student91 wrote:now we need quickfur or some kind of program that can process raw coordinates to polytopes in eg Stella format. Do you know someone who has time to build a convex-hull program, or do you know a way to make quickfur active again? I can't wait to see the full structure of these new things.
Klitzing wrote:... The matrices come out correctly, so at least some such structure should exist. But I cannot say much about the geometric realization.
E.g. meanwhile run through g = Fooo3oxoF3ofxo *b3xxFo&#zx as well. That one then has 32+96+32+24 = 184 vertices.Cells ought to be 8+32+96+96+192+32+8 = 464 tets, 32 tricues, 96 squippies, and 8 tuts.- But then I spotted an edge class in the matrix, which shall have 6 incident tets. - So either I made some further error in calculation, or it simply is not possible with unit edges only (e.g. by having longer lacing ones, and so dihedral angles would change accordingly). ...
That's a pitty indeed, but knowing something isn't CRF is also a result, so we're making progressKlitzing wrote:Finally to the remaining, so far not further evaluated case j = Fxoo3oooF3oFxo *b3xoFo&#zx.
Sadly in that case all lacing edges turn out to have a size of f, so this one too cannot become CRF.
I don't think that to be a wrong working method, I use it myself sometimes as well. On the other hand this method highly confuses me which things are and which aren't CRF. As far as got it, You have investigated the following:Sorry for all that to and fro of my recent mails. But I'd think this is a discussion forum, and so it should be allowed to announce promises, discuss troubles, and finally bury them as illusions.
Those renders are still a problem. is this listing right? it looks like a mostly exhaustive investigation of expansions of ex
is this listing right?
it looks like a mostly exhaustive investigation of expansions of ex
student91 wrote:I haven't tried to confirm the things you have suggested, but I have investigated the double axial expansion of ex. note that axial symmetry can be represented with a lace city, a lace tower, a lace-universe or a &#zx-thing, because (A2B2C2D)=(A B C D)(.)[&#zx]=(A B C)(D)[tower]=(A B)(C D)[city]=(A)(B C D)[universe].
The first axial expansion of ex has already been found, and is given by x3o5o||o3o5x||o3x5o||f3o5o||o3x5o||o3o5x||x3o5o. it's lace-city looks like this:But can also be given as
- Code: Select all
o2o
o2x f2o x2f f2o o2x
x2o f2f o2F F2x o2F f2f x2o
o2o f2x x2F F2f Vo2oV F2f x2F f2x o2o
o2f F2o f2F F2o o2f
o2o f2x x2F F2f Vo2oV F2f x2F f2x o2o
x2o f2f o2F F2x o2F f2f x2o
o2x f2o x2f f2o o2x
o2o
- Code: Select all
x2o
x2x F2o o2f F2o x2x
o2o F2f x2F A2x x2F F2f o2o
x2f A2o F2F A2o x2f
x2o F2x o2F A2f Bx2oV A2f o2F F2x x2o
x2f A2o F2F A2o x2f
o2o F2f x2F A2x x2F F2f o2o
x2x F2o o2f F2o x2x
x2o
A=F+x=f+2x
B=V+x=2f+x
This is a polytope with quite some tetrahedra, 24 icosahedra, 20 triangular prisms and 60 squippies.
...
ofo3xox5ooo&#xt = blend of x o3x5o (iddip) with 12 ox ox5oo&#x (pippy)
o..3o..5o.. & | 60 * | 4 2 1 | 2 2 4 4 2 | 1 2 2 4 id layers
.o.3.o.5.o. | * 12 | 0 10 0 | 0 0 10 0 5 | 0 2 0 5 f-ike layer > verf = pip
-----------------+-------+------------+-----------------+-----------
... x.. ... & | 2 0 | 120 * * | 1 1 1 1 0 | 1 1 1 1
oo.3oo.5oo.&#x & | 1 1 | * 120 * | 0 0 2 0 1 | 0 1 0 2
o.o3o.o5o.o&#x | 2 0 | * * 30 | 0 0 0 4 2 | 0 0 2 4
-----------------+-------+------------+-----------------+-----------
o..3x.. ... & | 3 0 | 3 0 0 | 40 * * * * | 1 0 1 0
... x..5o.. & | 5 0 | 5 0 0 | * 24 * * * | 1 1 0 0
... xo. ...&#x & | 2 1 | 1 2 0 | * * 120 * * | 0 1 0 1
... x.x ...&#x | 4 0 | 2 0 2 | * * * 60 * | 0 0 1 1
ooo3ooo5ooo&#x | 2 1 | 0 2 1 | * * * * 60 | 0 0 0 2
-----------------+-------+------------+-----------------+-----------
o..3x..5o.. & | 30 0 | 60 0 0 | 20 12 0 0 0 | 2 * * * id
... xo.5oo.&#x & | 5 1 | 5 5 0 | 0 1 5 0 0 | * 24 * * peppy
o.o3x.x ...&#x | 6 0 | 6 0 3 | 2 0 0 3 0 | * * 20 * trip
... xox ...&#x | 4 1 | 2 4 2 | 0 0 2 1 2 | * * * 60 squippy
oofoo3oxoxo5xooox&#xt
o....3o....5o.... & | 40 * * | 3 3 0 0 0 | 3 3 3 0 0 0 0 | 1 1 3 0 0 doe layers
.o...3.o...5.o... & | * 60 * | 0 2 4 2 1 | 0 4 1 2 4 4 2 | 0 2 2 2 4 id layers
..o..3..o..5..o.. | * * 12 | 0 0 0 10 0 | 0 0 0 0 10 0 5 | 0 0 2 0 5 f-ike layer > verf = pip
-----------------------+----------+-------------------+------------------------+--------------
..... ..... x.... & | 2 0 0 | 60 * * * * | 2 0 1 0 0 0 0 | 1 0 2 0 0
oo...3oo...5oo...&#x & | 1 1 0 | * 120 * * * | 0 2 1 0 0 0 0 | 0 1 2 0 0
..... .x... ..... & | 0 2 0 | * * 120 * * | 0 1 0 1 1 1 0 | 0 1 1 1 1
.oo..3.oo..5.oo..&#x & | 0 1 1 | * * * 120 * | 0 0 0 0 2 0 1 | 0 0 1 0 2
.o.o.3.o.o.5.o.o.&#x | 0 2 0 | * * * * 30 | 0 0 0 0 0 4 2 | 0 0 0 2 4
-----------------------+----------+-------------------+------------------------+--------------
..... o....5x.... & | 5 0 0 | 5 0 0 0 0 | 24 * * * * * * | 1 0 1 0 0
..... ox... .....&#x & | 1 2 0 | 0 2 1 0 0 | * 120 * * * * * | 0 1 1 0 0
..... ..... xo...&#x & | 2 1 0 | 1 2 0 0 0 | * * 60 * * * * | 0 0 2 0 0
.o...3.x... ..... & | 0 3 0 | 0 0 3 0 0 | * * * 40 * * * | 0 1 0 1 0
..... .xo.. .....&#x & | 0 2 1 | 0 0 1 2 0 | * * * * 120 * * | 0 0 1 0 1
..... .x.x. .....&#x | 0 4 0 | 0 0 2 0 2 | * * * * * 60 * | 0 0 0 1 1
.ooo.3.ooo.5.ooo.&#x | 0 2 1 | 0 0 0 2 1 | * * * * * * 60 | 0 0 0 0 2
-----------------------+----------+-------------------+------------------------+--------------
o....3o....5x.... & | 20 0 0 | 30 0 0 0 0 | 12 0 0 0 0 0 0 | 2 * * * * doe
oo...3ox... .....&#x & | 1 3 0 | 0 3 3 0 0 | 0 3 0 1 0 0 0 | * 40 * * * tet
..... oxo..5xoo..&#x & | 5 5 1 | 5 10 5 5 0 | 1 5 5 0 5 0 0 | * * 24 * * gyepip
.o.o.3.x.x. .....&#x | 0 6 0 | 0 0 6 0 3 | 0 0 0 2 0 3 0 | * * * 20 * trip
..... .xox. .....&#x | 0 4 1 | 0 0 2 4 2 | 0 0 0 0 2 1 2 | * * * * 60 squippy
oxoofooxo3oooxoxooo5ooxoooxoo&#xt
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 polar points > 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 ike layers > verf = gyepip
..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 doe layers > verf = teddi
...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 id layers
....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 f-ike layer > 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
though I quite literally said the opposite, I meant to say that it doesn't have bilbiro-cells. It does have the structure of a bilbiro: when you delete two corresponding vertices of the ids, you delete the apex of a bilbiro-pseudopyramid, revealing a bilbiro that has formerly been internal to the polytope.Klitzing wrote:[...]
Finally got some time to wrap my mind around that one (the first expansion it is). You first introduced that one in this post, beginning some sub-thread of following discussions. In that Initial post you state that this extension ought contain itself bilbiroes. - Now you state it doesn't. - We will see!
Great! You were also able to tell quite a bit about the demitesseractic expansions of ex, do you have incmats of these as well? (As long as we don't have renders I would really like to discover the structure of these in this manner, with polytopes I fully don't understand).In the sequel of this discussion then in this post it was proved that this suggested polychoron indeed would be convex at least. Moreover the structure of its bistratic tropical segment was understood to be nothing but "twau iddip", i.e. a twelf-augmented id-prism. Here each augmentation part then is a "pippy" (pip-pyramid).
Accordingly the incidence matrix of that deep paradiminishing of that axial mono-expanded ex then reads:
- Code: Select all
ofo3xox5ooo&#xt = blend of x o3x5o (iddip) with 12 ox ox5oo&#x (pippy)
o..3o..5o.. & | 60 * | 4 2 1 | 2 2 4 4 2 | 1 2 2 4 id layers
.o.3.o.5.o. | * 12 | 0 10 0 | 0 0 10 0 5 | 0 2 0 5 f-ike layer > verf = pip
-----------------+-------+------------+-----------------+-----------
... x.. ... & | 2 0 | 120 * * | 1 1 1 1 0 | 1 1 1 1
oo.3oo.5oo.&#x & | 1 1 | * 120 * | 0 0 2 0 1 | 0 1 0 2
o.o3o.o5o.o&#x | 2 0 | * * 30 | 0 0 0 4 2 | 0 0 2 4
-----------------+-------+------------+-----------------+-----------
o..3x.. ... & | 3 0 | 3 0 0 | 40 * * * * | 1 0 1 0
... x..5o.. & | 5 0 | 5 0 0 | * 24 * * * | 1 1 0 0
... xo. ...&#x & | 2 1 | 1 2 0 | * * 120 * * | 0 1 0 1
... x.x ...&#x | 4 0 | 2 0 2 | * * * 60 * | 0 0 1 1
ooo3ooo5ooo&#x | 2 1 | 0 2 1 | * * * * 60 | 0 0 0 2
-----------------+-------+------------+-----------------+-----------
o..3x..5o.. & | 30 0 | 60 0 0 | 20 12 0 0 0 | 2 * * * id
... xo.5oo.&#x & | 5 1 | 5 5 0 | 0 1 5 0 0 | * 24 * * peppy
o.o3x.x ...&#x | 6 0 | 6 0 3 | 2 0 0 3 0 | * * 20 * trip
... xox ...&#x | 4 1 | 2 4 2 | 0 0 2 1 2 | * * * 60 squippy
Next we will consider a medial paradiminishing. That one would contain the next layer on either side in addition. The added segmentochoron (on either side) then is nothing but doe||id, having prominent paps (pentagonal antiprisms) within. These will align with pentagonal pyramids (peppies) of the so far considered bistratic part. The question thus arises whether those are corealmic or not. - In fact they are, as can be seen like this. Within ex we have the same sorts of layers, just slightly rearranged at its central part. Esp. the doe layer of one side, its equatorial id layer, and the f-ike layer of the other side o have exactly the same relation as have the doe layer, the id layer of the same side, and the equatorial f-ike layer of this expanded one. But ex can be diminished to have ikes. Thus this gyroelongated pentagonal pyramid (gyepip) indeed is a single cell here, not 2 disjoint ones.
The incidence matrix of this one then reads:
- Code: Select all
oofoo3oxoxo5xooox&#xt
o....3o....5o.... & | 40 * * | 3 3 0 0 0 | 3 3 3 0 0 0 0 | 1 1 3 0 0 doe layers
.o...3.o...5.o... & | * 60 * | 0 2 4 2 1 | 0 4 1 2 4 4 2 | 0 2 2 2 4 id layers
..o..3..o..5..o.. | * * 12 | 0 0 0 10 0 | 0 0 0 0 10 0 5 | 0 0 2 0 5 f-ike layer > verf = pip
-----------------------+----------+-------------------+------------------------+--------------
..... ..... x.... & | 2 0 0 | 60 * * * * | 2 0 1 0 0 0 0 | 1 0 2 0 0
oo...3oo...5oo...&#x & | 1 1 0 | * 120 * * * | 0 2 1 0 0 0 0 | 0 1 2 0 0
..... .x... ..... & | 0 2 0 | * * 120 * * | 0 1 0 1 1 1 0 | 0 1 1 1 1
.oo..3.oo..5.oo..&#x & | 0 1 1 | * * * 120 * | 0 0 0 0 2 0 1 | 0 0 1 0 2
.o.o.3.o.o.5.o.o.&#x | 0 2 0 | * * * * 30 | 0 0 0 0 0 4 2 | 0 0 0 2 4
-----------------------+----------+-------------------+------------------------+--------------
..... o....5x.... & | 5 0 0 | 5 0 0 0 0 | 24 * * * * * * | 1 0 1 0 0
..... ox... .....&#x & | 1 2 0 | 0 2 1 0 0 | * 120 * * * * * | 0 1 1 0 0
..... ..... xo...&#x & | 2 1 0 | 1 2 0 0 0 | * * 60 * * * * | 0 0 2 0 0
.o...3.x... ..... & | 0 3 0 | 0 0 3 0 0 | * * * 40 * * * | 0 1 0 1 0
..... .xo.. .....&#x & | 0 2 1 | 0 0 1 2 0 | * * * * 120 * * | 0 0 1 0 1
..... .x.x. .....&#x | 0 4 0 | 0 0 2 0 2 | * * * * * 60 * | 0 0 0 1 1
.ooo.3.ooo.5.ooo.&#x | 0 2 1 | 0 0 0 2 1 | * * * * * * 60 | 0 0 0 0 2
-----------------------+----------+-------------------+------------------------+--------------
o....3o....5x.... & | 20 0 0 | 30 0 0 0 0 | 12 0 0 0 0 0 0 | 2 * * * * doe
oo...3ox... .....&#x & | 1 3 0 | 0 3 3 0 0 | 0 3 0 1 0 0 0 | * 40 * * * tet
..... oxo..5xoo..&#x & | 5 5 1 | 5 10 5 5 0 | 1 5 5 0 5 0 0 | * * 24 * * gyepip
.o.o.3.x.x. .....&#x | 0 6 0 | 0 0 6 0 3 | 0 0 0 2 0 3 0 | * * * 20 * trip
..... .xox. .....&#x | 0 4 1 | 0 0 2 4 2 | 0 0 0 0 2 1 2 | * * * * 60 squippy
Finally we will add the so far missing polar caps to that one. This then complete the gyepips into full ikes.
- Code: Select all
oxoofooxo3oooxoxooo5ooxoooxoo&#xt
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 polar points > 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 ike layers > verf = gyepip
..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 doe layers > verf = teddi
...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 id layers
....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 f-ike layer > 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
Thus indeed ikes, trips, and tets. No bilbiroes so far.
--- rk
student91 wrote:Besides, I haven't been able to understand the connection between [5,3,3] and [[3,3,3]]. It seems they don't have a connection. Ex does have tetrahedral antiprismatic symmetry though, which is quite similar though not the same. If this connection is not based on this symmetry, I have no idea what it is based on then. Could Wendy give me a hint regarding this connection?
student91 wrote:Great! You were also able to tell quite a bit about the demitesseractic expansions of ex, do you have incmats of these as well? (As long as we don't have renders I would really like to discover the structure of these in this manner, with polytopes I fully don't understand).
o...3o...3o... *b3o... & | 288 * | 2 1 4 1 | 2 1 4 3 1 2 4 | 1 3 2 3 2
...o3...o3...o *b3...o | * 24 | 0 0 0 12 | 0 0 0 0 6 0 24 | 0 0 0 12 8 verf = ike
----------------------------+--------+-----------------+----------------------------+-----------------
.... x... .... .... & | 2 0 | 288 * * * | 1 1 2 0 0 0 0 | 1 2 1 0 0
.... .... x... .... & | 2 0 | * 144 * * | 2 0 0 2 1 0 0 | 1 2 0 2 0
oo..3oo..3oo.. *b3oo..&#x & | 2 0 | * * 576 * | 0 0 1 1 0 1 1 | 0 1 1 1 1
o..o3o..o3o..o *b3o..o&#x & | 1 1 | * * * 288 | 0 0 0 0 1 0 4 | 0 0 0 3 2
----------------------------+--------+-----------------+----------------------------+-----------------
.... x...3x... .... & | 6 0 | 3 3 0 0 | 96 * * * * * * | 1 1 0 0 0
.... x... .... *b3o... & | 3 0 | 3 0 0 0 | * 96 * * * * * | 1 1 0 0 0
.... xx.. .... ....&#x & | 4 0 | 2 0 2 0 | * * 288 * * * * | 0 1 1 0 0
.... .... .... ox..&#x & | 3 0 | 0 1 2 0 | * * * 288 * * * | 0 1 0 1 0
.... .... x..o ....&#x & | 2 1 | 0 1 0 2 | * * * * 144 * * | 0 0 0 2 0
ooo.3ooo.3ooo. *b3ooo.&#x | 3 0 | 0 0 3 0 | * * * * * 192 * | 0 0 1 0 1
oo.o3oo.o3oo.o *b3oo.o&#x & | 2 1 | 0 0 1 2 | * * * * * * 576 | 0 0 0 1 1
----------------------------+--------+-----------------+----------------------------+-----------------
.... x...3x... *b3o... & | 12 0 | 12 6 0 0 | 4 4 0 0 0 0 0 | 24 * * * * tut
.... xx.. .... *b3ox..&#x & | 9 0 | 6 3 6 0 | 1 1 3 3 0 0 0 | * 96 * * * tricu
.... xxx. .... ....&#x | 6 0 | 3 0 6 0 | 0 0 3 0 0 2 0 | * * 96 * * trip
.... .... .... ox.o&#x & | 3 1 | 0 1 2 3 | 0 0 0 1 1 0 2 | * * * 288 * tet
oooo3oooo3oooo *b3oooo&#x | 3 1 | 0 0 3 3 | 0 0 0 0 0 1 3 | * * * * 192 tet
o...3o...3o... *b3o... | 96 * * * | 2 1 2 1 0 0 0 0 0 | 2 1 2 1 1 2 0 0 0 0 0 0 | 1 1 1 2 0 0 0
.o..3.o..3.o.. *b3.o.. | * 96 * * | 0 0 2 0 2 1 1 1 0 | 0 0 2 2 0 2 1 2 2 1 1 0 | 0 2 2 2 1 1 0
..o.3..o.3..o. *b3..o. | * * 32 * | 0 0 0 0 0 0 3 0 3 | 0 0 0 0 0 0 0 0 3 0 3 3 | 0 0 0 3 0 1 1
...o3...o3...o *b3...o | * * * 24 | 0 0 0 4 0 0 0 4 0 | 0 0 0 0 2 8 0 0 0 2 2 0 | 0 0 4 4 0 0 0
---------------------------+-------------+-----------------------------+--------------------------------------+------------------
.... x... .... .... | 2 0 0 0 | 96 * * * * * * * * | 1 1 1 0 0 0 0 0 0 0 0 0 | 1 1 0 1 0 0 0
.... .... x... .... | 2 0 0 0 | * 48 * * * * * * * | 2 0 0 0 1 0 0 0 0 0 0 0 | 1 0 0 2 0 0 0
oo..3oo..3oo.. *b3oo..&#x | 1 1 0 0 | * * 192 * * * * * * | 0 0 1 1 0 1 0 0 0 0 0 0 | 0 1 1 1 0 0 0
o..o3o..o3o..o *b3o..o&#x | 1 0 0 1 | * * * 96 * * * * * | 0 0 0 0 1 2 0 0 0 0 0 0 | 0 0 1 2 0 0 0
.... .x.. .... .... | 0 2 0 0 | * * * * 96 * * * * | 0 0 1 0 0 0 1 1 1 0 0 0 | 0 1 0 1 1 1 0
.... .... .... .x.. | 0 2 0 0 | * * * * * 48 * * * | 0 0 0 2 0 0 0 2 0 1 0 0 | 0 2 2 0 1 0 0
.oo.3.oo.3.oo. *b3.oo.&#x | 0 1 1 0 | * * * * * * 96 * * | 0 0 0 0 0 0 0 0 2 0 1 0 | 0 0 0 2 0 1 0
.o.o3.o.o3.o.o *b3.o.o&#x | 0 1 0 1 | * * * * * * * 96 * | 0 0 0 0 0 2 0 0 0 1 1 0 | 0 0 2 2 0 0 0
.... .... ..x. .... | 0 0 2 0 | * * * * * * * * 48 | 0 0 0 0 0 0 0 0 0 0 1 2 | 0 0 0 2 0 0 1
---------------------------+-------------+-----------------------------+--------------------------------------+------------------
.... x...3x... .... | 6 0 0 0 | 3 3 0 0 0 0 0 0 0 | 32 * * * * * * * * * * * | 1 0 0 1 0 0 0
.... x... .... *b3o... | 3 0 0 0 | 3 0 0 0 0 0 0 0 0 | * 32 * * * * * * * * * * | 1 1 0 0 0 0 0
.... xx.. .... ....&#x | 2 2 0 0 | 1 0 2 0 1 0 0 0 0 | * * 96 * * * * * * * * * | 0 1 0 1 0 0 0
.... .... .... ox..&#x | 1 2 0 0 | 0 0 2 0 0 1 0 0 0 | * * * 96 * * * * * * * * | 0 1 1 0 0 0 0
.... .... x..o ....&#x | 2 0 0 1 | 0 1 0 2 0 0 0 0 0 | * * * * 48 * * * * * * * | 0 0 0 2 0 0 0
oo.o3oo.o3oo.o *b3oo.o&#x | 1 1 0 1 | 0 0 1 1 0 0 0 1 0 | * * * * * 192 * * * * * * | 0 0 1 1 0 0 0
.o..3.x.. .... .... | 0 3 0 0 | 0 0 0 0 3 0 0 0 0 | * * * * * * 32 * * * * * | 0 0 0 0 1 1 0
.... .x.. .... *b3.x.. | 0 6 0 0 | 0 0 0 0 3 3 0 0 0 | * * * * * * * 32 * * * * | 0 1 0 0 1 0 0
.... .xo. .... ....&#x | 0 2 1 0 | 0 0 0 0 1 0 2 0 0 | * * * * * * * * 96 * * * | 0 0 0 1 0 1 0
.... .... .... .x.o&#x | 0 2 0 1 | 0 0 0 0 0 1 0 2 0 | * * * * * * * * * 48 * * | 0 0 2 0 0 0 0
.... .... .fxo ....&#zx | 0 2 2 1 | 0 0 0 0 0 0 2 2 1 | * * * * * * * * * * 48 * | 0 0 0 2 0 0 0
.... ..o.3..x. .... | 0 0 3 0 | 0 0 0 0 0 0 0 0 3 | * * * * * * * * * * * 32 | 0 0 0 1 0 0 1
---------------------------+-------------+-----------------------------+--------------------------------------+------------------
.... x...3x... *b3o... | 12 0 0 0 | 12 6 0 0 0 0 0 0 0 | 4 4 0 0 0 0 0 0 0 0 0 0 | 8 * * * * * * tut
.... xx.. .... *b3ox..&#x | 3 6 0 0 | 3 0 6 0 3 3 0 0 0 | 0 1 3 3 0 0 0 1 0 0 0 0 | * 32 * * * * * tricu
.... .... .... ox.o&#x | 1 2 0 1 | 0 0 2 1 0 1 0 2 0 | 0 0 0 1 0 2 0 0 0 1 0 0 | * * 96 * * * * tet
.... xxoF3xfxo ....&#zx | 6 6 3 3 | 3 3 6 6 3 0 6 6 3 | 1 0 3 0 3 6 0 0 3 0 3 1 | * * * 32 * * * thawro
.o..3.x.. .... *b3.x.. | 0 12 0 0 | 0 0 0 0 12 6 0 0 0 | 0 0 0 0 0 0 4 4 0 0 0 0 | * * * * 8 * * tut
.oo.3.xo. .... ....&#x | 0 3 1 0 | 0 0 0 0 3 0 3 0 0 | 0 0 0 0 0 0 1 0 3 0 0 0 | * * * * * 32 * tet
..o.3..o.3..x. .... | 0 0 4 0 | 0 0 0 0 0 0 0 0 6 | 0 0 0 0 0 0 0 0 0 0 0 4 | * * * * * * 8 tet
o...3o...3o... *b3o... | 48 * * * | 1 2 4 2 0 0 0 0 0 0 | 1 2 4 2 4 4 0 0 0 0 0 0 0 0 0 | 2 2 2 4 0 0 0 0 0
.o..3.o..3.o.. *b3.o.. | * 32 * * | 0 3 0 0 3 3 0 0 0 0 | 3 0 0 0 6 0 3 3 0 0 0 0 0 0 0 | 3 3 0 0 1 1 0 0 0
..o.3..o.3..o. *b3..o. | * * 96 * | 0 0 2 0 0 1 2 2 2 0 | 0 2 2 0 2 2 0 2 1 1 2 1 2 0 0 | 1 2 2 2 0 1 1 1 0
...o3...o3...o *b3...o | * * * 48 | 0 0 0 2 0 0 0 0 4 4 | 0 0 0 1 0 4 0 0 0 0 0 4 2 2 2 | 2 0 0 2 0 0 0 2 1
---------------------------+-------------+---------------------------------+------------------------------------------------+------------------------
.... .... .... x... | 2 0 0 0 | 24 * * * * * * * * * | 0 0 4 2 0 0 0 0 0 0 0 0 0 0 0 | 0 0 2 4 0 0 0 0 0
oo..3oo..3oo.. *b3oo..&#x | 1 1 0 0 | * 96 * * * * * * * * | 1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 | 2 1 0 0 0 0 0 0 0
o.o.3o.o.3o.o. *b3o.o.&#x | 1 0 1 0 | * * 192 * * * * * * * | 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 | 1 1 1 1 0 0 0 0 0
o..o3o..o3o..o *b3o..o&#x | 1 0 0 1 | * * * 96 * * * * * * | 0 0 0 1 0 2 0 0 0 0 0 0 0 0 0 | 1 0 0 2 0 0 0 0 0
.... .... .x.. .... | 0 2 0 0 | * * * * 48 * * * * * | 1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 | 2 0 0 0 1 0 0 0 0
.oo.3.oo.3.oo. *b3.oo.&#x | 0 1 1 0 | * * * * * 96 * * * * | 0 0 0 0 2 0 0 2 0 0 0 0 0 0 0 | 1 2 0 0 0 1 0 0 0
..x. .... .... .... | 0 0 2 0 | * * * * * * 96 * * * | 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 | 0 1 1 0 0 1 1 0 0
.... .... .... ..x. | 0 0 2 0 | * * * * * * * 96 * * | 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 | 0 0 1 1 0 0 1 1 0
..oo3..oo3..oo *b3..oo&#x | 0 0 1 1 | * * * * * * * * 192 * | 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 | 1 0 0 1 0 0 0 1 0
.... ...x .... .... | 0 0 0 2 | * * * * * * * * * 96 | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 | 1 0 0 0 0 0 0 1 1
---------------------------+-------------+---------------------------------+------------------------------------------------+------------------------
.... .... ox.. ....&#x | 1 2 0 0 | 0 2 0 0 1 0 0 0 0 0 | 48 * * * * * * * * * * * * * * | 2 0 0 0 0 0 0 0 0
o.x. .... .... ....&#x | 1 0 2 0 | 0 0 2 0 0 0 1 0 0 0 | * 96 * * * * * * * * * * * * * | 0 1 1 0 0 0 0 0 0
.... .... .... x.x.&#x | 2 0 2 0 | 1 0 2 0 0 0 0 1 0 0 | * * 96 * * * * * * * * * * * * | 0 0 1 1 0 0 0 0 0
.... .... .... x..o&#x | 2 0 0 1 | 1 0 0 2 0 0 0 0 0 0 | * * * 48 * * * * * * * * * * * | 0 0 0 2 0 0 0 0 0
ooo.3ooo.3ooo. *b3ooo.&#x | 1 1 1 0 | 0 1 1 0 0 1 0 0 0 0 | * * * * 192 * * * * * * * * * * | 1 1 0 0 0 0 0 0 0
o.oo3o.oo3o.oo *b3o.oo&#x | 1 0 1 1 | 0 0 1 1 0 0 0 0 1 0 | * * * * * 192 * * * * * * * * * | 1 0 0 1 0 0 0 0 0
.... .o..3.x.. .... | 0 3 0 0 | 0 0 0 0 3 0 0 0 0 0 | * * * * * * 32 * * * * * * * * | 1 0 0 0 1 0 0 0 0
.ox. .... .... ....&#x | 0 1 2 0 | 0 0 0 0 0 2 1 0 0 0 | * * * * * * * 96 * * * * * * * | 0 1 0 0 0 1 0 0 0
..x.3..o. .... .... | 0 0 3 0 | 0 0 0 0 0 0 3 0 0 0 | * * * * * * * * 32 * * * * * * | 0 0 0 0 0 1 1 0 0
.... ..o. .... *b3..x. | 0 0 3 0 | 0 0 0 0 0 0 0 3 0 0 | * * * * * * * * * 32 * * * * * | 0 0 0 0 0 0 1 1 0
..x. .... .... ..x. | 0 0 4 0 | 0 0 0 0 0 0 2 2 0 0 | * * * * * * * * * * 48 * * * * | 0 0 1 0 0 0 1 0 0
.... ..ox .... ....&#x | 0 0 1 2 | 0 0 0 0 0 0 0 0 2 1 | * * * * * * * * * * * 96 * * * | 1 0 0 0 0 0 0 1 0
.... .... .... ..xo&#x | 0 0 2 1 | 0 0 0 0 0 0 0 1 2 0 | * * * * * * * * * * * * 96 * * | 0 0 0 1 0 0 0 1 0
.... ...x3...o .... | 0 0 0 3 | 0 0 0 0 0 0 0 0 0 3 | * * * * * * * * * * * * * 32 * | 1 0 0 0 0 0 0 0 1
.... ...x3.... *b3...o | 0 0 0 3 | 0 0 0 0 0 0 0 0 0 3 | * * * * * * * * * * * * * * 32 | 0 0 0 0 0 0 0 1 1
---------------------------+-------------+---------------------------------+------------------------------------------------+------------------------
.... foox3oxfo ....&#zx | 3 3 3 3 | 0 6 6 3 3 3 0 0 6 3 | 3 0 0 0 6 6 1 0 0 0 0 3 0 1 0 | 32 * * * * * * * * ike
oox. .... .... ....&#x | 1 1 2 0 | 0 1 2 0 0 2 1 0 0 0 | 0 1 0 0 2 0 0 1 0 0 0 0 0 0 0 | * 96 * * * * * * * tet
o.x. .... .... x.x.&#x | 2 0 4 0 | 1 0 4 0 0 0 2 2 0 0 | 0 2 2 0 0 0 0 0 0 0 1 0 0 0 0 | * * 48 * * * * * * trip
.... .... .... x.xo&#x | 2 0 2 1 | 1 0 2 2 0 0 0 1 2 0 | 0 0 1 1 0 2 0 0 0 0 0 0 1 0 0 | * * * 96 * * * * * squippy
.o..3.o..3.x.. .... | 0 4 0 0 | 0 0 0 0 6 0 0 0 0 0 | 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 | * * * * 8 * * * * tet
.ox.3.oo. .... ....&#x | 0 1 3 0 | 0 0 0 0 0 3 3 0 0 0 | 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 | * * * * * 32 * * * tet
..x.3..o. .... *b3..x. | 0 0 12 0 | 0 0 0 0 0 0 12 12 0 0 | 0 0 0 0 0 0 0 0 4 4 6 0 0 0 0 | * * * * * * 8 * * co
.... ..ox .... *b3..xo&#x | 0 0 3 3 | 0 0 0 0 0 0 0 3 6 3 | 0 0 0 0 0 0 0 0 0 1 0 3 3 0 1 | * * * * * * * 32 * oct
.... ...x3...o *b3...o | 0 0 0 6 | 0 0 0 0 0 0 0 0 0 12 | 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 | * * * * * * * * 8 oct
o...3o...3o... *b3o... | 48 * * * | 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 0 | 1 2 2 0 0 0 0 0 0
.o..3.o..3.o.. *b3.o.. | * 96 * * | 0 0 2 2 2 2 0 0 0 0 0 | 0 0 1 0 1 2 1 2 1 2 2 2 0 0 0 0 0 | 0 1 1 1 1 2 2 0 0
..o.3..o.3..o. *b3..o. | * * 96 * | 0 0 0 0 2 0 2 1 2 0 0 | 0 0 0 0 0 0 0 1 2 0 0 2 1 2 1 2 0 | 0 0 2 0 1 0 1 1 1
...o3...o3...o *b3...o | * * * 96 | 0 1 0 0 0 2 0 0 2 1 1 | 0 0 2 1 0 0 0 0 0 2 1 2 0 0 2 2 1 | 0 1 2 0 0 1 2 0 2
---------------------------+-------------+-------------------------------------+-----------------------------------------------------+------------------------
.... x... .... .... | 2 0 0 0 | 96 * * * * * * * * * * | 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 0
o..o3o..o3o..o *b3o..o&#x | 1 0 0 1 | * 96 * * * * * * * * * | 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 2 0 0 0 0 0 0
.x.. .... .... .... | 0 2 0 0 | * * 96 * * * * * * * * | 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 | 0 0 0 1 1 1 1 0 0
.... .... .... .x.. | 0 2 0 0 | * * * 96 * * * * * * * | 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 | 0 1 0 1 0 1 0 0 0
.oo.3.oo.3.oo. *b3.oo.&#x | 0 1 1 0 | * * * * 192 * * * * * * | 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 | 0 0 1 0 1 0 1 0 0
.o.o3.o.o3.o.o *b3.o.o&#x | 0 1 0 1 | * * * * * 192 * * * * * | 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 | 0 1 1 0 0 1 1 0 0
.... ..x. .... .... | 0 0 2 0 | * * * * * * 96 * * * * | 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 | 0 0 1 0 1 0 0 1 0
.... .... ..x. .... | 0 0 2 0 | * * * * * * * 48 * * * | 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 | 0 0 2 0 0 0 0 1 1
..oo3..oo3..oo *b3..oo&#x | 0 0 1 1 | * * * * * * * * 192 * * | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 | 0 0 1 0 0 0 1 0 1
...x .... .... .... | 0 0 0 2 | * * * * * * * * * 48 * | 0 0 0 0 0 0 0 0 0 2 0 0 0 0 2 0 1 | 0 0 0 0 0 1 2 0 2
.... .... ...x .... | 0 0 0 2 | * * * * * * * * * * 48 | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 1 | 0 0 2 0 0 0 0 0 2
---------------------------+-------------+-------------------------------------+-----------------------------------------------------+------------------------
.... x...3o... .... | 3 0 0 0 | 3 0 0 0 0 0 0 0 0 0 0 | 32 * * * * * * * * * * * * * * * * | 1 0 1 0 0 0 0 0 0
.... x... .... *b3o... | 3 0 0 0 | 3 0 0 0 0 0 0 0 0 0 0 | * 32 * * * * * * * * * * * * * * * | 1 1 0 0 0 0 0 0 0
.... xo.f .... ....&#zx | 2 1 0 2 | 1 2 0 0 0 2 0 0 0 0 0 | * * 96 * * * * * * * * * * * * * * | 0 1 1 0 0 0 0 0 0
.... .... o..x ....&#x | 1 0 0 2 | 0 2 0 0 0 0 0 0 0 0 1 | * * * 48 * * * * * * * * * * * * * | 0 0 2 0 0 0 0 0 0
.x..3.o.. .... .... | 0 3 0 0 | 0 0 3 0 0 0 0 0 0 0 0 | * * * * 32 * * * * * * * * * * * * | 0 0 0 1 1 0 0 0 0
.x.. .... .... .x.. | 0 4 0 0 | 0 0 2 2 0 0 0 0 0 0 0 | * * * * * 48 * * * * * * * * * * * | 0 0 0 1 0 1 0 0 0
.... .o.. .... *b3.x.. | 0 3 0 0 | 0 0 0 3 0 0 0 0 0 0 0 | * * * * * * 32 * * * * * * * * * * | 0 1 0 1 0 0 0 0 0
.xo. .... .... ....&#x | 0 2 1 0 | 0 0 1 0 2 0 0 0 0 0 0 | * * * * * * * 96 * * * * * * * * * | 0 0 0 0 1 0 1 0 0
.... .ox. .... ....&#x | 0 1 2 0 | 0 0 0 0 2 0 1 0 0 0 0 | * * * * * * * * 96 * * * * * * * * | 0 0 1 0 1 0 0 0 0
.x.x .... .... ....&#x | 0 2 0 2 | 0 0 1 0 0 2 0 0 0 1 0 | * * * * * * * * * 96 * * * * * * * | 0 0 0 0 0 1 1 0 0
.... .... .... .x.o&#x | 0 2 0 1 | 0 0 0 1 0 2 0 0 0 0 0 | * * * * * * * * * * 96 * * * * * * | 0 1 0 0 0 1 0 0 0
.ooo3.ooo3.ooo *b3.ooo&#x | 0 1 1 1 | 0 0 0 0 1 1 0 0 1 0 0 | * * * * * * * * * * * 192 * * * * * | 0 0 1 0 0 0 1 0 1
..o.3..x. .... .... | 0 0 3 0 | 0 0 0 0 0 0 3 0 0 0 0 | * * * * * * * * * * * * 32 * * * * | 0 0 0 0 1 0 0 1 0
.... ..x.3..x. .... | 0 0 6 0 | 0 0 0 0 0 0 3 3 0 0 0 | * * * * * * * * * * * * * 32 * * * | 0 0 1 0 0 0 0 1 0
..ox .... .... ....&#x | 0 0 1 2 | 0 0 0 0 0 0 0 0 2 1 0 | * * * * * * * * * * * * * * 96 * * | 0 0 0 0 0 0 1 0 1
.... .... ..xx ....&#x | 0 0 2 2 | 0 0 0 0 0 0 0 1 2 0 1 | * * * * * * * * * * * * * * * 96 * | 0 0 1 0 0 0 0 0 1
...x .... ...x .... | 0 0 0 4 | 0 0 0 0 0 0 0 0 0 2 2 | * * * * * * * * * * * * * * * * 24 | 0 0 0 0 0 0 0 0 2
---------------------------+-------------+-------------------------------------+-----------------------------------------------------+------------------------
.... x...3o... *b3o... | 6 0 0 0 | 12 0 0 0 0 0 0 0 0 0 0 | 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 8 * * * * * * * * oct
.... xo.f3.... *b3ox.o&#zx | 3 3 0 3 | 3 3 0 3 0 6 0 0 0 0 0 | 0 1 3 0 0 0 1 0 0 0 3 0 0 0 0 0 0 | * 32 * * * * * * * teddi
.... xoxf3oFxx ....&#zx | 3 3 6 6 | 3 6 0 0 6 6 3 3 6 0 3 | 1 0 3 3 0 0 0 0 3 0 0 6 0 1 0 3 0 | * * 32 * * * * * * thawro
.x..3.o.. .... *b3.x.. | 0 12 0 0 | 0 0 12 12 0 0 0 0 0 0 0 | 0 0 0 0 4 6 4 0 0 0 0 0 0 0 0 0 0 | * * * 8 * * * * * co
.xo.3.ox. .... ....&#x | 0 3 3 0 | 0 0 3 0 6 0 3 0 0 0 0 | 0 0 0 0 1 0 0 3 3 0 0 0 1 0 0 0 0 | * * * * 32 * * * * oct
.x.x .... .... .x.o&#x | 0 4 0 2 | 0 0 2 2 0 4 0 0 0 1 0 | 0 0 0 0 0 1 0 0 0 2 2 0 0 0 0 0 0 | * * * * * 48 * * * trip
.xox .... .... ....&#x | 0 2 1 2 | 0 0 1 0 2 2 0 0 2 1 0 | 0 0 0 0 0 0 0 1 0 1 0 2 0 0 1 0 0 | * * * * * * 96 * * squippy
..o.3..x.3..x. .... | 0 0 12 0 | 0 0 0 0 0 0 12 6 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 4 4 0 0 0 | * * * * * * * 8 * tut
..ox .... ..xx ....&#x | 0 0 2 4 | 0 0 0 0 0 0 0 1 4 2 2 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 1 | * * * * * * * * 48 trip
wendy wrote:student91 asks about [[3,3,3]] in [3,3,5].
...
I imagaine one could build a x5o3o3o or even x3o3o5o as a kind of lace prism in 3,3,3. I never really tried this, but Coxeter's table of face-first {3,3,5} would be an ideal place to start.
student91 wrote:Tanks for these tips, they really helped me out.
I can now say that ex can be written as x3o3f3o + f3x3o3o + f3o3o3f + o3o3x3f + o3f3o3x, or xffoo3oxoof3fooxo3ooffx&#zx. All lacing edges are of length x, except the one between o3o3x3f and f3x3o3o, which is f.
xffoo3oxoof3fooxo3ooffx&#zx
o....3o....3o....3o.... & | 60 * * | 2 2 2 2 4 0 0 | 1 1 4 6 4 8 2 4 0 0 | 2 2 2 6 8 0 0
.o...3.o...3.o...3.o... & | * 40 * | 0 3 0 3 0 3 3 | 0 3 3 0 6 6 3 0 3 6 | 1 0 6 3 6 1 3
..o..3..o..3..o..3..o.. | * * 20 | 0 0 6 0 0 0 6 | 0 0 0 0 12 0 6 6 0 6 | 0 0 6 0 12 0 2
------------------------------+----------+---------------------------+--------------------------------------+------------------------
x.... ..... ..... ..... & | 2 0 0 | 60 * * * * * * | 1 0 2 2 0 0 0 0 0 0 | 2 1 0 2 0 0 0
oo...3oo...3oo...3oo...&#x & | 1 1 0 | * 120 * * * * * | 0 1 0 0 2 2 0 0 0 0 | 0 0 2 1 2 0 0
o.o..3o.o..3o.o..3o.o..&#x & | 1 0 1 | * * 120 * * * * | 0 0 0 0 2 0 1 2 0 0 | 0 0 1 0 4 0 0
o..o.3o..o.3o..o.3o..o.&#x & | 1 1 0 | * * * 120 * * * | 0 0 2 0 0 2 1 0 0 0 | 1 0 0 2 2 0 0
o...o3o...o3o...o3o...o&#x | 2 0 0 | * * * * 120 * * | 0 0 0 2 0 2 0 1 0 0 | 0 1 0 2 2 0 0
..... .x... ..... ..... & | 0 2 0 | * * * * * 60 * | 0 1 0 0 0 0 0 0 2 2 | 0 0 2 0 0 1 2
.oo..3.oo..3.oo..3.oo..&#x & | 0 1 1 | * * * * * * 120 | 0 0 0 0 2 0 1 0 0 2 | 0 0 2 0 2 0 1
------------------------------+----------+---------------------------+--------------------------------------+------------------------
x....3o.... ..... ..... & | 3 0 0 | 3 0 0 0 0 0 0 | 20 * * * * * * * * * | 2 0 0 0 0 0 0
..... ox... ..... .....&#x & | 1 2 0 | 0 2 0 0 0 1 0 | * 60 * * * * * * * * | 0 0 2 0 0 0 0
x..o. ..... ..... .....&#x & | 2 1 0 | 1 0 0 2 0 0 0 | * * 120 * * * * * * * | 1 0 0 1 0 0 0
x...o ..... ..... .....&#x & | 3 0 0 | 1 0 0 0 2 0 0 | * * * 120 * * * * * * | 0 1 0 1 0 0 0
ooo..3ooo..3ooo..3ooo..&#x & | 1 1 1 | 0 1 1 0 0 0 1 | * * * * 240 * * * * * | 0 0 1 0 1 0 0
oo..o3oo..o3oo..o3oo..o&#x & | 2 1 0 | 0 1 0 1 1 0 0 | * * * * * 240 * * * * | 0 0 0 1 1 0 0
o.oo.3o.oo.3o.oo.3o.oo.&#x & | 1 1 1 | 0 0 1 1 0 0 1 | * * * * * * 120 * * * | 0 0 0 0 2 0 0
o.o.o3o.o.o3o.o.o3o.o.o&#x | 2 0 1 | 0 0 2 0 1 0 0 | * * * * * * * 120 * * | 0 0 0 0 2 0 0
..... .x...3.o... ..... & | 0 3 0 | 0 0 0 0 0 3 0 | * * * * * * * * 40 * | 0 0 0 0 0 1 1
..... .xo.. ..... .....&#x & | 0 2 1 | 0 0 0 0 0 1 2 | * * * * * * * * * 120 | 0 0 1 0 0 0 1
------------------------------+----------+---------------------------+--------------------------------------+------------------------
x..o.3o..o. ..... .....&#x & | 3 1 0 | 3 0 0 3 0 0 0 | 1 0 3 0 0 0 0 0 0 0 | 40 * * * * * *
x...o ..... ..... o...x&#x | 4 0 0 | 2 0 0 0 4 0 0 | 0 0 0 4 0 0 0 0 0 0 | * 30 * * * * *
..... oxo.. ..... .....&#x & | 1 2 1 | 0 2 1 0 0 1 2 | 0 1 0 0 2 0 0 0 0 1 | * * 120 * * * *
..... ..... ..... oo..x&#x & | 3 1 0 | 1 1 0 2 2 0 0 | 0 0 1 1 0 2 0 0 0 0 | * * * 120 * * *
ooo.o3ooo.o3ooo.o3ooo.o&#x & | 2 1 1 | 0 1 2 1 1 0 1 | 0 0 0 0 1 1 1 1 0 0 | * * * * 240 * *
..... .x...3.o...3.o... & | 0 4 0 | 0 0 0 0 0 6 0 | 0 0 0 0 0 0 0 0 4 0 | * * * * * 10 *
..... .xo..3.oo.. .....&#x & | 0 3 1 | 0 0 0 0 0 3 3 | 0 0 0 0 0 0 0 0 1 3 | * * * * * * 40
Expansions now are also discoverable in pentachoric symmetry.
student91 wrote:Tanks for these tips, they really helped me out. I can now say that ex can be written as x3o3f3o + f3x3o3o + f3o3o3f + o3o3x3f + o3f3o3x, or xffoo3oxoof3fooxo3ooffx&#zx. All lacing edges are of length x, except the one between o3o3x3f and f3x3o3o, which is f. Expansions now are also discoverable in pentachoric symmetry.
Klitzing wrote:Then those to be checked single quirks would be:
xffoo3oxoof3fooxo3ooffx&#zx -> (-x)ffoo3xxoof3fooxo3ooffx&#zx -> oFFxx3xxoof3fooxo3ooffx&#zx
...
oFFxx3xxoof3fooxo3ooffx&#zx
o....3o....3o....3o.... | 30 * * * * | 2 4 4 0 0 0 0 0 0 0 0 0 0 | 1 2 4 2 2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 | 2 2 2 1 0 0 0 0 0 0 0 0
.o...3.o...3.o...3.o... | * 20 * * * | 0 0 0 3 3 3 0 0 0 0 0 0 0 | 0 0 0 0 0 0 3 6 3 3 0 0 0 0 0 0 0 0 0 | 3 0 0 0 1 3 1 0 0 0 0 0
..o..3..o..3..o..3..o.. | * * 20 * * | 0 0 0 0 3 0 3 3 0 0 0 0 0 | 0 0 0 0 0 0 0 3 0 3 3 6 0 0 0 0 0 0 0 | 3 0 0 0 0 1 0 1 3 0 0 0
...o.3...o.3...o.3...o. | * * * 60 * | 0 2 0 0 0 0 1 0 2 2 2 0 0 | 0 2 1 0 0 2 0 0 0 0 2 2 1 2 1 2 2 0 0 | 1 1 2 0 0 0 0 1 2 1 2 0
....o3....o3....o3....o | * * * * 60 | 0 0 2 0 0 1 0 1 0 0 2 1 2 | 0 0 0 2 2 2 0 0 2 1 0 2 0 0 0 2 1 2 1 | 2 0 2 2 0 0 1 0 1 0 1 1
----------------------------+----------------+-------------------------------------------+------------------------------------------------------------+----------------------------------
..... x.... ..... ..... | 2 0 0 0 0 | 30 * * * * * * * * * * * * | 1 0 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 2 0 0 0 0 0 0 0 0 0 0
o..o.3o..o.3o..o.3o..o.&#x | 1 0 0 1 0 | * 120 * * * * * * * * * * * | 0 1 1 0 0 1 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
o...o3o...o3o...o3o...o&#x | 1 0 0 0 1 | * * 120 * * * * * * * * * * | 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 0 1 1 0 0 0 0 0 0 0 0
..... .x... ..... ..... | 0 2 0 0 0 | * * * 30 * * * * * * * * * | 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 | 1 0 0 0 1 2 0 0 0 0 0 0
.oo..3.oo..3.oo..3.oo..&#x | 0 1 1 0 0 | * * * * 60 * * * * * * * * | 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 0 0 0 0 | 2 0 0 0 0 1 0 0 0 0 0 0
.o..o3.o..o3.o..o3.o..o&#x | 0 1 0 0 1 | * * * * * 60 * * * * * * * | 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 | 2 0 0 0 0 0 1 0 0 0 0 0
..oo.3..oo.3..oo.3..oo.&#x | 0 0 1 1 0 | * * * * * * 60 * * * * * * | 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 | 2 0 0 0 0 0 0 1 1 0 0 0
..o.o3..o.o3..o.o3..o.o&#x | 0 0 1 0 1 | * * * * * * * 60 * * * * * | 0 0 0 0 0 0 0 0 0 1 0 2 0 0 0 0 0 0 0 | 2 0 0 0 0 0 0 0 1 0 0 0
...x. ..... ..... ..... | 0 0 0 2 0 | * * * * * * * * 60 * * * * | 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 | 0 1 1 0 0 0 0 0 0 1 1 0
..... ..... ...x. ..... | 0 0 0 2 0 | * * * * * * * * * 60 * * * | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 0 | 0 0 0 0 0 0 0 1 1 1 1 0
...oo3...oo3...oo3...oo&#x | 0 0 0 1 1 | * * * * * * * * * * 120 * * | 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 | 1 0 1 0 0 0 0 0 1 0 1 0
....x ..... ..... ..... | 0 0 0 0 2 | * * * * * * * * * * * 30 * | 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 | 0 0 2 2 0 0 0 0 0 0 1 1
..... ..... ..... ....x | 0 0 0 0 2 | * * * * * * * * * * * * 60 | 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 1 | 1 0 0 1 0 0 1 0 0 0 0 1
----------------------------+----------------+-------------------------------------------+------------------------------------------------------------+----------------------------------
o....3x.... ..... ..... | 3 0 0 0 0 | 3 0 0 0 0 0 0 0 0 0 0 0 0 | 10 * * * * * * * * * * * * * * * * * * | 0 2 0 0 0 0 0 0 0 0 0 0
o..x. ..... ..... ..... | 1 0 0 2 0 | 0 2 0 0 0 0 0 0 1 0 0 0 0 | * 60 * * * * * * * * * * * * * * * * * | 0 1 1 0 0 0 0 0 0 0 0 0
..... x..o. ..... .....&#x | 2 0 0 1 0 | 1 2 0 0 0 0 0 0 0 0 0 0 0 | * * 60 * * * * * * * * * * * * * * * * | 1 1 0 0 0 0 0 0 0 0 0 0
o...x ..... ..... .....&#x | 1 0 0 0 2 | 0 0 2 0 0 0 0 0 0 0 0 1 0 | * * * 60 * * * * * * * * * * * * * * * | 0 0 1 1 0 0 0 0 0 0 0 0
..... ..... ..... o...x&#x | 1 0 0 0 2 | 0 0 2 0 0 0 0 0 0 0 0 0 1 | * * * * 60 * * * * * * * * * * * * * * | 1 0 0 1 0 0 0 0 0 0 0 0
o..oo3o..oo3o..oo3o..oo&#x | 1 0 0 1 1 | 0 1 1 0 0 0 0 0 0 0 1 0 0 | * * * * * 120 * * * * * * * * * * * * * | 1 0 1 0 0 0 0 0 0 0 0 0
..... .x...3.o... ..... | 0 3 0 0 0 | 0 0 0 3 0 0 0 0 0 0 0 0 0 | * * * * * * 20 * * * * * * * * * * * * | 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 | * * * * * * * 60 * * * * * * * * * * * | 1 0 0 0 0 1 0 0 0 0 0 0
..... ..... ..... .o..x&#x | 0 1 0 0 2 | 0 0 0 0 0 2 0 0 0 0 0 0 1 | * * * * * * * * 60 * * * * * * * * * * | 1 0 0 0 0 0 1 0 0 0 0 0
.oo.o3.oo.o3.oo.o3.oo.o&#x | 0 1 1 0 1 | 0 0 0 0 1 1 0 1 0 0 0 0 0 | * * * * * * * * * 60 * * * * * * * * * | 2 0 0 0 0 0 0 0 0 0 0 0
..... ..... ..ox. .....&#x | 0 0 1 2 0 | 0 0 0 0 0 0 2 0 0 1 0 0 0 | * * * * * * * * * * 60 * * * * * * * * | 0 0 0 0 0 0 0 1 1 0 0 0
..ooo3..ooo3..ooo3..ooo&#x | 0 0 1 1 1 | 0 0 0 0 0 0 1 1 0 0 1 0 0 | * * * * * * * * * * * 120 * * * * * * * | 1 0 0 0 0 0 0 0 1 0 0 0
...x.3...o. ..... ..... | 0 0 0 3 0 | 0 0 0 0 0 0 0 0 3 0 0 0 0 | * * * * * * * * * * * * 20 * * * * * * | 0 1 0 0 0 0 0 0 0 1 0 0
...x. ..... ...x. ..... | 0 0 0 4 0 | 0 0 0 0 0 0 0 0 2 2 0 0 0 | * * * * * * * * * * * * * 30 * * * * * | 0 0 0 0 0 0 0 0 0 1 1 0
..... ...o.3...x. ..... | 0 0 0 3 0 | 0 0 0 0 0 0 0 0 0 3 0 0 0 | * * * * * * * * * * * * * * 20 * * * * | 0 0 0 0 0 0 0 1 0 1 0 0
...xx ..... ..... .....&#x | 0 0 0 2 2 | 0 0 0 0 0 0 0 0 1 0 2 1 0 | * * * * * * * * * * * * * * * 60 * * * | 0 0 1 0 0 0 0 0 0 0 1 0
..... ..... ...xo .....&#x | 0 0 0 2 1 | 0 0 0 0 0 0 0 0 0 1 2 0 0 | * * * * * * * * * * * * * * * * 60 * * | 0 0 0 0 0 0 0 0 1 0 1 0
....x ..... ..... ....x | 0 0 0 0 4 | 0 0 0 0 0 0 0 0 0 0 0 2 2 | * * * * * * * * * * * * * * * * * 30 * | 0 0 0 1 0 0 0 0 0 0 0 1
..... ..... ....o3....x | 0 0 0 0 3 | 0 0 0 0 0 0 0 0 0 0 0 0 3 | * * * * * * * * * * * * * * * * * * 20 | 0 0 0 0 0 0 1 0 0 0 0 1
----------------------------+----------------+-------------------------------------------+------------------------------------------------------------+----------------------------------
..... xxoof ..... ooffx&#zx | 2 2 2 2 4 | 1 4 4 1 4 4 2 4 0 0 4 0 2 | 0 0 2 0 2 4 0 2 2 4 0 4 0 0 0 0 0 0 0 | 30 * * * * * * * * * * * ike
o..x.3x..o. ..... .....&#x | 3 0 0 3 0 | 3 6 0 0 0 0 0 0 3 0 0 0 0 | 1 3 3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | * 20 * * * * * * * * * * oct
o..xx ..... ..... .....&#x | 1 0 0 2 2 | 0 2 2 0 0 0 0 0 1 0 2 1 0 | 0 1 0 1 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 | * * 60 * * * * * * * * * squippy
o...x ..... ..... o...x&#x | 1 0 0 0 4 | 0 0 4 0 0 0 0 0 0 0 0 2 2 | 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | * * * 30 * * * * * * * * squippy
..... .x...3.o...3.o... | 0 4 0 0 0 | 0 0 0 6 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 | * * * * 5 * * * * * * * tet
..... .xo..3.oo.. .....&#x | 0 3 1 0 0 | 0 0 0 3 3 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 | * * * * * 20 * * * * * * tet
..... ..... .o..o3.o..x&#x | 0 1 0 0 3 | 0 0 0 0 0 3 0 0 0 0 0 0 3 | 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 1 | * * * * * * 20 * * * * * tet
..... ..oo.3..ox. .....&#x | 0 0 1 3 0 | 0 0 0 0 0 0 3 0 0 3 0 0 0 | 0 0 0 0 0 0 0 0 0 0 3 0 0 0 1 0 0 0 0 | * * * * * * * 20 * * * * tet
..... ..... ..oxo .....&#x | 0 0 1 2 1 | 0 0 0 0 0 0 2 1 0 1 2 0 0 | 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 1 0 0 | * * * * * * * * 60 * * * tet
...x.3...o.3...x. ..... | 0 0 0 12 0 | 0 0 0 0 0 0 0 0 12 12 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 4 6 4 0 0 0 0 | * * * * * * * * * 5 * * co
...xx ..... ...xo .....&#x | 0 0 0 4 2 | 0 0 0 0 0 0 0 0 2 2 4 1 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 2 0 0 | * * * * * * * * * * 30 * trip
....x ..... ....o3....x | 0 0 0 0 6 | 0 0 0 0 0 0 0 0 0 0 0 3 6 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 | * * * * * * * * * * * 10 trip
Nice. In the structure of the just discovered pSe are some thawro-pseudopyramids. The apices of these pyramids are the triangles of the coes. These triangles can be moved away in a further pSe when expanding on the third node. This would give you oFFxx3xxoof3foo(-x)o3oofFx&#zx => oFFxx3xxoof3Fxxox3oofFx&#zx. Of course you can see this pSe as a double pSe starting with ex.Klitzing wrote:[...]
most things here are correct, but not the last conclusion. You can easily apply triple quirk without getting back your starting position, e.g. foxo3ooof3xfoo*b3oxfo => fo(-x)o3ooxf3xfoo*b3oxfo => fooo3oo(-x)f3xfxo*b3oxFo => fooo3ooof3xf(-x)o*b3oxFo. This then gives a challenge for the third conclusion, as fooo3ooof3xf(-x)o*b3oxFo => fooo3xoof3(-x)f(-x)o*b3oxFo =E=> fooo3xoof3oFox*b3oxFo might be CRF, though I doubt that, as there's much stuff taken away by negative nodes, probably revealing some f-edges. This means the search here was still exhaustive, but for different reasons. Therefore I think we have been a bit too fast at concluding the thing to be exhaustive, and we should take more such care in the future.Klitzing wrote:it looks like a mostly exhaustive investigation of expansions of ex
... with respect to the demicubic subsymmetry of ex = foxo3ooof3xfoo *b3oxfo&#zx, for sure.Thus again yes, looks indeed exhaustive.
- Double quirks then stick always to the same layer. Multiple double quirks accordingly to multiple layers.
- "Side node expansions" use multiple layer applications of single quirks.
- Mixtures of single quirk and double quirk on different layers could be considered too, but then cannot be expanded back to convexity without producing an u-edge.
- Any triple quirk here would just undo the second.
--- rk
student91 wrote:You can easily apply triple quirk without getting back your starting position, e.g. foxo3ooof3xfoo*b3oxfo => fo(-x)o3ooxf3xfoo*b3oxfo => fooo3oo(-x)f3xfxo*b3oxFo => fooo3ooof3xf(-x)o*b3oxFo. This then gives a challenge for the third conclusion, as fooo3ooof3xf(-x)o*b3oxFo => fooo3xoof3(-x)f(-x)o*b3oxFo =E=> fooo3xoof3oFox*b3oxFo might be CRF, though I doubt that, as there's much stuff taken away by negative nodes, probably revealing some f-edges. This means the search here was still exhaustive, but for different reasons. Therefore I think we have been a bit too fast at concluding the thing to be exhaustive, and we should take more such care in the future.
Klitzing wrote:Then those to be checked single quirks would be:
[...]
xffoo3oxoof3fooxo3ooffx&#zx -> xFfoo3o(-x)oof3fxoxo3ooffx&#zx -> xFfoo3xoxxF3fxoxo3ooffx&#zx
[...]
student91 wrote:Nice. In the structure of the just discovered pSe are some thawro-pseudopyramids. The apices of these pyramids are the triangles of the coes. These triangles can be moved away in a further pSe when expanding on the third node. This would give you oFFxx3xxoof3foo(-x)o3oofFx&#zx => oFFxx3xxoof3Fxxox3oofFx&#zx. Of course you can see this pSe as a double pSe starting with ex.Klitzing wrote:[...]
Klitzing wrote:Btw. there will be a further CRF (to be checked) with just 2 quirks in that subsymmetry. (More precisely it even would have an additional inversiv one, which reflects itself in the mirror symmetry of the final symbol.)
ex = xffoo3oxoof3fooxo3ooffx&#zx
-> (-x)ffoo3xxoof3fooxo3ooffx&#zx (quirk in 1st layer, 1st node)
-> (-x)ffoo3xxoof3fooxx3ooff(-x)&#zx (quirk in 5th layer, 4th node)
-> oFFxx3xxoof3fooxx3xxFFo&#zx (Stott expansion in 1st and 4th node)
oFFxx3xxoof3fooxx3xxFFo&#zx
o....3o....3o....3o.... & | 120 * * | 2 1 2 2 0 0 0 | 1 2 1 2 3 0 0 0 0 0 | 1 1 1 3 0 0
.o...3.o...3.o...3.o... & | * 120 * | 0 0 2 0 2 2 1 | 0 0 2 1 0 1 2 1 2 2 | 0 1 0 3 1 1
..o..3..o..3..o..3..o.. | * * 20 | 0 0 0 0 0 0 6 | 0 0 0 0 0 0 0 0 6 6 | 0 0 0 6 0 2
------------------------------+------------+----------------------------+------------------------------------+------------------
..... x.... ..... ..... & | 2 0 0 | 120 * * * * * * | 1 1 0 1 0 0 0 0 0 0 | 1 1 0 1 0 0
..... ..... ..... x.... & | 2 0 0 | * 60 * * * * * | 0 2 0 0 2 0 0 0 0 0 | 1 0 1 2 0 0
o..o.3o..o.3o..o.3o..o.&#x & | 1 1 0 | * * 240 * * * * | 0 0 1 1 0 0 0 0 0 1 | 0 1 0 2 0 0
o...o3o...o3o...o3o...o&#x | 2 0 0 | * * * 120 * * * | 0 0 0 0 2 0 0 0 0 1 | 0 0 1 2 0 0
..... .x... ..... ..... & | 0 2 0 | * * * * 120 * * | 0 0 0 0 0 1 1 0 1 0 | 0 0 0 1 1 1
..... ..... ..... .x... & | 0 2 0 | * * * * * 120 * | 0 0 1 0 0 0 1 1 0 0 | 0 1 0 1 1 0
.oo..3.oo..3.oo..3.oo..&#x & | 0 1 1 | * * * * * * 120 | 0 0 0 0 0 0 0 0 2 2 | 0 0 0 3 0 1
------------------------------+------------+----------------------------+------------------------------------+------------------
o....3x.... ..... ..... & | 3 0 0 | 3 0 0 0 0 0 0 | 40 * * * * * * * * * | 1 1 0 0 0 0
..... x.... ..... x.... & | 4 0 0 | 2 2 0 0 0 0 0 | * 60 * * * * * * * * | 1 0 0 1 0 0
o..x. ..... ..... .....&#x & | 1 2 0 | 0 0 2 0 0 1 0 | * * 120 * * * * * * * | 0 1 0 1 0 0
..... x..o. ..... .....&#x & | 2 1 0 | 1 0 2 0 0 0 0 | * * * 120 * * * * * * | 0 1 0 1 0 0
o...x ..... ..... .....&#x & | 3 0 0 | 0 1 0 2 0 0 0 | * * * * 120 * * * * * | 0 0 1 1 0 0
..... .x...3.o... ..... & | 0 3 0 | 0 0 0 0 3 0 0 | * * * * * 40 * * * * | 0 0 0 0 1 1
..... .x... ..... .x... & | 0 4 0 | 0 0 0 0 2 2 0 | * * * * * * 60 * * * | 0 0 0 1 1 0
..... ..... .o...3.x... & | 0 3 0 | 0 0 0 0 0 3 0 | * * * * * * * 40 * * | 0 1 0 0 1 0
..... .xo.. ..... .....&#x & | 0 2 1 | 0 0 0 0 1 0 2 | * * * * * * * * 120 * | 0 0 0 1 0 1
ooooo3ooooo3ooooo3ooooo&#x | 2 2 1 | 0 0 2 1 0 0 2 | * * * * * * * * * 120 | 0 0 0 2 0 0
------------------------------+------------+----------------------------+------------------------------------+------------------
o....3x.... ..... x.... & | 6 0 0 | 6 3 0 0 0 0 0 | 2 3 0 0 0 0 0 0 0 0 | 20 * * * * * trip
o..x.3x..o. ..... .....&#x & | 3 3 0 | 3 0 6 0 0 3 0 | 1 0 3 3 0 0 0 1 0 0 | * 40 * * * * oct
o...x ..... ..... x...o&#x | 4 0 0 | 0 2 0 4 0 0 0 | 0 0 0 0 4 0 0 0 0 0 | * * 30 * * * tet
oFFxx ..... fooxx .....&#zx & | 6 6 2 | 2 2 8 4 2 2 6 | 0 1 2 2 2 0 1 0 2 4 | * * * 60 * * bilbiro (tower: 43125)
..... .x...3.o...3.x... & | 0 12 0 | 0 0 0 0 12 12 0 | 0 0 0 0 0 4 6 4 0 0 | * * * * 10 * co
..... .xo..3.oo.. .....&#x & | 0 3 1 | 0 0 0 0 3 0 3 | 0 0 0 0 0 1 0 0 3 0 | * * * * * 40 tet
Klitzing wrote:Klitzing wrote:Btw. there will be a further CRF [..]
ex = xffoo3oxoof3fooxo3ooffx&#zx -> [...] -> oFFxx3xxoof3fooxx3xxFFo&#zx (Stott expansion in 1st and 4th node)
Here it comes, finally
- Code: Select all
oFFxx3xxoof3fooxx3xxFFo&#zx
o....3o....3o....3o.... & | 120 * * | 2 1 2 2 0 0 0 | 1 2 1 2 3 0 0 0 0 0 | 1 1 1 3 0 0
.o...3.o...3.o...3.o... & | * 120 * | 0 0 2 0 2 2 1 | 0 0 2 1 0 1 2 1 2 2 | 0 1 0 3 1 1
..o..3..o..3..o..3..o.. | * * 20 | 0 0 0 0 0 0 6 | 0 0 0 0 0 0 0 0 6 6 | 0 0 0 6 0 2
------------------------------+------------+----------------------------+------------------------------------+------------------
..... x.... ..... ..... & | 2 0 0 | 120 * * * * * * | 1 1 0 1 0 0 0 0 0 0 | 1 1 0 1 0 0
..... ..... ..... x.... & | 2 0 0 | * 60 * * * * * | 0 2 0 0 2 0 0 0 0 0 | 1 0 1 2 0 0
o..o.3o..o.3o..o.3o..o.&#x & | 1 1 0 | * * 240 * * * * | 0 0 1 1 0 0 0 0 0 1 | 0 1 0 2 0 0
o...o3o...o3o...o3o...o&#x | 2 0 0 | * * * 120 * * * | 0 0 0 0 2 0 0 0 0 1 | 0 0 1 2 0 0
..... .x... ..... ..... & | 0 2 0 | * * * * 120 * * | 0 0 0 0 0 1 1 0 1 0 | 0 0 0 1 1 1
..... ..... ..... .x... & | 0 2 0 | * * * * * 120 * | 0 0 1 0 0 0 1 1 0 0 | 0 1 0 1 1 0
.oo..3.oo..3.oo..3.oo..&#x & | 0 1 1 | * * * * * * 120 | 0 0 0 0 0 0 0 0 2 2 | 0 0 0 3 0 1
------------------------------+------------+----------------------------+------------------------------------+------------------
o....3x.... ..... ..... & | 3 0 0 | 3 0 0 0 0 0 0 | 40 * * * * * * * * * | 1 1 0 0 0 0
..... x.... ..... x.... & | 4 0 0 | 2 2 0 0 0 0 0 | * 60 * * * * * * * * | 1 0 0 1 0 0
o..x. ..... ..... .....&#x & | 1 2 0 | 0 0 2 0 0 1 0 | * * 120 * * * * * * * | 0 1 0 1 0 0
..... x..o. ..... .....&#x & | 2 1 0 | 1 0 2 0 0 0 0 | * * * 120 * * * * * * | 0 1 0 1 0 0
o...x ..... ..... .....&#x & | 3 0 0 | 0 1 0 2 0 0 0 | * * * * 120 * * * * * | 0 0 1 1 0 0
..... .x...3.o... ..... & | 0 3 0 | 0 0 0 0 3 0 0 | * * * * * 40 * * * * | 0 0 0 0 1 1
..... .x... ..... .x... & | 0 4 0 | 0 0 0 0 2 2 0 | * * * * * * 60 * * * | 0 0 0 1 1 0
..... ..... .o...3.x... & | 0 3 0 | 0 0 0 0 0 3 0 | * * * * * * * 40 * * | 0 1 0 0 1 0
..... .xo.. ..... .....&#x & | 0 2 1 | 0 0 0 0 1 0 2 | * * * * * * * * 120 * | 0 0 0 1 0 1
ooooo3ooooo3ooooo3ooooo&#x | 2 2 1 | 0 0 2 1 0 0 2 | * * * * * * * * * 120 | 0 0 0 2 0 0
------------------------------+------------+----------------------------+------------------------------------+------------------
o....3x.... ..... x.... & | 6 0 0 | 6 3 0 0 0 0 0 | 2 3 0 0 0 0 0 0 0 0 | 20 * * * * * trip
o..x.3x..o. ..... .....&#x & | 3 3 0 | 3 0 6 0 0 3 0 | 1 0 3 3 0 0 0 1 0 0 | * 40 * * * * oct
o...x ..... ..... x...o&#x | 4 0 0 | 0 2 0 4 0 0 0 | 0 0 0 0 4 0 0 0 0 0 | * * 30 * * * tet
oFFxx ..... fooxx .....&#zx & | 6 6 2 | 2 2 8 4 2 2 6 | 0 1 2 2 2 0 1 0 2 4 | * * * 60 * * bilbiro (tower: 43125)
..... .x...3.o...3.x... & | 0 12 0 | 0 0 0 0 12 12 0 | 0 0 0 0 0 4 6 4 0 0 | * * * * 10 * co
..... .xo..3.oo.. .....&#x & | 0 3 1 | 0 0 0 0 3 0 3 | 0 0 0 0 0 1 0 0 3 0 | * * * * * 40 tet
Lacing edges 1-2 & 4-5, 1-3 & 3-5, and 2-4 all are of size f and thus discarded. The other ones are of unit size.
The local complexes here can be recognized as follows: Take a co and apply tetrahedral symmetry to it: Attach 4 tets and 4 octs onto its triangles. Then attach bilbiroes in the then obvious way to its squares. Onto the opposite side of the oct then connect a trip. To the squares of the trips then attach further bilbiros with the second square (those are NOT equivalent!). The lacing triangles of the octs, pointing away from the coes, already are connected to the first mentioned bilbiroes (to an id-like triangle). The lacing triangles of the octs, pointing away from the trips, now get connected to the second set/side of bilbiroes (to lune-type triangles). The pentagons of both "sets" of bilbiroes then connect. - Again, there is just one class of bilbiroes, but because of ist lacking mirror symmetrical surroundings the same bilbiro acts with the one square in the one way, with the other one in the second one!
--- rk
xxfoF3oxxFx3xFxxo3Fofxx&#zx
1-2 & 4-5, 1-4 & 2-5 and 1-5 are f, all others are x.
------------------------------+------------+--------------------------------+--------------------------------------------+--------------------+
v....3v....3v....3v.... & | 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 |
.v...3.v...3.v...3.v... & | * 120 * | 0 1 1 0 0 0 2 2 | 0 1 0 0 0 2 3 0 2 0 2 | 0 3 1 2 0 |
..v..3..v..3..v..3..v.. | * * 120| 0 0 0 1 0 2 2 0 | 0 0 0 0 1 1 0 2 2 2 1 | 0 2 2 0 2 |
------------------------------+------------+--------------------------------+--------------------------------------------+--------------------+
x....3.....3.....3..... & | 2 0 0 | 120 * * * * * * * | 1 0 1 0 0 1 0 0 0 0 0 | 1 1 1 0 0 |
.x...3.....3.....3..... & | 0 2 0 | * 60 * * * * * * | 0 1 0 0 0 0 2 0 0 0 0 | 0 2 0 2 0 |
.....3.x...3.....3..... & | 0 2 0 | * * 60 * * * * * | 0 1 0 0 0 0 0 0 2 0 0 | 0 2 1 0 0 |
.....3..x..3.....3..... & | 0 0 2 | * * * 120 * * * * | 0 0 0 0 1 0 0 1 1 1 0 | 0 1 1 0 2 |
.....3.....3x....3..... & | 2 0 0 | * * * * 120 * * * | 0 0 1 2 0 0 0 0 0 1 0 | 1 0 1 0 1 |
v.v..3v.v..3v.v..3v.v..&#zx & | 1 0 1 | * * * * * 240 * * | 0 0 0 0 0 1 0 1 0 1 0 | 0 1 1 0 1 |
.vv..3.vv..3.vv..3.vv..&#zx & | 0 1 1 | * * * * * * 240 * | 0 0 0 0 0 1 0 0 1 0 1 | 0 2 1 0 0 |
.v.v.3.v.v.3.v.v.3.v.v.&#zx | 0 2 0 | * * * * * * * 120| 0 0 0 0 0 0 2 0 0 0 1 | 0 2 0 2 0 |
------------------------------+------------+--------------------------------+--------------------------------------------+--------------------+
x....3o....3.....3..... & | 3 0 0 | 3 0 0 0 0 0 0 0 | 40 * * * * * * * * * * | 1 1 0 0 0 |
.x...3.x...3.....3..... & | 0 6 0 | 0 3 3 0 0 0 0 0 | * 20 * * * * * * * * * | 0 2 0 0 0 |
x....3.....3x....3..... & | 4 0 0 | 2 0 0 0 2 0 0 0 | * * 60 * * * * * * * * | 1 0 1 0 0 |
.....3o....3x....3..... & | 3 0 0 | 0 0 0 0 3 0 0 0 | * * * 40 * * * * * * * | 1 0 0 0 1 |
.....3..x..3..x..3..... | 0 0 6 | 0 0 0 6 0 0 0 0 | * * * * 20 * * * * * * | 0 0 0 0 2 |
x.fo.3.....3.....3.....&#zx & | 2 2 1 | 1 0 0 0 0 2 2 0 | * * * * * 120 * * * * * | 0 1 1 0 0 |
.x.o.3.....3.....3.....&#zx & | 0 3 0 | 0 1 0 0 0 0 0 2 | * * * * * * 120 * * * * | 0 1 0 1 0 |
.....3o.x..3.....3.....&#zx & | 1 0 2 | 0 0 0 1 0 2 0 0 | * * * * * * * 120 * * * | 0 1 0 0 1 |
.....3.xx..3.....3.....&#zx & | 0 2 2 | 0 0 1 1 0 0 2 0 | * * * * * * * * 120 * * | 0 1 1 0 0 |
.....3.....3x.x..3.....&#zx & | 2 0 2 | 0 0 0 1 1 2 0 0 | * * * * * * * * * 120 * | 0 0 1 0 1 |
.vvv.3.vvv.3.vvv.3.vvv.&#zx | 0 2 1 | 0 0 0 0 0 0 2 1 | * * * * * * * * * * 120| 0 2 0 0 0 |
------------------------------+------------+--------------------------------+--------------------------------------------+--------------------+
x....3o....3x....3..... & | 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.3.....3.....&#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.3.....3x.xx.3.....&#zx & | 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.3.....3.....3.o.x.&#zx & | 0 4 0 | 0 2 0 0 0 0 0 4 | 0 0 0 0 0 0 4 0 0 0 0 | * * * 60 * | tet
.....3o.x..3x.x..3.....&#zx & | 3 0 6 | 0 0 0 6 3 6 0 0 | 0 0 0 1 1 0 0 3 0 3 0 | * * * * 40 | tricu
Here that comes:student91 wrote:[...]
Btw, we should gain some systematics on these expansions, will think bout that tomorrow.
In summary, the ones that haven't been investigated and that don't produce (F+x)-edges are:student91 wrote:[extremely long post]
student91 wrote:In summary, the ones that haven't been investigated and that don't produce (F+x)-edges are:student91 wrote:[extremely long post]
[...]
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