5fold 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.
Unlike the 5coloured icosahedron and the 4coloured dodecahedron, for the 5coloured 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/virtualpolyhedra/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 2fold 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 2fold axes perpendicular to the first 2fold 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 2fold 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 3coloured 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 3coloured cubes, and six rhombic bisphenoids)
whose intersection corresponds to the central 9colour 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:
ALTERNATIVE CHOICES for POLYHEDRA COLOURING  
ncoloured polyhedron 
n colours  Chirality of ncoloured polyhedron 
Point group of polyhedron 
Point group of ncoloured polyhedron 
Chromatic point group of ncoloured 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' 
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