Miracle Octad Generator MOG

The Miracle Octad Generator or MOG is described in

Sphere Packings, Lattices and Groups by John H. Conway, Neil J.A. Sloane

The hexacode has 64 code words:

The 35 ways to partition 8 digits in sets of 4 can be used to connect to the 35 lines in the smallest projective space PG(3,2):

The hexacode is used to check that the 35 lines are valid as blocks in a Steiner system (5,8,24). The left block with 4 blue squares is used for the 35 lines, like  1238.4567 for the upper left one. This can be shortened to triples like  123 , because all 35 blocks have 8 in it. The other two blocks with black squares indicate a second labelling of the same 35 lines. In addtion, the other symbols + and O and empty square show a partition in 3 sets of 4. These 3 sets of 4 points complete the line to a fano plane incident with the line. As an example, the upper left one 1238.4567 , for which the 3 planes as point sets are [1 2 3 4 5 6 7], [1 2 3 8 9 10 11] and [1 2 3 12 13 14 15].

If you prefer numbers the same 35 lines in the same 5 x 7 rectangular ordering are shown as triples, then as lines in PG(3,2) and the last 4 again with 2 digit symbols explained below:

123   456 467 457 567       [2 3 1]    [7 11 12] [13 5 8] [15 6 9] [4 14 10]    34 05 1∞ 26

145   236 237 267 367       [1 5 4]    [13 10 7] [8 6 14] [12 15 3] [9 2 11]    25 06 4∞ 13

167   234 245 345 235         [6 7 1]    [5 12 9] [3 14 13] [10 8 2] [15 4 11]    14 56 02 3

246   135 137 157 357        [6 4 2]     [7 9 14] [5 10 15] [3 11 8] [1 13 12]    16 2∞ 03 45

257   134 146 346 136       [7 2 5]    [11 13 6] [9 3 10] [14 1 15] [12 8 4]    35 12 6∞ 04

347   126 125 256 156        [4 7 3]    [14 11 5] [10 12 6] [9 1 8] [15 2 13]     36 24 01 5

356   124 247 147 127       [6 3 5]     [8 15 7] [11 1 10] [2 14 12] [4 9 13]     0∞ 23 46 15

In each row of the table one sees a spread of 5 lines, where each vertex occurs in exactly one line. The smallest projective space with a labelling of the vertices used for the lines that corresponds to the black squares in the 5 x 7 ordering above is in the next picture.

The 28 lines outside the bottom plane can be given a shorter labelling with 2 digits as indicated in the next picture. In the table above these 28 labels are given at the right in the same ordering as the symbols left from them. The symbols for the vertices are taken as 9 to 15 mod 8 = 0 to 7, and 7→∞.

Keeping the first column of plane 8 as it stands, and doubling the 4 lines in each row, all 28 lines occuring twice, one can create 8 Aronhold sets Ti (i = 0, .. , 6, ) based on the 7 lines from plane 8, which is the bottom plane in the figure.

1238    4678 4578 5678 4568 4568 4678 5678 4578

1458    2378 3678 2368 3678 2678 2368 2378 2678

1678    3458 2348 3458 2358 2348 2458 2458 2358

2468    1578 1358 1378 1578 3578 3578 1358 1378

2578    1368 1468 1468 1348 1368 1348 3468 3468

3478    2568 2568 1258 1268 1258 1568 1268 1568

3568    1248 1278 2478 2478 1478 1278 1478 1248

   ↑        ↑       ↑      ↑      ↑     ↑      ↑     ↑      ↑

pl 8     T0    T1   T2    T3   T4   T5   T6   T

One can check that they are just the 8 fano pencils from the points 8 to 15, where T0 = pencil 8, T1 = pencil 9 etc. In the two digit symbols we see that in the order from top to bottom of each column

T0 = {05, 06, 02, 03, 04, 01, 0}

T1 = {1, 13, 14, 16, 12, 01, 15}

T2 = {26, 25, 02, 2, 12, 24, 23}

T3 = {34, 13, 3, 03, 35, 36, 23}

T4 = {34, 4, 14, 45, 04, 24, 46}

T5 = {05, 25, 56, 45, 35, 5, 15)

T6 = {26, 06, 56, 16, 6, 36, 46}

T = {1, 4, 3, 2, 6, 5, 0)

See also in arXiv.math:

Configurations of lines and models in Lie algebras (2005)   L. Manivel

 

 

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s