Scaliform polypeton

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

Scaliform polypeton

Postby polychoronlover » Wed Aug 05, 2015 5:02 am

Today I found that a polypeton could be constructed by taking the convex hull of three hexadecachora in different orientations, each having demitesseractic symmetry, in parallel flunes arranged like the vertices of an equilateral triangle. I think it is scaliform. It has 3 hemipenteracts and 24 (I think) hexadecachoric pyramids, and possibly other facets too. Here is its lace city:
Code: Select all
    A
   / \
  /   \
/     \
B-------C

A = x3o3o *b3o
B = o3o3x *b3o
C = o3o3o *b3x

Has this been found before? Similar shapes can be made from the rectified tesseract, truncated hexadecachoron, etc.
Climbing method and elemental naming scheme are good.
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Re: Scaliform polypeton

Postby wendy » Wed Aug 05, 2015 7:15 am

It is the tri-diminished 2_21, a creature that i first introduced Richard to the art of lace cities. It's three points shy of a 2_21.

Draw a edge-2 triangle, where A, B, C, and mark these D. D has a lace-stratum of o3o3o *b3o or o3o3oAo are the vertex midpoints, and you get the lace city for Gosset's 6-dimensional uniform polytope with 27 cross polytopes and 72 simplexes.
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Re: Scaliform polypeton

Postby Klitzing » Wed Aug 05, 2015 9:36 pm

polychoronlover wrote:Today I found that a polypeton could be constructed by taking the convex hull of three hexadecachora in different orientations, each having demitesseractic symmetry, in parallel flunes arranged like the vertices of an equilateral triangle.


So it is   xoo3ooo3oxo *b3oox&#x .

I think it is scaliform. It has 3 hemipenteracts and 24 (I think) hexadecachoric pyramids, and possibly other facets too.


The full incidence matrix here is:
Code: Select all
xoo3ooo3oxo *b3oox&#x

o..3o..3o.. *b3o..    & | 24 |  6  8 | 12  36 12 |  8  32 12  48 | 1  8 10 20  30 | 2  9  6
------------------------+----+-------+-----------+---------------+----------------+--------
x.. ... ...    ...    & |  2 | 72  * |  4   4  0 |  4   8  2   4 | 1  4  4  4   4 | 2  4  1
oo.3oo.3oo. *b3oo.&#x & |  2 |  * 96 |  0   6  3 |  0   6  3  15 | 0  3  2  7  12 | 1  5  3
------------------------+----+-------+-----------+---------------+----------------+--------
x..3o.. ...    ...    & |  3 |  3  0 | 96   *  * |  2   2  0   0 | 1  2  2  1   0 | 2  2  0
xo. ... ...    ...&#x & |  3 |  1  2 |  * 288  * |  0   2  1   2 | 0  2  1  2   3 | 1  3  1
ooo3ooo3ooo *b3ooo&#x   |  3 |  0  3 |  *   * 96 |  0   0  0   6 | 0  0  0  3   6 | 0  3  2
------------------------+----+-------+-----------+---------------+----------------+--------
x..3o..3o..    ...    & |  4 |  6  0 |  4   0  0 | 48   *  *   * | 1  1  1  0   0 | 2  1  0   tet
xo.3oo. ...    ...&#x & |  4 |  3  3 |  1   3  0 |  * 192  *   * | 0  1  1  1   0 | 1  2  0   tet
xo. ... ox.    ...&#x & |  4 |  2  4 |  0   4  0 |  *   * 72   * | 0  2  0  0   2 | 1  2  1   tet
xoo ... ...    ...&#x & |  4 |  1  5 |  0   2  2 |  *   *  * 288 | 0  0  0  1   2 | 0  2  1   tet
------------------------+----+-------+-----------+---------------+----------------+--------
x..3o..3o.. *b3o..    & |  8 | 24  0 | 32   0  0 | 16   0  0   0 | 3  *  *  *   * | 2  0  0   hex
xo.3oo.3ox.    ...&#x & |  8 | 12 12 |  8  24  0 |  2   8  6   0 | * 24  *  *   * | 1  1  0   hex
xo.3oo. ... *b3oo.&#x & |  5 |  6  4 |  4   6  0 |  1   4  0   0 | *  * 48  *   * | 1  1  0   pen
xoo3ooo ...    ...&#x & |  5 |  3  7 |  1   6  3 |  0   2  0   3 | *  *  * 96   * | 0  2  0   pen
xoo ... oxo    ...&#x & |  5 |  2  8 |  0   6  4 |  0   0  1   4 | *  *  *  * 144 | 0  1  1   pen
------------------------+----+-------+-----------+---------------+----------------+--------
xo.3oo.3ox. *b3oo.&#x & | 16 | 48 32 | 64  96  0 | 32  64 24   0 | 2  8 16  0   0 | 3  *  *   hin
xoo3ooo3oxo    ...&#x & |  9 | 12 20 |  8  36 12 |  2  16  6  24 | 0  1  2  8   6 | * 24  *   hexpy
xoo ... oxo    oox&#x   |  6 |  3 12 |  0  12  8 |  0   0  3  12 | 0  0  0  0   6 | *  * 24   hix

Esp. it has 3 hemipenteracts, 24 pyramids on top of hexadecachora, and 24 hexatera (= 5D simplex) for faces.

wendy wrote:It is the tri-diminished 2_21, a creature that i first introduced Richard to the art of lace cities. It's three points shy of a 2_21.


Accordingly Hedrondude might like to call that figure a "tedjak". The syllable "ted-" here stems from "tri-diminished". And "jak" is already his acronym for that Gosset polypeton 2_21.

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Re: Scaliform polypeton

Postby Klitzing » Mon Aug 17, 2015 6:16 pm

polychoronlover wrote:Today I found that a polypeton could be constructed by taking the convex hull of three hexadecachora in different orientations, each having demitesseractic symmetry, in parallel flunes arranged like the vertices of an equilateral triangle. I think it is scaliform. [...] Here is its lace city:
Code: Select all
    A
   / \
  /   \
 /     \
B-------C

A = x3o3o *b3o
B = o3o3x *b3o
C = o3o3o *b3x

Has this been found before? Similar shapes can be made from the [...] truncated hexadecachoron [...].


Well, though about looking into that one.

As can be seen from the lace city (for sure, using here truncated hexadecachora instead of simple hexadecachora for A, B, C), all these lace-simplectical figures (with equivalent bases) are not only scaliform, they also are convex segmentopeta. And as is known for segmentotopes in general, the lacing facets then have to be segmentotopes of one lesser dimension in turn. - The specific shape of using a triangle (or more generally a lace simplex) shows moreover, that there is no dominant base; instead anyone can be used with the same right. Thus the chosen base here as well has to be a segmentotope.

Therefore we have to look for the sequence of 1D, 2D, 3D, 4D, 5D boundary segmentotopes of that specific fellow to be investigated, i.e. of xoo3xxx3oxo *b3oox&#x.
  • The 1D segmentotope is unique. It is just a lacing edge.
  • There are only 2 segmentogons in 2D, the triangle (pt || line) and the square (line || line).
  • Segmentohedra in 3D which are used in here are the triangular cupola ({3} || {6} = tricu), the triangular prism ({3} || {3} = trip), and the tetrahedron (in here used as: line || perp line = tet). Further being used cells (e.g. within the 3 base polychora of that lace simplex or spanning more than just 2 bases themselves) are octs (octahedra), tuts (truncated tetrahedra), further tets, and further trips.
  • Used segmentochora in 4D are tutcup (tut || inv tut, itself scaliform!), octatut (oct || tut), and tricuf ({6} || trip). Other 4D cells are the 3 bases, that is thrice a thex, and some pens (pentachora).
  • The only in here as such being used segmentoteron is also scaliform. It is thexag thex (thex || gyro thex). And besides of hix (hexateron) a further segmentoteron occures in here slanted as a cell which joins all 3 base positions. That one is the octatutcup (oct || (tut || inv tut)) - obviously a true lace simplex itself.
Now let's come to the incidence matrix of that one. Using all the symbol-immanent 3fold symmetry in addition, this boils down to:
Code: Select all
xoo3xxx3oxo *b3oox&#x

o..3o..3o.. *b3o..    & | 144 |  1   4   4 |  4   4   6   8   4 |  4  1  12  2  4   8  4 | 1  4  6  8   5 | 2  5  1
------------------------+-----+------------+--------------------+------------------------+----------------+--------
x.. ... ...    ...    & |   2 | 72   *   * |  4   0   4   0   0 |  4  0   8  2  0   4  0 | 1  4  4  4   4 | 2  4  1
... x.. ...    ...    & |   2 |  * 288   * |  1   2   0   2   0 |  2  1   4  0  2   0  1 | 1  2  4  3   0 | 2  3  0
oo.3oo.3oo. *b3oo.&#x & |   2 |  *   * 288 |  0   0   2   2   2 |  0  0   4  1  1   5  2 | 0  2  2  5   4 | 1  4  1
------------------------+-----+------------+--------------------+------------------------+----------------+--------
x..3x.. ...    ...    & |   6 |  3   3   0 | 96   *   *   *   * |  2  0   2  0  0   0  0 | 1  2  2  1   0 | 2  2  0
... x..3o..    ...    & |   3 |  0   3   0 |  * 192   *   *   * |  1  1   1  0  1   0  0 | 1  1  3  1   0 | 2  2  0
xo. ... ...    ...&#x & |   3 |  1   0   2 |  *   * 288   *   * |  0  0   2  1  0   2  0 | 0  2  1  2   3 | 1  3  1
... xx. ...    ...&#x & |   4 |  0   2   2 |  *   *   * 288   * |  0  0   2  0  1   0  1 | 0  1  2  3   0 | 1  3  0
ooo3ooo3ooo *b3ooo&#x   |   3 |  0   0   3 |  *   *   *   * 192 |  0  0   0  0  0   3  1 | 0  0  0  3   3 | 0  3  1
------------------------+-----+------------+--------------------+------------------------+----------------+--------
x..3x..3o..    ...    & |  12 |  6  12   0 |  4   4   0   0   0 | 48  *   *  *  *   *  * | 1  1  1  0   0 | 2  1  0 tut
... x..3o.. *b3o..    & |   6 |  0  12   0 |  0   8   0   0   0 |  * 24   *  *  *   *  * | 1  0  2  0   0 | 2  1  0 oct
xo.3xx. ...    ...&#x & |   9 |  3   6   6 |  1   1   3   3   0 |  *  * 192  *  *   *  * | 0  1  1  1   0 | 1  2  0 tricu
xo. ... ox.    ...&#x & |   4 |  2   0   4 |  0   0   4   0   0 |  *  *   * 72  *   *  * | 0  2  0  0   2 | 1  2  1 tet
... xx. ... *b3oo.&#x & |   6 |  0   6   3 |  0   2   0   3   0 |  *  *   *  * 96   *  * | 0  0  2  1   0 | 1  2  0 trip
xoo ... ...    ...&#x & |   4 |  1   0   5 |  0   0   2   0   2 |  *  *   *  *  * 288  * | 0  0  0  1   2 | 0  2  1 tet
... xxx ...    ...&#x   |   6 |  0   3   6 |  0   0   0   3   2 |  *  *   *  *  *   * 96 | 0  0  0  3   0 | 0  3  0 trip
------------------------+-----+------------+--------------------+------------------------+----------------+--------
x..3x..3o.. *b3o..    & |  48 | 24  96   0 | 32  64   0   0   0 | 16  8   0  0  0   0  0 | 3  *  *  *   * | 2  0  0 thex
xo.3xx.3ox.    ...&#x & |  24 | 12  24  24 |  8   8  24  12   0 |  2  0   8  6  0   0  0 | * 24  *  *   * | 1  1  0 tutcup
xo.3xx. ... *b3oo.&#x & |  18 |  6  24  12 |  4  12   6  12   0 |  1  1   4  0  4   0  0 | *  * 48  *   * | 1  1  0 octatut
xoo3xxx ...    ...&#x & |  12 |  3   9  15 |  1   2   6   9   6 |  0  0   2  0  1   3  3 | *  *  * 96   * | 0  2  0 tricuf
xoo ... oxo    ...&#x & |   5 |  2   0   8 |  0   0   6   0   4 |  0  0   0  1  0   4  0 | *  *  *  * 144 | 0  1  1 pen
------------------------+-----+------------+--------------------+------------------------+----------------+--------
xo.3xx.3ox. *b3oo.&#x & |  96 | 48 192  96 | 64 128  96  96   0 | 32 16  64 24 32   0  0 | 2  8 16  0   0 | 3  *  * thexag thex
xoo3xxx3oxo    ...&#x & |  30 | 12  36  48 |  8  16  36  36  24 |  2  1  16  6  8  24 12 | 0  1  2  8   6 | * 24  * octatutcup
xoo ... oxo    oox&#x   |   6 |  3   0  12 |  0   0  12   0   8 |  0  0   0  3  0  12  0 | 0  0  0  0   6 | *  * 24 hix


Nice one, ain't it?

Btw., as can be seen from writing xoo3xxx3oxo *b3oox&#x, it becomes obvious that this fellow here is nothing but a partial Stott expansion of tedjak (tri-dim. jak, the one this thread started with): we just filled in the central node position simultanuously in all 3 layers A, B, C.


Jonathan, do you have some special OBSA building advice for such 3fold gyrated fellows?
It kind of is "thex || gyro (thex || gyro thex)" - so we might come up with "thexag thexagg thex", but that's way too clumsy, I fear. There ought to be some better (general) advice for any of these special lace-triangular scaliforms. (We already have "tedjak" (tri-dim. jak) for "hexag hexagg hex". But this might feature as an alternate name, as this just happens to be a coincidence there, I think. - There are still to be looked at further such gyrated lace-triangular scaliforms based on rit and on tah as well!) - This is why something like "pex tedjak" (partially expanded tedjak) would not work either.

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Re: Scaliform polypeton

Postby Klitzing » Sat Aug 22, 2015 10:30 am

polychoronlover wrote:... Similar shapes can be made from the rectified tesseract, ..., etc.

Next one: The one using rit = x3o3x *b3o in the 3 mutually gyrated positions.

The 4D cells here are rit, tetaco (= tet || co), tepe, hex, trippy, and bidrap (= bi-dim. rap = {4} || tet). And the 5D facets then are ritag rit (= rit || gyro rit), coahex (= co || hex), and tedrix (= tri-dim. rix = lace simplex of 3 ortho {4}).

The according incidence matrix elaborates as (again using the additional symbol-internal symmetry - then only pointing out its scaliformity):
Code: Select all
xxo3ooo3xox *b3oxx&#x

o..3o..3o.. *b3o..    & | 96 |   6   6 |   6  3  12  18  3 |  3  2  6  18   8   6  15 | 1  8  6  2  7  12 | 2  5  3
------------------------+----+---------+-------------------+--------------------------+-------------------+--------
x.. ... ...    ...    & |  2 | 288   * |   2  1   2   2  0 |  2  1  2   5   2   1   2 | 1  5  2  1  2   3 | 2  3  1
oo.3oo.3oo. *b3oo.&#x & |  2 |   * 288 |   0  0   2   4  1 |  0  0  1   4   2   2   6 | 0  2  2  1  3   6 | 1  3  2
------------------------+----+---------+-------------------+--------------------------+-------------------+--------
x..3o.. ...    ...    & |  3 |   3   0 | 192  *   *   *  * |  1  1  1   0   1   0   0 | 1  3  0  1  1   0 | 2  2  0
x.. ... x..    ...    & |  4 |   4   0 |   * 72   *   *  * |  2  0  0   4   0   0   0 | 1  4  2  0  0   2 | 2  2  1
xx. ... ...    ...&#x & |  4 |   2   2 |   *  * 288   *  * |  0  0  1   2   0   0   1 | 0  2  1  0  1   2 | 1  2  1
... ... xo.    ...&#x & |  3 |   1   2 |   *  *   * 576  * |  0  0  0   1   1   1   1 | 0  1  1  1  1   2 | 1  2  1
ooo3ooo3ooo *b3ooo&#x   |  3 |   0   3 |   *  *   *   * 96 |  0  0  0   0   0   0   6 | 0  0  0  0  3   6 | 0  3  2
------------------------+----+---------+-------------------+--------------------------+-------------------+--------
x..3o..3x..    ...    & | 12 |  24   0 |   8  6   0   0  0 | 24  *  *   *   *   *   * | 1  2  0  0  0   0 | 2  1  0 co
x..3o.. ... *b3o..    & |  4 |   6   0 |   4  0   0   0  0 |  * 48  *   *   *   *   * | 1  1  0  1  0   0 | 2  1  0 tet
xx.3oo. ...    ...&#x & |  6 |   6   3 |   2  0   3   0  0 |  *  * 96   *   *   *   * | 0  2  0  0  1   0 | 1  2  0 trip
xx. ... xo.    ...&#x & |  6 |   5   4 |   0  1   2   2  0 |  *  *  * 288   *   *   * | 0  1  1  0  0   1 | 1  1  1 trip
... oo.3xo.    ...&#x & |  4 |   3   3 |   1  0   0   3  0 |  *  *  *   * 192   *   * | 0  1  0  1  1   0 | 1  2  0 tet
... ... xo.    ox.&#x & |  4 |   2   4 |   0  0   0   4  0 |  *  *  *   *   * 144   * | 0  0  1  1  0   1 | 1  1  1 tet
xxo ... ...    ...&#x & |  5 |   2   6 |   0  0   1   2  2 |  *  *  *   *   *   * 288 | 0  0  0  0  1   2 | 0  2  1 squippy
------------------------+----+---------+-------------------+--------------------------+-------------------+--------
x..3o..3x.. *b3o..    & | 32 |  96   0 |  64 24   0   0  0 |  8 16  0   0   0   0   0 | 3  *  *  *  *   * | 2  0  0 rit
xx.3oo.3xo.    ...&#x & | 16 |  30  12 |  12  6  12  12  0 |  1  1  4   6   4   0   0 | * 48  *  *  *   * | 1  1  0 tetaco
xx. ... xo.    ox.&#x & |  8 |   8   8 |   0  2   4   8  0 |  0  0  0   4   0   2   0 | *  * 72  *  *   * | 1  0  1 tepe
... oo.3xo. *b3ox.&#x & |  8 |  12  12 |   8  0   0  24  0 |  0  2  0   0   8   6   0 | *  *  * 24  *   * | 1  1  0 hex
xxo3ooo ...    ...&#x & |  7 |   6   9 |   2  0   3   6  3 |  0  0  1   0   2   0   3 | *  *  *  * 96   * | 0  2  0 trippy
xxo ... xox    ...&#x & |  8 |   6  12 |   0  1   4   8  4 |  0  0  0   2   0   1   4 | *  *  *  *  * 144 | 0  1  1 bidrap
------------------------+----+---------+-------------------+--------------------------+-------------------+--------
xx.3oo.3xo. *b3ox.&#x & | 64 | 192  96 | 128 48  96 192  0 | 16 32 32  96  64  48   0 | 2 16 24  8  0   0 | 3  *  * ritag rit
xxo3ooo3xox    ...&#x & | 20 |  36  36 |  16  6  24  48 12 |  1  2  8  12  16   6  24 | 0  2  0  1  8   6 | * 24  * coahex
xxo ... xox    oxx&#x   | 12 |  12  24 |   0  3  12  24  8 |  0  0  0  12   0   6  12 | 0  0  3  0  0   6 | *  * 24 tedrix

(Still un-named. Waiting for according input from Jonathan - or others.)

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Re: Scaliform polypeton

Postby Klitzing » Sun Aug 23, 2015 3:28 pm

In a private reply Hedrondude wrote:
I thought of a better name for the 6D one, thex gyrotrigonism - thexgyt for short. Trigonism being a fusion of trigon and prism.
The other two could be called ritgyt and tahgyt for short.

So we have now:
  • xoo3ooo3oxo *b3oox&#x = hexadecachoral gyrotrigonism = hexgyt
    (while this one also can be seen as: tridiminished Gosset polytope 3_12 = tridim. icosiheptaheptacontidipeton = tridim. jak = tedjak)
  • xoo3xxx3oxo *b3oox&#x = truncated-hexadecachoral gyrotrigonism = thexgyt
    (accordingly to the previous this one can also be seen as its partial Stott expansion wrt. the central node each, thus we might alternatively write here too: partially expanded tridim. jak = pextedjak - or even: partially expanded hexadecachoral gyrotrigonism = pexhexgyt)
  • xxo3ooo3xox *b3oxx&#x = rectified-tesseractic gyrotrigonism = ritgyt
  • xxo3xxx3xox *b3oxx&#x = tesseracti-hexadecachoral gyrotrigonism = tahgyt
    (this one again is a partial Stott expansion of the former wrt. the central node each, thus we might write here too: partially expanded rectified-tesseractic gyrotrigonism = pexritgyt)

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Re: Scaliform polypeton

Postby Klitzing » Fri Aug 28, 2015 9:08 am

And today finally the incidence matrix of tahgyt, again according to the additional symmetry, showing up its scaliformity:

Code: Select all
xxo3xxx3xox *b3oxx&#x

o..3o..3o.. *b3o..    & | 288 |   2   2   4 |   4  1  1   4   4   6   2 |  2  2  4   6   6   2   5  2 | 1  6  2  2  5   4 | 2  4  1
------------------------+-----+-------------+---------------------------+-----------------------------+-------------------+--------
x.. ... ...    ...    & |   2 | 288   *   * |   2  1  0   2   0   2   0 |  2  1  2   5   2   1   2  0 | 1  5  2  1  2   3 | 2  3  1
... x.. ...    ...    & |   2 |   * 288   * |   2  0  1   0   2   0   0 |  1  2  2   0   4   0   0  1 | 1  4  0  2  3   0 | 2  3  0
oo.3oo.3oo. *b3oo.&#x & |   2 |   *   * 576 |   0  0  0   1   1   2   1 |  0  0  1   2   2   1   3  1 | 0  2  1  1  3   3 | 1  3  1
------------------------+-----+-------------+---------------------------+-----------------------------+-------------------+--------
x..3x.. ...    ...    & |   6 |   3   3   0 | 192  *  *   *   *   *   * |  1  1  1   0   1   0   0  0 | 1  3  0  1  1   0 | 2  2  0
x.. ... x..    ...    & |   4 |   4   0   0 |   * 72  *   *   *   *   * |  2  0  0   4   0   0   0  0 | 1  4  2  0  0   2 | 2  2  1
... x.. ... *b3o..    & |   3 |   0   3   0 |   *  * 96   *   *   *   * |  0  2  0   0   2   0   0  0 | 1  2  0  2  1   0 | 2  2  0
xx. ... ...    ...&#x & |   4 |   2   0   2 |   *  *  * 288   *   *   * |  0  0  1   2   0   0   1  0 | 0  2  1  0  1   2 | 1  2  1
... xx. ...    ...&#x & |   4 |   0   2   2 |   *  *  *   * 288   *   * |  0  0  1   0   2   0   0  1 | 0  2  0  1  3   0 | 1  3  0
... ... xo.    ...&#x & |   3 |   1   0   2 |   *  *  *   *   * 576   * |  0  0  0   1   1   1   1  0 | 0  1  1  1  1   2 | 1  2  1
ooo3ooo3ooo *b3ooo&#x   |   3 |   0   0   3 |   *  *  *   *   *   * 192 |  0  0  0   0   0   0   3  1 | 0  0  0  0  3   3 | 0  3  1
------------------------+-----+-------------+---------------------------+-----------------------------+-------------------+--------
x..3x..3x..    ...    & |  24 |  24  12   0 |   8  6  0   0   0   0   0 | 24  *  *   *   *   *   *  * | 1  2  0  0  0   0 | 2  1  0 toe
x..3x.. ... *b3o..    & |  12 |   6  12   0 |   4  0  4   0   0   0   0 |  * 48  *   *   *   *   *  * | 1  1  0  1  0   0 | 2  1  0 tut
xx.3xx. ...    ...&#x & |  12 |   6   6   6 |   2  0  0   3   3   0   0 |  *  * 96   *   *   *   *  * | 0  2  0  0  1   0 | 1  2  0 hip
xx. ... xo.    ...&#x & |   6 |   5   0   4 |   0  1  0   2   0   2   0 |  *  *  * 288   *   *   *  * | 0  1  1  0  0   1 | 1  1  1 trip
... xx.3xo.    ...&#x & |   9 |   3   6   6 |   1  0  1   0   3   3   0 |  *  *  *   * 192   *   *  * | 0  1  0  1  1   0 | 1  2  0 tricu
... ... xo.    ox.&#x & |   4 |   2   0   4 |   0  0  0   0   0   4   0 |  *  *  *   *   * 144   *  * | 0  0  1  1  0   1 | 1  1  1 tet
xxo ... ...    ...&#x & |   5 |   2   0   6 |   0  0  0   1   0   2   2 |  *  *  *   *   *   * 288  * | 0  0  0  0  1   2 | 0  2  1 squippy
... xxx ...    ...&#x   |   6 |   0   3   6 |   0  0  0   0   3   0   2 |  *  *  *   *   *   *   * 96 | 0  0  0  0  3   0 | 0  3  0 trip
------------------------+-----+-------------+---------------------------+-----------------------------+-------------------+--------
x..3x..3x.. *b3o..    & |  96 |  96  96   0 |  64 24 32   0   0   0   0 |  8 16  0   0   0   0   0  0 | 3  *  *  *  *   * | 2  0  0 tah
xx.3xx.3xo.    ...&#x & |  36 |  30  24  24 |  12  6  4  12  12  12   0 |  1  1  4   6   4   0   0  0 | * 48  *  *  *   * | 1  1  0 tutatoe
xx. ... xo.    ox.&#x & |   8 |   8   0   8 |   0  2  0   4   0   8   0 |  0  0  0   4   0   2   0  0 | *  * 72  *  *   * | 1  0  1 tepe
... xx.3xo. *b3ox.&#x & |  24 |  12  24  24 |   8  0  8   0  12  24   0 |  0  2  0   0   8   6   0  0 | *  *  * 24  *   * | 1  1  0 tutcup
xxo3xxx ...    ...&#x & |  15 |   6   9  18 |   2  0  1   3   9   6   6 |  0  0  1   0   2   0   3  3 | *  *  *  * 96   * | 0  2  0 tripuf
xxo ... xox    ...&#x & |   8 |   6   0  12 |   0  1  0   4   0   8   4 |  0  0  0   2   0   1   4  0 | *  *  *  *  * 144 | 0  1  1 bidrap
------------------------+-----+-------------+---------------------------+-----------------------------+-------------------+--------
xx.3xx.3xo. *b3ox.&#x & | 192 | 192 192 192 | 128 48 64  96  96 192   0 | 16 32 32  96  64  48   0  0 | 2 16 24  8  0   0 | 3  *  * tahagtah
xxo3xxx3xox    ...&#x & |  48 |  36  36  72 |  16  6  8  24  36  48  24 |  1  2  8  12  16   6  24 12 | 0  2  0  1  8   6 | * 24  * toa tutcup
xxo ... xox    oxx&#x   |  12 |  12   0  24 |   0  3  0  12   0  24   8 |  0  0  0  12   0   6  12  0 | 0  0  3  0  0   6 | *  * 24 tedrix


Note that all these 4 gyrotrigonisms have a common typical 3+24+24 polytera facetal structure! :nod:

--- rk
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Re: Scaliform polypeton

Postby polychoronlover » Sun Aug 30, 2015 5:21 am

I was thinking "thex-cupolitrigonism" for thexgyt, or "thex-cupolitrism" for short, similar to the truncated tetrahedral cupoliprism, to represent lace towers of the same shape in different orientations.
Climbing method and elemental naming scheme are good.
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Re: Scaliform polypeton

Postby Klitzing » Sun Aug 30, 2015 7:27 pm

polychoronlover wrote:I was thinking "thex-cupolitrigonism" for thexgyt, or "thex-cupolitrism" for short, similar to the truncated tetrahedral cupoliprism, to represent lace towers of the same shape in different orientations.

Well, yes, it is kind somthing "like a cupola", in fact: a lace prism or segmentopeton. But you probably did not get the true intend of PolyhedronDude's term "gyrotrigonism".

We have normal prisms, we have duoprisms, triprisms, etc. E.g. when P, Q, R are polytopes (usually not a mere edge, but any larger dimensional one), then PxQ is a duoprism, and PxQxR a triprism.
But here we do not have a triprism connotion, rather something like a trigon-type prism, i.e. somthing which happens not to elongate into a single direction, but rather like a lace city with trigonal shape. That is, more like a duoprism with one element being restricted to be a trigon: {3}xP. This is most probably his "trigonism" part. - But here we don't have true trigonisms (in that sense), as we don't have a prism type connection between any 2 of the 3 "bases". Rather we use some gyrated connection there. - This then is what becomes a "gyrotrigonism". - And this seems to apply here indeed: thus hex-gyt, thex-gyt, rit-gyt, and tah-gyt.

In the same right we then might call a 3D antiprism also a "gyro-digon-ism". - This then even would apply to higher dimensional polytopes with linear and (when undecorated) inversionally symmetrical Dynkin diagrams. - But already the term "antiprism" is not uniquely defined for such higher dimensions. - I for one in fact do use "antiprism" for any lace prism, where the bases define a dual pair of regular polytopes. E.g. a trigon (+ dual trigon) -antiprism, an octahedron+cube-antiprism, etc.

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