CUE polytopes?

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

CUE polytopes?

Postby quickfur » Mon Oct 27, 2014 11:13 pm

I've been so busy lately that I've hardly had any time to do anything 4D- or CRF-related... but today an idea occurred to me: has anyone ever considered a slight generalization of CRF polytopes, where the regular-polygon requirement is relaxed? That is, the polytope is still required to be strictly convex (no coplanar facets, all subpolytopes must be convex), but polygonal surtopes are permitted to be non-regular, the only restriction being that all edges must be unit length.

In 3D at least, this would expand the set of permissible polytopes to figures like the rhombic dodecahedron and the rhombic triacontahedron. Things like sheared cubes would also be permissible. But I'm not sure what consequences this would have for 4D and higher?

Any thoughts on these CUE (convex unit-edged) polytopes?
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Re: CUE polytopes?

Postby Klitzing » Tue Oct 28, 2014 6:45 am

Not only rhombs would have to be considered. Any even sided polygon with alternating angles would pass. And even other deformed flat polygons. Eg. the hexagon wx oq&#zx = pt || q-line || q-line || point (digonally elongated square). - The latter one even allow dimensional extrapolation:

Two specific polyhedra come to mind there too:
  • wx xo4oq&#zx = x4o || o4q || o4q || x4o (tetrafold axially elongated cuboctahedron)
  • qo xw4xo&#zx = x4x || w4o || x4x (tetrafold axially contracted truncated cube)

And these again occur quite often within 4D...
(If you'd like I could digg out some of them. I've already stumbled across some while applying student91's quirks onto icositetrachoral or hexadecachoral starting figures. But surely had to discard them when looking for CRFs in that context.)

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Re: CUE polytopes?

Postby quickfur » Fri Oct 31, 2014 3:10 pm

It would be interesting to explore some of the highly-symmetrical examples of such polytopes, e.g., something with 24-cell symmetry and rhombus ridges.
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Re: CUE polytopes?

Postby Klitzing » Fri Oct 31, 2014 7:20 pm

Okay, we will elaborate a first example of such a 4D figure, which incorporates such non-regular 90-135-135-90-135-135 degrees Hexagons.

To that aim first consider qo3oo3oq4xo&#zx, which is nothing but a o3o3o4o-subsymmetrical description of o3x4o3o (rico)

Code: Select all
qo3oo3oq4xo&#zx -> height = 0

o.3o.3o.4o.     | 64  * |  3   3 |  3  3  3 | 1  1  3 verf = x q3o
.o3.o3.o4.o     |  * 32 |  0   6 |  0  6  3 | 0  2  3 verf = x q3o
----------------+-------+--------+----------+--------
.. .. .. x.     |  2  0 | 96   * |  2  0  1 | 1  0  2
oo3oo3oo4oo&#x  |  1  1 |  * 192 |  0  2  1 | 0  1  2
----------------+-------+--------+----------+--------
.. .. o.4x.     |  4  0 |  4   0 | 48  *  * | 1  0  1
qo .. oq ..&#zx |  2  2 |  0   4 |  * 96  * | 0  1  1
.. .. .. xo&#x  |  2  1 |  1   2 |  *  * 96 | 0  0  2
----------------+-------+--------+----------+--------
.. o.3o.4x.     |  8  0 | 12   0 |  6  0  0 | 8  *  * cube
qo3oo3oq ..&#zx |  4  4 |  0  12 |  0  6  0 | * 16  * cube
qo .. oq4xo&#zx |  8  4 |  8  16 |  2  4  8 | *  * 24 co


Now consider a Stott expansion of that with respect to its first node, i.e. wx3oo3oq4xo&#zx
What would that thingy look like? - Okay, here it comes:

Code: Select all
wx3oo3oq4xo&#zx -> height = 0

o.3o.3o.4o.     | 64  * |  3   3  0 |  3  3  3  0 | 1  1  3 verf = x q3o
.o3.o3.o4.o     |  * 96 |  0   2  2 |  0  4  1  1 | 0  2  2
----------------+-------+-----------+-------------+--------
.. .. .. x.     |  2  0 | 96   *  * |  2  0  1  0 | 1  0  2
oo3oo3oo4oo&#x  |  1  1 |  * 192  * |  0  2  1  0 | 0  1  2
.x .. .. ..     |  0  2 |  *   * 96 |  0  2  0  1 | 0  2  1
----------------+-------+-----------+-------------+--------
.. .. o.4x.     |  4  0 |  4   0  0 | 48  *  *  * | 1  0  1
wx .. oq ..&#zx |  2  4 |  0   4  2 |  * 96  *  * | 0  1  1 non-regular 90-135-135-90-135-135 hexagons!!!
.. .. .. xo&#x  |  2  1 |  1   2  0 |  *  * 96  * | 0  0  2
.x3.o .. ..     |  0  3 |  0   0  3 |  *  *  * 32 | 0  2  0
----------------+-------+-----------+-------------+--------
.. o.3o.4x.     |  8  0 | 12   0  0 |  6  0  0  0 | 8  *  * cube
wx3oo3oq ..&#zx |  4 12 |  0  12 12 |  0  6  0  4 | * 16  * non-CRF tetrahedrally-truncated cube (1)
wx .. oq4xo&#zx |  8  8 |  8  16  4 |  2  4  8  0 | *  * 24 non-CRF tetragonal-elongated cuboctahedron (2)


(1) Looks like a cube which has been truncated only at 4 (= half) of its 8 vertices in a tetrahedral way. The former regular squares deform then into these non-regular hexagons (in a diametral orientation), the truncated polygons clearly are the vertex figures of the cube, i.e. regular triangles.

(2) Looks like a cuboctahedron which had been elongated in its forefold axial orientation. Thus the extremal squares and either set of polar triangles moves a bit apart, while the remaining 4 (lateral) squares become diametrally elongated into these non-regular hexagons.

Both these polyhedra are CUE on their own! (As is also that non-regular hexagon.) - But the much more interesting fact here is, that this CUE polygon can not only be used to produce these CUE polyhedra, but also the CUE polychoron, the matrix of which is provided above.

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Re: CUE polytopes?

Postby Klitzing » Fri Oct 31, 2014 9:42 pm

You well could Stott expand from rico (o3x4o3o) in its o3o3o4o-subsymmetrical description, i.e. from qo3oo3oq4xo&#zx, the third node with a similar effect. Then this would become qo3oo3xw4xo&#zx. Again we can spot qo .. xw ..&#zx, i.e. that non-regular 90-135-135-90-135-135 degrees hexagon. But what else is contained in here? - We'll elaborate:

Code: Select all
qo3oo3xw4xo&#zx -> height = 0

o.3o.3o.4o.     | 192  * |   2  1   1 |  1  2  2  1 | 1  1  2
.o3.o3.o4.o     |   * 32 |   0  0   6 |  0  0  6  3 | 0  2  3 verf = x q3o
----------------+--------+------------+-------------+--------
.. .. x. ..     |   2  0 | 192  *   * |  1  1  1  0 | 1  1  1
.. .. .. x.     |   2  0 |   * 96   * |  0  2  0  1 | 1  0  2
oo3oo3oo4oo&#x  |   1  1 |   *  * 192 |  0  0  2  1 | 0  1  2
----------------+--------+------------+-------------+--------
.. o.3x. ..     |   3  0 |   3  0   0 | 64  *  *  * | 1  1  0
.. .. x.4x.     |   8  0 |   4  4   0 |  * 48  *  * | 1  0  1
qo .. xw ..&#zx |   4  2 |   2  0   4 |  *  * 96  * | 0  1  1 non-regular 90-135-135-90-135-135 hexagons!!!
.. .. .. xo&#x  |   2  1 |   0  1   2 |  *  *  * 96 | 0  0  2
----------------+--------+------------+-------------+--------
.. o.3x.4x.     |  24  0 |  24 12   0 |  8  6  0  0 | 8  *  * truncated cube
qo3oo3xw ..&#zx |  12  4 |  12  0  12 |  4  0  6  0 | * 16  * non-CRF tetrahedrally-truncated cube (1)
qo .. xw4xo&#zx |  16  4 |   8  8  16 |  0  2  4  8 | *  * 24 non-CRF axially-contracted truncated cube (3)


(1) as within previous case.

(3) is new here. It can be derived from the uniform truncated cube by the withdrawel of its equatorial segment. Thus it contains still the 2 polar octagons and the attached triangles. But the 4 lateral octagons become these specific non-regular hexagons.

So again here we get a case, were that CUE Hexagon results not even in one old and one new type of CUE polyhedra, but all is contained furthermore within a CUE polychoron!



Btw., and what would be derived if both nodes would be expanded? I.e. what is wx3oo3xw4xo&#zx?

Well at first glance we can spot that our non-regular hexagon has been gone. Instead we have wx .. xw ..&#zx, the regular octagon again. - The other details as following:
Code: Select all
wx3oo3xw4xo&#zx -> height = 0

o.3o.3o.4o.     | 192  * |   2  1   1  0 |  1  2  2  1  0 | 1  1  2
.o3.o3.o4.o     |   * 96 |   0  0   2  2 |  0  0  4  1  1 | 0  2  2
----------------+--------+---------------+----------------+--------
.. .. x. ..     |   2  0 | 192  *   *  * |  1  1  1  0  0 | 1  1  1
.. .. .. x.     |   2  0 |   * 96   *  * |  0  2  0  1  0 | 1  0  2
oo3oo3oo4oo&#x  |   1  1 |   *  * 192  * |  0  0  2  1  0 | 0  1  2
.x .. .. ..     |   0  2 |   *  *   * 96 |  0  0  2  0  1 | 0  2  1
----------------+--------+---------------+----------------+--------
.. o.3x. ..     |   3  0 |   3  0   0  0 | 64  *  *  *  * | 1  1  0
.. .. x.4x.     |   8  0 |   4  4   0  0 |  * 48  *  *  * | 1  0  1
wx .. xw ..&#zx |   4  4 |   2  0   4  2 |  *  * 96  *  * | 0  1  1
.. .. .. xo&#x  |   2  1 |   0  1   2  0 |  *  *  * 96  * | 0  0  2
.x3.o .. ..     |   0  3 |   0  0   0  3 |  *  *  *  * 32 | 0  2  0
----------------+--------+---------------+----------------+--------
.. o.3x.4x.     |  24  0 |  24 12   0  0 |  8  6  0  0  0 | 8  *  * tic
wx3oo3xw ..&#zx |  12 12 |  12  0  12 12 |  4  0  6  0  4 | * 16  * tic
wx .. xw4xo&#zx |  16  8 |   8  8  16  4 |  0  2  4  8  0 | *  * 24 tic

That is, this then is nothing but the Wythoffian figure known as cont (o3x4x3o).

--- rk
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Re: CUE polytopes?

Postby quickfur » Fri Oct 31, 2014 10:33 pm

Ahh, so then the first figure you described is nothing but a non-CRF Stott-contracted o3x4x3o. That's interesting. Maybe when I find some free time I could construct the coordinates of this figure and run it through my polytope renderer. :)

This makes me wonder if there's other CUE Stott contractions (or expansions!) of o3x4x3o that retain the augmented 24-cell symmetry. Does the radial expansion of the 48 truncated cubes in o3x4x3o produce a CUE figure?
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Re: CUE polytopes?

Postby Klitzing » Fri Oct 31, 2014 11:19 pm

quickfur wrote:Ahh, so then the first figure you described is nothing but a non-CRF Stott-contracted o3x4x3o. That's interesting. Maybe when I find some free time I could construct the coordinates of this figure and run it through my polytope renderer. :)


Infact we have the following:
Code: Select all
     qo3oo3oq4xo&#zx   x . . . exp                     
     = rico            ------------>   wx3oo3oq4xo&#zx
     = o3x4o3o        <------------                   
                       x . . . contr                   
           ^                                 ^         
. . x .  | |  . . x .             . . x .  | |  . . x .
exp      | |  contr               exp      | |  contr 
         | |                               | |         
         v                                 v           
                       x . . . exp     wx3oo3xw4xo&#zx
     qo3oo3xw4xo&#zx   ------------>   = cont         
                      <------------    = o3x4x3o       
                       x . . . contr                   


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Re: CUE polytopes?

Postby Klitzing » Sat Nov 01, 2014 8:03 am

quickfur wrote:... Does the radial expansion of the 48 truncated cubes in o3x4x3o produce a CUE figure?


Actually not. Faces ought to be expanded into prisms - so far okay. Edges into edge-figure prisms, and vertices into vertex-figure polyhedra. And at least the latter are not CUE here!

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Re: CUE polytopes?

Postby wendy » Sat Nov 01, 2014 9:07 am

The radial expansion of o3x4x3o is.. xoox!o3x4x3o.

The face-consist is x4x3o (48), x4x2x (124) x3o2w (172), wo2xx&#x (496) and wo2ow&x (248), w = sqrt(2+q) = a(8).

The first three are wythoffian, the last three are not. All numbers in base 120.
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Re: CUE polytopes?

Postby Klitzing » Sat Nov 01, 2014 11:51 am

Okay, we further could consider to add the Stott expansion wrt. the second node onto the former commutative diagram as well.

Code: Select all
     qo3xx3oq4xo&#zx   x . . . exp                     
     = srico           ------------>   wx3xx3oq4xo&#zx
     = o3x4o3x        <------------                   
                       x . . . contr                   
           ^                                 ^         
. . x .  | |  . . x .             . . x .  | |  . . x .
exp      | |  contr               exp      | |  contr 
         | |                               | |         
         v                                 v           
                       x . . . exp     wx3xx3xw4xo&#zx
     qo3xx3xw4xo&#zx   ------------>   = grico         
                      <------------    = o3x4x3x       
                       x . . . contr                   


Here we start with qo3xx3oq4xo&#zx:
Code: Select all
qo3xx3oq4xo&#zx -> height = 0

o.3o.3o.4o.     | 192  * |   2   2   2  0 |  1  2  1  1   2   2  0 | 1  1  2  1
.o3.o3.o4.o     |   * 96 |   0   0   4  2 |  0  0  0  2   4   2  1 | 0  1  2  2
----------------+--------+----------------+------------------------+-----------
.. x. .. ..     |   2  0 | 192   *   *  * |  1  1  0  0   1   0  0 | 1  0  1  1
.. .. .. x.     |   2  0 |   * 192   *  * |  0  1  1  0   0   1  0 | 1  1  1  0
oo3oo3oo4oo&#x  |   1  1 |   *   * 384  * |  0  0  0  1   1   1  0 | 0  1  1  1
.. .x .. ..     |   0  2 |   *   *   * 96 |  0  0  0  0   2   0  1 | 0  0  1  2
----------------+--------+----------------+------------------------+-----------
.. x.3o. ..     |   3  0 |   3   0   0  0 | 64  *  *  *   *   *  * | 1  0  0  1
.. x. .. x.     |   4  0 |   2   2   0  0 |  * 96  *  *   *   *  * | 1  0  1  0
.. .. o.4x.     |   4  0 |   0   4   0  0 |  *  * 48  *   *   *  * | 1  1  0  0
qo .. oq ..&#zx |   2  2 |   0   0   4  0 |  *  *  * 96   *   *  * | 0  1  0  1
.. xx .. ..&#x  |   2  2 |   1   0   2  1 |  *  *  *  * 192   *  * | 0  0  1  1
.. .. .. xo&#x  |   2  1 |   0   1   2  0 |  *  *  *  *   * 192  * | 0  1  1  0
.o3.x .. ..     |   0  3 |   0   0   0  3 |  *  *  *  *   *   * 32 | 0  0  0  2
----------------+--------+----------------+------------------------+-----------
.. x.3o.4x.     |  24  0 |  24  24   0  0 |  8 12  6  0   0   0  0 | 8  *  *  * sirco
qo .. oq4xo&#zx |   8  4 |   0   8  16  0 |  0  0  2  4   0   8  0 | * 24  *  * co
.. xx .. xo&#x  |   4  2 |   2   2   4  1 |  0  1  0  0   2   2  0 | *  * 96  * trip
qo3xx3oq ..&#zx |  12 12 |  12   0  24 12 |  4  0  0  6  12   0  4 | *  *  * 16 sirco

which obviously is nothing but srico (o3x4o3x). - (Thus this addition of .. xx .. .. in that o3o34o subsymmetry corresponds to the addition of . . . x in o3o4o3o symmetry.)

Then we add again xx .. .. .. to the above. This results in wx3xx3oq4xo&#zx:
Code: Select all
wx3xx3oq4xo&#zx -> height = 0

o.3o.3o.4o.     | 192   * |   2   2   2  0  0 |  1  2  1  1   2   2  0 | 1  1  1  2
.o3.o3.o4.o     |   * 192 |   0   0   2  1  1 |  0  0  0  2   2   1  1 | 0  2  1  1
----------------+---------+-------------------+------------------------+-----------
.. x. .. ..     |   2   0 | 192   *   *  *  * |  1  1  0  0   1   0  0 | 1  1  0  1
.. .. .. x.     |   2   0 |   * 192   *  *  * |  0  1  1  0   0   1  0 | 1  0  1  1
oo3oo3oo4oo&#x  |   1   1 |   *   * 384  *  * |  0  0  0  1   1   1  0 | 0  1  1  1
.x .. .. ..     |   0   2 |   *   *   * 96  * |  0  0  0  2   0   0  1 | 0  2  1  0
.. .x .. ..     |   0   2 |   *   *   *  * 96 |  0  0  0  0   2   0  1 | 0  2  0  1
----------------+---------+-------------------+------------------------+-----------
.. x.3o. ..     |   3   0 |   3   0   0  0  0 | 64  *  *  *   *   *  * | 1  1  0  0
.. x. .. x.     |   4   0 |   2   2   0  0  0 |  * 96  *  *   *   *  * | 1  0  0  1
.. .. o.4x.     |   4   0 |   0   4   0  0  0 |  *  * 48  *   *   *  * | 1  0  1  0
wx .. oq ..&#zx |   2   4 |   0   0   4  2  0 |  *  *  * 96   *   *  * | 0  1  1  0 non-regular 90-135-135-90-135-135 hexagons!!!
.. xx .. ..&#x  |   2   2 |   1   0   2  0  1 |  *  *  *  * 192   *  * | 0  1  0  1
.. .. .. xo&#x  |   2   1 |   0   1   2  0  0 |  *  *  *  *   * 192  * | 0  0  1  1
.x3.x .. ..     |   0   6 |   0   0   0  3  3 |  *  *  *  *   *   * 32 | 0  2  0  0
----------------+---------+-------------------+------------------------+-----------
.. x.3o.4x.     |  24   0 |  24  24   0  0  0 |  8 12  6  0   0   0  0 | 8  *  *  * sirco
wx3xx3oq ..&#zx |  12  24 |  12   0  24 12 12 |  4  0  0  6  12   0  4 | * 16  *  * non-CRF tetrahedrally-lowered sirco (4)
wx .. oq4xo&#zx |   8   8 |   0   8  16  4  0 |  0  0  2  4   0   8  0 | *  * 24  * non-CRF tetragonal-elongated cuboctahedron (2)
.. xx .. xo&#x  |   4   2 |   2   2   4  0  1 |  0  1  0  0   2   2  0 | *  *  * 96 trip


Alternatively we could add .. .. xx .. to the today starting figure, resulting in qo3xx3xw4xo&#zx:
Code: Select all
qo3xx3xw4xo&#zx -> height = 0

o.3o.3o.4o.     | 384  * |   1   1   1   1  0 |  1  1  1  1   1   1  0 | 1  1  1  1
.o3.o3.o4.o     |   * 96 |   0   0   0   4  2 |  0  0  0  2   4   2  1 | 0  2  1  2
----------------+--------+--------------------+------------------------+-----------
.. x. .. ..     |   2  0 | 192   *   *   *  * |  1  1  0  0   1   0  0 | 1  1  0  1
.. .. x. ..     |   2  0 |   * 192   *   *  * |  1  0  1  1   0   0  0 | 1  1  1  0
.. .. .. x.     |   2  0 |   *   * 192   *  * |  0  1  1  0   0   1  0 | 1  0  1  1
oo3oo3oo4oo&#x  |   1  1 |   *   *   * 384  * |  0  0  0  1   1   1  0 | 0  1  1  1
.. .x .. ..     |   0  2 |   *   *   *   * 96 |  0  0  0  0   2   0  1 | 0  2  0  1
----------------+--------+--------------------+------------------------+-----------
.. x.3x. ..     |   6  0 |   3   3   0   0  0 | 64  *  *  *   *   *  * | 1  1  0  0
.. x. .. x.     |   4  0 |   2   0   2   0  0 |  * 96  *  *   *   *  * | 1  0  0  1
.. .. x.4x.     |   8  0 |   0   4   4   0  0 |  *  * 48  *   *   *  * | 1  0  1  0
qo .. xw ..&#zx |   4  2 |   0   2   0   4  0 |  *  *  * 96   *   *  * | 0  1  1  0 non-regular 90-135-135-90-135-135 hexagons!!!
.. xx .. ..&#x  |   2  2 |   1   0   0   2  1 |  *  *  *  * 192   *  * | 0  1  0  1
.. .. .. xo&#x  |   2  1 |   0   0   1   2  0 |  *  *  *  *   * 192  * | 0  0  1  1
.o3.x .. ..     |   0  3 |   0   0   0   0  3 |  *  *  *  *   *   * 32 | 0  2  0  0
----------------+--------+--------------------+------------------------+-----------
.. x.3x.4x.     |  48  0 |  24  24  24   0  0 |  8 12  6  0   0   0  0 | 8  *  *  * girco
qo3xx3xw ..&#zx |  24 12 |  12  12   0  24 12 |  4  0  0  6  12   0  4 | * 16  *  * non-CRF tetrahedrally-lowered sirco (4)
qo .. xw4xo&#zx |  16  4 |   0   8   8  16  0 |  0  0  2  4   0   8  0 | *  * 24  * non-CRF axially-contracted truncated cube (3)
.. xx .. xo&#x  |   4  2 |   2   0   2   4  1 |  0  1  0  0   2   2  0 | *  *  * 96 trip


Here (2) and (3) are again those CUE polyhedra mentioned yesterday.
(4) is also a CUE polyhedron. It can be derived from sirco (x3o4x), when a tetrahedral subset of triangles becomes lowered. Thereby These triangles become regular hexagons (while the other tetrahedral subset of triangles remains as is). The squares in cubical positions become those non-regular hexagons, and the lacing squares (after some rescaling) also remain as is.

Finally we can have all 3 Stott expansions simultanuously. This then results in wx3xx3xw4xo&#zx:
Code: Select all
wx3xx3xw4xo&#zx -> height = 0

o.3o.3o.4o.     | 384   * |   1   1   1   1  0  0 |  1  1  1  1   1   1  0 | 1  1  1  1
.o3.o3.o4.o     |   * 192 |   0   0   0   2  1  1 |  0  0  0  2   2   1  1 | 0  2  1  1
----------------+---------+-----------------------+------------------------+-----------
.. x. .. ..     |   2   0 | 192   *   *   *  *  * |  1  1  0  0   1   0  0 | 1  1  0  1
.. .. x. ..     |   2   0 |   * 192   *   *  *  * |  1  0  1  1   0   0  0 | 1  1  1  0
.. .. .. x.     |   2   0 |   *   * 192   *  *  * |  0  1  1  0   0   1  0 | 1  0  1  1
oo3oo3oo4oo&#x  |   1   1 |   *   *   * 384  *  * |  0  0  0  1   1   1  0 | 0  1  1  1
.x .. .. ..     |   0   2 |   *   *   *   * 96  * |  0  0  0  2   0   0  1 | 0  2  1  0
.. .x .. ..     |   0   2 |   *   *   *   *  * 96 |  0  0  0  0   2   0  1 | 0  2  0  1
----------------+---------+-----------------------+------------------------+-----------
.. x.3x. ..     |   6   0 |   3   3   0   0  0  0 | 64  *  *  *   *   *  * | 1  1  0  0
.. x. .. x.     |   4   0 |   2   0   2   0  0  0 |  * 96  *  *   *   *  * | 1  0  0  1
.. .. x.4x.     |   8   0 |   0   4   4   0  0  0 |  *  * 48  *   *   *  * | 1  0  1  0
wx .. xw ..&#zx |   4   4 |   0   2   0   4  2  0 |  *  *  * 96   *   *  * | 0  1  1  0
.. xx .. ..&#x  |   2   2 |   1   0   0   2  0  1 |  *  *  *  * 192   *  * | 0  1  0  1
.. .. .. xo&#x  |   2   1 |   0   0   1   2  0  0 |  *  *  *  *   * 192  * | 0  0  1  1
.x3.x .. ..     |   0   6 |   0   0   0   0  3  3 |  *  *  *  *   *   * 32 | 0  2  0  0
----------------+---------+-----------------------+------------------------+-----------
.. x.3x.4x.     |  48   0 |  24  24  24   0  0  0 |  8 12  6  0   0   0  0 | 8  *  *  * girco
wx3xx3xw ..&#zx |  24  24 |  12  12   0  24 12 12 |  4  0  0  6  12   0  4 | * 16  *  * girco
wx .. xw4xo&#zx |  16   8 |   0   8   8  16  4  0 |  0  0  2  4   0   8  0 | *  * 24  * tic
.. xx .. xo&#x  |   4   2 |   2   0   2   4  0  1 |  0  1  0  0   2   2  0 | *  *  * 96 trip

which thus happens to be Wythoffian again: it is nothing but grico (o3x4x3x), again the corresponding expansion of cont (o3x4x3o) wrt. to full o3o4o3o symmetry, as mentioned above.

Thus we have figured out here 2 more (i.e. in total 4) CUE polychora, all being related to that single CUE polygon, the 90-135-135-90-135-135 degrees Hexagon (in the following pic marked 'H'). And we found now a total 4 corresponding CUE polyhedra as well:
Code: Select all
(1) tetrahedrally    (2) fourfold-       (3) fourfold-            (4) tetrahedrally     
    truncated            axially             contracted               lowered           
    cube                 elongated           truncated                small rhombi-     
                         cuboctahedron       cube                     cuboctahedron     
        H                   4                      8                        H           
   +-----+---+          +---+---+           +---+-----+---+            +-----+---+       
   |       \ |          | /   \ |           | /         \ |          / |  4  |     \     
   |         +          +       +        H  +      H      +  H     +---+-----+   6   +   
H  +    H    |  H    H  |   H   |  H        | \         / |        | 4 |       \     |   
   | \       |          +       +           +---+-----+---+        |   |         +---+   
   +---+-----+          | \   / |                  8            H  +---+    H    |   |  H
        H               +---+---+                                  |     \       | 4 |   
                            4                                      +   6   +-----+---+   
                                                                     \     |  4  | /     
                                                                       +---+-----+       
                                                                            H           


--- rk
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Re: CUE polytopes?

Postby Klitzing » Sat Nov 01, 2014 12:07 pm

Btw., in a different mailing archive today I found a picture of a different CUE polyhedron, having 12 regular pentagons, 20 semiregular (of the second kind) shield hexagons (i.e. angle sequence a-b-a-b-a-b), and 60 diametrally streched hexagons (with some different-valued a-b-b-a-b-b angle sequence than in my previous posts, I'd suppose):
Image
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Re: CUE polytopes?

Postby Keiji » Mon Nov 03, 2014 7:39 pm

That image doesn't work:

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Re: CUE polytopes?

Postby Klitzing » Mon Nov 03, 2014 9:16 pm

Keiji wrote:That image doesn't work: ...


Haha, that proves the working privacy of that other mailing archive...

Okay, I suppose Adrian would not disallow showing off his pic here, provided it is cited correctly.
So both, him and the origin of that polyhedron can be gotten from
http://www.antiprism.com/examples/150_named_models/580_geodesic_duals/geo_3_d_Med.jpg.15.html.

The pic then was obtained from the linked one by relaxation to equal edge length, while keeping the faces flat.
12p60h20h.jpg
12p60h20h.jpg (28.29 KiB) Viewed 8969 times

(Btw., this discussion there was initiated by this article.)

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Re: CUE polytopes?

Postby quickfur » Mon Nov 03, 2014 9:30 pm

That's a really beautiful little 3D CUE! Now it's making me wonder if there isn't a 120-cell symmetry analogue of this in 4D... :D
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