Uniform Polyteron Sections and Verfs

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

Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Sun Oct 20, 2013 2:18 pm

Polyhedron Dude wrote:... Chad - cellihemidodecateron. Its facets are 30 tepes (cyan and yellow) and 6 spids (red).

Image
Image
http://pages.suddenlink.net/hedrondude/chad.png


Matrix of its vertex figure (using the same vertex pattern as before):
Code: Select all
8 |  3  3 | 3  9 | 4 3
--+-------+------+----
2 | 12  * | 2  2 | 2 2  ab x
2 |  * 12 | 0  4 | 2 2  ae q
--+-------+------+----
3 |  3  0 | 8  * | 1 1  abc xxx verf(tet)
3 |  1  2 | * 24 | 1 1  abf xqq verf(trip)
--+-------+------+----
4 |  3  3 | 1  3 | 8 *  abcf xo3oo&#q = verf(tepe)
6 |  6  6 | 2  6 | * 4  abcegh xo3ox&#q = verf(spid)


And thus of chad:
Code: Select all
30 |   8 |  12 12 |  8  24 |  8 4
---+-----+--------+--------+-----
 2 | 120 |   3  3 |  3   9 |  4 3
---+-----+--------+--------+-----
 3 |   3 | 120  * |  2   2 |  2 2
 4 |   4 |   * 90 |  0   4 |  2 2
---+-----+--------+--------+-----
 4 |   6 |   4  0 | 60   * |  1 1  (tet)
 6 |   9 |   2  3 |  * 120 |  1 1  (trip)
---+-----+--------+--------+-----
 8 |  16 |   8  6 |  2   4 | 30 *  (tepe)
20 |  60 |  40 30 | 10  20 |  * 6  (spid)


--- rk
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Re: Uniform Polyteron Sections and Verfs

Postby Polyhedron Dude » Mon Oct 21, 2013 1:17 pm

Arrgh!!! got bad news folks - my monitor messed up and I don't know how long it might be before I get it fixed - that means there's no way I can generate the graphics. I can still get online with my Playstation 3 -so I can still visit the forums.
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Re: Uniform Polyteron Sections and Verfs

Postby Polyhedron Dude » Fri Dec 06, 2013 9:21 pm

BOOYA! - got my monitor fixed! Now for new polytera - this time there are four of them, the remaining members of the dot regiment. Dot has five members in its regiment: Dot, Bad, Bend, Cabbix, and Caddix. Bad was mentioned in September.

Dot - dodecateron - symbol is ooxoo. Its facets are 12 raps (red and yellow).

Image
Image
http://pages.suddenlink.net/hedrondude/dot.png

Bend - biprismatointercepted dodecateron. Its facets are 12 pinnips (green and blue) and 20 triddips (yellow).

Image
Image
http://pages.suddenlink.net/hedrondude/bend.png

Cabbix - cellibiprismatohexateron. Its facets are 6 raps (light wood), 15 opes (hot pink), and 20 triddips (brown).

Image
Image
http://pages.suddenlink.net/hedrondude/cabbix.png

Caddix - cellidishexateron. Its facets are 6 firps (purple), 6 pinnips (red-orange), and 15 opes (yellow-green).

Image
Image
http://pages.suddenlink.net/hedrondude/caddix.png
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Re: Uniform Polyteron Sections and Verfs

Postby ICN5D » Sat Dec 07, 2013 2:03 am

Those are some wild looking cross sections. Reminds me of snowflakes, the intricate structure. You've got some funny names for them, too!
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Sat Dec 07, 2013 9:29 am

Polyhedron Dude wrote:BOOYA! - got my monitor fixed!

Ey, you seem to have been quite well-behaved, when Nikolaus brought you a new monitor! :D

Now for new polytera - this time there are four of them, the remaining members of the dot regiment. Dot has five members in its regiment: Dot, Bad, Bend, Cabbix, and Caddix. Bad was mentioned in September.

Dot - dodecateron - symbol is ooxoo. Its facets are 12 raps (red and yellow).

Image
Image
http://pages.suddenlink.net/hedrondude/dot.png

And here is its incidence matrix:
Code: Select all
o3o3x3o3o

. . . . . | 20 |  9 |  9  9 |  3  9  3 | 3 3  verf: triddip
----------+----+----+-------+----------+----
. . x . . |  2 | 90 |  2  2 |  1  4  1 | 2 2
----------+----+----+-------+----------+----
. o3x . . |  3 |  3 | 60  * |  1  2  0 | 2 1
. . x3o . |  3 |  3 |  * 60 |  0  2  1 | 1 2
----------+----+----+-------+----------+----
o3o3x . . |  4 |  6 |  4  0 | 15  *  * | 2 0  tet
. o3x3o . |  6 | 12 |  4  4 |  * 30  * | 1 1  oct
. . x3o3o |  4 |  6 |  0  4 |  *  * 15 | 0 2  tet
----------+----+----+-------+----------+----
o3o3x3o . | 10 | 30 | 20 10 |  5  5  0 | 6 *  rap
. o3x3o3o | 10 | 30 | 10 20 |  0  5  5 | * 6  rap
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Sat Dec 07, 2013 10:43 am

Polyhedron Dude wrote:Bend - biprismatointercepted dodecateron. Its facets are 12 pinnips (green and blue) and 20 triddips (yellow).

Image
Image
http://pages.suddenlink.net/hedrondude/bend.png


Next the matrix for that fellow
Code: Select all
20 |  9 |  18 18 |  36  9 |  6  9
---+----+--------+--------+------
 2 | 90 |   4  4 |  12  4 |  4  4
---+----+--------+--------+------
 3 |  3 | 120  * |   2  2 |  3  1
 4 |  4 |   * 90 |   4  0 |  2  2
---+----+--------+--------+------
 6 |  9 |   2  3 | 120  * |  1  1  (trip)
 6 | 12 |   8  0 |   * 30 |  2  0  (oct)
---+----+--------+--------+------
10 | 30 |  30 15 |  10  5 | 12  *  (pinnip)
 9 | 18 |   6  9 |   6  0 |  * 20  (triddip)


(Being derived from that of its vertex figure:
Code: Select all
Vertex pattern of verf:
a   b
 c            g   h
               i
d   e
 f

9 |  4  4 | 12 4 | 4 4
--+-------+------+----
2 | 18  * |  2 2 | 3 1  ab,ad x
2 |  * 18 |  4 0 | 2 2  ae q
--+-------+------+----
3 |  1  2 | 36 * | 1 1  abf xqq verf(trip)
4 |  4  0 |  * 9 | 2 0  abed xxxx verf(oct)
--+-------+------+----
6 |  9  6 |  6 3 | 6 *  abcdef verf(pinnip)
4 |  2  4 |  4 0 | * 9  abfi verf(triddip)
)

--- rk
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Sat Dec 07, 2013 12:52 pm

Polyhedron Dude wrote:Cabbix - cellibiprismatohexateron. Its facets are 6 raps (light wood), 15 opes (hot pink), and 20 triddips (brown).

Image
Image
http://pages.suddenlink.net/hedrondude/cabbix.png


Vertex figure (same vertex pattern as before):
Code: Select all
9 | 2 2  4 | 1  6  6 4 | 2 5 4
--+--------+-----------+------
2 | 9 *  * | 1  2  0 2 | 2 2 1  ab x
2 | * 9  * | 0  0  2 2 | 1 2 1  ad x
2 | * * 18 | 0  2  2 0 | 0 2 2  ae q
--+--------+-----------+------
3 | 3 0  0 | 3  *  * * | 2 0 0  abc xxx verf(tet)
3 | 1 0  2 | * 18  * * | 0 1 1  abf xqq verf(trip)
3 | 0 1  2 | *  * 18 * | 0 1 1  adg xqq verf(trip)
4 | 2 2  0 | *  *  * 9 | 1 1 0  abed xxxx verf(oct)
--+--------+-----------+------
6 | 6 3  0 | 2  0  0 3 | 3 * *  abcdef verf(rap)
5 | 2 2  4 | 0  2  2 1 | * 9 *  abdei verf(ope)
4 | 1 1  4 | 0  2  2 0 | * * 9  abfi verf(triddip)


And thus for cabbix itself:
Code: Select all
20 |  9 |  9  9 18 |  3 18 18  9 | 3  9  9
---+----+----------+-------------+--------
 2 | 90 |  2  2  4 |  1  6  6  4 | 2  5  4
---+----+----------+-------------+--------
 3 |  3 | 60  *  * |  1  2  0  2 | 2  2  1
 3 |  3 |  * 60  * |  0  0  2  2 | 1  2  1
 4 |  4 |  *  * 90 |  0  2  2  0 | 0  2  2
---+----+----------+-------------+--------
 4 |  6 |  4  0  0 | 15  *  *  * | 2  0  0  (tet)
 6 |  9 |  2  0  4 |  * 60  *  * | 0  1  1  (trip)
 6 |  9 |  0  2  4 |  *  * 60  * | 0  1  1  (trip)
 6 | 12 |  4  4  0 |  *  *  * 30 | 1  1  0  (oct)
---+----+----------+-------------+--------
10 | 30 | 20 10  0 |  5  0  0  5 | 6  *  *  (rap)
12 | 30 |  8  8 12 |  0  4  4  2 | * 15  *  (ope)
 9 | 18 |  3  3  9 |  0  3  3  0 | *  * 20  (triddip)


--- rk
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Sat Dec 07, 2013 4:19 pm

Polyhedron Dude wrote:Caddix - cellidishexateron. Its facets are 6 firps (purple), 6 pinnips (red-orange), and 15 opes (yellow-green).

Image
Image
http://pages.suddenlink.net/hedrondude/caddix.png


And here now the last one of the dot regiment: caddix

Its verf (according to the same pattern):
Code: Select all
9 | 2 2  4 | 1  6  6 4 | 2 2 5
--+--------+-----------+------
2 | 9 *  * | 0  2  0 2 | 0 2 2  ab x
2 | * 9  * | 1  0  2 2 | 2 1 2  ad x
2 | * * 18 | 0  2  2 0 | 1 1 2  ae q
--+--------+-----------+------
3 | 0 3  0 | 3  *  * * | 2 0 0  adg xxx verf(tet)
3 | 1 0  2 | * 18  * * | 0 1 1  abf xqq verf(trip)
3 | 0 1  2 | *  * 18 * | 1 0 1  adg xqq verf(trip)
4 | 2 2  0 | *  *  * 9 | 0 1 1  abed xxxx verf(oct)
--+--------+-----------+------
6 | 0 6  6 | 2  0  6 0 | 3 * *  abdegh verf(firp)
6 | 6 3  6 | 0  6  0 3 | * 3 *  abcdef verf(pinnip)
5 | 2 2  4 | 0  2  2 1 | * * 9  abdei verf(ope)


Thus caddix itself:
Code: Select all
20 |  9 |  9  9 18 |  3 18 18  9 | 3 3  9
---+----+----------+-------------+-------
 2 | 90 |  2  2  4 |  1  6  6  4 | 2 2  5
---+----+----------+-------------+-------
 3 |  3 | 60  *  * |  0  2  0  2 | 0 2  2
 3 |  3 |  * 60  * |  1  0  2  2 | 2 1  2
 4 |  4 |  *  * 90 |  0  2  2  0 | 1 1  2
---+----+----------+-------------+-------
 4 |  6 |  0  4  0 | 15  *  *  * | 2 0  0  (tet)
 6 |  9 |  2  0  3 |  * 60  *  * | 0 1  1  (trip)
 6 |  9 |  0  2  3 |  *  * 60  * | 1 0  1  (trip)
 6 | 12 |  4  4  0 |  *  *  * 30 | 0 1  1  (oct)
---+----+----------+-------------+-------
10 | 30 |  0 20 15 |  5  0 10  0 | 6 *  *  (firp)
10 | 30 | 20 10 15 |  0 10  0  5 | * 6  *  (pinnip)
12 | 30 |  8  8 12 |  0  4  4  2 | * * 15  (ope)


--- rk
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Re: Uniform Polyteron Sections and Verfs

Postby Polyhedron Dude » Mon Dec 09, 2013 2:13 pm

Now for three from the ginnont regiment:

Ginnont - great penteractipenteracti32teron - it is the 5-D version of gocco (3-D) and gittith (4-D). Its symbol is oo(o'x"x) = ooG and its regiment has 15 members. Its facets are 10 gittiths (cyan), 10 tesses (yellow), and 32 pens (red). My short names for higher dimensional analogues are goxaxog (6-D), gososaz (7-D), gook (8-D), ganinov (9-D), godedak (10-D), gafefer (11-D), and gizazac (12-D) - they have symbols like ooo...oooG.
Image
Image
http://pages.suddenlink.net/hedrondude/ginnont.png

Quacant - quasicellated penteracti32teron - it is the 5-D version of querco (3-D) and quidpith (4-D). Its symbol is xooo"x. Its facets are 10 tesses (yellow), 40 tesses as cube prisms (cyan), 80 tisdips (light wood), 80 tepes (red), and 32 pens (blue). My short names for higher dimensional analogues are quatoxog (6-D), quoposaz (7-D), quaxoke (8-D), quezanov (9-D), queyedak (10-D), quoxofer (11-D), and quekazac (12-D) - they have symbols like xoo...ooo"x and belong to the gocco analogue regiments. After looking at the sections - imagine how busy quekazac sections would look - they would be 128 times busy in each dimension than quacant!.
Image
Image
http://pages.suddenlink.net/hedrondude/quacant.png

Gabdacan - great biprismatodiscellipenteract - it is non-orientable, unlike its 4-D orientable analogue picnut - but more like its 3-D version groh. Its facets are 10 gittiths (gold), 40 tesses (red), 80 tisdips (green), and 80 tepes (blue). I gave the verf a bit of transparency to better see its interior. Even dimensional analogues will be orientable, while odd dimensional ones would be non-orientable.
Image
Image
http://pages.suddenlink.net/hedrondude/gabdacan.png
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Mon Dec 09, 2013 8:37 pm

Polyhedron Dude wrote:Ginnont - great penteractipenteracti32teron - it is the 5-D version of gocco (3-D) and gittith (4-D). Its symbol is oo(o'x"x) = ooG and its regiment has 15 members. Its facets are 10 gittiths (cyan), 10 tesses (yellow), and 32 pens (red). My short names for higher dimensional analogues are goxaxog (6-D), gososaz (7-D), gook (8-D), ganinov (9-D), godedak (10-D), gafefer (11-D), and gizazac (12-D) - they have symbols like ooo...oooG.
Image
Image
http://pages.suddenlink.net/hedrondude/ginnont.png


Ginnont can be directly derived, as it is Wythoffian:
Code: Select all
o3o3o3x4/3x4*c

. . . .   .    | 160 |   4   4 |   6   6  4 |   4  4  6 |  1  1  4
---------------+-----+---------+------------+-----------+---------
. . . x   .    |   2 | 320   * |   3   0  1 |   3  0  3 |  1  0  3
. . . .   x    |   2 |   * 320 |   0   3  1 |   0  3  3 |  0  1  3
---------------+-----+---------+------------+-----------+---------
. . o3x   .    |   3 |   3   0 | 320   *  * |   2  0  1 |  1  0  2
. . o .   x4*c |   4 |   0   4 |   * 240  * |   0  2  1 |  0  1  2
. . . x4/3x    |   8 |   4   4 |   *   * 80 |   0  0  3 |  0  0  3
---------------+-----+---------+------------+-----------+---------
. o3o3x   .    |   4 |   6   0 |   4   0  0 | 160  *  * |  1  0  1  tet
. o3o .   x4*c |   8 |   0  12 |   0   6  0 |   * 80  * |  0  1  1  cube
. . o3x4/3x4*c |  24 |  24  24 |   8   6  6 |   *  * 40 |  0  0  2  gocco
---------------+-----+---------+------------+-----------+---------
o3o3o3x   .    |   5 |  10   0 |  10   0  0 |   5  0  0 | 32  *  *  pen
o3o3o .   x4*c |  16 |   0  32 |   0  24  0 |   0  8  0 |  * 10  *  tes
. o3o3x4/3x4*c |  64 |  96  96 |  64  48 24 |  16  8  8 |  *  * 10  gittith


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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Mon Dec 09, 2013 9:18 pm

Polyhedron Dude wrote:Quacant - quasicellated penteracti32teron - it is the 5-D version of querco (3-D) and quidpith (4-D). Its symbol is xooo"x. Its facets are 10 tesses (yellow), 40 tesses as cube prisms (cyan), 80 tisdips (light wood), 80 tepes (red), and 32 pens (blue). My short names for higher dimensional analogues are quatoxog (6-D), quoposaz (7-D), quaxoke (8-D), quezanov (9-D), queyedak (10-D), quoxofer (11-D), and quekazac (12-D) - they have symbols like xoo...ooo"x and belong to the gocco analogue regiments. After looking at the sections - imagine how busy quekazac sections would look - they would be 128 times busy in each dimension than quacant!.
Image
Image
http://pages.suddenlink.net/hedrondude/quacant.png


That one can likewise easily be derived directly, as it is Wythoffian too.
Code: Select all
x3o3o3o4/3x

. . . .   . | 160 |   4   4 |   6  12   6 |   4  12  12  4 |  1  4  6  4  1
------------+-----+---------+-------------+----------------+---------------
x . . .   . |   2 | 320   * |   3   3   0 |   3   6   3  0 |  1  3  3  1  0
. . . .   x |   2 |   * 320 |   0   3   3 |   0   3   6  3 |  0  1  3  3  1
------------+-----+---------+-------------+----------------+---------------
x3o . .   . |   3 |   3   0 | 320   *   * |   2   2   0  0 |  1  2  1  0  0
x . . .   x |   4 |   2   2 |   * 480   * |   0   2   2  0 |  0  1  2  1  0
. . . o4/3x |   4 |   0   4 |   *   * 240 |   0   0   2  2 |  0  0  1  2  1
------------+-----+---------+-------------+----------------+---------------
x3o3o .   . |   4 |   6   0 |   4   0   0 | 160   *   *  * |  1  1  0  0  0  tet
x3o . .   x |   6 |   6   3 |   2   3   0 |   * 320   *  * |  0  1  1  0  0  trip
x . . o4/3x |   8 |   4   8 |   0   4   2 |   *   * 240  * |  0  0  1  1  0  cube
. . o3o4/3x |   8 |   0  12 |   0   0   6 |   *   *   * 80 |  0  0  0   1 1  cube
------------+-----+---------+-------------+----------------+---------------
x3o3o3o   . |   5 |  10   0 |  10   0   0 |   5   0   0  0 | 32  *  *  *  *  pen
x3o3o .   x |   8 |  12   4 |   8   6   0 |   2   4   0  0 |  * 80  *  *  *  tepe
x3o . o4/3x |  12 |  12  12 |   4  12   3 |   0   4   3  0 |  *  * 80  *  *  tisdip
x . o3o4/3x |  16 |   8  24 |   0  12  12 |   0   0   6  2 |  *  *  * 40  *  tes
. o3o3o4/3x |  16 |   0  32 |   0   0  24 |   0   0   0  8 |  *  *  *  * 10  tes


Btw., speaking of ease, it well could be derived even faster: it is just the conjugate of scant (= x3o3o3o4x), i.e. a well-known convex polyteron.

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Re: Uniform Polyteron Sections and Verfs

Postby wendy » Tue Dec 10, 2013 7:12 am

You could even calculate its density too from the dynkin symbol. I suspect that it's "31", which is a lot shy of the densitiy of the octagrammy at 73.
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Tue Dec 10, 2013 10:54 am

Polyhedron Dude wrote:Gabdacan - great biprismatodiscellipenteract - it is non-orientable, unlike its 4-D orientable analogue picnut - but more like its 3-D version groh. Its facets are 10 gittiths (gold), 40 tesses (red), 80 tisdips (green), and 80 tepes (blue). I gave the verf a bit of transparency to better see its interior. Even dimensional analogues will be orientable, while odd dimensional ones would be non-orientable.
Image
Image
http://pages.suddenlink.net/hedrondude/gabdacan.png


Okay, and this morn' the final one follows.

First the incidence matrix of its vertex figure. - It should be noted here, that both the base tetrahedra on either right- and lefthand end of Hedrondude's sectioning sequence do not belong to the figure, i.e. gabdacan has no fully symmetrical tesseracts nor penteracts for facets. Those will be open ends of the vertex figure (in that orientation), kind of like in a tube. Sure, dihedrality condition still is met. This is why Hedrondude did produce this vertex figure sectioning sequence as partially transparent...
Code: Select all
Vertex figure pattern:
    a
                  e
                  f
    b           g   h
c       d

4 * | 3 1  3 0 | 3 3  6  3 0 | 3 3 3 1  a
* 4 | 0 1  3 3 | 0 3  3  6 3 | 3 1 3 3  e
----+----------+-------------+--------
2 0 | 6 *  * * | 2 1  2  0 0 | 2 2 1 0  ab q
1 1 | * 4  * * | 0 3  0  0 0 | 3 0 0 0  ae x(8/3)
1 1 | * * 12 * | 0 0  2  2 0 | 0 1 2 1  af q
0 2 | * *  * 6 | 0 1  0  2 2 | 2 0 1 2  ef x
----+----------+-------------+--------
3 0 | 3 0  0 0 | 4 *  *  * * | 1 1 0 0  abc q3o = verf(cube)
2 2 | 1 2  0 1 | * 6  *  * * | 2 0 0 0  abfe xq&#x(8/3) = verf(gocco)
2 1 | 1 0  2 0 | * * 12  * * | 0 1 1 0  abg q3o = verf(cube)
1 2 | 0 0  2 1 | * *  * 12 * | 0 0 1 1  cef xo&#q = verf(trip)
0 3 | 0 0  0 3 | * *  *  * 4 | 1 0 0 1  efg x3o = verf(tet)
----+----------+-------------+--------
3 3 | 3 3  0 3 | 1 3  0  0 1 | 4 * * *  abcefg xq3oo&#x(8/3) verf(gittith)
3 1 | 3 0  3 0 | 1 0  3  0 0 | * 4 * *  abch q3o3o = verf(tes)
2 2 | 1 0  4 1 | 0 0  2  2 0 | * * 6 *  cdef xo oq&#q = verf(tisdip)
1 3 | 0 0  3 3 | 0 0  0  3 1 | * * * 4  efgd xo3oo&#q = verf(tepe)


From this matrix now follows the incidence matrix of gabdacan then:
Code: Select all
160 |   4   4 |   6  4  12   6 |  4  6  12  12   4 |  4  4  6  4
----+---------+----------------+-------------------+------------
  2 | 320   * |   3  1   3   0 |  3  3   6   3   0 |  3  3  3  1
  2 |   * 320 |   0  1   3   3 |  0  3   3   6   3 |  3  1  3  3
----+---------+----------------+-------------------+------------
  4 |   4   0 | 240  *   *   * |  2  1   2   0   0 |  2  2  1  0
  8 |   4   4 |   * 80   *   * |  0  3   0   0   0 |  3  0  0  0  {8/3}
  4 |   2   2 |   *  * 480   * |  0  0   2   2   0 |  0  1  2  1
  3 |   0   3 |   *  *   * 320 |  0  1   0   2   2 |  2  0  1  2
----+---------+----------------+-------------------+------------
  8 |  12   0 |   6  0   0   0 | 80  *   *   *   * |  1  1  0  0  cube
 24 |  24  24 |   6  6   0   8 |  * 40   *   *   * |  2  0  0  0  gocco
  8 |   8   4 |   2  0   4   0 |  *  * 240   *   * |  0  1  1  0  cube
  6 |   3   6 |   0  0   3   2 |  *  *   * 320   * |  0  0  1  1  trip
  4 |   0   6 |   0  0   0   4 |  *  *   *   * 160 |  1  0  0  1  tet
----+---------+----------------+-------------------+------------
 64 |  96  96 |  48 24   0  64 |  8  8   0   0  16 | 10  *  *  *  gittith
 16 |  24   8 |  12  0  12   0 |  2  0   6   0   0 |  * 40  *  *  tes
 12 |  12  12 |   3  0  12   4 |  0  0   3   4   0 |  *  * 80  *  tisdip
  8 |   4  12 |   0  0   6   8 |  0  0   0   4   2 |  *  *  * 80  tepe


--- rk
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Re: Uniform Polyteron Sections and Verfs

Postby Polyhedron Dude » Thu Dec 12, 2013 8:01 am

Two members of the fawdint regiment coming up:

Fawdint - frustosphenary dispenteracti32teron. The symbol is oGo = o(o'x"x)o. Its regiment has 37 members and one fissary. This one is half way between ginnont - ooG and wavinant - Goo. Its verf is a trigonal fastegium. It has 320 vertices and its facets are 10 wavitoths (blue), 10 gittiths (red), and 32 raps (yellow).

Image
Image
http://pages.suddenlink.net/hedrondude/fawdint.png

Gatopin - great pteroprismated penteract. It is the "chosen one" in the fawdint regiment. What I mean by "chosen one" is the non-orientable member(s) with the least number of facet types and its facets are either chosen ones or maximized polytopes, examples include garpop, nat, cho, groh. The best definition would be one that is produced using a maximized CD diagram that blends together completely - many chosen ones have choes in them - but this one doesn't. Not all regiments have chosen ones and some have several. Gatopin's facets are 10 gaqripts (purple) and 40 grohps (light wood).

Image
Image
http://pages.suddenlink.net/hedrondude/gatopin.png
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Re: Uniform Polyteron Sections and Verfs

Postby wendy » Thu Dec 12, 2013 8:17 am

Many of the names sound like they're from 'Bored of the Rings'. None the less, i do appreciate your need for strange names.
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the dream we dream together is reality.
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Re: Uniform Polyteron Sections and Verfs

Postby Polyhedron Dude » Thu Dec 12, 2013 11:08 am

wendy wrote:Many of the names sound like they're from 'Bored of the Rings'. None the less, i do appreciate your need for strange names.


Now if only the IAU would hire me to coin names for the exoplanets ;) .
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Thu Dec 12, 2013 12:22 pm

Polyhedron Dude wrote:Two members of the fawdint regiment coming up:

Fawdint - frustosphenary dispenteracti32teron. The symbol is oGo = o(o'x"x)o. Its regiment has 37 members and one fissary. This one is half way between ginnont - ooG and wavinant - Goo. Its verf is a trigonal fastegium. It has 320 vertices and its facets are 10 wavitoths (blue), 10 gittiths (red), and 32 raps (yellow).

Image
Image
http://pages.suddenlink.net/hedrondude/fawdint.png


Fawdint is Wythoffian, thus its IncMat can be derived directly as:
Code: Select all
o3x3o3o *b4/3x4*c

. . . .      .    | 320 |   6   3 |   3   6   6   3 |   3  3   2  6  1 |  1  3  2
------------------+-----+---------+-----------------+------------------+---------
. x . .      .    |   2 | 960   * |   1   2   1   0 |   2  1   1  2  0 |  1  2  1
. . . .      x    |   2 |   * 480 |   0   0   2   2 |   0  1   0  4  1 |  0  2  2
------------------+-----+---------+-----------------+------------------+---------
o3x . .      .    |   3 |   3   0 | 320   *   *   * |   2  1   0  0  0 |  1  2  0
. x3o .      .    |   3 |   3   0 |   * 640   *   * |   1  0   1  1  0 |  1  1  1
. x . . *b4/3x    |   8 |   4   4 |   *   * 240   * |   0  1   0  2  0 |  0  2  1
. . o .      x4*c |   4 |   0   4 |   *   *   * 240 |   0  0   0  2  1 |  0  1  2
------------------+-----+---------+-----------------+------------------+---------
o3x3o .      .    |   6 |  12   0 |   4   4   0   0 | 160  *   *  *  * |  1  1  0  oct
o3x . . *b4/3x    |  24 |  24  12 |   8   0   6   0 |   * 40   *  *  * |  0  2  0  quith
. x3o3o      .    |   4 |   6   0 |   0   4   0   0 |   *  * 160  *  * |  1  0  1  tet
. x3o . *b4/3x4*c |  24 |  24  24 |   0   8   6   6 |   *  *   * 80  * |  0  1  1  gocco
. . o3o      x4*c |   8 |   0  12 |   0   0   0   6 |   *  *   *  * 40 |  0  0  2  cube
------------------+-----+---------+-----------------+------------------+---------
o3x3o3o      .    |  10 |  30   0 |  10  20   0   0 |   5  0   5  0  0 | 32  *  *  rap
o3x3o . *b4/3x4*c |  96 | 192  96 |  64  64  48  24 |  16  8   0  8  0 |  * 10  *  wavitoth
. x3o3o *b4/3x4*c |  64 |  96  96 |   0  64  24  48 |   0  0  16  8  8 |  *  * 10  gittith


--- rk
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Thu Dec 12, 2013 12:25 pm

Polyhedron Dude wrote:
wendy wrote:Many of the names sound like they're from 'Bored of the Rings'. None the less, i do appreciate your need for strange names.


Now if only the IAU would hire me to coin names for the exoplanets ;) .


You would have my opt :D
--- rk
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Thu Dec 12, 2013 12:34 pm

Polyhedron Dude wrote:Gatopin - great pteroprismated penteract. It is the "chosen one" in the fawdint regiment. What I mean by "chosen one" is the non-orientable member(s) with the least number of facet types and its facets are either chosen ones or maximized polytopes, examples include garpop, nat, cho, groh. The best definition would be one that is produced using a maximized CD diagram that blends together completely - many chosen ones have choes in them - but this one doesn't. Not all regiments have chosen ones and some have several. Gatopin's facets are 10 gaqripts (purple) and 40 grohps (light wood).

Image
Image
http://pages.suddenlink.net/hedrondude/gatopin.png


That one has to be derived again in 2 steps.

First the matrix for its verf:
Code: Select all
Pattern of Vertices:
a   b
 c                 g      h
 
d   e                i
 f

6 * | 2 1  2 |  4 2 2 | 2 3  a
* 3 | 0 2  4 |  4 2 4 | 2 4  g
----+--------+--------+----
2 0 | 6 *  * |  2 1 0 | 1 2  ae q
1 1 | * 6  * |  2 0 2 | 2 2  ag x(8/3)
1 1 | * * 12 |  1 1 1 | 1 2  ah q
----+--------+--------+----
2 1 | 1 1  1 | 12 * * | 1 1  aeh x(8/3)qq = verf(stop)
2 1 | 1 0  2 |  * 6 * | 0 2  aei qqq = verf(cube)
2 2 | 0 2  2 |  * * 6 | 1 1  agbh x(8/3)q(-x(8/3))(-q) = verf(groh)
----+--------+--------+----
4 2 | 2 4  4 |  4 0 2 | 3 *  abdegh verf(gaqript)
3 2 | 2 2  4 |  2 2 1 | * 6  abfgh verf(grohp)

Note that this vertex figure polychoron would be self-dual, at least when being considered as abstract polytope.
(Didn't consider the geometric realisation for this, just its matrix.)

From that then the IncMat of gatopin itself follows as:
Code: Select all
320 |   6   3 |   6   6  12 |  12   6  6 |  3  6
----+---------+-------------+------------+------
  2 | 960   * |   2   1   2 |   4   2  2 |  2  3
  2 |   * 480 |   0   2   4 |   4   2  4 |  2  4
----+---------+-------------+------------+------
  4 |   4   0 | 480   *   * |   2   1  0 |  1  2
  8 |   4   4 |   * 240   * |   2   0  2 |  2  2  {8/3}
  4 |   2   2 |   *   * 960 |   1   1  1 |  1  2
----+---------+-------------+------------+------
 16 |  16   8 |   4   2   4 | 240   *  * |  1  1  stop
  8 |   8   4 |   2   0   4 |   * 240  * |  0  2  cube
 24 |  24  24 |   0   6  12 |   *   * 80 |  1  1  groh
----+---------+-------------+------------+------
 96 | 192  96 |  48  48  96 |  24   0  8 | 10  *  gaqript
 48 |  72  48 |  24  12  48 |   6  12  2 |  * 40  grohp


--- rk
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Re: Uniform Polyteron Sections and Verfs

Postby Polyhedron Dude » Fri Dec 13, 2013 11:22 am

Now for two from the wavinant regiment:

Wavinant - sphenoverted penteractipenteracti32teron. Its symbol is (o'x"x)oo = Goo, although gooey it is not. There are 15 members in its regiment, it is the 5-D version of wavitoth. Its facets are 10 quitits (gold), 10 wavitoths (magenta), and 32 raps (cyan).

Image
Image
http://pages.suddenlink.net/hedrondude/wavinant.png

Gafwan - great facetosphenary penteract. It is the 5-D version of gapript, but unlike it, gafwan is non-orientable. Its facets are 10 quitits (lavender), 40 quithips (red), 80 tistodips (yellow), and 80 tepes (blue).

Image
Image
http://pages.suddenlink.net/hedrondude/gafwan.png
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Fri Dec 13, 2013 2:37 pm

Polyhedron Dude wrote:Wavinant - sphenoverted penteractipenteracti32teron. Its symbol is (o'x"x)oo = Goo, although gooey it is not. There are 15 members in its regiment, it is the 5-D version of wavitoth. Its facets are 10 quitits (gold), 10 wavitoths (magenta), and 32 raps (cyan).

Image
Image
http://pages.suddenlink.net/hedrondude/wavinant.png


Again Wythoffian. Thus directly:
Code: Select all
o3o3x3o4x4/3*c

. . . . .      | 320 |   6   2 |   6   3   6  1 |   2   3  6  3 |  1  2  3
---------------+-----+---------+----------------+---------------+---------
. . x . .      |   2 | 960   * |   2   1   1  0 |   1   2  2  1 |  1  1  2
. . . . x      |   2 |   * 320 |   0   0   3  1 |   0   0  3  3 |  0  1  3
---------------+-----+---------+----------------+---------------+---------
. o3x . .      |   3 |   3   0 | 640   *   *  * |   1   1  1  0 |  1  1  1
. . x3o .      |   3 |   3   0 |   * 320   *  * |   0   2  0  1 |  1  0  2
. . x . x4/3*c |   8 |   4   4 |   *   * 240  * |   0   0  2  1 |  0  1  2
. . . o4x      |   4 |   0   4 |   *   *   * 80 |   0   0  0  3 |  0  0  3
---------------+-----+---------+----------------+---------------+---------
o3o3x . .      |   4 |   6   0 |   4   0   0  0 | 160   *  *  * |  1  1  0  tet
. o3x3o .      |   6 |  12   0 |   4   4   0  0 |   * 160  *  * |  1  0  1  trip
. o3x . x4/3*c |  24 |  24  12 |   8   0   6  0 |   *   * 80  * |  0  1  1  quith
. . x3o4x4/3*c |  24 |  24  24 |   0   8   6  6 |   *   *  * 40 |  0  0  2  gocco
---------------+-----+---------+----------------+---------------+---------
o3o3x3o .      |  10 |  30   0 |  20  10   0  0 |   5   5  0  0 | 32  *  *  rap
o3o3x . x4/3*c |  64 |  96  32 |  64   0  24  0 |  16   0  8  0 |  * 10  *  quitit
. o3x3o4x4/3*c |  96 | 192  96 |  64  64  48 24 |   0  16  8  8 |  *  * 10  wavitoth


--- rk

PS @Hedrondude: found a typo at http://www.polytope.net/hedrondude/truncates.htm: within truncverf4.png you provide the edge lengths for quitit as being the shortchords of {3} and {10/3} (instead of {8/3})...
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Fri Dec 13, 2013 3:00 pm

Polyhedron Dude wrote:Gafwan - great facetosphenary penteract. It is the 5-D version of gapript, but unlike it, gafwan is non-orientable. Its facets are 10 quitits (lavender), 40 quithips (red), 80 tistodips (yellow), and 80 tepes (blue).

Image
Image
http://pages.suddenlink.net/hedrondude/gafwan.png


Here again first the vertex figure:
Code: Select all
Pattern of vertices:
a   b          g
  c
 
d   e
  f            h

6 * | 2 2 1 1 | 1 3 2 2  4 | 1 1 3 3  a
* 2 | 0 0 3 3 | 0 0 3 3  6 | 1 1 3 3  g
----+---------+------------+--------
2 0 | 6 * * * | 1 1 1 1  0 | 1 1 1 1  ab x
2 0 | * 6 * * | 0 2 0 0  2 | 0 0 2 2  ae q
1 1 | * * 6 * | 0 0 2 0  2 | 1 0 2 1  aq x(8/3)
1 1 | * * * 6 | 0 0 0 2  2 | 0 1 1 2  ah q
----+---------+------------+--------
3 0 | 3 0 0 0 | 2 * * *  * | 1 1 0 0  abc x3o = verf(tet)
3 0 | 1 2 0 0 | * 6 * *  * | 0 0 1 1  abf xo&#q = verf(trip)
2 1 | 1 0 2 0 | * * 6 *  * | 1 0 1 0  abg xo&#x(8/3) = verf(quith)
2 1 | 1 0 0 2 | * * * 6  * | 0 1 0 1  abh xo&#q = verf(trip)
2 1 | 0 1 1 1 | * * * * 12 | 0 0 1 1  aeg ox(8/3)&#q = verf(stop)
----+---------+------------+--------
3 1 | 3 0 3 0 | 1 0 3 0  0 | 2 * * *  abcg verf(quitit)
3 1 | 3 0 0 3 | 1 0 0 3  0 | * 2 * *  abch verf(tepe)
3 1 | 1 2 2 1 | 0 1 1 0  2 | * * 6 *  abfg verf(quithip)
3 1 | 1 2 1 2 | 0 1 0 1  2 | * * * 6  abfh verf(tistodip)


and thus then gafwan itself:
Code: Select all
320 |   6   2 |   6   6   6   6 |   2   6  6   6  12 |  2  2  6  6
----+---------+-----------------+--------------------+------------
  2 | 960   * |   2   2   1   1 |   1   3  2   2   4 |  1  1  3  3
  2 |   * 320 |   0   0   3   3 |   0   0  3   3   6 |  1  1  3  3
----+---------+-----------------+--------------------+------------
  3 |   3   0 | 640   *   *   * |   1   1  1   1   0 |  1  1  1  1
  4 |   4   0 |   * 480   *   * |   0   2  0   0   2 |  0  0  2  2
  8 |   4   4 |   *   * 240   * |   0   0  2   0   2 |  1  0  2  1  {8/3}
  4 |   2   2 |   *   *   * 480 |   0   0  0   2   2 |  0  1  1  2
----+---------+-----------------+--------------------+------------
  4 |   6   0 |   4   0   0   0 | 160   *  *   *   * |  1  1  0  0  tet
  6 |   9   0 |   2   3   0   0 |   * 320  *   *   * |  0  0  1  1  trip
 24 |  24  12 |   8   0   6   0 |   *   * 80   *   * |  1  0  1  0  quith
  6 |   6   3 |   2   0   0   3 |   *   *  * 320   * |  0  1  0  1  trip
 16 |  16   8 |   0   4   2   4 |   *   *  *   * 240 |  0  0  1  1  stop
----+---------+-----------------+--------------------+------------
 64 |  96  32 |  64   0  24   0 |  16   0  8   0   0 | 10  *  *  *  quitit
  8 |  12   4 |   8   0   0   6 |   2   0  0   4   0 |  * 80  *  *  tepe
 48 |  72  24 |  16  24  12  12 |   0   8  2   0   6 |  *  * 40  *  quithip
 24 |  36  12 |   8  12   3  12 |   0   4  0   4   3 |  *  *  * 80  tistodip


It should be noted, that the leftmost section of its vertex figure in fact is nothing but the vertex figure of firp (in rap regiment). But, as the provided matrix shows as well, that a firp vertex figure there just occures as a pseudo element: all faces of the left section would be already used by 2 different cells running through the whole sequence, and thus, by dyadicity, leave no possibility for a further connected cell.

Therefore in consequence firp itself would serve for pseudo cells of gafwan. There is one firp per vertex figure and here are 320 vertices in total - while firp has 10 itself, thus we'd get 320/10 = 32 pseudo firps here. - But those firps surely would occur in other regiment members for cells, I'd suppose ...

--- rk
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Re: Uniform Polyteron Sections and Verfs

Postby Polyhedron Dude » Tue Dec 17, 2013 12:25 pm

So whats for dinner - these are:

Sadinnert - small dispenteractirhombated 32teron - o(x'x"x)x = oCx. It is a lone operative and is quite cool looking. Its facets are 10 thaquitpaths (blue), 10 thatoths (peach), 40 ticcups (ice), and 32 grips (magenta).

Image
Image
http://pages.suddenlink.net/hedrondude/sadinnert.png

Gadinnert - great dispenteractirhombated 32teron - x(x'x"x)o = xCo. It is also a lone operative. Its facets are 10 thatpaths (aqua), 10 thaquitoths (purple), 40 quithips (pink), and 32 grips (gold). Notice that there is a detail that got missed in the small pic, but got seen in the big pic - towards the center of the polyteron (lower right corner) girco shaped sections barely show up.

Image
Image
http://pages.suddenlink.net/hedrondude/gadinnert.png - big pic

That was a nice dinner - could use more barbecue sauce though :XP: .
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Wed Dec 18, 2013 11:42 pm

Polyhedron Dude wrote:So whats for dinner - these are:

Sadinnert - small dispenteractirhombated 32teron - o(x'x"x)x = oCx. It is a lone operative and is quite cool looking. Its facets are 10 thaquitpaths (blue), 10 thatoths (peach), 40 ticcups (ice), and 32 grips (magenta).

Image
Image
http://pages.suddenlink.net/hedrondude/sadinnert.png


As being Wythoffian once more, the matrix can be derived directly:
Code: Select all
o3x3x3x *b4x4/3*c

. . . .    .      | 1920 |    2   1   1   1 |   1   2   2   2   1   1   1 |   1   1  1   2  2   2  1 |  1  1  1  2
------------------+------+------------------+-----------------------------+--------------------------+------------
. x . .    .      |    2 | 1920   *   *   * |   1   1   1   1   0   0   0 |   1   1  1   1  1   1  0 |  1  1  1  1
. . x .    .      |    2 |    * 960   *   * |   0   2   0   0   1   1   0 |   1   0  0   2  2   0  1 |  1  1  0  2
. . . x    .      |    2 |    *   * 960   * |   0   0   2   0   1   0   1 |   0   1  0   2  0   2  1 |  1  0  1  2
. . . .    x      |    2 |    *   *   * 960 |   0   0   0   2   0   1   1 |   0   0  1   0  2   2  1 |  0  1  1  2
------------------+------+------------------+-----------------------------+--------------------------+------------
o3x . .    .      |    3 |    3   0   0   0 | 640   *   *   *   *   *   * |   1   1  1   0  0   0  0 |  1  1  1  0
. x3x .    .      |    6 |    3   3   0   0 |   * 640   *   *   *   *   * |   1   0  0   1  1   0  0 |  1  1  0  1
. x . x    .      |    4 |    2   0   2   0 |   *   * 960   *   *   *   * |   0   1  0   1  0   1  0 |  1  0  1  1
. x . . *b4x      |    8 |    4   0   0   4 |   *   *   * 480   *   *   * |   0   0  1   0  1   1  0 |  0  1  1  1
. . x3x    .      |    6 |    0   3   3   0 |   *   *   *   * 320   *   * |   0   0  0   2  0   0  1 |  1  0  0  2
. . x .    x4/3*c |    8 |    0   4   0   4 |   *   *   *   *   * 240   * |   0   0  0   0  2   0  1 |  0  1  0  2
. . . x    x      |    4 |    0   0   2   2 |   *   *   *   *   *   * 480 |   0   0  0   0  0   2  1 |  0  0  1  2
------------------+------+------------------+-----------------------------+--------------------------+------------
o3x3x .    .      |   12 |   12   6   0   0 |   4   4   0   0   0   0   0 | 160   *  *   *  *   *  * |  1  1  0  0  tut
o3x . x    .      |    6 |    6   0   3   0 |   2   0   3   0   0   0   0 |   * 320  *   *  *   *  * |  1  0  1  0  trip
o3x . . *b4x      |   24 |   24   0   0  12 |   8   0   0   6   0   0   0 |   *   * 80   *  *   *  * |  0  1  1  0  tic
. x3x3x    .      |   24 |   12  12  12   0 |   0   4   6   0   4   0   0 |   *   *  * 160  *   *  * |  1  0  0  1  toe
. x3x . *b4x4/3*c |   48 |   24  24   0  24 |   0   8   0   6   0   6   0 |   *   *  *   * 80   *  * |  0  1  0  1  cotco
. x . x *b4x      |   16 |    8   0   8   8 |   0   0   4   2   0   0   4 |   *   *  *   *  * 240  * |  0  0  1  1  op
. . x3x    x4/3*c |   48 |    0  24  24  24 |   0   0   0   0   8   6  12 |   *   *  *   *  *   * 40 |  0  0  0  2  quitco
------------------+------+------------------+-----------------------------+--------------------------+------------
o3x3x3x    .      |   60 |   60  30  30   0 |  20  20  30   0  10   0   0 |   5  10  0   5  0   0  0 | 32  *  *  *  grip
o3x3x . *b4x4/3*c |  192 |  192  96   0  96 |  64  64   0  48   0  24   0 |  16   0  8   0  8   0  0 |  * 10  *  *  thatoth
o3x . x *b4x      |   48 |   48   0  24  24 |  16   0  24  12   0   0  12 |   0   8  2   0  0   6  0 |  *  * 40  *  ticcup
. x3x3x *b4x4/3*c |  384 |  192 192 192 192 |   0  64  96  48  64  48  96 |   0   0  0  16  8  24  8 |  *  *  * 10  thaquitpath


--- rk
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Wed Dec 18, 2013 11:54 pm

Polyhedron Dude wrote:Gadinnert - great dispenteractirhombated 32teron - x(x'x"x)o = xCo. It is also a lone operative. Its facets are 10 thatpaths (aqua), 10 thaquitoths (purple), 40 quithips (pink), and 32 grips (gold). Notice that there is a detail that got missed in the small pic, but got seen in the big pic - towards the center of the polyteron (lower right corner) girco shaped sections barely show up.

Image
Image
http://pages.suddenlink.net/hedrondude/gadinnert.png - big pic

That was a nice dinner - could use more barbecue sauce though :XP: .


Matrix being derived either directly again, or, even easier, as the conjugate of the former:
Code: Select all
o3x3x3x *b4/3x4*c

. . . .      .    | 1920 |    2   1   1   1 |   1   2   2   2   1   1   1 |   1   1  1   2  2   2  1 |  1  1  1  2
------------------+------+------------------+-----------------------------+--------------------------+------------
. x . .      .    |    2 | 1920   *   *   * |   1   1   1   1   0   0   0 |   1   1  1   1  1   1  0 |  1  1  1  1
. . x .      .    |    2 |    * 960   *   * |   0   2   0   0   1   1   0 |   1   0  0   2  2   0  1 |  1  1  0  2
. . . x      .    |    2 |    *   * 960   * |   0   0   2   0   1   0   1 |   0   1  0   2  0   2  1 |  1  0  1  2
. . . .      x    |    2 |    *   *   * 960 |   0   0   0   2   0   1   1 |   0   0  1   0  2   2  1 |  0  1  1  2
------------------+------+------------------+-----------------------------+--------------------------+------------
o3x . .      .    |    3 |    3   0   0   0 | 640   *   *   *   *   *   * |   1   1  1   0  0   0  0 |  1  1  1  0
. x3x .      .    |    6 |    3   3   0   0 |   * 640   *   *   *   *   * |   1   0  0   1  1   0  0 |  1  1  0  1
. x . x      .    |    4 |    2   0   2   0 |   *   * 960   *   *   *   * |   0   1  0   1  0   1  0 |  1  0  1  1
. x . . *b4/3x    |    8 |    4   0   0   4 |   *   *   * 480   *   *   * |   0   0  1   0  1   1  0 |  0  1  1  1
. . x3x      .    |    6 |    0   3   3   0 |   *   *   *   * 320   *   * |   0   0  0   2  0   0  1 |  1  0  0  2
. . x .      x4*c |    8 |    0   4   0   4 |   *   *   *   *   * 240   * |   0   0  0   0  2   0  1 |  0  1  0  2
. . . x      x    |    4 |    0   0   2   2 |   *   *   *   *   *   * 480 |   0   0  0   0  0   2  1 |  0  0  1  2
------------------+------+------------------+-----------------------------+--------------------------+------------
o3x3x .      .    |   12 |   12   6   0   0 |   4   4   0   0   0   0   0 | 160   *  *   *  *   *  * |  1  1  0  0  tut
o3x . x      .    |    6 |    6   0   3   0 |   2   0   3   0   0   0   0 |   * 320  *   *  *   *  * |  1  0  1  0  trip
o3x . . *b4/3x    |   24 |   24   0   0  12 |   8   0   0   6   0   0   0 |   *   * 80   *  *   *  * |  0  1  1  0  quith
. x3x3x      .    |   24 |   12  12  12   0 |   0   4   6   0   4   0   0 |   *   *  * 160  *   *  * |  1  0  0  1  toe
. x3x . *b4/3x4*c |   48 |   24  24   0  24 |   0   8   0   6   0   6   0 |   *   *  *   * 80   *  * |  0  1  0  1  cotco
. x . x *b4/3x    |   16 |    8   0   8   8 |   0   0   4   2   0   0   4 |   *   *  *   *  * 240  * |  0  0  1  1  stop
. . x3x      x4*c |   48 |    0  24  24  24 |   0   0   0   0   8   6  12 |   *   *  *   *  *   * 40 |  0  0  0  2  girco
------------------+------+------------------+-----------------------------+--------------------------+------------
o3x3x3x      .    |   60 |   60  30  30   0 |  20  20  30   0  10   0   0 |   5  10  0   5  0   0  0 | 32  *  *  *  grip
o3x3x . *b4/3x4*c |  192 |  192  96   0  96 |  64  64   0  48   0  24   0 |  16   0  8   0  8   0  0 |  * 10  *  *  thaquitoth
o3x . x *b4/3x    |   48 |   48   0  24  24 |  16   0  24  12   0   0  12 |   0   8  2   0  0   6  0 |  *  * 40  *  quithip
. x3x3x *b4/3x4*c |  384 |  192 192 192 192 |   0  64  96  48  64  48  96 |   0   0  0  16  8  24  8 |  *  *  * 10  thatpath


PS: Ended both courses still in dinner time! :XP:

--- rk
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Re: Uniform Polyteron Sections and Verfs

Postby quickfur » Fri Dec 20, 2013 3:53 am

These are all impressive renders. I think this topic deserves a sticky. Keiji? :)

I'm in awe at the intricacy of some of these uniform polytera.
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Re: Uniform Polyteron Sections and Verfs

Postby Polyhedron Dude » Tue Dec 24, 2013 12:20 pm

Today's polyteron is Irohlohn - invertiretro16lepidohemipenteract. It is from the nit regiment. It is pronounced "I roll own" which sounds like a great name for a country song. Its facets are 16 firps (red), 16 pinnips (purple), 5 firts (green), 16 srips (ice), and 16 sirdops (peach).

Image
Image
http://pages.suddenlink.net/hedrondude/irohlohn.png

Attempt at country song:
Life's battles may knock me down, but I get up and Irohlohn.
Machine gun fire all around batatadit I stand up and Irohlohn.
- Batatadit is another nit member.
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Re: Uniform Polyteron Sections and Verfs

Postby Klitzing » Wed Dec 25, 2013 12:11 pm

Polyhedron Dude wrote:Today's polyteron is Irohlohn - invertiretro16lepidohemipenteract. It is from the nit regiment. It is pronounced "I roll own" which sounds like a great name for a country song. Its facets are 16 firps (red), 16 pinnips (purple), 5 firts (green), 16 srips (ice), and 16 sirdops (peach).

Image
Image
http://pages.suddenlink.net/hedrondude/irohlohn.png

Attempt at country song:
Life's battles may knock me down, but I get up and Irohlohn.
Machine gun fire all around batatadit I stand up and Irohlohn.
- Batatadit is another nit member.


First the vertex figure thereof:
Code: Select all
Vertex pattern:
a     b                  g     h
  c                        i   
                               
d     e                  j     k
  f                        l   

12 |  2 1  2  2  2 | 1  3  3  3 2 2 2 | 1 1 1 3 3
---+---------------+------------------+----------
 2 | 12 *  *  *  * | 1  1  1  1 1 0 0 | 1 1 1 1 1  ab x
 2 |  * 6  *  *  * | 0  0  0  0 2 2 0 | 0 0 1 0 3  ag x
 2 |  * * 12  *  * | 0  2  0  0 0 1 0 | 1 0 0 0 2  ae q
 2 |  * *  * 12  * | 0  0  2  0 0 0 1 | 0 0 1 2 0  ah q
 2 |  * *  *  * 12 | 0  0  0  2 0 0 1 | 0 1 0 2 0  ak h
---+---------------+------------------+----------
 3 |  3 0  0  0  0 | 4  *  *  * * * * | 1 1 0 0 0  abc xxx = verf(tet)
 3 |  1 0  2  0  0 | * 12  *  * * * * | 1 0 0 0 1  abf xqq = verf(trip)
 3 |  1 0  0  2  0 | *  * 12  * * * * | 0 0 1 1 0  abi xqq = verf(trip)
 3 |  1 0  0  0  2 | *  *  * 12 * * * | 0 1 0 1 0  abl xhh = verf(tut)
 4 |  2 2  0  0  0 | *  *  *  * 6 * * | 0 0 1 0 1  abgh xxxx = verf(oct)
 4 |  0 2  2  0  0 | *  *  *  * * 6 * | 0 0 0 0 2  aekg xqxq = verf(co)
 4 |  0 0  0  2  2 | *  *  *  * * * 6 | 0 0 0 2 0  ahdk qh(-q)h = verf(cho)
---+---------------+------------------+----------
 6 |  6 0  6  0  0 | 2  6  0  0 0 0 0 | 2 * * * *  abcdef verf(firp)
 6 |  6 0  0  0  6 | 2  0  0  6 0 0 0 | * 2 * * *  abcjkl verf(firt)
 6 |  6 3  0  6  0 | 0  0  6  0 3 0 0 | * * 2 * *  abcghi verf(pinnip)
 6 |  2 0  0  4  4 | 0  0  2  2 0 0 2 | * * * 6 *  abdeil verf(sirdop)
 6 |  2 3  4  0  0 | 0  2  0  0 1 2 0 | * * * * 6  abfghl verf(srip)


Using that the incidence matrix of irohlohn follows then as:
Code: Select all
80 |  12 |  12   6  12  12  12 |  4  12  12 12  6  6  6 |  2 2  2  6  6
---+-----+---------------------+------------------------+--------------
 2 | 480 |   2   1   2   2   2 |  1   3   3  3  2  2  2 |  1 1  1  3  3
---+-----+---------------------+------------------------+--------------
 3 |   3 | 320   *   *   *   * |  1   1   1  1  1  0  0 |  1 1  1  1  1
 3 |   3 |   * 160   *   *   * |  0   0   0  0  2  2  0 |  0 0  1  0  3
 4 |   4 |   *   * 240   *   * |  0   2   0  0  0  1  0 |  1 0  0  0  2
 4 |   4 |   *   *   * 240   * |  0   0   2  0  0  0  1 |  0 0  1  2  0
 6 |   6 |   *   *   *   * 160 |  0   0   0  2  0  0  1 |  0 1  0  2  0
---+-----+---------------------+------------------------+--------------
 4 |   6 |   4   0   0   0   0 | 80   *   *  *  *  *  * |  1 1  0  0  0  tet
 6 |   9 |   2   0   3   0   0 |  * 160   *  *  *  *  * |  1 0  0  0  1  trip
 6 |   9 |   2   0   0   3   0 |  *   * 160  *  *  *  * |  0 0  1  1  0  trip
12 |  18 |   4   0   0   0   4 |  *   *   * 80  *  *  * |  0 1  0  1  0  tut
 6 |  12 |   4   4   0   0   0 |  *   *   *  * 80  *  * |  0 0  1  0  1  oct
12 |  24 |   0   8   6   0   0 |  *   *   *  *  * 40  * |  0 0  0  0  2  co
12 |  24 |   0   0   0   6   4 |  *   *   *  *  *  * 40 |  0 0  0  2  0  cho
---+-----+---------------------+------------------------+--------------
10 |  30 |  20   0  15   0   0 |  5  10   0  0  0  0  0 | 16 *  *  *  *  firp
32 |  96 |  64   0   0   0  32 | 16   0   0 16  0  0  0 |  * 5  *  *  *  firt
10 |  30 |  20  10   0  15   0 |  0   0  10  0  5  0  0 |  * * 16  *  *  pinnip
30 |  90 |  20   0   0  30  20 |  0   0  10  5  0  0  5 |  * *  * 16  *  sirdop
30 |  90 |  20  30  30   0   0 |  0  10   0  0  5  5  0 |  * *  *  * 16  srip


- BTW: merry Xmas to all! -

--- rk
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Re: Uniform Polyteron Sections and Verfs

Postby Polyhedron Dude » Wed Dec 25, 2013 12:46 pm

Merry Christmas! :mrgreen:
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Re: Uniform Polyteron Sections and Verfs

Postby Polyhedron Dude » Tue Dec 31, 2013 5:40 am

These three are from the siphin regiment:

Siphin - small prismated demipenteract - symbol = xo8x. The siphin regiment has 56 members plus 20 fissaries. Its facets are 10 hexes (blue), 16 pens (pink), 16 spids (light wood), and 40 tepes (light green).

Image
Image
http://pages.suddenlink.net/hedrondude/siphin.png

Fidoh - facetoinverted decahexadecateron. Its facets are 10 hinnits (gold), and 16 pippindips (red). Something interesting about fidoh is that it is the facet of a noble uniform polypeton in the trim regiment (trim = xo8ox), trim actually has five nobles in it!

Image
Image
http://pages.suddenlink.net/hedrondude/fidoh.png

Fenandoh - facetospinopenteractidishexateron. Its facets are 10 gottoes (red-orange), 16 garpops (gold), and 16 pens (green).

Image
Image
http://pages.suddenlink.net/hedrondude/fenandoh.png
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