x4oPo4x (M=2 for P=3; resp. M->oo for P>3)
. . . . | 12NM | 2P | 2P 2P | 2 2P vf: sPs2s
-----------+------+-------+-----------+--------
x . . . & | 2 | 12NMP | 2 2 | 1 3
-----------+------+-------+-----------+--------
x4o . . & | 4 | 4 | 6NMP * | 1 1
x . . x | 4 | 4 | * 6NMP | 0 2
-----------+------+-------+-----------+--------
x4oPo . & | 4M | 2MP | MP 0 | 6N *
x4o . x & | 8 | 12 | 2 4 | * 3NMP
xxoo!x4oPo4x (M=2 for P=3; resp. M->oo for P>3)
24NMP | 1 2 2 | 2 2 1 3 | 1 3 1
------+-------------------+----------------------+-------------
2 | 12NMP * * | 2 2 0 0 | 1 3 0
2 | * 24NMP * | 1 0 1 1 | 1 1 1
2 | * * 24NMP | 0 1 0 2 | 0 2 1
------+-------------------+----------------------+-------------
8 | 4 4 0 | 6NMP * * * | 1 1 0
8 | 4 0 4 | * 6NMP * * | 0 2 0
P | 0 P 0 | * * 24NM * | 1 0 1
3 | 0 1 2 | * * * 24NMP | 0 1 1
------+-------------------+----------------------+-------------
4MP | 2MP 4MP 0 | MP 0 4M 0 | 6N * * x4xPo
24 | 12 8 16 | 2 4 0 8 | * 3NMP * x4x3o
2P | 0 2P 2P | 0 0 2 2P | * * 12NM sPs2s
oxoo!x4oPo4x (M=2 for P=3; resp. M->oo for P>3)
12NMP | 4 4 | 2 2 2 6 | 1 3 2
------+-------------+----------------------+-------------
2 | 24NMP * | 1 0 1 1 | 1 1 1
2 | * 24NMP | 0 1 0 2 | 0 2 1
------+-------------+----------------------+-------------
4 | 4 0 | 6NMP * * * | 1 1 0
4 | 0 4 | * 6NMP * * | 0 2 0
P | P 0 | * * 24NM * | 1 0 1
3 | 1 2 | * * * 24NMP | 0 1 1
------+-------------+----------------------+-------------
2MP | 4MP 0 | MP 0 4M 0 | 6N * * o4xPo
12 | 8 16 | 2 4 0 8 | * 3NMP * o4x3o
2P | 2P 2P | 0 0 2 2P | * * 12NM sPs2s
sooo!x4oPo4x (M=2 for P=3; resp. M->oo for P>3)
6NM | 2P 2P | 2P 6P | 2 2P 2P
----+-----------+------------+------------
2 | 6NMP * | 2 2 | 1 1 2
2 | * 6NMP | 0 4 | 0 2 2
----+-----------+------------+------------
P | P 0 | 12NM * | 1 0 1
3 | 1 2 | * 12NMP | 0 1 1
----+-----------+------------+------------
2M | PM 0 | 2M 0 | 6N * * xPoPo
4 | 2 4 | 0 4 | * 3NMP * x3o3o
2P | 2P 2P | 2 2P | * * 6NM sPs2s
h^2 = (c + 1/2) / (c + 1),
c = cos(pi/P)
wendy wrote:... Regarding Richard's comments over the xo4ox&#xt ...
... etc, one might note that the equators of the vertex-figures never cross at any instance, and thus must continue throughout. ...
... In the cubic case, this amounts to the equators giving rise to layers, being only in planes like z=2n. What's more statling is the appearence of a Z4 system in the form of a x3o8o, as the etching on these planes.
wendy wrote:... One notes by the laws of symmetry that the equality: saPsePsaAPse = saPsePsa4o = sePsa4o3o exists. ...
sPsPs *bPs = sPsPs4o = sPs4o3o
sPoPs *bPo = sPoPs4o = oPs4o3o
oPsPo *bPs = oPsPo4o = sPo4o3o
o4x4o4o (N,M,K->oo)
. . . . | NMK | 8 | 4 8 | 4 2 vf:q-cube
--------+-----+------+----------+--------
. x . . | 2 | 4NMK | 1 2 | 2 1
--------+-----+------+----------+--------
o4x . . | 4 | 4 | NMK * | 2 0
. x4o . | 4 | 4 | * 2NMK | 1 1
--------+-----+------+----------+--------
o4x4o . | 2M | 4M | M M | 2NK * squat
. x4o4o | K | 2K | 0 K | * 2NM squat
x4o4x4o (N,M,K->oo)
. . . . | 4NMK | 2 4 | 1 4 2 2 | 2 2 1 vf:q-trip
--------+------+-----------+--------------------+------------
x . . . | 2 | 4NMK * | 1 2 0 0 | 2 1 0
. . x . | 2 | * 8NMK | 0 1 1 1 | 1 1 1
--------+------+-----------+--------------------+------------
x4o . . | 4 | 4 0 | NMK * * * | 2 0 0
x . x . | 4 | 2 2 | * 4NMK * * | 1 1 0
. o4x . | 4 | 0 4 | * * 2NMK * | 1 0 1
. . x4o | 4 | 0 4 | * * * 2NMK | 0 1 1
--------+------+-----------+--------------------+------------
x4o4x . | 4M | 4M 4M | M 2M M 0 | 2NK * * squat
x . x4o | 8 | 4 8 | 0 4 0 2 | * NMK * cube
. o4x4o | 2K | 0 4K | 0 0 K K | * * 2NM squat
x4x4x4o (N,M,K->oo)
. . . . | 8NMK | 1 1 2 | 1 2 2 1 | 2 1 1 vf:oq3oo&#k
--------+------+----------------+--------------------+------------
x . . . | 2 | 4NMK * * | 1 2 0 0 | 2 1 0
. x . . | 2 | * 4NMK * | 1 0 2 0 | 2 0 1
. . x . | 2 | * * 8NMK | 0 1 1 1 | 1 1 1
--------+------+----------------+--------------------+------------
x4x . . | 8 | 4 4 0 | NMK * * * | 2 0 0
x . x . | 4 | 2 0 2 | * 4NMK * * | 1 1 0
. x4x . | 8 | 0 4 4 | * * 2NMK * | 1 0 1
. . x4o | 4 | 0 0 4 | * * * 2NMK | 0 1 1
--------+------+----------------+--------------------+------------
x4x4x . | 8M | 4M 4M 4M | M 2M M 0 | 2NK * * tosquat
x . x4o | 8 | 4 0 8 | 0 4 0 2 | * NMK * cube
. x4x4o | 4K | 0 2K 4K | 0 0 K K | * * 2NM tosquat
o4s4o4o (N,M,K->oo)
demi( . . . . ) | NMK | 4 8 | 16 8 | 4 2 8 vf:q-co
----------------+-----+-----------+-----------+------------
o4s . . | 2 | 2NMK * | 4 0 | 2 0 2
. s4o . | 2 | * 4NMK | 2 2 | 1 1 2
----------------+-----+-----------+-----------+------------
sefa( o4s4o . ) | 4 | 2 2 | 4NMK * | 1 0 1
sefa( . s4o4o ) | 4 | 0 4 | * 2NMK | 0 1 1
----------------+-----+-----------+-----------+------------
o4s4o . | M | M M | M 0 | 4NK * * squat
. s4o4o | K | 0 2K | 0 K | * 2NM * squat
sefa( o4s4o4o ) | 8 | 4 8 | 4 2 | * * NMK cube
s4o4s4o (N,M,K->oo)
demi( . . . . ) | 2NMK | 1 4 2 2 | 8 6 4 | 2 2 1 6
----------------+------+--------------------+----------------+-----------------
s4o . . | 2 | NMK * * * | 4 0 0 | 2 0 0 2
s . s . | 2 | * 4NMK * * | 2 2 0 | 1 1 0 1
. o4s . | 2 | * * 2NMK * | 2 0 2 | 1 0 1 2
. . s4o | 2 | * * * 2NMK | 0 2 2 | 0 1 1 2
----------------+------+--------------------+----------------+-----------------
sefa( s4o4s . ) | 4 | 1 2 1 0 | 4NMK * * | 1 0 0 1
sefa( s . s4o ) | 3 | 0 2 0 1 | * 4NMK * | 0 1 0 1
sefa( . o4s4o ) | 4 | 0 0 2 2 | * * 2NMK | 0 0 1 1
----------------+------+--------------------+----------------+-----------------
s4o4s . | 2M | M 2M M 0 | 2M 0 0 | 2NK * * * squat
s . s4o | 4 | 0 4 0 2 | 0 4 0 | * NMK * * tet
. o4s4o | K | 0 0 K K | 0 0 K | * * 2NM * squat
sefa( s4o4s4o ) | 6 | 1 4 2 2 | 2 2 1 | * * * 2NMK trip
s4s4s4o (N,M,K->oo)
demi( . . . . ) | 4NMK | 2 1 2 4 | 1 2 6 3 3 | 2 1 1 4
----------------+------+---------------------+-------------------------+-----------------
s . s . | 2 | 4NMK * * * | 0 0 2 2 0 | 1 1 0 2
. . s4o | 2 | * 2NMK * * | 0 0 0 2 2 | 0 1 1 2
sefa( s4s . . ) | 2 | * * 4NMK * | 1 0 2 0 0 | 2 0 0 1
sefa( . s4s . ) | 2 | * * * 8NMK | 0 1 1 0 1 | 1 0 1 1
----------------+------+---------------------+-------------------------+-----------------
s4s . . | 4 | 0 0 4 0 | NMK * * * * | 2 0 0 0
. s4s . | 4 | 0 0 0 4 | * 2NMK * * * | 1 0 1 0
sefa( s4s4s . ) | 3 | 1 0 1 1 | * * 8NMK * * | 1 0 0 1
sefa( s . s4o ) | 3 | 2 1 0 0 | * * * 4NMK * | 0 1 0 1
sefa( . s4s4o ) | 3 | 0 1 0 2 | * * * * 4NMK | 0 0 1 1
----------------+------+---------------------+-------------------------+-----------------
s4s4s . | 4M | 2M 0 4M 4M | M M 4M 0 0 | 2NK * * * non-uniform snasquat
s . s4o | 4 | 4 2 0 0 | 0 0 0 4 0 | * NMK * * regular tet
. s4s4o | 2K | 0 K 0 4K | 0 K 0 0 2K | * * 2NM * non-uniform snasquat
sefa( s4s4s4o ) | 4 | 2 1 1 2 | 0 0 2 1 1 | * * * 4NMK non-uniform tet
Klitzing wrote:
- Code: Select all
s4o4s4o (N,M,K->oo)
demi( . . . . ) | 2NMK | 1 4 2 2 | 8 6 4 | 2 2 1 6
----------------+------+--------------------+----------------+-----------------
s4o . . | 2 | NMK * * * | 4 0 0 | 2 0 0 2
s . s . | 2 | * 4NMK * * | 2 2 0 | 1 1 0 1
. o4s . | 2 | * * 2NMK * | 2 0 2 | 1 0 1 2
. . s4o | 2 | * * * 2NMK | 0 2 2 | 0 1 1 2
----------------+------+--------------------+----------------+-----------------
sefa( s4o4s . ) | 4 | 1 2 1 0 | 4NMK * * | 1 0 0 1
sefa( s . s4o ) | 3 | 0 2 0 1 | * 4NMK * | 0 1 0 1
sefa( . o4s4o ) | 4 | 0 0 2 2 | * * 2NMK | 0 0 1 1
----------------+------+--------------------+----------------+-----------------
s4o4s . | 2M | M 2M M 0 | 2M 0 0 | 2NK * * * squat
s . s4o | 4 | 0 4 0 2 | 0 4 0 | * NMK * * tet
. o4s4o | K | 0 0 K K | 0 0 K | * * 2NM * squat
sefa( s4o4s4o ) | 6 | 1 4 2 2 | 2 2 1 | * * * 2NMK trip
Here too all edges are of the same length (diagonals of original squares) and there is a single vertex class. Faces are all regular (triangles and squares) and cells are all uniform again. Thus this structure would be uniform itself. (Whether this even would be a Wythoffian structure eludes me in the moment.)
2NM | 3 6 | 12 6 | 3 2 6
----+---------+---------+----------
2 | 3NM * | 4 0 | 2 0 2
2 | * 6NM | 2 2 | 1 1 2
----+---------+---------+----------
4 | 2 2 | 6NM * | 1 0 1
3 | 0 3 | * 4NM | 0 1 1
----+---------+---------+----------
M | M M | M 0 | 6N * * squat
4 | 0 6 | 0 4 | * NM * tet
6 | 3 6 | 3 2 | * * 2NM trip
Klitzing wrote:Klitzing wrote:
- Code: Select all
s4o4s4o (N,M,K->oo)
demi( . . . . ) | 2NMK | 1 4 2 2 | 8 6 4 | 2 2 1 6
----------------+------+--------------------+----------------+-----------------
s4o . . | 2 | NMK * * * | 4 0 0 | 2 0 0 2
s . s . | 2 | * 4NMK * * | 2 2 0 | 1 1 0 1
. o4s . | 2 | * * 2NMK * | 2 0 2 | 1 0 1 2
. . s4o | 2 | * * * 2NMK | 0 2 2 | 0 1 1 2
----------------+------+--------------------+----------------+-----------------
sefa( s4o4s . ) | 4 | 1 2 1 0 | 4NMK * * | 1 0 0 1
sefa( s . s4o ) | 3 | 0 2 0 1 | * 4NMK * | 0 1 0 1
sefa( . o4s4o ) | 4 | 0 0 2 2 | * * 2NMK | 0 0 1 1
----------------+------+--------------------+----------------+-----------------
s4o4s . | 2M | M 2M M 0 | 2M 0 0 | 2NK * * * squat
s . s4o | 4 | 0 4 0 2 | 0 4 0 | * NMK * * tet
. o4s4o | K | 0 0 K K | 0 0 K | * * 2NM * squat
sefa( s4o4s4o ) | 6 | 1 4 2 2 | 2 2 1 | * * * 2NMK trip
Here too all edges are of the same length (diagonals of original squares) and there is a single vertex class. Faces are all regular (triangles and squares) and cells are all uniform again. Thus this structure would be uniform itself. (Whether this even would be a Wythoffian structure eludes me in the moment.)
There might be some further class unifications by applying an higher symmetry, resulting in just
[...]
o3o4s4o (N,M->oo)
demi( . . . . ) | 2NM | 6 3 | 6 12 | 2 3 6 vf:xox3ouo&#qt
----------------+-----+---------+---------+----------
. o4s . | 2 | 6NM * | 2 2 | 1 1 2
. . s4o | 2 | * 3NM | 0 4 | 0 2 2
----------------+-----+---------+---------+----------
sefa( o3o4s . ) | 3 | 3 0 | 4NM * | 1 0 1
sefa( . o4s4o ) | 4 | 2 2 | * 6NM | 0 1 1
----------------+-----+---------+---------+----------
o3o4s . | 4 | 6 0 | 4 0 | NM * * tet
. o4s4o | M | M M | 0 M | * 6N * squat
sefa( o3o4s4o ) | 6 | 6 3 | 2 3 | * * 2NM trip
[...]
It further comes out that the vertex figure here is a clear relative to the cuboctahedron:
The latter one is xox4oqo&#xt = x4o || o4q || x4o, while the vertex figure here is xox3ouo&#qt = x3o |q| o3u |q| x3o, that is, just a (streched) three-fold version of the former!
I further realized that this snub is truely a non-Wythoffian uniform. Because the symbols x3o3o (tet), x3o . x (trip), and x4o4o (squat - or any other according symbol) cannot be combined into a single 4-node diagram in such a way, that any hiding of a single node would just re-produce one of those given symbols (possibly in addition one with un-ringed nodes only).
--- rk
sPsPs *bPs = sPsPs4o = sPs4o3o
sPoPs *bPo = sPoPs4o = oPs4o3o
oPsPo *bPs = oPsPo4o = sPo4o3o
*a o----4---(o) *b
| \ |
4 Q 4
| \ |
*d (o)---4----o *c
*a o----4---( ) *b
| \ |
4 Q 4
| \ |
*d ( )---4----o *c
o4x4o2Qo (Q parametrisable, N,M,K->oo)
. . . . | 2NMK | 4Q | 2Q 4Q | 2Q 2
---------+------+-------+------------+---------
. x . . | 2 | 4QNMK | 1 2 | 2 1
---------+------+-------+------------+---------
o4x . . | 4 | 4 | QNMK * | 2 0
. x4o . | 4 | 4 | * 2QNMK | 1 1
---------+------+-------+------------+---------
o4x4o . | 2M | 4M | M M | 2QNK *
. x4o2Qo | 2K | 2QK | 0 QK | * 2NM
o4s4o2Qo (Q parametrisable, N,M,K->oo)
demi( . . . . ) | NMK | 2Q 4Q | 8Q 4Q | 2Q 2 4Q
-----------------+-----+------------+------------+-------------
o4s . . | 2 | QNMK * | 4 0 | 2 0 2
. s4o . | 2 | * 2QNMK | 2 2 | 1 1 2
-----------------+-----+------------+------------+-------------
sefa( o4s4o . ) | 4 | 2 2 | 2QNMK * | 1 0 1
sefa( . s4o2Qo ) | 2Q | 0 2Q | * 2NMK | 0 1 1
-----------------+-----+------------+------------+-------------
o4s4o . | M | M M | M 0 | 2QNK * *
. s4o2Qo | K | 0 QK | 0 K | * 2NM *
sefa( o4s4o2Qo ) | 4Q | 2Q 4Q | 2Q 2 | * * NMK
wendy wrote:Richard K is indeed right about the vertex figure. What i gave belongs to s4s4oPo, not o4s4oPo.
In fact, the is no reqiorement for P to be even: it works with all values of P, such as o4s4oPo
o4s4oPo verf = popPouo&#qt p = shortchord of P edge2 = 4p/(4-p).
Cells are o4s4o = x4o4o, s4oPo = xPoPo, and q2qPo = pgonal prism.
When p=4, the outcome is x4o4oAo.
When p=U, the outcome is a slice of x4o4o4o.
For P=3 and 4, the tiling is type 2 (ie have finite content). This means they have horospheric cells.
For P>4, the tiling is type 4, ie might occur in a frieze that has horospheric cells.
It is interesting to note that the shape of the vertex figure is what i was originally thinking of when i was thinking of 'exotic prisms' all those years ago. These became 'lace prisms' in the end.
wendy wrote:Of course, while xPoQoPx to have a uniform vertex, requires P=4, the general form is xPoQoPxP/2z, (z=*a), which has a uniform Q-antiprism as a vertex figure.
Most notable is x6o3o6x3z, which is the same as x6o3o4o. In fact, this equity is used to devolve the symmetries of the latter into something more.
One could look at the groups like x8o4o8xPz, because these are likely to occur in the frieze-groups based on o8o3o4o.
But there are folk coming in from the outback soon, which will generally curb my activities for a week or so.
Klitzing wrote:wendy wrote:Most notable is x6o3o6x3z, which is the same as x6o3o4o. In fact, this equity is used to devolve the symmetries of the latter into something more.
Interesting. Will come back to that later, I suppose...
s6o3o3o
demi( . . . . ) | NM | 12 | 6 12 | 4 4 tut
----------------+----+-----+---------+------
sefa( s6o . . ) | 2 | 6NM | 1 2 | 2 1
----------------+----+-----+---------+------
s6o . . | 3 | 3 | 2NM * | 2 0
sefa( s6o3o . ) | 3 | 3 | * 4NM | 1 1
----------------+----+-----+---------+------
s6o3o . | M | 3M | M M | 4N * x3o6o
sefa( s6o3o3o ) | 4 | 6 | 0 4 | * NM tet
s6o3o4o
demi( . . . . ) | NM | 24 | 12 24 | 8 6 toe
----------------+----+------+---------+------
sefa( s6o . . ) | 2 | 12NM | 1 2 | 2 1
----------------+----+------+---------+------
s6o . . | 3 | 3 | 4NM * | 2 0
sefa( s6o3o . ) | 3 | 3 | * 8NM | 1 1
----------------+----+------+---------+------
s6o3o . | M | 3M | M M | 8N * x3o6o
sefa( s6o3o4o ) | 6 | 12 | 0 8 | * NM oct
s6o3o5o
demi( . . . . ) | NM | 60 | 30 60 | 20 12 ti
----------------+----+------+-----------+-------
sefa( s6o . . ) | 2 | 30NM | 1 2 | 2 1
----------------+----+------+-----------+-------
s6o . . | 3 | 3 | 10NM * | 2 0
sefa( s6o3o . ) | 3 | 3 | * 20NM | 1 1
----------------+----+------+-----------+-------
s6o3o . | M | 3M | M M | 20N * x3o6o
sefa( s6o3o5o ) | 12 | 30 | 0 20 | * NM ike
s8o4o3o
demi( . . . . ) | NM | 24 | 12 24 | 6 8 tic
----------------+----+------+---------+------
sefa( s8o . . ) | 2 | 12NM | 1 2 | 2 1
----------------+----+------+---------+------
s8o . . | 4 | 4 | 3NM * | 2 0
sefa( s8o4o . ) | 4 | 4 | * 6NM | 1 1
----------------+----+------+---------+------
s8o4o . | M | 4M | M M | 6N * x4o8o
sefa( s8o4o3o ) | 8 | 12 | 0 6 | * NM cube
s10o5o3o
demi( . . . . ) | NM | 60 | 30 60 | 12 20 tid
-----------------+----+------+----------+-------
sefa( s10o . . ) | 2 | 30NM | 1 2 | 2 1
-----------------+----+------+----------+-------
s10o . . | 5 | 5 | 6NM * | 2 0
sefa( s10o5o . ) | 5 | 5 | * 12NM | 1 1
-----------------+----+------+----------+-------
s10o5o . | M | 5M | M M | 12N * x5o10o
sefa( s10o5o3o ) | 20 | 30 | 0 12 | * NM doe
wendy wrote:The last five incidence matrices describe the polytopes s2PoPoQo = oPxPoPzQo, which comes from the laws of symmetry.
I'm not really sure if there are references. It's all pretty much new ground here, because we made available to ourselves, some pretty fancy tools. Coxeter did much the same when he loaded Wythoff's construction onto the Lie-group symbols we call Coxeter-Dynkin symbols, and Galileo did when he poked his spyglass at the skys. Lots to see, but you got to go outback to see it, so to speak.
One of the main reasons that you won't find many references to what we're doing is because the bulk of what's here are "type four" tesselations. This means that they have both bollotopic and horocyclic cells. There's plenty of them out there, many can not be rendered uniform.
Coxeter, Johnson and others do not stray past type two, because the way they approach hyperbolic mathematics conspires against random forays that lead to dead ends in the main (which is what type 3 and higher do). We have a pretty fancy approacy to this which gets rid of all of that ugly mathematics, and so it's not too hard to find these things. So the sort of thing that Johnson had to resort to for days to find my calculations were correct, were in response to pretty ordinary mental arithmetic. I don't use things like sinh or cosh or wotnots. Most of it comes pretty much by euclidean geometry.
The group o6o3o6o3z is size 40, consists of two identical mirror-sets of size "240", which corresponds to the cells of x4o3o6o. Apart from the regular division into 48 that x4o3o confers, there is a second division of order 5, because the cube is made of five regular tetrahedra, (rather like diminishing the cube to get a tetrahedron, by alternating corners, but the bits that come off are also regular. So this mirror group divides into five sets of cells of x3o3o6o, which in turn divides into 24 of the symmetry [3,3,6] (of size '2'). I however misread 240 as an octahedron, and missed bits of it until i revisited.
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