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The structure of βMn crystallizes in space group P4_{1}32. The pseudo 8fold nature of the 4_{1} axes makes it constitute an approximant to the octagonal quasicrystals. In this paper we analyze why a fivedimensional super space group containing mutually perpendicular 8fold axes cannot generate P4_{1}32 on projection to 3d space and how this may instead be accomplished from a sixdimensional model. A procedure for generating the actual structure of βMn lifted to sixdimensional space is given.
The octagonal quasi crystals (oQC) discovered by Kuo and his coworkers [
The structure of βMn represented as a packing of three mutually perpendicular sets of tetrahelices. Each tetrahelix has a symmetry close to 8_{3}.
This approximant is enigmatic since βMn is acentric, cubic (P4_{1}32). By lifting βMn to higher space, we may generate a structure where the three mutually perpendicular, pseudo 8fold axes are truly 8fold. An axial QC would be expected to be describable as a projection from a fivedimensional, hyperspace. As it turns out, however, a simple analysis shows that this cannot be done for βMn. As we attempt to show in this paper, this conundrum is resolved by recourse to a sixdimensional, tri
The purpose of this paper is to show why a fivedimensional space description of the oQC approximant βMn is unsatisfactory and why a six dimensional description much better captures the nature of this structure. We will do this by considering the implications of five and sixdimensional descriptions on this structure
An 8fold axis in fivedimensional space is most easily described as the operation that permutes positive and negative axial direction with respect to one invariant axis. This is perfectly analogous to the 4fold rotation in threedimensional space. Using the notation introduced by Deonarine and Birman [
And the 8fold rotation in fivedimensional space as
There are obviously 5 independent and mutually perpendicular axes in fivedimensional space, and we may chose any set of three to attempt to generate the point group that will project onto threedimensional space as 432. Interestingly, this turns out to be impossible, as we show below.
Using two orthogonal 8fold axes as generators induces a total of 1920 symmetry operators in fivedimensional space, a factor of 2 short of the fivedimensional hyper cubic holohedry 2^{5} × 5! = 3840. This means that all permutations of indices are realized and half of the permutations of sign (those with even parity):
12345;
12345; 12345; 12345; 12345; 12345;
12345; 12345; 12345; 12345; 12345; 12345; 12345; 12345; 12345; 12345;
Axial projection of these operators along any two axes will yield an inversion, e.g., dropping the last two indices yields:
123;
123; 123; 123; 123;
123; 123; 123;
and additionally some redundancies. Since all permutations in fivedimensional space are covered, the same will be true in threedimensional space, and therefore the projection onto threedimensional space of the fivedimensional superspace point group generated by two orthogonal 8fold axes will be the cubic holohedry in threedimensional space, m3m. Thus we find that it is not possible to project the point group of βMn, 432 from a fivedimensional super space point group containing orthogonal 8fold axes and hence not the threedimensional space group P4_{1}32 from any fivedimensional super space group containing orthogonal 8fold axes.
The holohedral hypercubic point group in six dimensional space is a bit of a monster with an order corresponding to the full set of permutations of positions and signs of all six indices, amounting to 2^{6} × 6! possible symmetry operations (=46080), a rather unwieldy number. Nevertheless, a full analysis of the character table for this group has been published [
The holohedral hypercubic six dimensional pointgroup B_{6} contains five distinct sets of 8 fold operations. Basically these are quite similar. The 8fold operation in six dimensional space is a hyper rotation that permutes four indices out of six, and changes the sign of one of those to produce an 8 fold operation. Simultaneously, the remaining two indices may be involved in another operation, creating a double rotation. The different possibilities are classified in the paper by Deonarine and Birman [
8fold hyperrotations about the plane given by indices 1 and 2.
Operation in12 plane  Invariant  Reflection in 1 or 2  C_{2} rotation perp 12  C_{2} rotation within 12  C_{4} rotation within 12 


1 2 4 5 63  12 4 5 63  12 4 5 63  2 1 4 5 63  21 4 5 63 

1 2 5 634  1 2 5 634  1 2 5 634  1 2 5 634  12 5 634 

1 2 6345  12 6345  12 6345  2 1 6345  2 1 6345 

1 23456  1 23456  1 23456  1 23456  1 23456 

1 2456 3  12456 3  12456 3  2 1456 3  21456 3 

1 256 3 4  1 256 3 4  1 256 3 4  1 256 3 4  1256 3 4 

1 26 3 4 5  126 3 4 5  126 3 4 5  2 16 3 4 5  2 16 3 4 5 

1 2 3 4 5 6  1 2 3 4 5 6  1 2 3 4 5 6  1 2 3 4 5 6  1 2 3 4 5 6 
Projecting these operations into five dimensional space and three dimensional space, respectively, making the operations axial may be achieved using the projection matrices:
Now it can be shown by direct computation that a basis set consisting of three mutually perpendicular 8 fold axes from class 43, that is, the set will generate a group of order 3840 that contains a hyper space inversion centre 123456:
This is again incompatible with the point group 432. Since classes 45 and 47 generate the same inversion operation, all these cases may be discarded for our purposes. The remaining choices are class 49 and 51. The projection matrix P_{6>5} should result in an axial 5 dimensional group with 8fold rotational symmetry. For class 49 this requires that A + B = B + A, which is trivially true. For class 51, that requirement is instead A + B = B − A = − A − B = − B + A which is equally trivially incompatible with any other solution than A = B = 0, rendering the class 51 option unhelpful.
The only remaining choice for a point group is that generated by a set of three mutually perpendicular 8 fold axes of class 49:
Direct computation yields a point group with only 576 elements. This group contains no axes of 12 10 or 5fold symmetry. To generate the corresponding super space group that contains 8_{3} operations as reported for the oQC, we simply add a translational part to the point group operations so that the three generators become
Again a direct computation shows that this yields a closed group with 576 elements. In the table below, the following information is given. The different classes of the rotational part of the super space group operations as defined by Deonarine and Birman, the general type of operation such as single rotation (C
Summary of the symmetry operators in the super space group.
DeonarineBirman class  Type  Number  Typical form  Translational part (* 1/8)  Generating form


50  C_{4}, C_{4}  144  21 4 5 6 3  3 7 3 3 3 7  111111212 
49  C_{8}, C_{2}  144  2 1 4 5 63  3 3 3 7 3 7  1 
39  C_{3}, C_{3}  64  3 4 5 6 1 2  0 0 0 0 0 0  2113 
35  C_{3}, C_{2}  48  1 43 65 2  4 0 4 0 0 0  2111 
31  C_{3}  16  1 4 3 6 5 2  0 0 0 0 0 0  112112112111 
24  C_{2}, C_{2 },C_{2}  36  12 5 6 3 4  2 4 6 6 2 2  222113 
22  C_{4}, C_{2},  72  125 6 3 4  2 4 2 6 6 2  212111 
20  C_{4}, C_{4}  36  1 2 5 634  6 6 2 2 2 4  11 
5  C_{2},C_{2}  9  1 23456  4 4 4 6 0 2  1111 
3  C_{2}  6  1 2 34 56  0 4 0 6 0 2  21111121 
1  C_{1}  1  1 2 3 4 5 6  0 0 0 0 0 0  11111111 
The general projection matrix to 3D is given by
And the simplest alternative is A = 1, B = 0. Allowing this projector to act on the different classes of the 6D point group we find that those generate the much reduced set given in
Projected symmetry operations in threedimensional space.
Classes  Typical form  Translational part eights  3D symmetry 

1,3,31  1 3 5  0 0 0 

5,35  135  4 4 0  ½ +

20  1 53  6 2 2  ¾ +

22  15 3  2 2 6  ¼ +

24  1 5 3  2 6 2  ¼

39  3 5 1  0 0 0 

49,50  2 4 6  3 3 3  Creation of a second independent position 
The 3D space group may be easily identified directly from this as P4_{1}32. Each position in 6D space will generate two separate orbits depending on the values of the odd and even coordinates respectively.
For this case, it is trivial to determine the single position in 6D space that generates the βMn structure upon projection into 3D. The values of coordinates 1, 3 and 5 are given by the
It is of course to be expected that the 3fold degeneracy of position 8c in βMn must be generated by a 3fold degenerate position in sixdimensional space.
To preserve the 8fold nature of the unique axis, we need to fix the values A and B of the projection matrix:
Since the 8fold rotation also permutes A and B, those must be equal, or the 8 fold axis will degenerate into two separate orbits. Putting A = B = 0.5 generates a projection according to
Preservation of the unique 8fold axis upon projection into fivedimensional space.
Operation  Rotational part  Translational part  Projection rotation part  Projection transational part 


2 1 4 5 63  3 3 3 7 3 7  1' 4 5 63  3 3 7 3 7 

1 2 5 634  6 6 2 2 2 4  1' 5 634  6 2 2 2 4 

2 1 6345  1 1 5 1 7 5  1' 6345  1 5 1 7 5 

1 23456  4 4 4 6 0 2  1' 3456  4 4 6 0 2 

2 1456 3  7 7 1 7 5 3  1' 456 3  7 1 7 5 3 

1 256 3 4  2 2 2 4 6 6  1' 256 3 4  2 2 4 6 6 

2 16 3 4 5  5 5 7 5 1 5  1'6 3 4 5  5 7 5 1 5 

1 2 3 4 5 6  0 0 0 0 0 0  1' 3 4 5 6  0 0 0 0 0 
This means that the 8_{3} axis is preserved under the projection. Examining another 8fold axis of rotation is enlightening. The orbit splits into 4 pairs of operations that leave the plane spanned by the basis vectors 3 and 4 invariant according to
Although the super space group of the oQC may be generated directly in fivedimensional space, without recourse to the procedure outlined above, we believe that it is useful to involve an analysis of the relationship between the octagonal QC and the structure of βMn in sixdimensional space if the cubic approximant is going to be used in the modelling of the structure of the oQC.
Orbit splitting of an 8fold axis upon projection into fivedimensional space.
Operation  Rotational part  Translational part  Projection rotation part  Projection translatinal part  New orbit 


25 4 3 6 1  3 7 3 3 3 7  1' 4 3 6 1  5 3 3 3 7  I 

56 3 4 1 2  2 4 6 6 2 2  2' 3 4 1 2  3 6 6 2 2  II 

61 4 3 25  7 5 1 1 5 1  3' 4 3 25  6 1 1 5 1  II 

12 3 456  0 2 4 4 4 6  4' 3 456  1 4 4 4 6  IV 

2 5 4 361  5 3 7 7 1 7  1' 4 361  4 7 7 1 7  I 

5 6 3 412  6 6 2 2 2 4  2' 3 412  6 2 2 2 4  II 

6 1 4 32 5  1 5 5 5 7 5  3' 4 32 5  3 5 5 7 5  III 

1 2 3 4 5 6  0 0 0 0 0 0  4' 3 4 5 6  0 0 0 0 0  IV 
This work was supported by the Swedish Research Council, VR. The authors are indebted to Bernd Souvignier for enlightening discussions.