The 6-D Case
We obtain the 6-D convergent ratio vector by successive transformations of a base vector by the 6-D triangular matrix:
| 0 | 0 | 0 | 0 | 0 | 1 | | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 | | 2 |
| 0 | 0 | 0 | 1 | 1 | 1 | × | 3 |
| 0 | 0 | 1 | 1 | 1 | 1 | | 4 |
| 0 | 1 | 1 | 1 | 1 | 1 | | 5 |
| 1 | 1 | 1 | 1 | 1 | 1 | | 6 |
This leads to the sequence
6-D Sequence
|
| 1 | |
6 | |
21 | |
91 | |
371 | |
1547 | |
6405 | |
26585 | |
110254 | |
| |
s1 |
| 2 | |
11 | |
41 | |
176 | |
721 | |
3003 | |
12439 | |
51623 | |
214103 | |
| |
s2 |
| 3 | |
15 | |
59 | |
250 | |
1030 | |
4283 | |
17752 | |
73658 | |
305513 | |
… | |
s3 |
| 4 | |
18 | |
74 | |
309 | |
1280 | |
5313 | |
22035 | |
91410 | |
379171 | |
| |
s4 |
| 5 | |
20 | |
85 | |
350 | |
1456 | |
6034 | |
25038 | |
103849 | |
430794 | |
| |
s5 |
| 6 | |
21 | |
91 | |
371 | |
1547 | |
6405 | |
26585 | |
110254 | |
457379 | |
| |
s6 |
The Matrix Aspect
The 6-D convergent ratio includes 15 relations between components. Together with inverses this creates 30 core matrices.
These are arranged in the table below in groups to show the relations between them. The first row, s6/s1 to s2/s1, comprises the 5 biggest. In the second row we see many of the matrices obtained by adding together matrices from the third row. The third row shows s6 mapping to other components.
There are visual clues to the relations between components. The second half of the table is more or less the inverses of the first half.
Table of 6-D Core Matrices
s1→s6 (s6/s1)
|
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 |
|
s1→s5 (s5/s1)
|
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 1 | 1 | 1 | 1 |
|
s1→s4 (s4/s1)
|
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 1 | 0 | 1 | 0 | 1 |
| 1 | 0 | 1 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 | 1 |
| 0 | 0 | 1 | 1 | 1 | 1 |
|
s1→s3 (s3/s1)
|
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 |
| 1 | 0 | 1 | 0 | 1 | 0 |
| 0 | 1 | 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 1 | 1 |
|
s1→s2 (s2/s1)
|
| 0 | 1 | 0 | -0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 |
|
s6→s1 (s1/s6)
|
| 0 | 0 | 0 | 0 | -1 | 1 |
| 0 | 0 | 0 | -1 | 1 | 0 |
| 0 | 0 | -1 | 1 | 0 | 0 |
| 0 | -1 | -1 | 0 | 0 | 0 |
| -1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
|
s6→s2 (s2/s6)
|
| 0 | 0 | 0 | -1 | 1 | 0 |
| 0 | 0 | -1 | 1 | -1 | 1 |
| 0 | -1 | 1 | -1 | 1 | 0 |
| -1 | 1 | -1 | 1 | 0 | 0 |
| 1 | -1 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
|
s6→s3 (s3/s6)
|
| 0 | 0 | -1 | 1 | 0 | 0 |
| 0 | -1 | 1 | -1 | 1 | 0 |
| -1 | 1 | -1 | 1 | -1 | 1 |
| 1 | -1 | 1 | -1 | 1 | 0 |
| 0 | 1 | -1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
|
s6→s4 (s4/s6)
|
| 0 | -1 | 1 | 0 | 0 | 0 |
| -1 | 1 | -1 | 1 | 0 | 0 |
| 1 | -1 | 1 | -1 | 1 | 0 |
| 0 | 1 | -1 | 1 | -1 | 1 |
| 0 | 0 | 1 | -1 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
|
s6→s5 (s5/s6)
|
| -1 | -1 | 0 | 0 | 0 | 0 |
| 1 | -1 | 1 | 0 | 0 | 0 |
| 0 | 1 | -1 | 1 | 0 | 0 |
| 0 | 0 | 1 | -1 | 1 | 0 |
| 0 | 0 | 0 | 1 | -1 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
|
s5→s3 (s3/s5)
|
| 0 | 0 | 0 | -1 | 0 | 1 |
| 0 | 0 | -1 | 0 | 0 | 1 |
| 0 | -1 | 0 | 0 | 1 | 0 |
| -1 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
|
s4→s5 (s5/s4)
|
| 0 | 0 | -1 | 0 | 1 | 0 |
| 0 | -1 | 0 | 0 | 0 | 1 |
| -1 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 0 |
|
s3→s6 (s6/s3)
|
| 0 | -1 | 0 | 1 | 0 | -0 |
| -1 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
|
s2→s4 (s4/s2)
|
| -1 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 1 | 0 |
|
s3→s4 (s4/s3)
|
| 1 | 0 | 0 | 0 | -1 | 1 |
| 0 | 1 | 0 | -1 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | -1 | 1 | 1 | 0 | 0 |
| -1 | 1 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 1 |
|
s5→s1 (s1/s5)
|
| -1 | -1 | 0 | 1 | 1 | -1 |
| -1 | -1 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 1 |
| 1 | 1 | -1 | -1 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| -1 | 0 | 1 | 1 | 0 | -1 |
|
s2→s5 (s5/s2)
|
| -1 | 0 | 1 | 0 | -1 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | -1 | 1 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 1 |
| -1 | 1 | 1 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 1 | 0 |
|
s2→s6 (s6/s2)
|
| 1 | 0 | -1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 1 | 0 | 0 | 1 |
|
s5→s2 (s2/s5)
|
| -1 | -1 | 0 | 1 | 0 | 0 |
| -1 | -1 | 0 | 0 | 1 | 0 |
| 0 | 0 | -1 | 0 | 0 | 1 |
| 1 | 0 | 0 | -1 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | -1 |
|
s4→s6 (s6/s4)
|
| -1 | -1 | 0 | 0 | 0 | 1 |
| -1 | -1 | -1 | 0 | 1 | 1 |
| 0 | -1 | -1 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 0 | -1 |
|
s3→s2 (s2/s3)
|
| -1 | 1 | 0 | 0 | 1 | -1 |
| 1 | -1 | 1 | 1 | -1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | -1 | 1 | 0 |
| 1 | -1 | 0 | 1 | -1 | 1 |
| -1 | 0 | 0 | 0 | 1 | 0 |
|
s2→s3 (s3/s2)
|
| 1 | 0 | -1 | 1 | 1 | -1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| -1 | 1 | 1 | 0 | -1 | 1 |
| 1 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | -1 | 1 | 1 | 0 |
| -1 | 0 | 1 | 0 | 0 | 1 |
|
s4→s2 (s2/s4)
|
| 0 | 1 | 1 | 0 | 0 | -1 |
| 1 | 1 | 1 | 1 | -1 | -1 |
| 1 | 1 | 1 | 0 | 0 | -1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 1 |
| -1 | -1 | -1 | 0 | 1 | 1 |
|
s5→s4 (s4/s5)
|
| 1 | 1 | -1 | -1 | 0 | 1 |
| 1 | 0 | 0 | -1 | 0 | 1 |
| -1 | 0 | 0 | 1 | 0 | 0 |
| -1 | -1 | 1 | 1 | 1 | -1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | -1 | 0 | 1 |
|
s3→s5 (s5/s3)
|
| -1 | 1 | 0 | -1 | 1 | 0 |
| 1 | -1 | 0 | 1 | -1 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| -1 | 1 | 0 | 0 | 1 | 0 |
| 1 | -1 | 1 | 1 | -1 | 1 |
| 0 | 1 | 0 | 0 | 1 | 0 |
|
s4→s3 (s3/s4)
|
| 0 | 1 | 1 | 0 | -1 | 0 |
| 1 | 1 | 1 | 0 | 0 | -1 |
| 1 | 1 | 0 | 1 | 0 | -1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 1 |
| 0 | -1 | -1 | 0 | 1 | 1 |
|
s2→s1 (s1/s2)
|
| -1 | 1 | 1 | -1 | -1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | -1 | 1 | 1 | -1 |
| -1 | 0 | 1 | 0 | 0 | 0 |
| -1 | 0 | 1 | 0 | -1 | 1 |
| 1 | 0 | -1 | 0 | 1 | 0 |
|
s3→s1 (s1/s3)
|
| 0 | -1 | 1 | 1 | -1 | 0 |
| -1 | 1 | 0 | 0 | 1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | -1 | 1 |
| -1 | 1 | 0 | -1 | 1 | 0 |
| 0 | -1 | 0 | 1 | 0 | 0 |
|
s4→s1 (s1/s4)
|
| 1 | 0 | 0 | 1 | 0 | -1 |
| 0 | 1 | 1 | 0 | 0 | -1 |
| 0 | 1 | 1 | 0 | -1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 1 | 0 |
| -1 | -1 | 0 | 0 | 0 | 1 |
|
s5→s6 (s6/s5)
|
| -1 | 0 | 1 | 1 | 0 | -1 |
| 0 | 0 | 1 | 1 | 0 | -1 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | -1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | -1 | 0 | 1 | 1 | 0 |
|
Matrices by Size
Component Relations
Visual appraisal of the first three rows of the table above reveals the relations in the first two columns below. Closer investigation will confirm the others. The list is not exhaustive. The patterns may be extrapolated to higher dimensions.
- s6/s1 - s5/s1 = s1/s6
- s5/s1 - s4/s1 = s2/s6
- s4/s1 - s3/s1 = s3/s6
- s3/s1 - s2/s1 = s4/s6
- s2/s1 - 1 = s5/s6
- s1/s6 + s2/s6 = s3/s5
- s2/s6 + s3/s6 = s5/s4
- s3/s6 + s4/s6 = s6/s3
- s4/s6 + s5/s6 = s4/s2
- (s5/s6 + 1 = s2/s1)
- s1/s6 + 1 = s4/s3
- s2/s5+ 1 = s6/s3
- s1/s6 + s2/s3 = s5/s6
- s6/s2 - 1 = s5/s4
- s/s + s/s = s/s
The 6-D Equations
Since we have 30 core matrices which, together with their negatives, comprise the solutions to the 10 equations of degree 6 in question, there is probably a way to select 10 groups of 6 solutions to build the equations. This was trivial in the 3-D case but the 5-D and higher cases are harder. At least we can arrange the solutions in size order. Meanwhile plan B to obtain the first equation relies on the relations listed above:
(s1 + s2 + s3 + s4 + s5 + s6) / (s1 + s2 + s3 + s4 + s5 + s6) = 1
s6/(s1 + s2 + s3 + s4 + s5 + s6) = s1/s6
s1 / (s1 + s2 + s3 + s4 + s5 + s6) = s1 / s6 * s1 / s6 = s1 / s62
s2 / (s1 + s2 + s3 + s4 + s5 + s6) = s2 / s6 * s1 / s6 = s1 / s62
s3 / (s1 + s2 + s3 + s4 + s5 + s6) = s3 / s6 * s1 / s6 = s1 / s62
s4 / (s1 + s2 + s3 + s4 + s5 + s6) = s4 / s6 * s1 / s6 = s2 / s62
s5 / (s1 + s2 + s3 + s4 + s5 + s6) = s5 / s6 * s1 / s6 = s1 / s62
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