Representations of the Icosahedral Group
with Application to Quasicrystals
NICOLAE COTFAS
Faculty of Physics, University of Bucharest, Bucharest, Romania
Abstract
An icosahedral quasicrystal can be regarded as a packing of copies of a welldefined atomic cluster C . On the other hand, by starting from any cluster C
with icosahedral symmetry we can define in terms of the strip projection
method a quasiperiodic packing of copies of C . The dimension of the
superspace we have to use in this definition can be reduced by using the notion
of multi-component model set (a generalization of the notion of model set,
proposed by Baake and Moody).
1.
Introduction
The set of the Bragg peaks with intensity above a certain threshold occurring in
the diffraction diagram of a quasicrystal is a discrete set, invariant under a finite
group G , and the high-resolution electron microscopy suggests that the
quasicrystal can be regarded as a packing of (partially occupied) copies of a
well-defined G -invariant atomic cluster C . From a mathematical point of view,
the cluster C can be defined as a finite union of orbits of G , and there exists an
algorithm[2,3,4] which leads from C directly to a pattern Q which can be
regarded as a union of interpenetrating partially occupied translations of C (the
neighbours of each point x  Q belong to the set x  C  { x  y | y  C} ).
This algorithm, based on the strip projection method and group theory,
represents an extended version of the model proposed by Katz & Duneau [6] and
independently by Elser[5] for the icosahedral quasicrystals.
2.
Strip projection method and multi-component model sets
Let E k  (R k ,  ,  ) be the usual k -dimensional Euclidean space, E and E 
be two orthogonal subspaces such that E k  E  E  , and let
L   Zk
K  [0,  ] k  { ( x1 , x 2 ,..., x k ) | 0  xi   }
where   (0, ) is a fixed constant. For each x  E k there exist x ||  E and
x   E  uniquely determined such that x  x ||  x  . The mappings
 ,   : E k  E k ,  x  x || ,   x  x  are the corresponding orthogonal
projectors. By using the bounded set K    (K ) we define in terms of the
strip projection method[6] the discrete set
Q  { x | x  L,   x  K }
formed by the projection on E of all the points of L lying in the strip
K  E  { x  y | x  K, y  E }.
It is known[7] that any Z -module M  R l is the direct sum of a lattice
M d of rank d and a Z -module M s dense in a vector subspace of dimension
s , where d  s is the dimension of the subspace generated by L in R l . In
view of this result the Z -module L    (L) is the direct sum L  L'  D
of a lattice D of rank d and a Z -module L ' dense in a subspace E'  E 
of dimension s , where d  s  dim E  . In this decomposition the space E ' is
uniquely determined and we denote by E ' ' its orthogonal complement
E ' '  { x  E  |  x, y  0 for any y  E ' } . We get E k  E  E ' E ' ' .
For each x  E k there exist x ||  E ,
x' E ' and x' ' E ' ' uniquely
determined such that x  x  x' x' ' . The mappings  ' : E k  E k ,  ' x  x'
||
and  ' ' : E k  E k ,  ' ' x  x' ' are the orthogonal projectors corresponding to
E ' and E ' ' .
One can prove[2] that the projection IL  (   ' )(L) of the lattice L
on the space IE  E  E ' is a lattice in IE ,  restricted to IL is injective
and  ' (IL) is dense in E ' . It follows that the collection of spaces and mappings

'
:E 

IE 
E': x   ' x

IL
[1]
is a cut and project scheme . The lattice L  L  IE is a sublattice of IL ,
and necessarily [IL : L] is finite. The projection L ' '   ' ' (L) of L on E ' ' is
a discrete countable set. Let Z  { z i | i  Z } be a subset of L such that
L' '   ' ' ( Z ) and  ' ' z i   ' ' z j for i  j . The lattice L is contained in the
xx
union

iZ
IE i of the cosets
IE i  z i  IE  { z i  x | x  IE } . Since
L  IE i  z i  L the set IL i  (   ' )(L  IE i )  (   ' ) z i  L is a
coset of L in IL for any i  Z .
Only for a finite number of cosets IE i the intersection K i  K  IE i
   (K  IE i ) is non-empty. By changing the indexation of the elements of
Z if necessary, we can assume that the subset IK i   ' ( K i )   ' (K  IE i )
of E ' has a non-empty interior only for i  {1,2,..., m } . The ‘polyhedral’ set
IK i satisfies for any i  {1,2,..., m } the conditions: IK i  E ' is compact,
IK i  int (IK i ) and the boundary of IK i has Lebesgue measure 0. This allows
us to define the multi-component model set[1]
  i 1{  x | x  IL i ,  ' x  IK i } .
m
One can remark that   Q , and hence we have re-defined our pattern Q as a
multi-component model set by using the superspace IE of dimension,
generally, smaller than the dimension k of the initial superspace E k . The main
difficulty in this approach is the determination of the ‘atomic surfaces’ IK i .
3.
A quasiperiodic packing of dodecahedra
Consider the 3D representation of the icosahedral group Y generated by[3,4]
 1 1  1  
1 
1
  1 
a( ,  ,  )  
  ,   , 
  
2
2 2
2
2
2
2
2 
 2
b( ,  ,  )  ( ,  ,  )
where   (1  5 ) / 2 , and the Y -cluster (vertices of a regular dodecahedron)
C  Y (1,1,1)  {g (1,1,1) | g  Y }  {e1 , e2 ,..., e10 ,e1 ,e2 ,...,e10 ) .
By starting from C we can define in a canonical way[2,3,4] a permutation
representation of Y in E10 . The orthogonal projectors corresponding to the Y invariant
orthogonal
subspace
E  { ( r , e1 ,  r , e2 ,...,  r , e10 ) | r  E 3 } and to its
complement
  M (3 / 10, 5 / 10,1 / 10) and


 



 
 





 
 


M ( ,  ,  )  
   

 
 
 





 
   


E   { x |  x, y  0 for all y  E }
are
  M (7 / 10, 5 / 10,1 / 10) , where




   


 


  

  






  





  
 
.



  






  
   


 

     





  

 

Let   10 , L   Z 10 , K  [0,  ]10 , and let K    (K ) . By using
the strip projection method we define the icosahedral pattern
Q  {  x | x  L,   x  K } .
Since the arithmetic neighbours of a point  x  Q are distributed on the
vertices of the regular dodecahedron  x  C , the pattern Q can be regarded as
a quasiperiodic packing of interpenetrating dodecahedra[3]. The pattern Q can
be re-defined[3,4] as a multi-component model set by using the decomposition
E   E ' E ' ' , where E ' and E ' ' are the subspaces corresponding to the
orthogonal projectors  '  M (3 / 10, 5 / 10,1 / 10) ,  ' '  M (2 / 5,0,1 / 5) .
The superspace IE  E  E ' used in this approach is only six-dimensional[3],
but we have to determine the corresponding ‘atomic surfaces’ IK i .
4.
Concluding remarks
If we replace the starting cluster Y (1,1,1) by the icosidodecahedron Y (1,0,0)
then we get a quasiperiodic packing of icosidodecahedra[3]. If we replace it by
the two-shell Y -cluster
C  Y ( ,  ,0)  Y ( ,  ,  )  {e1 ,..., e16 ,e1 ,...,e16 }
where  ,  are rational positive numbers, then we get[3,4] a pattern Q such that
the arithmetic neighbours of each point are distributed on two shells, namely, on
the vertices of a regular icosahedron of radius    2 and on the vertices of a
regular dodecahedron of radius  3 . The pattern Q can be regarded as a
quasiperiodic packing of interpenetrating copies of C . It can be re-defined as a
multi-component model set[1] by using only a 6-dimensional superspace[3,4], but
we have to determine some rather complicated ‘atomic surfaces’ IK i .
Acknowledgment This research was supported by the grant CNCSIS 630/2003.
References
1.
2.
3.
4.
5.
6.
7.
Baake, M., Moody, R. V., in Proc. Int. Conf. Aperiodic’97, ed. M. de
Boissieu et al., World Scientific, Singapore, 9-20 (1999).
Cotfas, N., J. Phys. A:Math. Gen., 32, 8079-93 (1999).
Cotfas, N., Preprint math-ph/0309017 (2003).
Cotfas, N., http://fpcm5.fizica.unibuc.ro/~ncotfas (2003).
Elser, V., Acta Cryst. A, 42, 36-43 (1986).
Katz, A., Duneau, M., J. Phys. (France), 47, 181-96 (1986).
Senechal, M., Quasicrystals and Geometry, Cambridge University Press,
Cambridge 264-6 (1995).