Appendix III

PR and TR representations for order 5 LS’s (par. 3.4)

We wish to determine the number of LS’s representing the full order 5 set if we count all point symmetry related squares as one (PR), as well as the similar number when counting all translation-related members as one (TR).

We will collect the dihedral-group related LS’s and the translation related LS’s in distinct “manyfolds”, symbolized in Chapter 2 by ( ) and [ ] respectively, but now for each structure type. The numbers we are looking for are the numbers of both types of these manyfolds summed over all 18 structure types.

To show how to do this we collect the (generally) 200 LS’s for a structure type on multiplication tables IIa and IIb). with dihedral group operations index l running left to right from 0 to 7 and 25 translations with indices hk running from (00) to (44) downwards. Assessing (in)equalities of the products is facilitated by expressing them as HP 48 checksums (par. 3.3 and 4.4).

As an example we will obtain the PR-contribution for structure type, 6.1, starting out with the preferred member for this type given in Table 3.2.1. We first prepare column hk0 by letting all hk translations act on structure type 6.1 and next obtain rows hkn horizontally by applying the dihedral group operations on each member of the hk0 series. The PR-number contribution of each independent row (manyfold) is 1 and the full contribution of the entire spreadsheets of the symmetry-free structures 4 and 11 would have been 25 (the default multiplicity). In our spread sheet for type 6.1 we observe that the content of some of the rows is equal except for order. The numbers of different rows (the actual multiplicity) goes down to 1 in structure types 1 and 14. The number in our case , 6.1, is 15 (see subscript of Table 3.4.1a), given already in List 3.2.2) .

To obtain the TR-contribution we proceeded in the reverse way, preparing rows l00 first and obtaining the translation related LS’s in columns lhk.

Table IIa
l 0 1 2 3 4 5 6 7
hk
00 34405 13677 52357 42908 13677 42908 34405 52537
10 46442 22504 13426 07383 31621 54370 54417 07730
01 54417 31621 07730 54370 22504 07383 46442 13426
20 65333 00734 46276 02407 46551 47924 46002 29839
02 46002 46551 29838 47924 00734 02407 65377 46276
21 19631 04054 43428 36523 11587 19900 05128 13856
12 05128 11587 13856 19900 04054 36523 19631 43428
11 07739 22398 22875 56078 22398 56078 07739 22875
22 48690 15621 04455 44627 15621 44627 48690 04455
30 53631 54093 43791 11790 36510 52397 03704 41530
Table IIIa. Sections of the multiplication table with intermediate results hk0 ( column 0) when translations hk have acted on a member of structure type 6.1 shown in Table 3.2.1 and with final results hkn when next point group operations act on the intermediate results. Data are HP48 checksums of the resulting LS’s.

We see that the content of each row pair (hk-n, kh-n, three pairs shown) is identical except for the order; the pair can be represented by one LS. There are 10 pairs of such rows; this brings, together with the contribution of five hh-n rows (two rows shown), the contribution of this structure on the PG- number to 15 LS. Corresponding contributions of the other structure types are given in List 3.2.2 as well as their sum: 192.

Let us next prepare 00-l first and compute the checksums of the hk-l products:

Table IIb
n 0 1 2 3 4 5 6 7
hk
00 34405 13677 52357 42908 13677 42908 34405 52357
10 46442 31621 05542 45371 31621 45371 46442 05542
20 65377 46551 43791 52397 46551 54370 65377 43791
30 53631 36510 46276 47924 36510 47924 53631 46276
40 61145 49583 13426 54370 49583 54370 61145 13426
01 54417 63835 01095 07383 63835 07383 54417 01095
11 07739 05809 23649 etc. 05809 07739 23649
21 19631 60070 63348 60070 19631 63348
41 04605 57898 etc. 57898 04605
02 46002 54093 54093 46002
12 05128 etc. 05128
22 48690 48690
etc. etc.
Table IIIb. Sections of the multiplication table with intermediate results l00(first row) when point group movements n act on the member of structure type 6.1 shown in table 3.2.1 and with final results after action of the translation operations on the intermediate results, obtaining the same abbreviated HP 48 checksums for the LS’s as in Table IIIa but in different order.

Checksums along the 00-n series are clearly two by two identical for structure type 6.1, (0,6), (1,4), etc., and the same will hold of course for the corresponding columns, meaning we have only 4 distinct columns so the contribution of this structure type to the TG-number is 4 . See List 3.2.2 for the contribution of the other structure types and their sum: 60.

Note that successive point group – and translation operations are non Abelian: the products hk-n and n-hk are generally unequal, as shown in the Tables with 20-2 and 2-20.