student91 wrote:... Anyway, I was thinking that if we could have a pSe with [[3,3,3]]-symmetry and bilbiroes, then maybe we might also have a pSe with [[3,3,3]]-symmetry and thawroes. This would be made like this:
ex = xffoo3oxoof3fooxo3ooffx&#zx
-> xFfoo3o(-x)oof3fxoxo3ooffx&#zx quirk at 2'nd layer, 2'nd node
-> xFfoo3oooof3f(-x)oxo3oxffx&#zx doubled the quirk at 3'rd node, 2'nd layer
-> xFfxo3ooo(-x)f3f(-x)ooo3oxfFx&#zx repeated this at the 4'th layer.
-E-> xFfxo3xxxof3foxxx3oxfFx&#zx expansion.
Of course, I didn't completely understand it's structure, but that's why I made its incmat, to be sure its structure is CRF. ...
xxfoF3oxxFx3xFxxo3Fofxx&#zx
1-2 & 4-5, 1-4 & 2-5 and 1-5 are f, all others are x.
o....3o....3o....3o.... & | 120 * * | 2 0 0 0 2 2 0 0 | 1 0 2 1 0 2 0 1 0 2 0 | 1 1 2 0 1
.o...3.o...3.o...3.o... & | * 120 * | 0 1 1 0 0 0 2 2 | 0 1 0 0 0 1 3 0 2 0 2 | 0 3 1 1 0
..o..3..o..3..o..3..o.. | * * 120 | 0 0 0 2 0 2 2 0 | 0 0 0 0 1 2 0 2 2 2 1 | 0 2 2 0 2
------------------------------+-------------+-------------------------------+----------------------------------------+---------------
x.... ..... ..... ..... & | 2 0 0 | 120 * * * * * * * | 1 0 1 0 0 1 0 0 0 0 0 | 1 1 1 0 0
.x... ..... ..... ..... & | 0 2 0 | * 60 * * * * * * | 0 1 0 0 0 0 2 0 0 0 0 | 0 2 0 1 0
..... .x... ..... ..... & | 0 2 0 | * * 60 * * * * * | 0 1 0 0 0 0 0 0 2 0 0 | 0 2 1 0 0
..... ..x.. ..... ..... & | 0 0 2 | * * * 120 * * * * | 0 0 0 0 1 0 0 1 1 1 0 | 0 1 1 0 2
..... ..... x.... ..... & | 2 0 0 | * * * * 120 * * * | 0 0 1 1 0 0 0 0 0 1 0 | 1 0 1 0 1
o.o..3o.o..3o.o..3o.o..&#x & | 1 0 1 | * * * * * 240 * * | 0 0 0 0 0 1 0 1 0 1 0 | 0 1 1 0 1
.oo..3.oo..3.oo..3.oo..&#x & | 0 1 1 | * * * * * * 240 * | 0 0 0 0 0 1 0 0 1 0 1 | 0 2 1 0 0
.o.o.3.o.o.3.o.o.3.o.o.&#x | 0 2 0 | * * * * * * * 120 | 0 0 0 0 0 0 2 0 0 0 1 | 0 2 0 1 0
------------------------------+-------------+-------------------------------+----------------------------------------+---------------
x....3o.... ..... ..... & | 3 0 0 | 3 0 0 0 0 0 0 0 | 40 * * * * * * * * * * | 1 1 0 0 0
.x...3.x... ..... ..... & | 0 6 0 | 0 3 3 0 0 0 0 0 | * 20 * * * * * * * * * | 0 2 0 0 0
x.... ..... x.... ..... & | 4 0 0 | 2 0 0 0 2 0 0 0 | * * 60 * * * * * * * * | 1 0 1 0 0
..... o....3x.... ..... & | 3 0 0 | 0 0 0 0 3 0 0 0 | * * * 40 * * * * * * * | 1 0 0 0 1
..... ..x..3..x.. ..... | 0 0 6 | 0 0 0 6 0 0 0 0 | * * * * 20 * * * * * * | 0 0 0 0 2
x.fo. ..... ..... .....&#xt & | 2 1 2 | 1 0 0 0 0 2 2 0 | * * * * * 120 * * * * * | 0 1 1 0 0
.x.o. ..... ..... .....&#x & | 0 3 0 | 0 1 0 0 0 0 0 2 | * * * * * * 120 * * * * | 0 1 0 1 0
..... o.x.. ..... .....&#x & | 1 0 2 | 0 0 0 1 0 2 0 0 | * * * * * * * 120 * * * | 0 1 0 0 1
..... .xx.. ..... .....&#x & | 0 2 2 | 0 0 1 1 0 0 2 0 | * * * * * * * * 120 * * | 0 1 1 0 0
..... ..... x.x.. .....&#x & | 2 0 2 | 0 0 0 1 1 2 0 0 | * * * * * * * * * 120 * | 0 0 1 0 1
.ooo.3.ooo.3.ooo.3.ooo.&#x | 0 2 1 | 0 0 0 0 0 0 2 1 | * * * * * * * * * * 120 | 0 2 0 0 0
------------------------------+-------------+-------------------------------+----------------------------------------+---------------
x....3o....3x.... ..... & | 12 0 0 | 12 0 0 0 12 0 0 0 | 4 0 6 4 0 0 0 0 0 0 0 | 10 * * * * co
xxfo.3oxxF. ..... .....&#zx & | 3 9 6 | 3 3 3 3 0 6 12 6 | 1 1 0 0 0 3 3 3 3 0 6 | * 40 * * * thawro
x.fo. ..... x.xx. .....&#xt & | 4 2 4 | 2 0 1 2 2 4 4 0 | 0 0 1 0 0 2 0 0 2 2 0 | * * 60 * * pip
.x.o. ..... ..... .o.x.&#x | 0 4 0 | 0 2 0 0 0 0 0 4 | 0 0 0 0 0 0 4 0 0 0 0 | * * * 30 * tet
..... o.x..3x.x.. .....&#x & | 3 0 6 | 0 0 0 6 3 6 0 0 | 0 0 0 1 1 0 0 3 0 3 0 | * * * * 40 tricu
student91 wrote:** here means that an f is changed in a (F+x).
student91 wrote:(+x)3.3.3.
ooFFx3xooxf3foooo3oFfox&#zx [=: A]
...
(+x)3.3(+x)3.
ooFFx3xooxf3Fxxxx3oFfox&#zx (= ooFFx3xooxf3foooo3oFfox&#zx [= A] with .3.3(+x)3.)
...
To be honest, I investigated all these things from the "summary" in this way. the only one that didn't have such isolated vertices was this:Klitzing wrote:
student91 wrote:[...]
Of these, fxFxo3xxooF3ofxoo3fooFx&#zx is the only one where every part has at least one x-lacing. This means that possibly the only thing my long post has given is fxFxo3xxooF3ofxoo3fooFx&#zx. it might be though that the (F+x)-things become CRF.
fxFxo3xxooF3ofxoo3fooFx&#zx is highly similar to the watery thawrochoron (When fxoo would have been changed to Fo(-x)x instead of F(-x)xo, and then expansion was done on both nodes, you would have had the watery thawrochoron). It has 20 thawroes, and some other stuff. I'm pretty sure it will be CRF.
This means that the (F+x)-things are the last one before we have exhaustively investigated this one.
Of course there are polytopes with a u at the outside. this u might very well be part of a hexagon. The problem only arises when an x is epanded into the u. if you take the hexagon xux, this must be an expansion of oxo, two triangles. such two triangles never occur on finite polytopes, and thus a u-edge at the outside is difficult to be valid. a u-edge on the inside will probably not occur, as x-edges mostly lay at the outside. Similarly, (F+x)-edges can very well occur at the inside of a polytope. even an expansion F=>(F+x) might very well be valid (you have such an expansion in the single axial expansion of ex). I just thought it to be very unlikely that an f-edge could be promoted to a F-edge by node-changing, and further expanded to (F+x). that's a bit like making a (x+u) out of a x. Note that I don't say they are non-CRF for sure, I just say it is very unlikely to be soKlitzing wrote:student91 wrote:** here means that an f is changed in a (F+x).
Well in fact neither the nodes of type (x+x) nor those of type (F+x) are invalid per se.
u = (x+x) even might occur at the outside, provided we have some local xux&#xt (hexagon) structure there.
(F+x) supposedly would not occur at the outside. Its validity then clearly depends on the existance of connecting unit lacings.
Wrt. your longer lists:
It is good to have those.
Those lists were exactly aiming to do so. If you look at these lists very carefully, you see I first list the possible quirks. Then I list all possible expansions. This is done by first saying what nodes I will be expanding, and then what quirks can/must be used in every kind of expansion. This is done by using (-x)'s only where the expansion will be, and using x's only where no expansion will be. I think that is quite exhaustive, at least it exhausts all things that don't get u-edges. Then the summary gave the (worked out) expansions of the things that were most likely to be CRF.But still I have no good feeling onto a systematical approach for exhausting each to be chosen subsymmetry.
I don't have any idea on how to do that whilst keeping the article not too unwieldy. I guess such a system might occur when we have found all valid things.Neither on to order all those finds somehow systematically - e.g. for future display in our articel, hehe.
--- rk
Klitzing wrote:student91 wrote:[...]
Checked your incmat. Mainly its correct. But some numbers had been wrong so. E.g. the count of the tets!
Here comes the corrected one:
[...]
student91 wrote:...
.3(+x)3.3.
fxFxo3xxooF3ofxoo3fooFx&#zx
...
student91 wrote:...
.3(+x)3.3.
...
foFxo3xoooF3oFxoo3fooFx&#zx
(+x)3.3(+x)3.
...
FoFox3oxxxF3xFxoo3fooFo&#zx
...
student91 wrote:In summary, the ones that haven't been investigated and that don't produce (F+x)-edges are:
...
(+x)3.3(+x)3.
...
ooFFx3xooxF3Fxxxo3oFfoo&#zx
Klitzing wrote:Next we will check which of these remaining 11 cases would allow for unit lacings (inter-layer edges of unity), where the respective stratos height is given by the ex. This might reduce that count then a bit.
(AAD) xoxFoFxox 3 oxoooooxo 5 ooxofoxoo &#xt
(A) oxofxfoxo 3 xxxxoxxxx 5 ooxofoxoo &#xt
(ADD) ooofxfooo 3 xoxxoxxox 5 ofxofoxfo &#xt
(C) oxofofoxo 3 oofoxofoo 5 xxoxxxoxx &#xt
(CD) xoxFxFxox 3 oxfoxofxo 5 xxoxxxoxx &#xt
(AC) oxofxfoxo 3 xxFxoxFxx 5 xxoxFxoxx &#xt
Klitzing wrote:
- Code: Select all
(A) oxofxfoxo 3 xxxxoxxxx 5 ooxofoxoo &#xt
- Or does anyone recognise some of those to be already known??? (I'd doubt, but who knows...)
--- rk
C then is D4.9. The part away from the equator is of course different. EDIT: without the vertices laying on (I guess) F3o5o.Klitzing wrote:Thus we then are finally left for this subsymmetry with the following 6 valid cases
- Code: Select all
(C) oxofofoxo 3 oofoxofoo 5 xxoxxxoxx &#xt
(Spaces introduced for better readability. - Will elaborate 'em in the sequel...)
- Or does anyone recognise some of those to be already known??? (I'd doubt, but who knows...)
--- rk
AAD is the castellated x5o3x-prism, with some more stuff on the top. To be exact x5o3x || o5x3o || o5o3x, on both sides.Klitzing wrote:Thus we then are finally left for this subsymmetry with the following 6 valid cases(Spaces introduced for better readability. - Will elaborate 'em in the sequel...)
- Code: Select all
(AAD) xoxFoFxox 3 oxoooooxo 5 ooxofoxoo &#xt
- Or does anyone recognise some of those to be already known??? (I'd doubt, but who knows...)
--- rk
I already did this too, then using [5,3,2]-symmetry instead of [5,3]. This allows one more transition: B: f3o5o x -> f3o5o (-x)Klitzing wrote:So far have not gone completely thru your longer mails, aiming to provide a systematic access to demicubic resp. pentic subsymmetric cases. Remeins to be done, esp. to be checked independently.
In the meantime i considered the same access to axial icosahedral subsymmetry.
.3.5. (+x):
(B): oxofo3oooox5ooxoo BCFox (B=A+x=2f+x, C=F+x=f+2x) (Just axial expansion, already known)
(+x)3.5. (+x):
(BD): xoxFx3oxoox5ooxoo BCFox
(AABD): xoxFo3oxooo5ooxof BCFox
.3(+x)5. (+x):
(AB): oxofx3xxxxo5ooxof BCFox
(ABDD): ooofx3xoxxo5ofxof BCFox
.3.5(+x) (+x):
(BC): oxofx3oofox5xFoxo BCFox
.3(+x)5(+x) (+x):
(ABC): oxofx3xxFxo5xFoxF BCFox
(ABCDD): ooofx3xoFxo5xFoxF BCFox
(+x)3.5(+x) (+x):
(BCD): xoxFx3oxfox5xxoxx BCFox
(AABCD): xoxFo3oxfoo5xxoxF BCFox
(+x)3(+x)5. (+x):
(AABDD): xxxFo3xoxxo5ofxof BCFox
(+x)3(+x)5(+x) (+x):
(AABCDD): xxxFo3xoFxx5xFoxF BCFox
(AABD):xoxFo3oxooo5ooxof BCFox
(AB):oxofx3xxxxo5ooxof BCFox
(ABDD): ooofx3xoxxo5ofxof BCFox
(BC):oxofx3oofox5xFoxo BCFox
(BCD):xoxFx3oxfox5xxoxx BCFox
(ABC):oxofx3xxFxo5xFoxF BCFox
Klitzing wrote:Thus we then are finally left for this subsymmetry with the following 6 valid cases
- Code: Select all
(AAD) xoxFoFxox 3 oxoooooxo 5 ooxofoxoo &#xt
(A) oxofxfoxo 3 xxxxoxxxx 5 ooxofoxoo &#xt
(ADD) ooofxfooo 3 xoxxoxxox 5 ofxofoxfo &#xt
(C) oxofofoxo 3 oofoxofoo 5 xxoxxxoxx &#xt
(CD) xoxFxFxox 3 oxfoxofxo 5 xxoxxxoxx &#xt
(AC) oxofxfoxo 3 xxFxoxFxx 5 xxoxFxoxx &#xt
xoxFoFxox3oxoooooxo5ooxofoxoo&#xt (AAD)
ED(03)=1
EC(08)=f
DC(05)=1
DB(08)=f
CB(03)=1
CA(08)=1
BA(05)=f
Bb(10)=1
E-D = ike || id, D-C = id || srid
C-c = castellated rhombicosidodecahedral prism
o........3o........5o........ & | 24 * * * * | 5 5 0 0 0 0 0 0 0 | 5 5 5 0 0 0 0 0 0 0 0 0 | 1 5 1 0 0 0 0 verf = pip
.o.......3.o.......5.o....... & | * 60 * * * | 0 2 4 4 0 0 0 0 0 | 0 1 4 2 2 4 2 0 0 0 0 0 | 0 2 2 2 1 0 0
..o......3..o......5..o...... & | * * 120 * * | 0 0 0 2 2 2 1 1 0 | 0 0 0 0 2 1 2 1 2 2 2 1 | 0 0 1 1 2 1 2
...o.....3...o.....5...o..... & | * * * 24 * | 0 0 0 0 0 0 5 0 1 | 0 0 0 0 0 0 0 0 0 5 0 5 | 0 0 1 0 0 0 5
....o....3....o....5....o.... | * * * * 20 | 0 0 0 0 0 0 0 6 0 | 0 0 0 0 0 0 0 0 0 0 6 3 | 0 0 0 0 0 2 3 verf = f x3o
------------------------------------+-----------------+-----------------------------------+-------------------------------------------+--------------------
x........ ......... ......... & | 2 0 0 0 0 | 60 * * * * * * * * | 2 1 0 0 0 0 0 0 0 0 0 0 | 1 2 0 0 0 0 0
oo.......3oo.......5oo.......&#x & | 1 1 0 0 0 | * 120 * * * * * * * | 0 1 2 0 0 0 0 0 0 0 0 0 | 0 2 1 0 0 0 0
......... .x....... ......... & | 0 2 0 0 0 | * * 120 * * * * * * | 0 0 1 1 0 1 0 0 0 0 0 0 | 0 1 1 1 0 0 0
.oo......3.oo......5.oo......&#x & | 0 1 1 0 0 | * * * 240 * * * * * | 0 0 0 0 1 1 1 0 0 0 0 0 | 0 0 1 1 1 0 0
..x...... ......... ......... & | 0 0 2 0 0 | * * * * 120 * * * * | 0 0 0 0 1 0 0 1 1 0 1 0 | 0 0 0 1 1 1 1
......... ......... ..x...... & | 0 0 2 0 0 | * * * * * 120 * * * | 0 0 0 0 0 0 1 0 1 1 0 0 | 0 0 1 0 1 0 1
..oo.....3..oo.....5..oo.....&#x & | 0 0 1 1 0 | * * * * * * 120 * * | 0 0 0 0 0 0 0 0 0 2 0 1 | 0 0 1 0 0 0 2
..o.o....3..o.o....5..o.o....&#x & | 0 0 1 0 1 | * * * * * * * 120 * | 0 0 0 0 0 0 0 0 0 0 2 1 | 0 0 0 0 0 1 2
...o.o...3...o.o...5...o.o...&#x | 0 0 0 2 0 | * * * * * * * * 12 | 0 0 0 0 0 0 0 0 0 0 0 5 | 0 0 0 0 0 0 5
------------------------------------+-----------------+-----------------------------------+-------------------------------------------+--------------------
x........3o........ ......... & | 3 0 0 0 0 | 3 0 0 0 0 0 0 0 0 | 40 * * * * * * * * * * * | 1 1 0 0 0 0 0
xo....... ......... .........&#x & | 2 1 0 0 0 | 1 2 0 0 0 0 0 0 0 | * 60 * * * * * * * * * * | 0 2 0 0 0 0 0
......... ox....... .........&#x & | 1 2 0 0 0 | 0 2 1 0 0 0 0 0 0 | * * 120 * * * * * * * * * | 0 1 1 0 0 0 0
.o.......3.x....... ......... & | 0 3 0 0 0 | 0 0 3 0 0 0 0 0 0 | * * * 40 * * * * * * * * | 0 1 0 1 0 0 0
.ox...... ......... .........&#x & | 0 1 2 0 0 | 0 0 0 2 1 0 0 0 0 | * * * * 120 * * * * * * * | 0 0 0 1 1 0 0
......... .xo...... .........&#x & | 0 2 1 0 0 | 0 0 1 2 0 0 0 0 0 | * * * * * 120 * * * * * * | 0 0 1 1 0 0 0
......... ......... .ox......&#x & | 0 1 2 0 0 | 0 0 0 2 0 1 0 0 0 | * * * * * * 120 * * * * * | 0 0 1 0 1 0 0
..x......3..o...... ......... & | 0 0 3 0 0 | 0 0 0 0 3 0 0 0 0 | * * * * * * * 40 * * * * | 0 0 0 1 0 1 0
..x...... ......... ..x...... & | 0 0 4 0 0 | 0 0 0 0 2 2 0 0 0 | * * * * * * * * 60 * * * | 0 0 0 0 1 0 1
......... ......... ..xo.....&#x & | 0 0 2 1 0 | 0 0 0 0 0 1 2 0 0 | * * * * * * * * * 120 * * | 0 0 1 0 0 0 1
..x.o.... ......... .........&#x & | 0 0 2 0 1 | 0 0 0 0 1 0 0 2 0 | * * * * * * * * * * 120 * | 0 0 0 0 0 1 1
..ooooo..3..ooooo..5..ooooo..&# | 0 0 2 2 1 | 0 0 0 0 0 0 2 2 1 | * * * * * * * * * * * 60 | 0 0 0 0 0 0 2
------------------------------------+-----------------+-----------------------------------+-------------------------------------------+--------------------
x........3o........5o........ & | 12 0 0 0 0 | 30 0 0 0 0 0 0 0 0 | 20 0 0 0 0 0 0 0 0 0 0 0 | 2 * * * * * * ike
xo.......3ox....... .........&#x & | 3 3 0 0 0 | 3 6 3 0 0 0 0 0 0 | 1 3 3 1 0 0 0 0 0 0 0 0 | * 40 * * * * * oct
......... oxoo.....5ooxo.....&#xt & | 1 5 5 1 0 | 0 5 5 10 0 5 5 0 0 | 0 0 5 0 0 5 5 0 0 5 0 0 | * * 24 * * * * ike
.ox......3.xo...... .........&#x & | 0 3 3 0 0 | 0 0 3 6 3 0 0 0 0 | 0 0 0 1 3 3 0 1 0 0 0 0 | * * * 40 * * * oct
.ox...... ......... .ox......&#x & | 0 1 4 0 0 | 0 0 0 4 2 2 0 0 0 | 0 0 0 0 2 0 2 0 1 0 0 0 | * * * * 60 * * squippy
..x.o....3..o.o.... .........&#x & | 0 0 3 0 1 | 0 0 0 0 3 0 0 3 0 | 0 0 0 0 0 0 0 1 0 0 3 0 | * * * * * 40 * tet
..xFoFx.. ......... ..xofox..&#xt & | 0 0 8 4 2 | 0 0 0 0 4 4 8 8 2 | 0 0 0 0 0 0 0 0 2 4 4 4 | * * * * * * 30 bilbiro
Klitzing wrote:Klitzing wrote:Thus we then are finally left for this subsymmetry with the following 6 valid cases
- Code: Select all
(AAD) xoxFoFxox 3 oxoooooxo 5 ooxofoxoo &#xt
(A) oxofxfoxo 3 xxxxoxxxx 5 ooxofoxoo &#xt
(ADD) ooofxfooo 3 xoxxoxxox 5 ofxofoxfo &#xt
(C) oxofofoxo 3 oofoxofoo 5 xxoxxxoxx &#xt
(CD) xoxFxFxox 3 oxfoxofxo 5 xxoxxxoxx &#xt
(AC) oxofxfoxo 3 xxFxoxFxx 5 xxoxFxoxx &#xt
Here comes the first, (AAD). ...
oxofxfoxo3xxxxoxxxx5ooxofoxoo&#xt (A)
ED(03)=1
EC(08)=f
DC(05)=1
DB(08)=1
CB(03)=1
CA(08)=1
BA(05)=1
Bb(10)=1
E-D = id||ti, D-C = ti||tid
o........3o........5o........ & | 60 * * * * | 4 2 0 0 0 0 0 0 0 0 0 0 0 0 | 2 2 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 2 2 0 0 0 0 0 0 0 0 0 0
.o.......3.o.......5.o....... & | * 120 * * * | 0 1 1 2 2 1 0 0 0 0 0 0 0 0 | 0 0 1 2 2 1 2 2 1 2 2 0 0 0 0 0 0 0 0 0 0 0 | 0 2 1 2 1 1 2 1 0 0 0 0 0
..o......3..o......5..o...... & | * * 120 * * | 0 0 0 0 2 0 2 1 2 2 0 0 0 0 | 0 0 0 0 0 0 1 2 2 0 2 1 2 2 1 2 2 0 0 0 0 0 | 0 0 0 1 1 0 2 2 1 1 2 0 0
...o.....3...o.....5...o..... & | * * * 120 * | 0 0 0 0 0 1 0 0 2 0 2 2 1 0 | 0 0 0 0 0 0 0 0 0 2 2 0 2 1 0 0 2 1 2 2 2 0 | 0 0 0 0 0 1 2 1 0 1 2 1 2
....o....3....o....5....o.... | * * * * 60 | 0 0 0 0 0 0 0 0 0 4 0 4 0 2 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 2 4 0 2 0 2 1 | 0 0 0 0 0 0 0 0 2 2 2 0 1
------------------------------------+-------------------+-----------------------------------------------------+--------------------------------------------------------------------------------+-----------------------------------------
......... x........ ......... & | 2 0 0 0 0 | 120 * * * * * * * * * * * * * | 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0 0 0 0 0 0 0 0
oo.......3oo.......5oo.......&#x & | 1 1 0 0 0 | * 120 * * * * * * * * * * * * | 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 2 1 0 0 0 0 0 0 0 0 0 0
.x....... ......... ......... & | 0 2 0 0 0 | * * 60 * * * * * * * * * * * | 0 0 1 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 2 0 2 1 0 0 0 0 0 0 0 0
......... .x....... ......... & | 0 2 0 0 0 | * * * 120 * * * * * * * * * * | 0 0 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 1 1 0 1 1 0 0 0 0 0 0
.oo......3.oo......5.oo......&#x & | 0 1 1 0 0 | * * * * 240 * * * * * * * * * | 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 1 1 0 1 1 0 0 0 0 0
.o.o.....3.o.o.....5.o.o.....&#x & | 0 1 0 1 0 | * * * * * 120 * * * * * * * * | 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 1 2 1 0 0 0 0 0
......... ..x...... ......... & | 0 0 2 0 0 | * * * * * * 120 * * * * * * * | 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 | 0 0 0 1 0 0 1 0 1 0 1 0 0
......... ......... ..x...... & | 0 0 2 0 0 | * * * * * * * 60 * * * * * * | 0 0 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 | 0 0 0 0 1 0 0 2 0 1 0 0 0
..oo.....3..oo.....5..oo.....&#x & | 0 0 1 1 0 | * * * * * * * * 240 * * * * * | 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 0 | 0 0 0 0 0 0 1 1 0 1 1 0 0
..o.o....3..o.o....5..o.o....&#x & | 0 0 1 0 1 | * * * * * * * * * 240 * * * * | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 | 0 0 0 0 0 0 0 0 1 1 1 0 0
......... ...x..... ......... & | 0 0 0 2 0 | * * * * * * * * * * 120 * * * | 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 0 | 0 0 0 0 0 1 1 0 0 0 1 1 1
...oo....3...oo....5...oo....&#x & | 0 0 0 1 1 | * * * * * * * * * * * 240 * * | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 | 0 0 0 0 0 0 0 0 0 1 1 0 1
...o.o...3...o.o...5...o.o...&#x | 0 0 0 2 0 | * * * * * * * * * * * * 60 * | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 | 0 0 0 0 0 0 0 0 0 1 0 1 2
....x.... ......... ......... | 0 0 0 0 2 | * * * * * * * * * * * * * 60 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 | 0 0 0 0 0 0 0 0 2 1 0 0 0
------------------------------------+-------------------+-----------------------------------------------------+--------------------------------------------------------------------------------+-----------------------------------------
o........3x........ ......... & | 3 0 0 0 0 | 3 0 0 0 0 0 0 0 0 0 0 0 0 0 | 40 * * * * * * * * * * * * * * * * * * * * * | 1 1 0 0 0 0 0 0 0 0 0 0 0
......... x........5o........ & | 5 0 0 0 0 | 5 0 0 0 0 0 0 0 0 0 0 0 0 0 | * 24 * * * * * * * * * * * * * * * * * * * * | 1 0 1 0 0 0 0 0 0 0 0 0 0
ox....... ......... .........&#x & | 1 2 0 0 0 | 0 2 1 0 0 0 0 0 0 0 0 0 0 0 | * * 60 * * * * * * * * * * * * * * * * * * * | 0 2 0 0 0 0 0 0 0 0 0 0 0
......... xx....... .........&#x & | 2 2 0 0 0 | 1 2 0 1 0 0 0 0 0 0 0 0 0 0 | * * * 120 * * * * * * * * * * * * * * * * * * | 0 1 1 0 0 0 0 0 0 0 0 0 0
.x.......3.x....... ......... & | 0 6 0 0 0 | 0 0 3 3 0 0 0 0 0 0 0 0 0 0 | * * * * 40 * * * * * * * * * * * * * * * * * | 0 1 0 1 0 0 0 0 0 0 0 0 0
......... .x.......5.o....... & | 0 5 0 0 0 | 0 0 0 5 0 0 0 0 0 0 0 0 0 0 | * * * * * 24 * * * * * * * * * * * * * * * * | 0 0 1 0 0 1 0 0 0 0 0 0 0
.xo...... ......... .........&#x & | 0 2 1 0 0 | 0 0 1 0 2 0 0 0 0 0 0 0 0 0 | * * * * * * 120 * * * * * * * * * * * * * * * | 0 0 0 1 1 0 0 0 0 0 0 0 0
......... .xx...... .........&#x & | 0 2 2 0 0 | 0 0 0 1 2 0 1 0 0 0 0 0 0 0 | * * * * * * * 120 * * * * * * * * * * * * * * | 0 0 0 1 0 0 1 0 0 0 0 0 0
......... ......... .ox......&#x & | 0 1 2 0 0 | 0 0 0 0 2 0 0 1 0 0 0 0 0 0 | * * * * * * * * 120 * * * * * * * * * * * * * | 0 0 0 0 1 0 0 1 0 0 0 0 0
......... .x.x..... .........&#x & | 0 2 0 2 0 | 0 0 0 1 0 2 0 0 0 0 1 0 0 0 | * * * * * * * * * 120 * * * * * * * * * * * * | 0 0 0 0 0 1 1 0 0 0 0 0 0
.ooo.....3.ooo.....5.ooo.....&#x & | 0 1 1 1 0 | 0 0 0 0 1 1 0 0 1 0 0 0 0 0 | * * * * * * * * * * 240 * * * * * * * * * * * | 0 0 0 0 0 0 1 1 0 0 0 0 0
..o......3..x...... ......... & | 0 0 3 0 0 | 0 0 0 0 0 0 3 0 0 0 0 0 0 0 | * * * * * * * * * * * 40 * * * * * * * * * * | 0 0 0 1 0 0 0 0 1 0 0 0 0
......... ..xx..... .........&#x & | 0 0 2 2 0 | 0 0 0 0 0 0 1 0 2 0 1 0 0 0 | * * * * * * * * * * * * 120 * * * * * * * * * | 0 0 0 0 0 0 1 0 0 0 1 0 0
......... ......... ..xo.....&#x & | 0 0 2 1 0 | 0 0 0 0 0 0 0 1 2 0 0 0 0 0 | * * * * * * * * * * * * * 120 * * * * * * * * | 0 0 0 0 0 0 0 1 0 1 0 0 0
..o.x.... ......... .........&#x & | 0 0 1 0 2 | 0 0 0 0 0 0 0 0 0 2 0 0 0 1 | * * * * * * * * * * * * * * 120 * * * * * * * | 0 0 0 0 0 0 0 0 1 1 0 0 0
......... ..x.o.... .........&#x & | 0 0 2 0 1 | 0 0 0 0 0 0 1 0 0 2 0 0 0 0 | * * * * * * * * * * * * * * * 120 * * * * * * | 0 0 0 0 0 0 0 0 1 0 1 0 0
..ooo....3..ooo....5..ooo....&#x & | 0 0 1 1 1 | 0 0 0 0 0 0 0 0 1 1 0 1 0 0 | * * * * * * * * * * * * * * * * 240 * * * * * | 0 0 0 0 0 0 0 0 0 1 1 0 0
......... ...x.....5...o..... & | 0 0 0 5 0 | 0 0 0 0 0 0 0 0 0 0 5 0 0 0 | * * * * * * * * * * * * * * * * * 24 * * * * | 0 0 0 0 0 1 0 0 0 0 0 1 0
......... ...xo.... .........&#x & | 0 0 0 2 1 | 0 0 0 0 0 0 0 0 0 0 1 2 0 0 | * * * * * * * * * * * * * * * * * * 120 * * * | 0 0 0 0 0 0 0 0 0 0 1 0 1
......... ...x.x... .........&#x | 0 0 0 4 0 | 0 0 0 0 0 0 0 0 0 0 2 0 2 0 | * * * * * * * * * * * * * * * * * * * 60 * * | 0 0 0 0 0 0 0 0 0 0 0 1 1
...ooo...3...ooo...5...ooo...&#x | 0 0 0 2 1 | 0 0 0 0 0 0 0 0 0 0 0 2 1 0 | * * * * * * * * * * * * * * * * * * * * 120 * | 0 0 0 0 0 0 0 0 0 1 0 0 1
....x....3....o.... ......... | 0 0 0 0 3 | 0 0 0 0 0 0 0 0 0 0 0 0 0 3 | * * * * * * * * * * * * * * * * * * * * * 20 | 0 0 0 0 0 0 0 0 2 0 0 0 0
------------------------------------+-------------------+-----------------------------------------------------+--------------------------------------------------------------------------------+-----------------------------------------
o........3x........5o........ & | 30 0 0 0 0 | 60 0 0 0 0 0 0 0 0 0 0 0 0 0 | 20 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 * * * * * * * * * * * * id
ox.......3xx....... .........&#x & | 3 6 0 0 0 | 3 6 3 3 0 0 0 0 0 0 0 0 0 0 | 1 0 3 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | * 40 * * * * * * * * * * * tricu
......... xx.......5oo.......&#x & | 5 5 0 0 0 | 5 5 0 5 0 0 0 0 0 0 0 0 0 0 | 0 1 0 5 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | * * 24 * * * * * * * * * * pip
.xo......3.xx...... .........&#x & | 0 6 3 0 0 | 0 0 3 3 6 0 3 0 0 0 0 0 0 0 | 0 0 0 0 1 0 3 3 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | * * * 40 * * * * * * * * * tricu
.xo...... ......... .ox......&#x & | 0 2 2 0 0 | 0 0 1 0 4 0 0 1 0 0 0 0 0 0 | 0 0 0 0 0 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 | * * * * 60 * * * * * * * * tet
......... .x.x.....5.o.o.....&#x & | 0 5 0 5 0 | 0 0 0 5 0 5 0 0 0 0 5 0 0 0 | 0 0 0 0 0 1 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 | * * * * * 24 * * * * * * * pip
......... .xxx..... .........&#x & | 0 2 2 2 0 | 0 0 0 1 2 2 1 0 2 0 1 0 0 0 | 0 0 0 0 0 0 0 1 0 1 2 0 1 0 0 0 0 0 0 0 0 0 | * * * * * * 120 * * * * * * trip
......... ......... .oxo.....&#x & | 0 1 2 1 0 | 0 0 0 0 2 1 0 1 2 0 0 0 0 0 | 0 0 0 0 0 0 0 0 1 0 2 0 0 1 0 0 0 0 0 0 0 0 | * * * * * * * 120 * * * * * tet
..o.x....3..x.o.... .........&#x & | 0 0 3 0 3 | 0 0 0 0 0 0 3 0 0 6 0 0 0 3 | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 3 3 0 0 0 0 0 1 | * * * * * * * * 40 * * * * oct
..ofxfo.. ......... ..xofox..&#xt | 0 0 4 4 4 | 0 0 0 0 0 0 0 2 8 8 0 8 2 2 | 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 0 8 0 0 0 4 0 | * * * * * * * * * 30 * * * ike
......... ..xxo.... .........&#x & | 0 0 2 2 1 | 0 0 0 0 0 0 1 0 2 2 1 2 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 2 0 1 0 0 0 | * * * * * * * * * * 120 * * squippy
......... ...x.x...5...o.o...&#x | 0 0 0 10 0 | 0 0 0 0 0 0 0 0 0 0 10 0 5 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 5 0 0 | * * * * * * * * * * * 12 * pip
......... ...xox... .........&#x | 0 0 0 4 1 | 0 0 0 0 0 0 0 0 0 0 2 4 2 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 2 0 | * * * * * * * * * * * * 60 squippy
Klitzing wrote:Klitzing wrote:Klitzing wrote:Thus we then are finally left for this subsymmetry with the following 6 valid cases
- Code: Select all
(AAD) xoxFoFxox 3 oxoooooxo 5 ooxofoxoo &#xt
(A) oxofxfoxo 3 xxxxoxxxx 5 ooxofoxoo &#xt
(ADD) ooofxfooo 3 xoxxoxxox 5 ofxofoxfo &#xt
(C) oxofofoxo 3 oofoxofoo 5 xxoxxxoxx &#xt
(CD) xoxFxFxox 3 oxfoxofxo 5 xxoxxxoxx &#xt
(AC) oxofxfoxo 3 xxFxoxFxx 5 xxoxFxoxx &#xt
Here comes the first, (AAD). ...
Next then, (A): ...
ooofxfooo3xoxxoxxox5ofxofoxfo&#xt (ADD)
ED(03)=1
EC(08)=f
DC(05)=1
DB(08)=f
CB(03)=1
CA(08)=1
BA(05)=1
Bb(10)=1
E-C = bistratic id cap of rahi (id || f-doe || tid)
o........3o........5o........ & | 60 * * * * | 4 2 0 0 0 0 0 0 0 0 0 | 2 2 4 1 0 0 0 0 0 0 0 0 0 0 0 0 | 1 2 2 0 0 0 0 0 0 verf = f x3o
.o.......3.o.......5.o....... & | * 40 * * * | 0 3 3 0 0 0 0 0 0 0 0 | 0 0 3 3 3 0 0 0 0 0 0 0 0 0 0 0 | 0 1 3 1 0 0 0 0 0 verf = f x3o
..o......3..o......5..o...... & | * * 120 * * | 0 0 1 2 1 2 2 0 0 0 0 | 0 0 0 1 2 1 2 2 1 2 2 0 0 0 0 0 | 0 0 2 1 1 1 2 0 0
...o.....3...o.....5...o..... & | * * * 120 * | 0 0 0 0 0 2 0 2 2 1 0 | 0 0 0 0 0 0 2 1 0 0 2 1 2 2 2 0 | 0 0 1 0 0 1 2 1 2
....o....3....o....5....o.... | * * * * 60 | 0 0 0 0 0 0 4 0 4 0 2 | 0 0 0 0 0 0 0 0 4 2 4 0 2 0 2 1 | 0 0 0 0 2 2 2 0 1
------------------------------------+------------------+------------------------------------------+----------------------------------------------------------+---------------------------
......... x........ ......... & | 2 0 0 0 0 | 120 * * * * * * * * * * | 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0 0 0 0
oo.......3oo.......5oo.......&#x & | 1 1 0 0 0 | * 120 * * * * * * * * * | 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 2 0 0 0 0 0 0
.oo......3.oo......5.oo......&#x & | 0 1 1 0 0 | * * 120 * * * * * * * * | 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 | 0 0 2 1 0 0 0 0 0
......... ..x...... ......... & | 0 0 2 0 0 | * * * 120 * * * * * * * | 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 | 0 0 1 1 1 0 1 0 0
......... ......... ..x...... & | 0 0 2 0 0 | * * * * 60 * * * * * * | 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0 0 | 0 0 2 0 0 1 0 0 0
..oo.....3..oo.....5..oo.....&#x & | 0 0 1 1 0 | * * * * * 240 * * * * * | 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 | 0 0 1 0 0 1 1 0 0
..o.o....3..o.o....5..o.o....&#x & | 0 0 1 0 1 | * * * * * * 240 * * * * | 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 | 0 0 0 0 1 1 1 0 0
......... ...x..... ......... & | 0 0 0 2 0 | * * * * * * * 120 * * * | 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 | 0 0 1 0 0 0 1 1 1
...oo....3...oo....5...oo....&#x & | 0 0 0 1 1 | * * * * * * * * 240 * * | 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 | 0 0 0 0 0 1 1 0 1
...o.o...3...o.o...5...o.o...&#x | 0 0 0 2 0 | * * * * * * * * * 60 * | 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 | 0 0 0 0 0 1 0 1 2
....x.... ......... ......... | 0 0 0 0 2 | * * * * * * * * * * 60 | 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 | 0 0 0 0 2 1 0 0 0
------------------------------------+------------------+------------------------------------------+----------------------------------------------------------+---------------------------
o........3x........ ......... & | 3 0 0 0 0 | 3 0 0 0 0 0 0 0 0 0 0 | 40 * * * * * * * * * * * * * * * | 1 1 0 0 0 0 0 0 0
......... x........5o........ & | 5 0 0 0 0 | 5 0 0 0 0 0 0 0 0 0 0 | * 24 * * * * * * * * * * * * * * | 1 0 1 0 0 0 0 0 0
......... xo....... .........&#x & | 2 1 0 0 0 | 1 2 0 0 0 0 0 0 0 0 0 | * * 120 * * * * * * * * * * * * * | 0 1 1 0 0 0 0 0 0
......... ......... ofx......&#x & | 1 2 2 0 0 | 0 2 2 0 1 0 0 0 0 0 0 | * * * 60 * * * * * * * * * * * * | 0 0 2 0 0 0 0 0 0
......... .ox...... .........&#x & | 0 1 2 0 0 | 0 0 2 1 0 0 0 0 0 0 0 | * * * * 120 * * * * * * * * * * * | 0 0 1 1 0 0 0 0 0
..o......3..x...... ......... & | 0 0 3 0 0 | 0 0 0 3 0 0 0 0 0 0 0 | * * * * * 40 * * * * * * * * * * | 0 0 0 1 1 0 0 0 0
......... ..xx..... .........&#x & | 0 0 2 2 0 | 0 0 0 1 0 2 0 1 0 0 0 | * * * * * * 120 * * * * * * * * * | 0 0 1 0 0 0 1 0 0
......... ......... ..xo.....&#x & | 0 0 2 1 0 | 0 0 0 0 1 2 0 0 0 0 0 | * * * * * * * 120 * * * * * * * * | 0 0 1 0 0 1 0 0 0
..o.x.... ......... .........&#x & | 0 0 1 0 2 | 0 0 0 0 0 0 2 0 0 0 1 | * * * * * * * * 120 * * * * * * * | 0 0 0 0 1 1 0 0 0
......... ..x.o.... .........&#x & | 0 0 2 0 1 | 0 0 0 1 0 0 2 0 0 0 0 | * * * * * * * * * 120 * * * * * * | 0 0 0 0 1 0 1 0 0
..ooo....3..ooo....5..ooo....&#x & | 0 0 1 1 1 | 0 0 0 0 0 1 1 0 1 0 0 | * * * * * * * * * * 240 * * * * * | 0 0 0 0 0 1 1 0 0
......... ...x.....5...o..... & | 0 0 0 5 0 | 0 0 0 0 0 0 0 5 0 0 0 | * * * * * * * * * * * 24 * * * * | 0 0 1 0 0 0 0 1 0
......... ...xo.... .........&#x & | 0 0 0 2 1 | 0 0 0 0 0 0 0 1 2 0 0 | * * * * * * * * * * * * 120 * * * | 0 0 0 0 0 0 1 0 1
......... ...x.x... .........&#x | 0 0 0 4 0 | 0 0 0 0 0 0 0 2 0 2 0 | * * * * * * * * * * * * * 60 * * | 0 0 0 0 0 0 0 1 1
...ooo...3...ooo...5...ooo...&#x | 0 0 0 2 1 | 0 0 0 0 0 0 0 0 2 1 0 | * * * * * * * * * * * * * * 120 * | 0 0 0 0 0 1 0 0 1
....x....3....o.... ......... | 0 0 0 0 3 | 0 0 0 0 0 0 0 0 0 0 3 | * * * * * * * * * * * * * * * 20 | 0 0 0 0 2 0 0 0 0
------------------------------------+------------------+------------------------------------------+----------------------------------------------------------+---------------------------
o........3x........5o........ & | 30 0 0 0 0 | 60 0 0 0 0 0 0 0 0 0 0 | 20 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 * * * * * * * * id
oo.......3xo....... .........&#x & | 3 1 0 0 0 | 3 3 0 0 0 0 0 0 0 0 0 | 1 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 | * 40 * * * * * * * tet
......... xoxx.....5ofxo.....&#xt & | 5 5 10 5 0 | 5 10 10 5 5 10 0 5 0 0 0 | 0 1 5 5 5 0 5 5 0 0 0 1 0 0 0 0 | * * 24 * * * * * * pocuro
.oo......3.ox...... .........&#x & | 0 1 3 0 0 | 0 0 3 3 0 0 0 0 0 0 0 | 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 | * * * 40 * * * * * tet
..o.x....3..x.o.... .........&#x & | 0 0 3 0 3 | 0 0 0 3 0 0 6 0 0 0 3 | 0 0 0 0 0 1 0 0 3 3 0 0 0 0 0 1 | * * * * 40 * * * * oct
..ofxfo.. ......... ..xofox..&#xt | 0 0 4 4 4 | 0 0 0 0 2 8 8 0 8 2 2 | 0 0 0 0 0 0 0 4 4 0 8 0 0 0 4 0 | * * * * * 30 * * * ike
......... ..xxo.... .........&#x & | 0 0 2 2 1 | 0 0 0 1 0 2 2 1 2 0 0 | 0 0 0 0 0 0 1 0 0 1 2 0 1 0 0 0 | * * * * * * 120 * * squippy
......... ...x.x...5...o.o...&#x | 0 0 0 10 0 | 0 0 0 0 0 0 0 10 0 5 0 | 0 0 0 0 0 0 0 0 0 0 0 2 0 5 0 0 | * * * * * * * 12 * pip
......... ...xox... .........&#x | 0 0 0 4 1 | 0 0 0 0 0 0 0 2 4 2 0 | 0 0 0 0 0 0 0 0 0 0 0 0 2 1 2 0 | * * * * * * * * 60 squippy
Klitzing wrote:Klitzing wrote:Klitzing wrote:Klitzing wrote:Thus we then are finally left for this subsymmetry with the following 6 valid cases
- Code: Select all
(AAD) xoxFoFxox 3 oxoooooxo 5 ooxofoxoo &#xt
(A) oxofxfoxo 3 xxxxoxxxx 5 ooxofoxoo &#xt
(ADD) ooofxfooo 3 xoxxoxxox 5 ofxofoxfo &#xt
(C) oxofofoxo 3 oofoxofoo 5 xxoxxxoxx &#xt
(CD) xoxFxFxox 3 oxfoxofxo 5 xxoxxxoxx &#xt
(AC) oxofxfoxo 3 xxFxoxFxx 5 xxoxFxoxx &#xt
Here comes the first, (AAD). ...
Next then, (A): ...
And now (ADD): ...
oxofofoxo3oofoxofoo5xxoxxxoxx&#xt (C)
ED(03)=1
EC(08)=f
DC(05)=1
DB(08)=1
CB(03)=1
CA(08)=1
BA(05)=1
Bb(10)=1
E-D = doe || srid
o........3o........5o........ & | 40 * * * * | 3 3 0 0 0 0 0 0 0 0 0 0 0 | 3 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 3 3 0 0 0 0 0 0 0 0
.o.......3.o.......5.o....... & | * 120 * * * | 0 1 2 2 2 1 0 0 0 0 0 0 0 | 0 2 2 1 2 1 2 2 2 2 0 0 0 0 0 0 0 0 0 | 0 1 2 1 1 2 1 2 0 0 0 0
..o......3..o......5..o...... & | * * 60 * * | 0 0 0 0 4 0 4 2 0 0 0 0 0 | 0 0 0 0 0 0 2 2 0 4 2 1 4 0 0 0 0 0 0 | 0 0 0 0 2 1 0 2 2 0 0 0
...o.....3...o.....5...o..... & | * * * 120 * | 0 0 0 0 0 1 2 0 2 2 1 0 0 | 0 0 0 0 0 0 0 0 2 2 2 0 2 1 1 2 2 2 0 | 0 0 0 0 1 0 1 2 2 1 1 2
....o....3....o....5....o.... | * * * * 60 | 0 0 0 0 0 0 0 2 0 4 0 2 1 | 0 0 0 0 0 0 0 0 0 0 0 2 4 0 4 4 0 2 1 | 0 0 0 0 2 0 0 0 4 0 2 2
------------------------------------+------------------+-------------------------------------------------+---------------------------------------------------------------------+-------------------------------------
......... ......... x........ & | 2 0 0 0 0 | 60 * * * * * * * * * * * * | 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 0 1 2 0 0 0 0 0 0 0 0
oo.......3oo.......5oo.......&#x & | 1 1 0 0 0 | * 120 * * * * * * * * * * * | 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 2 1 0 0 0 0 0 0 0 0
.x....... ......... ......... & | 0 2 0 0 0 | * * 120 * * * * * * * * * * | 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 1 0 1 1 0 0 0 0 0 0
......... ......... .x....... & | 0 2 0 0 0 | * * * 120 * * * * * * * * * | 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 | 0 0 1 1 0 1 1 1 0 0 0 0
.oo......3.oo......5.oo......&#x & | 0 1 1 0 0 | * * * * 240 * * * * * * * * | 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 | 0 0 0 0 1 1 0 1 0 0 0 0
.o.o.....3.o.o.....5.o.o.....&#x & | 0 1 0 1 0 | * * * * * 120 * * * * * * * | 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 | 0 0 0 0 1 0 1 2 0 0 0 0
..oo.....3..oo.....5..oo.....&#x & | 0 0 1 1 0 | * * * * * * 240 * * * * * * | 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 | 0 0 0 0 1 0 0 1 1 0 0 0
..o.o....3..o.o....5..o.o....&#x & | 0 0 1 0 1 | * * * * * * * 120 * * * * * | 0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 | 0 0 0 0 1 0 0 0 2 0 0 0
......... ......... ...x..... & | 0 0 0 2 0 | * * * * * * * * 120 * * * * | 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 | 0 0 0 0 0 0 1 1 1 1 0 1
...oo....3...oo....5...oo....&#x & | 0 0 0 1 1 | * * * * * * * * * 240 * * * | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 | 0 0 0 0 1 0 0 0 1 0 1 1
...o.o...3...o.o...5...o.o...&#x | 0 0 0 2 0 | * * * * * * * * * * 60 * * | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 | 0 0 0 0 0 0 0 0 0 1 1 2
......... ....x.... ......... | 0 0 0 0 2 | * * * * * * * * * * * 60 * | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 | 0 0 0 0 2 0 0 0 0 0 1 0
......... ......... ....x.... | 0 0 0 0 2 | * * * * * * * * * * * * 30 | 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 4 0 0 0 | 0 0 0 0 0 0 0 0 4 0 0 2
------------------------------------+------------------+-------------------------------------------------+---------------------------------------------------------------------+-------------------------------------
......... o........5x........ & | 5 0 0 0 0 | 5 0 0 0 0 0 0 0 0 0 0 0 0 | 24 * * * * * * * * * * * * * * * * * * | 1 0 0 1 0 0 0 0 0 0 0 0
ox....... ......... .........&#x & | 1 2 0 0 0 | 0 2 1 0 0 0 0 0 0 0 0 0 0 | * 120 * * * * * * * * * * * * * * * * * | 0 1 1 0 0 0 0 0 0 0 0 0
......... ......... xx.......&#x & | 2 2 0 0 0 | 1 2 0 1 0 0 0 0 0 0 0 0 0 | * * 120 * * * * * * * * * * * * * * * * | 0 0 1 1 0 0 0 0 0 0 0 0
.x.......3.o....... ......... & | 0 3 0 0 0 | 0 0 3 0 0 0 0 0 0 0 0 0 0 | * * * 40 * * * * * * * * * * * * * * * | 0 1 0 0 1 0 0 0 0 0 0 0
.x....... ......... .x....... & | 0 4 0 0 0 | 0 0 2 2 0 0 0 0 0 0 0 0 0 | * * * * 60 * * * * * * * * * * * * * * | 0 0 1 0 0 1 0 0 0 0 0 0
......... .o.......5.x....... & | 0 5 0 0 0 | 0 0 0 5 0 0 0 0 0 0 0 0 0 | * * * * * 24 * * * * * * * * * * * * * | 0 0 0 1 0 0 1 0 0 0 0 0
.xo...... ......... .........&#x & | 0 2 1 0 0 | 0 0 1 0 2 0 0 0 0 0 0 0 0 | * * * * * * 120 * * * * * * * * * * * * | 0 0 0 0 1 1 0 0 0 0 0 0
......... ......... .xo......&#x & | 0 2 1 0 0 | 0 0 0 1 2 0 0 0 0 0 0 0 0 | * * * * * * * 120 * * * * * * * * * * * | 0 0 0 0 0 1 0 1 0 0 0 0
......... ......... .x.x.....&#x & | 0 2 0 2 0 | 0 0 0 1 0 2 0 0 1 0 0 0 0 | * * * * * * * * 120 * * * * * * * * * * | 0 0 0 0 0 0 1 1 0 0 0 0
.ooo.....3.ooo.....5.ooo.....&#x & | 0 1 1 1 0 | 0 0 0 0 1 1 1 0 0 0 0 0 0 | * * * * * * * * * 240 * * * * * * * * * | 0 0 0 0 1 0 0 1 0 0 0 0
......... ......... ..ox.....&#x & | 0 0 1 2 0 | 0 0 0 0 0 0 2 0 1 0 0 0 0 | * * * * * * * * * * 120 * * * * * * * * | 0 0 0 0 0 0 0 1 1 0 0 0
......... ......... ..o.x....&#x & | 0 0 1 0 2 | 0 0 0 0 0 0 0 2 0 0 0 0 1 | * * * * * * * * * * * 60 * * * * * * * | 0 0 0 0 0 0 0 0 2 0 0 0
..ooo....3..ooo....5..ooo....&#x & | 0 0 1 1 1 | 0 0 0 0 0 0 1 1 0 1 0 0 0 | * * * * * * * * * * * * 240 * * * * * * | 0 0 0 0 1 0 0 0 1 0 0 0
......... ...o.....5...x..... & | 0 0 0 5 0 | 0 0 0 0 0 0 0 0 5 0 0 0 0 | * * * * * * * * * * * * * 24 * * * * * | 0 0 0 0 0 0 1 0 0 1 0 0
......... ...ox.... .........&#x & | 0 0 0 1 2 | 0 0 0 0 0 0 0 0 0 2 0 1 0 | * * * * * * * * * * * * * * 120 * * * * | 0 0 0 0 1 0 0 0 0 0 1 0
......... ......... ...xx....&#x & | 0 0 0 2 2 | 0 0 0 0 0 0 0 0 1 2 0 0 1 | * * * * * * * * * * * * * * * 120 * * * | 0 0 0 0 0 0 0 0 1 0 0 1
......... ......... ...x.x...&#x | 0 0 0 4 0 | 0 0 0 0 0 0 0 0 2 0 2 0 0 | * * * * * * * * * * * * * * * * 60 * * | 0 0 0 0 0 0 0 0 0 1 0 1
...ooo...3...ooo...5...ooo...&#x | 0 0 0 2 1 | 0 0 0 0 0 0 0 0 0 2 1 0 0 | * * * * * * * * * * * * * * * * * 120 * | 0 0 0 0 0 0 0 0 0 0 1 1
....o....3....x.... ......... | 0 0 0 0 3 | 0 0 0 0 0 0 0 0 0 0 0 3 0 | * * * * * * * * * * * * * * * * * * 20 | 0 0 0 0 2 0 0 0 0 0 0 0
------------------------------------+------------------+-------------------------------------------------+---------------------------------------------------------------------+-------------------------------------
o........3o........5x........ & | 20 0 0 0 0 | 30 0 0 0 0 0 0 0 0 0 0 0 0 | 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 * * * * * * * * * * * doe
ox.......3oo....... .........&#x & | 1 3 0 0 0 | 0 3 3 0 0 0 0 0 0 0 0 0 0 | 0 3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | * 40 * * * * * * * * * * tet
ox....... ......... xx.......&#x & | 2 4 0 0 0 | 1 4 2 2 0 0 0 0 0 0 0 0 0 | 0 2 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | * * 60 * * * * * * * * * trip
......... oo.......5xx.......&#x & | 5 5 0 0 0 | 5 5 0 5 0 0 0 0 0 0 0 0 0 | 1 0 5 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | * * * 24 * * * * * * * * pip
.xofo.... .ofox.... .........&#xt & | 0 3 3 3 3 | 0 0 3 0 6 3 6 3 0 6 0 3 0 | 0 0 0 1 0 0 3 0 0 6 0 0 6 0 3 0 0 0 1 | * * * * 40 * * * * * * * ike
.xo...... ......... .xo......&#x & | 0 4 1 0 0 | 0 0 2 2 4 0 0 0 0 0 0 0 0 | 0 0 0 0 1 0 2 2 0 0 0 0 0 0 0 0 0 0 0 | * * * * * 60 * * * * * * squippy
......... .o.o.....5.x.x.....&#x & | 0 5 0 5 0 | 0 0 0 5 0 5 0 0 5 0 0 0 0 | 0 0 0 0 0 1 0 0 5 0 0 0 0 1 0 0 0 0 0 | * * * * * * 24 * * * * * pip
......... ......... .xox.....&#x & | 0 2 1 2 0 | 0 0 0 1 2 2 2 0 1 0 0 0 0 | 0 0 0 0 0 0 0 1 1 2 1 0 0 0 0 0 0 0 0 | * * * * * * * 120 * * * * squippy
......... ......... ..oxx....&#x & | 0 0 1 2 2 | 0 0 0 0 0 0 2 2 1 2 0 0 1 | 0 0 0 0 0 0 0 0 0 0 1 1 2 0 0 1 0 0 0 | * * * * * * * * 120 * * * squippy
......... ...o.o...5...x.x...&#x | 0 0 0 10 0 | 0 0 0 0 0 0 0 0 10 0 5 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 5 0 0 | * * * * * * * * * 12 * * pip
......... ...oxo... .........&#x | 0 0 0 2 2 | 0 0 0 0 0 0 0 0 0 4 1 1 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 2 0 | * * * * * * * * * * 60 * tet
......... ......... ...xxx...&#x | 0 0 0 4 2 | 0 0 0 0 0 0 0 0 2 4 2 0 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 2 0 | * * * * * * * * * * * 60 trip
This remark just gave me a sudden realization of great truth!Klitzing wrote:[...] Esp. all hyperplanes of vertex layers still are unchanged!
Right this latter fact surely is valid for the other 5 such derived figures too. But would not hold for the ones student91 brought in, i.e. which do contain some (B)-transformation.
--- rk
Klitzing wrote:Klitzing wrote:Klitzing wrote:Klitzing wrote:Klitzing wrote:Thus we then are finally left for this subsymmetry with the following 6 valid cases
- Code: Select all
(AAD) xoxFoFxox 3 oxoooooxo 5 ooxofoxoo &#xt
(A) oxofxfoxo 3 xxxxoxxxx 5 ooxofoxoo &#xt
(ADD) ooofxfooo 3 xoxxoxxox 5 ofxofoxfo &#xt
(C) oxofofoxo 3 oofoxofoo 5 xxoxxxoxx &#xt
(CD) xoxFxFxox 3 oxfoxofxo 5 xxoxxxoxx &#xt
(AC) oxofxfoxo 3 xxFxoxFxx 5 xxoxFxoxx &#xt
Here comes the first, (AAD). ...
Next then, (A): ...
And now (ADD): ...
Furthermore (C): ...
xoxFxFxox3oxfoxofxo5xxoxxxoxx&#xt (CD)
ED(03)=1
EC(08)=f
DC(05)=1
DB(08)=f
CB(03)=1
CA(08)=1
BA(05)=1
Bb(10)=1
E-D = srid || tid
o........3o........5o........ & | 120 * * * * | 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 | 1 2 1 2 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 2 1 0 0 0 0 0 0 0
.o.......3.o.......5.o....... & | * 120 * * * | 0 0 2 2 1 2 0 0 0 0 0 0 0 0 0 | 0 0 0 1 2 2 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 | 0 1 1 2 1 1 0 0 0 0 0
..o......3..o......5..o...... & | * * 120 * * | 0 0 0 0 0 2 1 2 2 0 0 0 0 0 0 | 0 0 0 0 0 0 0 2 2 1 1 2 1 2 0 0 0 0 0 0 0 | 0 0 0 1 2 1 1 1 0 0 0
...o.....3...o.....5...o..... & | * * * 120 * | 0 0 0 0 0 0 0 2 0 2 2 1 0 0 0 | 0 0 0 0 0 0 0 0 1 0 2 0 0 2 1 1 2 2 2 0 0 | 0 0 0 1 1 0 0 2 1 1 2
....o....3....o....5....o.... | * * * * 120 | 0 0 0 0 0 0 0 0 2 0 2 0 1 1 1 | 0 0 0 0 0 0 0 0 0 0 0 2 2 2 0 2 2 0 1 1 1 | 0 0 0 0 2 0 2 2 0 1 1
------------------------------------+---------------------+-------------------------------------------------------+-----------------------------------------------------------------------------+---------------------------------
x........ ......... ......... & | 2 0 0 0 0 | 120 * * * * * * * * * * * * * * | 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0 0 0 0 0 0
......... ......... x........ & | 2 0 0 0 0 | * 120 * * * * * * * * * * * * * | 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 0 1 1 0 0 0 0 0 0 0
oo.......3oo.......5oo.......&#x & | 1 1 0 0 0 | * * 240 * * * * * * * * * * * * | 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 1 1 0 0 0 0 0 0 0
......... .x....... ......... & | 0 2 0 0 0 | * * * 120 * * * * * * * * * * * | 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 0 1 1 0 0 0 0 0 0
......... ......... .x....... & | 0 2 0 0 0 | * * * * 60 * * * * * * * * * * | 0 0 0 0 0 2 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 | 0 0 1 2 0 1 0 0 0 0 0
.oo......3.oo......5.oo......&#x & | 0 1 1 0 0 | * * * * * 240 * * * * * * * * * | 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 1 1 1 0 0 0 0 0
..x...... ......... ......... & | 0 0 2 0 0 | * * * * * * 60 * * * * * * * * | 0 0 0 0 0 0 0 2 0 0 0 2 0 0 0 0 0 0 0 0 0 | 0 0 0 0 2 1 1 0 0 0 0
..oo.....3..oo.....5..oo.....&#x & | 0 0 1 1 0 | * * * * * * * 240 * * * * * * * | 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 | 0 0 0 1 1 0 0 1 0 0 0
..o.o....3..o.o....5..o.o....&#x & | 0 0 1 0 1 | * * * * * * * * 240 * * * * * * | 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 | 0 0 0 0 1 0 1 1 0 0 0
......... ......... ...x..... & | 0 0 0 2 0 | * * * * * * * * * 120 * * * * * | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 | 0 0 0 1 0 0 0 1 1 0 1
...oo....3...oo....5...oo....&#x & | 0 0 0 1 1 | * * * * * * * * * * 240 * * * * | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 | 0 0 0 0 1 0 0 1 0 1 1
...o.o...3...o.o...5...o.o...&#x & | 0 0 0 2 0 | * * * * * * * * * * * 60 * * * | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 | 0 0 0 0 0 0 0 0 1 1 2
....x.... ......... ......... | 0 0 0 0 2 | * * * * * * * * * * * * 60 * * | 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 2 0 0 0 1 1 | 0 0 0 0 2 0 2 0 0 0 0
......... ....x.... ......... | 0 0 0 0 2 | * * * * * * * * * * * * * 60 * | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 0 | 0 0 0 0 2 0 0 0 0 1 0
......... ......... ....x.... | 0 0 0 0 2 | * * * * * * * * * * * * * * 60 | 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 2 0 0 0 1 | 0 0 0 0 0 0 2 2 0 0 1
------------------------------------+---------------------+-------------------------------------------------------+-----------------------------------------------------------------------------+---------------------------------
x........3o........ ......... & | 3 0 0 0 0 | 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 40 * * * * * * * * * * * * * * * * * * * * | 1 1 0 0 0 0 0 0 0 0 0
x........ ......... x........ & | 4 0 0 0 0 | 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 | * 60 * * * * * * * * * * * * * * * * * * * | 1 0 1 0 0 0 0 0 0 0 0
......... o........5x........ & | 5 0 0 0 0 | 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 | * * 24 * * * * * * * * * * * * * * * * * * | 1 0 0 1 0 0 0 0 0 0 0
xo....... ......... .........&#x & | 2 1 0 0 0 | 1 0 2 0 0 0 0 0 0 0 0 0 0 0 0 | * * * 120 * * * * * * * * * * * * * * * * * | 0 1 1 0 0 0 0 0 0 0 0
......... ox....... .........&#x & | 1 2 0 0 0 | 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 | * * * * 120 * * * * * * * * * * * * * * * * | 0 1 0 1 0 0 0 0 0 0 0
......... ......... xx.......&#x & | 2 2 0 0 0 | 0 1 2 0 1 0 0 0 0 0 0 0 0 0 0 | * * * * * 120 * * * * * * * * * * * * * * * | 0 0 1 1 0 0 0 0 0 0 0
.o.......3.x....... ......... & | 0 3 0 0 0 | 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 | * * * * * * 40 * * * * * * * * * * * * * * | 0 1 0 0 1 0 0 0 0 0 0
.ox...... ......... .........&#x & | 0 1 2 0 0 | 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 | * * * * * * * 120 * * * * * * * * * * * * * | 0 0 0 0 1 1 0 0 0 0 0
......... .xfo..... .........&#xt & | 0 2 2 1 0 | 0 0 0 1 0 2 0 2 0 0 0 0 0 0 0 | * * * * * * * * 120 * * * * * * * * * * * * | 0 0 0 1 1 0 0 0 0 0 0
......... ......... .xo......&#x & | 0 2 1 0 0 | 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 | * * * * * * * * * 120 * * * * * * * * * * * | 0 0 0 1 0 1 0 0 0 0 0
......... ......... ..ox.....&#x & | 0 0 1 2 0 | 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 | * * * * * * * * * * 120 * * * * * * * * * * | 0 0 0 1 0 0 0 1 0 0 0
..x.x.... ......... .........&#x & | 0 0 2 0 2 | 0 0 0 0 0 0 1 0 2 0 0 0 1 0 0 | * * * * * * * * * * * 120 * * * * * * * * * | 0 0 0 0 1 0 1 0 0 0 0
......... ......... ..o.x....&#x & | 0 0 1 0 2 | 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1 | * * * * * * * * * * * * 120 * * * * * * * * | 0 0 0 0 0 0 1 1 0 0 0
..ooo....3..ooo....5..ooo....&#x & | 0 0 1 1 1 | 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 | * * * * * * * * * * * * * 240 * * * * * * * | 0 0 0 0 1 0 0 1 0 0 0
......... ...o.....5...x..... & | 0 0 0 5 0 | 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 | * * * * * * * * * * * * * * 24 * * * * * * | 0 0 0 1 0 0 0 0 1 0 0
......... ...ox.... .........&#x & | 0 0 0 1 2 | 0 0 0 0 0 0 0 0 0 0 2 0 0 1 0 | * * * * * * * * * * * * * * * 120 * * * * * | 0 0 0 0 1 0 0 0 0 1 0
......... ......... ...xx....&#x & | 0 0 0 2 2 | 0 0 0 0 0 0 0 0 0 1 2 0 0 0 1 | * * * * * * * * * * * * * * * * 120 * * * * | 0 0 0 0 0 0 0 1 0 0 1
......... ......... ...x.x...&#x | 0 0 0 4 0 | 0 0 0 0 0 0 0 0 0 2 0 2 0 0 0 | * * * * * * * * * * * * * * * * * 60 * * * | 0 0 0 0 0 0 0 0 1 0 1
...ooo...3...ooo...5...ooo...&#x | 0 0 0 2 1 | 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 | * * * * * * * * * * * * * * * * * * 120 * * | 0 0 0 0 0 0 0 0 0 1 1
....x....3....x.... ......... | 0 0 0 0 6 | 0 0 0 0 0 0 0 0 0 0 0 0 3 3 0 | * * * * * * * * * * * * * * * * * * * 20 * | 0 0 0 0 2 0 0 0 0 0 0
....x.... ......... ....x.... | 0 0 0 0 4 | 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 | * * * * * * * * * * * * * * * * * * * * 30 | 0 0 0 0 0 0 2 0 0 0 0
------------------------------------+---------------------+-------------------------------------------------------+-----------------------------------------------------------------------------+---------------------------------
x........3o........5x........ & | 60 0 0 0 0 | 60 60 0 0 0 0 0 0 0 0 0 0 0 0 0 | 20 30 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 * * * * * * * * * * srid
xo.......3ox....... .........&#x & | 3 3 0 0 0 | 3 0 6 3 0 0 0 0 0 0 0 0 0 0 0 | 1 0 0 3 3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | * 40 * * * * * * * * * oct
xo....... ......... xx.......&#x & | 4 2 0 0 0 | 2 2 4 0 1 0 0 0 0 0 0 0 0 0 0 | 0 1 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | * * 60 * * * * * * * * trip
......... oxfo.....5xxox.....&#xt & | 5 10 5 5 0 | 0 5 10 5 5 10 0 10 0 5 0 0 0 0 0 | 0 0 1 0 5 5 0 0 5 5 5 0 0 0 1 0 0 0 0 0 0 | * * * 24 * * * * * * * pocuro
.oxFx....3.xfox.... .........&#x & | 0 3 6 3 6 | 0 0 0 3 0 6 3 6 6 0 6 0 3 3 0 | 0 0 0 0 0 0 1 3 3 0 0 3 0 6 0 3 0 0 0 1 0 | * * * * 40 * * * * * * thawro
.ox...... ......... .xo......&#x & | 0 2 2 0 0 | 0 0 0 0 1 4 1 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 | * * * * * 60 * * * * * tet
..x.x.... ......... ..o.x....&#x & | 0 0 2 0 4 | 0 0 0 0 0 0 1 0 4 0 0 0 2 0 2 | 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 1 | * * * * * * 60 * * * * trip
......... ......... ..oxx....&#x & | 0 0 1 2 2 | 0 0 0 0 0 0 0 2 2 1 2 0 0 0 1 | 0 0 0 0 0 0 0 0 0 0 1 0 1 2 0 0 1 0 0 0 0 | * * * * * * * 120 * * * squippy
......... ...o.o...5...x.x...&#x | 0 0 0 10 0 | 0 0 0 0 0 0 0 0 0 10 0 5 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 5 0 0 0 | * * * * * * * * 12 * * pip
......... ...oxo... .........&#x | 0 0 0 2 2 | 0 0 0 0 0 0 0 0 0 0 4 1 0 1 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 2 0 0 | * * * * * * * * * 60 * tet
......... ......... ...xxx...&#x | 0 0 0 4 2 | 0 0 0 0 0 0 0 0 0 2 4 2 0 0 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 2 0 0 | * * * * * * * * * * 60 trip
Klitzing wrote:Klitzing wrote:Klitzing wrote:Klitzing wrote:Klitzing wrote:Klitzing wrote:Thus we then are finally left for this subsymmetry with the following 6 valid cases
- Code: Select all
(AAD) xoxFoFxox 3 oxoooooxo 5 ooxofoxoo &#xt
(A) oxofxfoxo 3 xxxxoxxxx 5 ooxofoxoo &#xt
(ADD) ooofxfooo 3 xoxxoxxox 5 ofxofoxfo &#xt
(C) oxofofoxo 3 oofoxofoo 5 xxoxxxoxx &#xt
(CD) xoxFxFxox 3 oxfoxofxo 5 xxoxxxoxx &#xt
(AC) oxofxfoxo 3 xxFxoxFxx 5 xxoxFxoxx &#xt
Here comes the first, (AAD). ...
Next then, (A): ...
And now (ADD): ...
Furthermore (C): ...
Now continuing with (CD): ...
oxofxfoxo3xxFxoxFxx5xxoxFxoxx&#xt (AC)
ED(03)=1
EC(08)=f
DC(05)=1
DB(08)=1
CB(03)=1
CA(08)=f
BA(05)=1
Bb(10)=1
E-D = tid || grid
D-d = squashed hexaconta-sphenated id-first deep parabidiminished rahi
o........3o........5o........ & | 120 * * * * | 2 1 2 0 0 0 0 0 0 0 0 0 0 0 | 1 2 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 2 0 0 0 0 0 0 0
.o.......3.o.......5.o....... & | * 240 * * * | 0 0 1 1 1 1 1 1 0 0 0 0 0 0 | 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 | 0 1 1 1 1 1 1 1 0 0 0
..o......3..o......5..o...... & | * * 60 * * | 0 0 0 0 0 0 4 0 4 0 0 0 0 0 | 0 0 0 0 0 0 0 0 2 2 0 0 4 2 2 0 0 0 0 0 0 | 0 0 0 0 2 1 0 2 1 0 0
...o.....3...o.....5...o..... & | * * * 240 * | 0 0 0 0 0 0 0 1 1 1 1 1 1 0 | 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 | 0 0 0 0 1 0 1 1 1 1 1
....o....3....o....5....o.... | * * * * 60 | 0 0 0 0 0 0 0 0 0 0 0 4 0 2 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 2 0 0 2 1 | 0 0 0 0 2 0 0 0 2 1 0
------------------------------------+-------------------+-------------------------------------------------------+----------------------------------------------------------------------------+---------------------------------
......... x........ ......... & | 2 0 0 0 0 | 120 * * * * * * * * * * * * * | 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 0 1 0 0 0 0 0 0 0
......... ......... x........ & | 2 0 0 0 0 | * 60 * * * * * * * * * * * * | 0 2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 0 1 2 0 0 0 0 0 0 0
oo.......3oo.......5oo.......&#x & | 1 1 0 0 0 | * * 240 * * * * * * * * * * * | 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 1 1 0 0 0 0 0 0 0
.x....... ......... ......... & | 0 2 0 0 0 | * * * 120 * * * * * * * * * * | 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 1 0 1 1 0 0 0 0 0
......... .x....... ......... & | 0 2 0 0 0 | * * * * 120 * * * * * * * * * | 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 | 0 1 0 1 1 0 1 0 0 0 0
......... ......... .x....... & | 0 2 0 0 0 | * * * * * 120 * * * * * * * * | 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 | 0 0 1 1 0 1 1 1 0 0 0
.oo......3.oo......5.oo......&#x & | 0 1 1 0 0 | * * * * * * 240 * * * * * * * | 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 | 0 0 0 0 1 1 0 1 0 0 0
.o.o.....3.o.o.....5.o.o.....&#x & | 0 1 0 1 0 | * * * * * * * 240 * * * * * * | 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 | 0 0 0 0 1 0 1 1 0 0 0
..oo.....3..oo.....5..oo.....&#x & | 0 0 1 1 0 | * * * * * * * * 240 * * * * * | 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 | 0 0 0 0 1 0 0 1 1 0 0
......... ...x..... ......... & | 0 0 0 2 0 | * * * * * * * * * 120 * * * * | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 | 0 0 0 0 1 0 1 0 0 1 1
......... ......... ...x..... & | 0 0 0 2 0 | * * * * * * * * * * 120 * * * | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 | 0 0 0 0 0 0 1 1 1 0 1
...oo....3...oo....5...oo....&#x & | 0 0 0 1 1 | * * * * * * * * * * * 240 * * | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 | 0 0 0 0 1 0 0 0 1 1 0
...o.o...3...o.o...5...o.o...&#x | 0 0 0 2 0 | * * * * * * * * * * * * 120 * | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 | 0 0 0 0 0 0 0 0 1 1 1
....x.... ......... ......... | 0 0 0 0 2 | * * * * * * * * * * * * * 60 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1 | 0 0 0 0 2 0 0 0 1 0 0
------------------------------------+-------------------+-------------------------------------------------------+----------------------------------------------------------------------------+---------------------------------
o........3x........ ......... & | 3 0 0 0 0 | 3 0 0 0 0 0 0 0 0 0 0 0 0 0 | 40 * * * * * * * * * * * * * * * * * * * * | 1 1 0 0 0 0 0 0 0 0 0
......... x........5x........ & | 10 0 0 0 0 | 5 5 0 0 0 0 0 0 0 0 0 0 0 0 | * 24 * * * * * * * * * * * * * * * * * * * | 1 0 0 1 0 0 0 0 0 0 0
ox....... ......... .........&#x & | 1 2 0 0 0 | 0 0 2 1 0 0 0 0 0 0 0 0 0 0 | * * 120 * * * * * * * * * * * * * * * * * * | 0 1 1 0 0 0 0 0 0 0 0
......... xx....... .........&#x & | 2 2 0 0 0 | 1 0 2 0 1 0 0 0 0 0 0 0 0 0 | * * * 120 * * * * * * * * * * * * * * * * * | 0 1 0 1 0 0 0 0 0 0 0
......... ......... xx.......&#x & | 2 2 0 0 0 | 0 1 2 0 0 1 0 0 0 0 0 0 0 0 | * * * * 120 * * * * * * * * * * * * * * * * | 0 0 1 1 0 0 0 0 0 0 0
.x.......3.x....... ......... & | 0 6 0 0 0 | 0 0 0 3 3 0 0 0 0 0 0 0 0 0 | * * * * * 40 * * * * * * * * * * * * * * * | 0 1 0 0 1 0 0 0 0 0 0
.x....... ......... .x....... & | 0 4 0 0 0 | 0 0 0 2 0 2 0 0 0 0 0 0 0 0 | * * * * * * 60 * * * * * * * * * * * * * * | 0 0 1 0 0 1 0 0 0 0 0
......... .x.......5.x....... & | 0 10 0 0 0 | 0 0 0 0 5 5 0 0 0 0 0 0 0 0 | * * * * * * * 24 * * * * * * * * * * * * * | 0 0 0 1 0 0 1 0 0 0 0
.xo...... ......... .........&#x & | 0 2 1 0 0 | 0 0 0 1 0 0 2 0 0 0 0 0 0 0 | * * * * * * * * 120 * * * * * * * * * * * * | 0 0 0 0 1 1 0 0 0 0 0
......... ......... .xo......&#x & | 0 2 1 0 0 | 0 0 0 0 0 1 2 0 0 0 0 0 0 0 | * * * * * * * * * 120 * * * * * * * * * * * | 0 0 0 0 0 1 0 1 0 0 0
......... .x.x..... .........&#x & | 0 2 0 2 0 | 0 0 0 0 1 0 0 2 0 1 0 0 0 0 | * * * * * * * * * * 120 * * * * * * * * * * | 0 0 0 0 1 0 1 0 0 0 0
......... ......... .x.x.....&#x & | 0 2 0 2 0 | 0 0 0 0 0 1 0 2 0 0 1 0 0 0 | * * * * * * * * * * * 120 * * * * * * * * * | 0 0 0 0 0 0 1 1 0 0 0
.ooo.....3.ooo.....5.ooo.....&#x & | 0 1 1 1 0 | 0 0 0 0 0 0 1 1 1 0 0 0 0 0 | * * * * * * * * * * * * 240 * * * * * * * * | 0 0 0 0 1 0 0 1 0 0 0
......... ......... ..ox.....&#x & | 0 0 1 2 0 | 0 0 0 0 0 0 0 0 2 0 1 0 0 0 | * * * * * * * * * * * * * 120 * * * * * * * | 0 0 0 0 0 0 0 1 1 0 0
..ofx.... ......... .........&#x & | 0 0 1 2 2 | 0 0 0 0 0 0 0 0 2 0 0 2 0 1 | * * * * * * * * * * * * * * 120 * * * * * * | 0 0 0 0 1 0 0 0 1 0 0
......... ...x.....5...x.....&#x & | 0 0 0 10 0 | 0 0 0 0 0 0 0 0 0 5 5 0 0 0 | * * * * * * * * * * * * * * * 24 * * * * * | 0 0 0 0 0 0 1 0 0 0 1
......... ...xo.... .........&#x & | 0 0 0 2 1 | 0 0 0 0 0 0 0 0 0 1 0 2 0 0 | * * * * * * * * * * * * * * * * 120 * * * * | 0 0 0 0 1 0 0 0 0 1 0
......... ...x.x... .........&#x | 0 0 0 4 0 | 0 0 0 0 0 0 0 0 0 2 0 0 2 0 | * * * * * * * * * * * * * * * * * 60 * * * | 0 0 0 0 0 0 0 0 0 1 1
......... ......... ...x.x...&#x | 0 0 0 4 0 | 0 0 0 0 0 0 0 0 0 0 2 0 2 0 | * * * * * * * * * * * * * * * * * * 60 * * | 0 0 0 0 0 0 0 0 1 0 1
...ooo...3...ooo...5...ooo...&#x | 0 0 0 2 1 | 0 0 0 0 0 0 0 0 0 0 0 2 1 0 | * * * * * * * * * * * * * * * * * * * 120 * | 0 0 0 0 0 0 0 0 1 1 0
....x....3....o.... ......... | 0 0 0 0 3 | 0 0 0 0 0 0 0 0 0 0 0 0 0 3 | * * * * * * * * * * * * * * * * * * * * 20 | 0 0 0 0 2 0 0 0 0 0 0
------------------------------------+-------------------+-------------------------------------------------------+----------------------------------------------------------------------------+---------------------------------
o........3x........5x........ & | 60 0 0 0 0 | 60 30 0 0 0 0 0 0 0 0 0 0 0 0 | 20 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 * * * * * * * * * * tid
ox.......3xx....... .........&#x & | 3 6 0 0 0 | 3 0 6 3 3 0 0 0 0 0 0 0 0 0 | 1 0 3 3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | * 40 * * * * * * * * * tricu
ox....... ......... xx.......&#x & | 2 4 0 0 0 | 0 1 4 2 0 2 0 0 0 0 0 0 0 0 | 0 0 2 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | * * 60 * * * * * * * * trip
......... xx.......5xx.......&#x & | 10 10 0 0 0 | 5 5 10 0 5 5 0 0 0 0 0 0 0 0 | 0 1 0 5 5 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | * * * 24 * * * * * * * dip
.xofx....3.xFxo.... .........&#xt & | 0 6 3 6 3 | 0 0 0 3 3 0 6 6 6 3 0 6 0 3 | 0 0 0 0 0 1 0 0 3 0 3 0 6 0 3 0 3 0 0 0 1 | * * * * 40 * * * * * * thawro
.xo...... ......... .xo......&#x & | 0 4 1 0 0 | 0 0 0 2 0 2 4 0 0 0 0 0 0 0 | 0 0 0 0 0 0 1 0 2 2 0 0 0 0 0 0 0 0 0 0 0 | * * * * * 60 * * * * * squippy
......... .x.x.....5.x.x.....&#x & | 0 10 0 10 0 | 0 0 0 0 5 5 0 10 0 5 5 0 0 0 | 0 0 0 0 0 0 0 1 0 0 5 5 0 0 0 1 0 0 0 0 0 | * * * * * * 24 * * * * dip
......... ......... .xox.....&#x & | 0 2 1 2 0 | 0 0 0 0 0 1 2 2 2 0 1 0 0 0 | 0 0 0 0 0 0 0 0 0 1 0 1 2 1 0 0 0 0 0 0 0 | * * * * * * * 120 * * * squippy
..ofxfo.. ......... ..oxFxo..&#xt | 0 0 2 8 4 | 0 0 0 0 0 0 0 0 8 0 4 8 4 2 | 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 0 0 0 2 4 0 | * * * * * * * * 30 * * bilbiro
......... ...xox... .........&#x | 0 0 0 4 1 | 0 0 0 0 0 0 0 0 0 2 0 4 2 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 2 0 | * * * * * * * * * 60 * squippy
......... ...x.x...5...x.x...&#x | 0 0 0 20 0 | 0 0 0 0 0 0 0 0 0 10 10 0 10 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 5 5 0 0 | * * * * * * * * * * 12 dip
Klitzing wrote:student91 wrote:... You take some representation of ex, e.g. [[5,2,5]] oxofox5ooxofx xofoxo5xfoxoo&#zx, ...
The mentioned representation seems to be wrong, I fear.
Just calculated the inter-layer lacing edge lengths. For your representation I'd get:
- Code: Select all
1-2 = 5-6 = 1
1-3 = 4-6 = 1
1-4 = 3-6 = f
1-5 = 2-6 = f
1-6 = fq = 2.288246
2-3 = 4-5 = 1
2-4 = 3-5 = rt[1-1/rt5] = 0.743496
2-5 = rt[1+1/rt5] = 1.203002
3-4 = rt[1+1/rt5]
Esp. because we were looking onto a convex unit-edged polychoron here (ex), lacing "edges" (or vertex distances) of (absolute) size <1 cannot be allowed.
--- rk
student91 wrote:I already did this too, then using [5,3,2]-symmetry instead of [5,3]. This allows one more transition: B: f3o5o x -> f3o5o (-x)Klitzing wrote:So far have not gone completely thru your longer mails, aiming to provide a systematic access to demicubic resp. pentic subsymmetric cases. Remeins to be done, esp. to be checked independently.
In the meantime i considered the same access to axial icosahedral subsymmetry.
This then gives some more possible expansions:
[...]
Though I'm not sure a non-CRF thing might give a CRF-thing, the things that are derived from CRF-things are:
- Code: Select all
(AABD):xoxFo3oxooo5ooxof BCFox
(AB):oxofx3xxxxo5ooxof BCFox
(ABDD): ooofx3xoxxo5ofxof BCFox
(BC):oxofx3oofox5xFoxo BCFox
(BCD):xoxFx3oxfox5xxoxx BCFox
(ABC):oxofx3xxFxo5xFoxF BCFox
A=a = o
B-b = o+x = x
C-c = x+v = f
D-d = f+x = F
E-e = F+v=2f = V
E: V2o3o5o
D: F2x3o5o -> F2(-x)3x5o (D) -> F2o3(-x)5f (DD)
C: f2o3o5x -> f2o3f5(-x) (C)
B: x2f3o5o -> (-x)2f3o5o (B)
A: o2o3x5o -> o2x3(-x)5f (A) -> o2(-x)3o5f (AA)
That one is just the same as the single axial expansion we discovered earlier. Thus it is CRF, but not newKlitzing wrote:student91 wrote:[...]
Okay, let's have a look into that slightly broader setup, which uses a further edge flip within the orthogonal direction of the axis of symmetry.
First we remind onto the corresponding heights between corresponding layers in ex:
- Code: Select all
A=a = o
B-b = o+x = x
C-c = x+v = f
D-d = f+x = F
E-e = F+v=2f = V
additionally let's use here then the following elongated sizes A:=F+x=f+2x, B:=V+x=2f+x.
Now consider the transition from [3,5] to [2,3,5]. It thus translates into a notational change as follows:
ex = oxofofoxo3ooooxoooo5ooxoooxoo&#xt = VFfxo2oxofo3oooox5ooxoo&#zx
Then remind the possible quirks of each layer, rewritten to that subsymmetry:
- Code: Select all
E: V2o3o5o
D: F2x3o5o -> F2(-x)3x5o (D) -> F2o3(-x)5f (DD)
C: f2o3o5x -> f2o3f5(-x) (C)
B: x2f3o5o -> (-x)2f3o5o (B)
A: o2o3x5o -> o2x3(-x)5f (A) -> o2(-x)3o5f (AA)
The cases without (B) already have been done in my former mails. So here we only will have to consider cases where (B) always is applied. This provides us the following further possibilities:
(+x)2.3.5.
BAFox2oxofo3oooox5ooxoo&#zx (B)
This one needs to be investigated, as I discarted all things that weren't CRF without the B(+x)2(+x)3.5.
BAFox2xoxFx3oxoox5ooxoo&#zx (BD)
This one is not CRF: the thing without B is the (augmented) castellated prism, and these bilbiroes don't allow a center-switch x2x||o2F||f2o||o2F||x2x => x2x||f2o||o2F||f2o||x2x, this is not CRF. Alternately one can see that the layer 4 vertices will become isolated-> u (AAB)
BAFox2xoxFo3oxooo5ooxof&#zx (AABD)
this one is CRF. Lacing edges are: length x: 1-2, 2-3 3-5 and 4-5, length f:1-3, 2-5 and 3-4, length F:1-4, q*f:1-5, length sqrt(f+u):2-4(+x)2.3(+x)5.
BAFox2oxofx3xxxxo5ooxof&#zx (AB)
This one is CRF again, its lacing edges are: length x: 1-2, 2-3, 3-5 and 5-4, length f: 1-3, 2-5 and 3-4, length F: 1-4 and length q*f: 1-5 2-4.-> u (BDD)
BAFox2ooofx3xoxxo5ofxof&#zx (ABDD)
This one is CRF, it has the same lacings as (AB)(+x)2.3.5(+x)
BAFox2oxofo3oofox5xxoxx&#zx (BC)
This one hasn't been investigated by me again, as (AADD) isn't CRF(+x)2(+x)3(+x)5.
-> u (ABD)
BAFox2xxxFo3xoxxx5ofxof&#zx (AABDD)
This one is CRF, it has the same lacings as (ABDD)(+x)2(+x)3.5(+x)
BAFox2xoxFx3oxfox5xxoxx&#zx (BCD)
again, not investigated-> u (AABC)
BAFox2xoxFo3oxfoo5xxoxF&#zx (AABCD)
how does (ABC) give u-edges? as far as I see it, it would be BAFox 2 oxofxfoxo 3 xxFxoxFxx 5 xxoxFxoxx &#zx. This one isn't CRF, as there are no lacing edges between layers {1,2,3} and {4,5}, but it doesn't produce u-edges though.(+x)2.3(+x)5(+x)
-> u (ABC)
again, I haven't investigated these-> u (BCDD)
BAFox2ooofx3xoFxo5xFoxF&#zx (ABCDD)
(+x)2(+x)3(+x)5(+x)
-> u (ABCD)
BAFox2xxxFo3xoFxx5xFoxF&#zx (AABCDD)
Thus so far, the onesThese again are just all combinations.
Next we will check which of these remaining (further) 11 cases would allow for unit lacings (inter-layer edges of unity), where the respective "stratos height" is (now) always zero. This might reduce that count then a bit again.
BD
AABDD
AABCD
ABCDD
AABCDD
AB
ABDD
BC
BCD
AABD
(ABC)
--- rk
student91 wrote:That one is just the same as the single axial expansion we discovered earlier. Thus it is CRF, but not newKlitzing wrote:(+x)2.3.5.
BAFox2oxofo3oooox5ooxoo&#zx (B)
o........3o........5o........ & | 2 * * * * | 12 0 0 0 0 0 0 0 | 30 0 0 0 0 0 0 0 0 0 | 20 0 0 0 0 0 0 verf=ike
.o.......3.o.......5.o....... & | * 24 * * * | 1 5 5 0 0 0 0 0 | 5 5 10 5 0 0 0 0 0 0 | 5 5 5 1 0 0 0 verf=gyepip(J11)
..o......3..o......5..o...... & | * * 40 * * | 0 0 3 3 3 0 0 0 | 0 0 3 6 3 3 0 0 0 0 | 0 1 3 3 1 0 0 verf=teddi(J63)
...o.....3...o.....5...o..... & | * * * 60 * | 0 0 0 0 2 4 2 1 | 0 0 0 0 4 1 2 4 4 2 | 0 0 0 2 2 2 4
....o....3....o....5....o.... | * * * * 12 | 0 0 0 0 0 0 10 0 | 0 0 0 0 0 0 0 10 0 5 | 0 0 0 2 0 0 5 verf=pip
------------------------------------+---------------+-----------------------------+-----------------------------------+---------------------
oo.......3oo.......5oo.......&#x & | 1 1 0 0 0 | 24 * * * * * * * | 5 0 0 0 0 0 0 0 0 0 | 5 0 0 0 0 0 0
.x....... ......... ......... & | 0 2 0 0 0 | * 60 * * * * * * | 1 2 2 0 0 0 0 0 0 0 | 2 2 1 0 0 0 0
.oo......3.oo......5.oo......&#x & | 0 1 1 0 0 | * * 120 * * * * * | 0 0 2 2 0 0 0 0 0 0 | 0 1 2 1 0 0 0
......... ......... ..x...... & | 0 0 2 0 0 | * * * 60 * * * * | 0 0 0 2 0 1 0 0 0 0 | 0 0 1 2 0 0 0
..oo.....3..oo.....5..oo.....&#x & | 0 0 1 1 0 | * * * * 120 * * * | 0 0 0 0 2 1 0 0 0 0 | 0 0 0 2 1 0 0
......... ...x..... ......... & | 0 0 0 2 0 | * * * * * 120 * * | 0 0 0 0 1 0 1 1 1 0 | 0 0 0 1 1 1 1
...oo....3...oo....5...oo....&#x & | 0 0 0 1 1 | * * * * * * 120 * | 0 0 0 0 0 0 0 2 0 1 | 0 0 0 1 0 0 2
...o.o...3...o.o...5...o.o...&#x | 0 0 0 2 0 | * * * * * * * 30 | 0 0 0 0 0 0 0 0 4 2 | 0 0 0 0 0 2 4
------------------------------------+---------------+-----------------------------+-----------------------------------+---------------------
ox....... ......... .........&#x & | 1 2 0 0 0 | 2 1 0 0 0 0 0 0 | 60 * * * * * * * * * | 2 0 0 0 0 0 0
.x.......3.o....... ......... & | 0 3 0 0 0 | 0 3 0 0 0 0 0 0 | * 40 * * * * * * * * | 1 1 0 0 0 0 0
.xo...... ......... .........&#x & | 0 2 1 0 0 | 0 1 2 0 0 0 0 0 | * * 120 * * * * * * * | 0 1 1 0 0 0 0
......... ......... .ox......&#x & | 0 1 2 0 0 | 0 0 2 1 0 0 0 0 | * * * 120 * * * * * * | 0 0 1 1 0 0 0
......... ..ox..... .........&#x & | 0 0 1 2 0 | 0 0 0 0 2 1 0 0 | * * * * 120 * * * * * | 0 0 0 1 1 0 0
......... ......... ..xo.....&#x & | 0 0 2 1 0 | 0 0 0 1 2 0 0 0 | * * * * * 60 * * * * | 0 0 0 2 0 0 0
...o.....3...x..... ......... & | 0 0 0 3 0 | 0 0 0 0 0 3 0 0 | * * * * * * 40 * * * | 0 0 0 0 1 1 0
......... ...xo.... .........&#x & | 0 0 0 2 1 | 0 0 0 0 0 1 2 0 | * * * * * * * 120 * * | 0 0 0 1 0 0 1
......... ...x.x... .........&#x | 0 0 0 4 0 | 0 0 0 0 0 2 0 2 | * * * * * * * * 60 * | 0 0 0 0 0 1 1
...ooo...3...ooo...5...ooo...&#x | 0 0 0 2 1 | 0 0 0 0 0 0 2 1 | * * * * * * * * * 60 | 0 0 0 0 0 0 2
------------------------------------+---------------+-----------------------------+-----------------------------------+---------------------
ox.......3oo....... .........&#x & | 1 3 0 0 0 | 3 3 0 0 0 0 0 0 | 3 1 0 0 0 0 0 0 0 0 | 40 * * * * * * tet
.xo......3.oo...... .........&#x & | 0 3 1 0 0 | 0 3 3 0 0 0 0 0 | 0 1 3 0 0 0 0 0 0 0 | * 40 * * * * * tet
.xo...... ......... .ox......&#x & | 0 2 2 0 0 | 0 1 4 1 0 0 0 0 | 0 0 2 2 0 0 0 0 0 0 | * * 60 * * * * tet
......... .ooxo....5.oxoo....&#x & | 0 1 5 5 1 | 0 0 5 5 10 5 5 0 | 0 0 0 5 5 5 0 5 0 0 | * * * 24 * * * ike
..oo.....3..ox..... .........&#x & | 0 0 1 3 0 | 0 0 0 0 3 3 0 0 | 0 0 0 0 3 0 1 0 0 0 | * * * * 40 * * tet
...o.o...3...x.x... .........&#x | 0 0 0 6 0 | 0 0 0 0 0 6 0 3 | 0 0 0 0 0 0 2 0 3 0 | * * * * * 20 * trip
......... ...xox... .........&#x | 0 0 0 4 1 | 0 0 0 0 0 2 4 2 | 0 0 0 0 0 0 0 2 1 2 | * * * * * * 60 squippy
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