## Uniform Polyteron Sections and Verfs

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

### Re: Uniform Polyteron Sections and Verfs

Is this thing in the list yet? It's the gosset polytope 1_21 in five dimensions or the same as the half-cube. The lace city for it follows. 2_11 is bleadingly obvious.

Code: Select all
`               x3o3o                      o3o3x      o3o3o              o3x3o                      o3o3o`

I watch the list, there are plenty of pretty pictures, but time constraints etc limit me to what is mayly on the box.
The dream you dream alone is only a dream
the dream we dream together is reality.

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wendy
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### Re: Uniform Polyteron Sections and Verfs

wendy wrote:Is this thing in the list yet? It's the gosset polytope 1_21 in five dimensions or the same as the half-cube. The lace city for it follows. 2_11 is bleadingly obvious.

Code: Select all
`               x3o3o                      o3o3x      o3o3o              o3x3o                      o3o3o`

I watch the list, there are plenty of pretty pictures, but time constraints etc limit me to what is mayly on the box.

No, Wendy, neither in here nor on his polyteron of the day website. But it is provided (with an older sectioning field rendering) at his polyteron website. He calls that one "hin", for hemi- (h) -penteract (in longer abbreviations represented simply by n).

The incidence matrix of hin is easy:
Code: Select all
`x3o3o *b3o3o. . .    . . | 16 | 10 |  30 | 10 20 |  5  5  verf=rap-------------+----+----+-----+-------+------x . .    . . |  2 | 80 |   6 |  3  6 |  3  2  edge-fig.=trip-------------+----+----+-----+-------+------x3o .    . . |  3 |  3 | 160 |  1  2 |  2  1-------------+----+----+-----+-------+------x3o3o    . . |  4 |  6 |   4 | 40  * |  2  0  tetx3o . *b3o . |  4 |  6 |   4 |  * 80 |  1  1  tet-------------+----+----+-----+-------+------x3o3o *b3o . |  8 | 24 |  32 |  8  8 | 10  *  hexx3o . *b3o3o |  5 | 10 |  10 |  0  5 |  * 16  pen`

Your lace city diagram shows, that hin can be represented as lace prism "hex || gyro hex" = xo3oo3ox *b3oo&#x, but also as lace tower "point || pseudo rap || inverted pen" = ooo3oxo3ooo3oox&#xt, cf. my website.

--- rk
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### Re: Uniform Polyteron Sections and Verfs

Polyhedron Dude wrote:... Quittin - quasitruncated penteract, or stellatruncated penteract. Its symbol is ooox"x, it is the 5-D version of quith. Interestingly, this one has holes in it due to its quitit facets having zero density regions inside. Facets are 10 quitits (lavender) and 32 pens (aqua).

Again a Wythoffian:
Code: Select all
`o3o3o3x4/3x. . . .   . | 160 |   4  1 |   6  4 |   4  6 |  1  4------------+-----+--------+--------+--------+------. . . x   . |   2 | 320  * |   3  1 |   3  3 |  1  3. . . .   x |   2 |   * 80 |   0  4 |   0  6 |  0  4  edge-fig.=tet------------+-----+--------+--------+--------+------. . o3x   . |   3 |   3  0 | 320  * |   2  1 |  1  2. . . x4/3x |   8 |   4  4 |   * 80 |   0  3 |  0  3------------+-----+--------+--------+--------+------. o3o3x   . |   4 |   6  0 |   4  0 | 160  * |  1  1  tet. . o3x4/3x |  24 |  24 12 |   8  6 |   * 40 |  0  2  quith------------+-----+--------+--------+--------+------o3o3o3x   . |   5 |  10  0 |  10  0 |   5  0 | 32  *  pen. o3o3x4/3x |  64 |  96 32 |  64 24 |  16  8 |  * 10  quitit`

--- rk
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### Re: Uniform Polyteron Sections and Verfs

Polyhedron Dude wrote:... Gaqrin - great quasirhombated penteract. Its symbol is ooxx"x. Its facets are 10 gaqrits (blue), 32 tips (golden), and 80 tepes (red).

Again Wythoffian. Its incidence matrix is the same as that of girn, which is its convex isomorph (i.e. ooxx'x).

Here is girn:
Code: Select all
`o3o3x3x4x. . . . . | 640 |   3   1   1 |   3   3   3  1 |   1   3   3  3 |  1  1  3----------+-----+-------------+----------------+----------------+---------. . x . . |   2 | 960   *   * |   2   1   1  0 |   1   2   2  1 |  1  1  2. . . x . |   2 |   * 320   * |   0   3   0  1 |   0   3   0  3 |  1  0  3. . . . x |   2 |   *   * 320 |   0   0   3  1 |   0   0   3  3 |  0  1  3----------+-----+-------------+----------------+----------------+---------. o3x . . |   3 |   3   0   0 | 640   *   *  * |   1   1   1  0 |  1  1  1. . x3x . |   6 |   3   3   0 |   * 320   *  * |   0   2   0  1 |  1  0  2. . x . x |   4 |   2   0   2 |   *   * 480  * |   0   0   2  1 |  0  1  2. . . x4x |   8 |   0   4   4 |   *   *   * 80 |   0   0   0  3 |  0  0  3----------+-----+-------------+----------------+----------------+---------o3o3x . . |   4 |   6   0   0 |   4   0   0  0 | 160   *   *  * |  1  1  0  tet. o3x3x . |  12 |  12   6   0 |   4   4   0  0 |   * 160   *  * |  1  0  1  tut. o3x . x |   6 |   6   0   3 |   2   0   3  0 |   *   * 320  * |  0  1  1  trip. . x3x4x |  48 |  24  24  24 |   0   8  12  6 |   *   *   * 40 |  0  0  2  girco----------+-----+-------------+----------------+----------------+---------o3o3x3x . |  20 |  30  10   0 |  20  10   0  0 |   5   5   0  0 | 32  *  *  tipo3o3x . x |   8 |   6   0   4 |   8   0   6  0 |   2   0   4  0 |  * 80  *  tepe. o3x3x4x | 192 | 192  96  96 |  64  64  96 24 |   0  16  32  8 |  *  * 10  grit`

And thus gaqrin:
Code: Select all
`o3o3x3x4/3x. . . .   . | 640 |   3   1   1 |   3   3   3  1 |   1   3   3  3 |  1  1  3------------+-----+-------------+----------------+----------------+---------. . x .   . |   2 | 960   *   * |   2   1   1  0 |   1   2   2  1 |  1  1  2. . . x   . |   2 |   * 320   * |   0   3   0  1 |   0   3   0  3 |  1  0  3. . . .   x |   2 |   *   * 320 |   0   0   3  1 |   0   0   3  3 |  0  1  3------------+-----+-------------+----------------+----------------+---------. o3x .   . |   3 |   3   0   0 | 640   *   *  * |   1   1   1  0 |  1  1  1. . x3x   . |   6 |   3   3   0 |   * 320   *  * |   0   2   0  1 |  1  0  2. . x .   x |   4 |   2   0   2 |   *   * 480  * |   0   0   2  1 |  0  1  2. . . x4/3x |   8 |   0   4   4 |   *   *   * 80 |   0   0   0  3 |  0  0  3------------+-----+-------------+----------------+----------------+---------o3o3x .   . |   4 |   6   0   0 |   4   0   0  0 | 160   *   *  * |  1  1  0  tet. o3x3x   . |  12 |  12   6   0 |   4   4   0  0 |   * 160   *  * |  1  0  1  tut. o3x .   x |   6 |   6   0   3 |   2   0   3  0 |   *   * 320  * |  0  1  1  trip. . x3x4/3x |  48 |  24  24  24 |   0   8  12  6 |   *   *   * 40 |  0  0  2  quitco------------+-----+-------------+----------------+----------------+---------o3o3x3x   . |  20 |  30  10   0 |  20  10   0  0 |   5   5   0  0 | 32  *  *  tipo3o3x .   x |   8 |   6   0   4 |   8   0   6  0 |   2   0   4  0 |  * 80  *  tepe. o3x3x4/3x | 192 | 192  96  96 |  64  64  96 24 |   0  16  32  8 |  *  * 10  gaqrit`

--- rk
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### Re: Uniform Polyteron Sections and Verfs

wendy wrote:Is this thing in the list yet? It's the gosset polytope 1_21 in five dimensions or the same as the half-cube. The lace city for it follows. 2_11 is bleadingly obvious.

Code: Select all
`               x3o3o                      o3o3x      o3o3o              o3x3o                      o3o3o`

I watch the list, there are plenty of pretty pictures, but time constraints etc limit me to what is mayly on the box.

It is now!

Hin - aka the demipenteract or 121. Its symbol is oo8x and its regiment contains 6 members which includes a noble one. Its verf is a rap and its facets are 10 hexes (green) and 16 pens (yellow).

Klitzing provided the incmat.

Code: Select all
`x3o3o *b3o3o. . .    . . | 16 | 10 |  30 | 10 20 |  5  5  verf=rap-------------+----+----+-----+-------+------x . .    . . |  2 | 80 |   6 |  3  6 |  3  2  edge-fig.=trip-------------+----+----+-----+-------+------x3o .    . . |  3 |  3 | 160 |  1  2 |  2  1-------------+----+----+-----+-------+------x3o3o    . . |  4 |  6 |   4 | 40  * |  2  0  tetx3o . *b3o . |  4 |  6 |   4 |  * 80 |  1  1  tet-------------+----+----+-----+-------+------x3o3o *b3o . |  8 | 24 |  32 |  8  8 | 10  *  hexx3o . *b3o3o |  5 | 10 |  10 |  0  5 |  * 16  pen`

Next up is Girn - great rhombated penteract or cantitruncated penteract. It is a lone operative with the symbol ooxx'x. Its facets are 10 grits (red), 32 tips (wood), and 80 tepes (cyan).

Klitzing provided its incmat as well.

Code: Select all
`o3o3x3x4x. . . . . | 640 |   3   1   1 |   3   3   3  1 |   1   3   3  3 |  1  1  3----------+-----+-------------+----------------+----------------+---------. . x . . |   2 | 960   *   * |   2   1   1  0 |   1   2   2  1 |  1  1  2. . . x . |   2 |   * 320   * |   0   3   0  1 |   0   3   0  3 |  1  0  3. . . . x |   2 |   *   * 320 |   0   0   3  1 |   0   0   3  3 |  0  1  3----------+-----+-------------+----------------+----------------+---------. o3x . . |   3 |   3   0   0 | 640   *   *  * |   1   1   1  0 |  1  1  1. . x3x . |   6 |   3   3   0 |   * 320   *  * |   0   2   0  1 |  1  0  2. . x . x |   4 |   2   0   2 |   *   * 480  * |   0   0   2  1 |  0  1  2. . . x4x |   8 |   0   4   4 |   *   *   * 80 |   0   0   0  3 |  0  0  3----------+-----+-------------+----------------+----------------+---------o3o3x . . |   4 |   6   0   0 |   4   0   0  0 | 160   *   *  * |  1  1  0  tet. o3x3x . |  12 |  12   6   0 |   4   4   0  0 |   * 160   *  * |  1  0  1  tut. o3x . x |   6 |   6   0   3 |   2   0   3  0 |   *   * 320  * |  0  1  1  trip. . x3x4x |  48 |  24  24  24 |   0   8  12  6 |   *   *   * 40 |  0  0  2  girco----------+-----+-------------+----------------+----------------+---------o3o3x3x . |  20 |  30  10   0 |  20  10   0  0 |   5   5   0  0 | 32  *  *  tipo3o3x . x |   8 |   6   0   4 |   8   0   6  0 |   2   0   4  0 |  * 80  *  tepe. o3x3x4x | 192 | 192  96  96 |  64  64  96 24 |   0  16  32  8 |  *  * 10  grit`
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### Re: Uniform Polyteron Sections and Verfs

- got nothing to do ...

--- rk
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### Re: Uniform Polyteron Sections and Verfs

The 'hin' section is rather interesting, as there is lots of activity going on in it. Wendy thanks Polyhedron Dude and Klitzing for the show.

It is interesting, because in part, this is how i ended up convieving the stott operator to work. Wythoff mirror-edge figures can have any size edge, but only the uniforms (ie 0, 1) are considered. I used letters as commas in a coordinate system, where numbers were the size of the figure, and the branches were reduced effectively to commas. The hin section gives a portion of polytope-space, where each point becomes a 'position polytope', (like 'position vector')

Straight lines of polytopes are 'progressions'.

The column in the left of the hin represents the equal-sum runcinate, ie (x)3(0)3(1-x) as x varies from 0 to 1 (here in steps of 0.1). Note that the sloping sides of the antiprism are pyramids of a surtope and its orthosurtope. For the vertices, we have x pt and 1-x of triangle, which leads to an ordinary triangular pyramid. For the line-line we have a series of rectangles (x)2(1-x), which pass through (0.5)2(0.5) square at the cuboctahedron.

The middle row shows an equal-sum truncation, as one goes from (x sqrt(2) )4(1-x)3(0), with x going from 0.5 to 1 (left to centre).

If one supposes the cube is a tetrahedral affair, then it's actually a convex hull over two figures (1)3(0)3(0) and (0)3(0)3(1). The diagonal from the corner to the centres are variously (1)3(0)3(0), and (0)3(0)3(x), and the reverse. This causes an alternate diminishing of the vertices of a cube (ie a progression that goes from x4o3o to s4o3o, or an edge-bevel of the tetrahedron, ie x3o3o to o3b3o.

The diagrams i played around with were a bit more complex (like let's have a line representing 7 dimensions), but the overall idea can be seen quite nicely in this figure. It's a kind of pictorial lace city cum position-polytope diagram.
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### Re: Uniform Polyteron Sections and Verfs

Klitzing wrote::( - got nothing to do ...

--- rk

wendy wrote:The 'hin' section is rather interesting, as there is lots of activity going on in it. Wendy thanks Polyhedron Dude and Klitzing for the show.

Stay tuned for next time, for "Hin" is gonna lay a few eggs!
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### Re: Uniform Polyteron Sections and Verfs

Recently I (re)found a very nice and quite useful formula!

In the run of collecting the incidence matrices of non-Wythoffian polytera it did serve useful to do that same stuff for its vertex figure first. But then in general that one comes out to be non-uniform itself, e.g. having quite different sized edges. When I became interested in their circumradii (which always do exists, being the vertex figures of uniform polytopes), I got heavy troubles. In rare cases I managed to calculate that one by hand or sometimes by means of computer aided algebraical solver algorithms. But that surely was disappointing.

So i looked whether there could be some general formula here. And indeed it is!

How does that work? Consider a general uniform polytope, having the center C and one vertex V0. Consider further a neighbouring vertex V1. Then you obviously would have an isogonal triangle V0-C-V1, the 2 long sides being R (the polytopal circumradius) and the short side being x (the edge length).

Next consider the orthogonal projection c of V1 onto V0-C. Clearly, that c then is nothing but the center of the vertex figure. Call the distance V0-c to be p. And the distance V1-c obviously is r, the searched for circumradius of the vertex figure.

From the right triangles you get r^2 = x^2 - p^2, but also r^2 = R^2 - (R - p)^2 = 2Rp - p^2. Therefore x^2 = 2Rp. Solving that for p and introducing back into the first equation, you would get r^2 = (4R^2 - x^2)/(4R^2), or, when using unit sized edges (x=1) the desired, and very simple formula: r^2 = 1 - 1/(2R)^2.

Note that for spherical cases R generally is real. - Accordingly (as 2R always is > x = 1) r will be real and < 1.
For euclidean cases R gets infinite. - Accordingly r will be = 1.
For hyperbolic cases R is purely imaginary. - Then r would get > 1.
In rare hyperbolic cases R is itself 0 i. - Then we get r being positive infinite.

--- rk
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### Re: Uniform Polyteron Sections and Verfs

It's not too rare that one gets an infinite 'r'. The trouble is that you can have a polytope like {6,3,6}, which has an indicated size, but the face-radius is infinite. It's one of the reasons i ditched R in favour of using E = 4/R.
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### Re: Uniform Polyteron Sections and Verfs

Time for hin to lay a few eggs:

First up:

Dah - dishexadecateron. It is a member of the hin regiment and its verf is a firp. It is semiregular. Its facets are 16 raps (pink) and 16 pens (blue).

Next egg:

Han - hexadecapenteract. Also a member of the hin regiment, its verf is a pinnip and it is also semiregular. Its facets are 10 hexes (yellow) and 16 raps (blue grey).

And there's a third polyteron!

Hit - hexadecateron. This is the noble member of the hin regiment. Its facets are 16 pinnips. I bet this one will be a hit!

The hin regiment has 2 more members to be revealed (even though they are on my polyteron website ) - rinah and radah - coming soon!
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### Re: Uniform Polyteron Sections and Verfs

Polyhedron Dude wrote:... Dah - dishexadecateron. It is a member of the hin regiment and its verf is a firp. It is semiregular. Its facets are 16 raps (pink) and 16 pens (blue).

...

Incidence matrix
Code: Select all
`16 | 10 |  30 | 20 15 |  5 10  verf=firp---+----+-----+-------+------ 2 | 80 |   6 |  6  6 |  2  6---+----+-----+-------+------ 3 |  3 | 160 |  2  2 |  1  3---+----+-----+-------+------ 4 |  6 |   4 | 80  * |  1  1  tet 6 | 12 |   8 |  * 40 |  0  2  oct---+----+-----+-------+------ 5 | 10 |  10 |  5  0 | 16  *  pen10 | 30 |  30 |  5  5 |  * 16  rap`

--- rk
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### Re: Uniform Polyteron Sections and Verfs

Polyhedron Dude wrote:... Han - hexadecapenteract. Also a member of the hin regiment, its verf is a pinnip and it is also semiregular. Its facets are 10 hexes (yellow) and 16 raps (blue grey).

...

Incidence matrix

Code: Select all
`16 | 10 |  30 | 10 20 15 |  5 10  verf=pinnip---+----+-----+----------+------ 2 | 80 |   6 |  3  6  6 |  3  6---+----+-----+----------+------ 3 |  3 | 160 |  1  2  2 |  2  3---+----+-----+----------+------ 4 |  6 |   4 | 40  *  * |  2  0  tet 4 |  6 |   4 |  * 80  * |  1  1  tet 6 | 12 |   8 |  *  * 40 |  0  2  oct---+----+-----+----------+------ 8 | 24 |  32 |  8  8  0 | 10  *  hex10 | 30 |  30 |  0  5  5 |  * 16  rap`

--- rk
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### Re: Uniform Polyteron Sections and Verfs

Polyhedron Dude wrote:... Hit - hexadecateron. This is the noble member of the hin regiment. Its facets are 16 pinnips. I bet this one will be a hit!

The hin regiment has 2 more members to be revealed (even though they are on my polyteron website ) - rinah and radah - coming soon!

Incidence matrix
Code: Select all
`16 | 10 |  30 !20 | 15 30 | 10---+----+---------+-------+--- 2 | 80 |   6  !4 |  6  9 |  6---+----+---------+-------+--- 3 |  3 | 160   * |  2  1 |  3 4 |  4 |   * !80 |  0 !3 | !3---+----+---------+-------+--- 6 | 12 |   8   0 | 40  * |  2  oct 6 |  9 |   2   3 |  * 80 |  2  trip---+----+---------+-------+---10 | 30 |  30  15 |  5 10 | 16  pinnip`

--- rk

Edit: found some error in the number of squares, cf. the corrected matrix in the following posts.
Last edited by Klitzing on Sun Oct 13, 2013 11:08 pm, edited 1 time in total.
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### Re: Uniform Polyteron Sections and Verfs

Now for the final two members of the hin regiment:

Radah - retrodishexadecateron - facets are 16 pens (cyan) and 16 firps (lavender). This one has a nice spiky look to it.

Rinah - retropenteractihexadecateron - facets are 10 hexes (transparent red) and 16 firps (green). This one is a copycat to han and its only difference is its internal structure, so I rendered the hexes as transparent to get a glimpse inside.

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### Re: Uniform Polyteron Sections and Verfs

Klitzing wrote:
Polyhedron Dude wrote:... Hit - hexadecateron. This is the noble member of the hin regiment. Its facets are 16 pinnips. I bet this one will be a hit!

The hin regiment has 2 more members to be revealed (even though they are on my polyteron website ) - rinah and radah - coming soon!

Incidence matrix ...

Outch, got the number of squares wrong. (And therby some few derived numbers too.) One should don not make 2 steps by one...

Okay, here is the matrix of the verf:
Code: Select all
`Vertex Pattern:    a                    g    b    c             h    e    d               i    j    f                      6 * |  4 1  2  2 0 | 2  4  6  4 1 | 4 2  a* 4 |  0 0  3  3 3 | 0  6  3  3 3 | 3 3  g----+--------------+--------------+----2 0 | 12 *  *  * * | 1  1  1  0 0 | 2 1  ab x2 0 |  * 3  *  * * | 0  0  0  4 0 | 4 0  af q1 1 |  * * 12  * * | 0  2  0  1 0 | 2 1  ag x1 1 |  * *  * 12 * | 0  0  2  1 1 | 2 2  ai q0 2 |  * *  *  * 6 | 0  2  0  0 1 | 1 2  gh x----+--------------+--------------+----4 0 |  4 0  0  0 0 | 3  *  *  * * | 2 0  abfd xxxx = verf(oct)2 2 |  1 0  2  0 1 | * 12  *  * * | 1 1  abih xxxx = verf(oct)2 1 |  1 0  0  2 0 | *  * 12  * * | 1 1  abj xqq = verf(trip)2 1 |  0 1  1  1 0 | *  *  * 12 * | 2 0  agf xqq = verf(trip)1 2 |  0 0  0  2 1 | *  *  *  * 6 | 0 2  aij xqq = verf(trip)----+--------------+--------------+----4 2 |  4 2  4  4 1 | 1  2  2  4 0 | 6 *  abdfgj verf(pinnip)3 3 |  3 0  3  6 3 | 0  3  3  0 3 | * 4  abchij verf(pinnip)or10 |  6  3 |  6  9 |  6---+-------+-------+--- 2 | 30  * |  2  1 |  3  x 2 |  * 15 |  0  4 |  4  q---+-------+-------+--- 4 |  4  0 | 15  * |  2  verf(oct) 3 |  1  2 |  * 30 |  2  verf(trip)---+-------+-------+--- 6 |  9  6 |  3  6 | 10  verf(pinnip)`

And thus the (corrected) matrix of hit:
Code: Select all
`16 | 10 |  30 15 | 15 30 | 10---+----+--------+-------+--- 2 | 80 |   6  3 |  6  9 |  6---+----+--------+-------+--- 3 |  3 | 160  * |  2  1 |  3 4 |  4 |   * 60 |  0  4 |  4---+----+--------+-------+--- 6 | 12 |   8  0 | 40  * |  2  (oct) 6 |  9 |   2  3 |  * 80 |  2  (trip)---+----+--------+-------+---10 | 30 |  30 15 |  5 10 | 16  (pinnip)`

--- rk
Klitzing
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### Re: Uniform Polyteron Sections and Verfs

Polyhedron Dude wrote:... Radah - retrodishexadecateron - facets are 16 pens (cyan) and 16 firps (lavender). This one has a nice spiky look to it.

...

Verf matrix:
Code: Select all
`Vertex Pattern:    a                    g    b    c             h    e    d               i    j    f                      6 * |  4 1  2  2 0 | 2  4  4  4 1 0 | 4 2 2 0  a* 4 |  0 0  3  3 3 | 0  3  3  3 3 3 | 3 3 1 1  g----+--------------+----------------+--------2 0 | 12 *  *  * * | 1  1  1  0 0 0 | 1 1 1 0  ab x2 0 |  * 3  *  * * | 0  0  0  4 0 0 | 4 0 0 0  af q1 1 |  * * 12  * * | 0  2  0  1 0 0 | 2 0 1 0  ag x1 1 |  * *  * 12 * | 0  0  2  1 1 0 | 2 2 0 0  ai q0 2 |  * *  *  * 6 | 0  0  0  0 1 2 | 0 2 0 1  gh x----+--------------+----------------+--------3 0 |  3 0  0  0 0 | 4  *  *  * * * | 0 1 1 0  abc xxx = verf(tet)2 1 |  1 0  2  0 0 | * 12  *  * * * | 1 0 1 0  abg xxx = verf(tet)2 1 |  1 0  0  2 0 | *  * 12  * * * | 1 1 0 0  abj xqq = verf(trip)2 1 |  0 1  1  1 0 | *  *  * 12 * * | 2 0 0 0  afg xqq = verf(trip)1 2 |  0 0  0  2 1 | *  *  *  * 6 * | 0 2 0 0  aij xqq = verf(trip)0 3 |  0 0  0  0 3 | *  *  *  * * 4 | 0 1 0 1  ghi xxx = verf(tet)----+--------------+----------------+--------4 2 |  2 2  4  4 0 | 0  2  2  4 0 0 | 6 * * *  abdfgj verf(firp)3 3 |  3 0  0  6 3 | 1  0  3  0 3 1 | * 4 * *  abchij verf(firp)3 1 |  3 0  3  0 0 | 1  3  0  0 0 0 | * * 4 *  abcg tet = verf(pen)0 4 |  0 0  0  0 6 | 0  0  0  0 0 4 | * * * 1  ghij tet = verf(pen)or10 |  6  3 |  6  9 |  6 2---+-------+-------+----- 2 | 30  * |  2  1 |  2 1  x 2 |  * 15 |  0  4 |  4 0  q---+-------+-------+----- 3 |  3  0 | 20  * |  1 1  xxx = verf(tet) 3 |  1  2 |  * 30 |  2 0  xqq = verf(trip)---+-------+-------+----- 6 |  6  6 |  2  6 | 10 *  verf(firp) 4 |  6  0 |  4  0 |  * 5  tet = verf(pen)`

Code: Select all
`16 | 10 |  30 15 | 20 30 | 10  5---+----+--------+-------+------ 2 | 80 |   6  3 |  6  9 |  6  2---+----+--------+-------+------ 3 |  3 | 160  * |  2  1 |  2  1 4 |  4 |   * 60 |  0  4 |  4  0---+----+--------+-------+------ 4 |  6 |   4  0 | 80  * |  1  1  tet 6 |  9 |   2  3 |  * 80 |  2  0  trip---+----+--------+-------+------10 | 30 |  20 15 |  5 10 | 16  *  firp 5 | 10 |  10  0 |  5  0 |  * 16  pen`

(Stay tuned for rinah ...   )

--- rk
Klitzing
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### Re: Uniform Polyteron Sections and Verfs

Polyhedron Dude wrote:Now for the final [two] member[s] of the hin regiment:

... Rinah - retropenteractihexadecateron - facets are 10 hexes (transparent red) and 16 firps (green). This one is a copycat to han and its only difference is its internal structure, so I rendered the hexes as transparent to get a glimpse inside.

Here is the verf:
Code: Select all
`Vertex Pattern:    a                    g    b    c             h    e    d               i    j    f                      6 * |  4 1  2  2 0 | 2 2  4  4  4 1 1 0 | 1 4 2 2  a* 4 |  0 0  3  3 3 | 0 0  3  3  3 3 3 3 | 0 3 3 3  g----+--------------+--------------------+--------2 0 | 12 *  *  * * | 1 1  1  1  0 0 0 0 | 1 1 1 1  ab x2 0 |  * 3  *  * * | 0 0  0  0  4 0 0 0 | 0 4 0 0  af q1 1 |  * * 12  * * | 0 0  2  0  1 1 0 0 | 0 2 0 2  ag x1 1 |  * *  * 12 * | 0 0  0  2  1 0 1 0 | 0 2 2 0  ai q0 2 |  * *  *  * 6 | 0 0  0  0  0 1 1 2 | 0 0 2 2  gh x----+--------------+--------------------+--------3 0 |  3 0  0  0 0 | 4 *  *  *  * * * * | 1 0 1 0  abc xxx = verf(tet) hf3 0 |  3 0  0  0 0 | * 4  *  *  * * * * | 1 0 0 1  abe xxx = verf(tet) hh2 1 |  1 0  2  0 0 | * * 12  *  * * * * | 0 1 0 1  abg xxx = verf(tet) hf2 1 |  1 0  0  2 0 | * *  * 12  * * * * | 0 1 1 0  abj xqq = verf(trip)2 1 |  0 1  1  1 0 | * *  *  * 12 * * * | 0 2 0 0  afg xqq = verf(trip)1 2 |  0 0  2  0 1 | * *  *  *  * 6 * * | 0 0 0 2  agh xxx = verf(tet) hh1 2 |  0 0  0  2 1 | * *  *  *  * * 6 * | 0 0 2 0  aij xqq = verf(trip)0 3 |  0 0  0  0 3 | * *  *  *  * * * 4 | 0 0 1 1  ghi xxx = verf(tet) hf----+--------------+--------------------+--------6 0 | 12 0  0  0 0 | 4 4  0  0  0 0 0 0 | 1 * * *  abcdef oct = verf(hex)4 2 |  2 2  4  4 0 | 0 0  2  2  4 0 0 0 | * 6 * *  abdfgj verf(firp)3 3 |  3 0  0  6 3 | 1 0  0  3  0 0 3 1 | * * 4 *  abchij verf(firp)3 3 |  3 0  6  0 3 | 0 1  3  0  0 3 0 1 | * * * 4  abeghi oct = verf(hex)or10 |  6  3 |  3  6  9 | 3  6---+-------+----------+----- 2 | 30  * |  1  2  1 | 2  2  x 2 |  * 15 |  0  0  4 | 0  4  q---+-------+----------+----- 3 |  3  0 | 10  *  * | 2  0  xxx = verf(tet) hh 3 |  3  0 |  * 20  * | 1  1  xxx = verf(tet) hf 3 |  1  2 |  *  * 30 | 0  2  xqq = verf(trip)---+-------+----------+----- 6 | 12  0 |  4  4  0 | 5  *  oct = verf(hex) 6 |  6  6 |  0  2  6 | * 10  verf(firp)`

And here then comes rinah itself:
Code: Select all
`16 | 10 |  30 15 | 10 20 30 |  5 10---+----+--------+----------+------ 2 | 80 |   6  3 |  3  6  9 |  3  6---+----+--------+----------+------ 3 |  3 | 160  * |  1  2  1 |  2  2 4 |  4 |   * 60 |  0  0  4 |  0  4---+----+--------+----------+------ 4 |  6 |   4  0 | 40  *  * |  2  0  tet 4 |  6 |   4  0 |  * 80  * |  1  1  tet 6 |  9 |   2  3 |  *  * 80 |  0  2  trip---+----+--------+----------+------ 8 | 24 |  32  0 |  8  8  0 | 10  *  hex10 | 30 |  20 15 |  0  5 10 |  * 16  firp`

Hedron Dude pointed out, that
This one is a copycat to han and its only difference is its internal structure
(copycat = same external surtope). This is quite evident from his field of sectionings, but else would be rather surprising. Han has for facets octs and trip, while rinah has for facets octs and firps. Sure, firp is a faceting of trip, but the prominent external squares would be missing. That is, han would hide those squares from external vision.

Sure the squares themselves are one dimension lower than belonging to the surtope, but this shows, that from the trips would be just the polar parts visible to a 4D spectator. The remainder would be burried within.

This kind of reminds me to the hollow pentagrams (center pentagon is missing, aka mod-density pentagram), which then are used to build a stellar dodecahedron. This building then has the same surhedron as the usual sissid. Only the internal structure differs. ...

--- rk
Klitzing
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### Re: Uniform Polyteron Sections and Verfs

Btw., dealing with all that hin folks, I just realised that not only the hin-verf, i.e.
rap can be described as "tet || oct", but that too
firp can be described as "tet || pseudo thah", and also
pinnip can be described as "pseudo tet || oct".

(Well, the latter case would be a bit cheeted, because the base octahedron uses additionally its 3 diametral squares ...)

--- rk
Klitzing
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### Re: Uniform Polyteron Sections and Verfs

Our next two polytera are from the nit regiment, notice how different they look:

Irl - invertiretrolepidoteron - its facets are 16 pinnips (red), 16 rawvtips (purple), and 16 sirdops (cyan).

http://pages.suddenlink.net/hedrondude/irl.png -- the url to irl

Raccoth - retrocelli32-16teron - its facets are 32 firps (purple and green), 16 garpops (yellow), and 40 ohopes (red).

Whale Kumtu Dedge Ungol.
Polyhedron Dude
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### Re: Uniform Polyteron Sections and Verfs

Polyhedron Dude wrote:Our next [two] polytera are from the nit regiment, notice how different they look:

Irl - invertiretrolepidoteron - its facets are 16 pinnips (red), 16 rawvtips (purple), and 16 sirdops (cyan).

http://pages.suddenlink.net/hedrondude/irl.png -- the url to irl ...

That one looks cool!

Here the incidence matrix of its verf:
Code: Select all
`Vertex Pattern:a     b                  g     h  c                        i                                    d     e                  j     k  f                        l    12 |  2 1  2  2 | 2  3  3 2 2 | 1 3 3---+------------+-------------+------ 2 | 12 *  *  * | 1  1  1 0 0 | 1 1 1  ab x 2 |  * 6  *  * | 2  0  0 2 0 | 1 3 0  ag x 2 |  * * 12  * | 0  2  0 0 1 | 1 0 2  ah q 2 |  * *  * 12 | 0  0  2 1 1 | 0 2 2  ak h---+------------+-------------+------ 4 |  2 2  0  0 | 6  *  * * * | 1 1 0  abgh xxxx = verf(oct) 3 |  1 0  2  0 | * 12  * * * | 1 0 1  abi xqq = verf(trip) 3 |  1 0  0  2 | *  * 12 * * | 0 1 1  abl xhh = verf(tut) 4 |  0 2  0  2 | *  *  * 6 * | 0 2 0  aegk xh(-x)h = verf(oho) 4 |  0 0  2  2 | *  *  * * 6 | 0 0 2  adhk qh(-q)h = verf(cho)---+------------+-------------+------ 6 |  6 3  6  0 | 3  6  0 0 0 | 2 * *  abcghi verf(pinnip) 6 |  2 3  0  4 | 1  0  2 2 0 | * 6 *  abfghl verf(rawvtip) 6 |  2 0  4  4 | 0  2  2 0 2 | * * 6  abdeil verf(sirdop)`

And then that of irl itself:
Code: Select all
`80 |  12 |  12   6  12  12 |  6  12 12  6  6 |  2  6  6---+-----+-----------------+-----------------+--------- 2 | 480 |   2   1   2   2 |  2   3  3  2  2 |  1  3  3---+-----+-----------------+-----------------+--------- 3 |   3 | 320   *   *   * |  1   1  1  0  0 |  1  1  1 3 |   3 |   * 160   *   * |  2   0  0  2  0 |  1  3  0 4 |   4 |   *   * 240   * |  0   2  0  0  1 |  1  0  2 6 |   6 |   *   *   * 160 |  0   0  2  1  1 |  0  2  2---+-----+-----------------+-----------------+--------- 6 |  12 |   4   4   0   0 | 80   *  *  *  * |  1  1  0  oct 6 |   9 |   2   0   3   0 |  * 160  *  *  * |  1  0  1  trip12 |  18 |   4   0   0   4 |  *   * 80  *  * |  0  1  1  tut12 |  24 |   0   8   0   4 |  *   *  * 40  * |  0  2  0  oho12 |  24 |   0   0   6   4 |  *   *  *  * 40 |  0  0  2  cho---+-----+-----------------+-----------------+---------10 |  30 |  20  10  15   0 |  5  10  0  0  0 | 16  *  *  pinnip30 |  90 |  20  30   0  20 |  5   0  5  5  0 |  * 16  *  rawvtip30 |  90 |  20   0  30  20 |  0  10  5  0  5 |  *  * 16  sirdop`

--- rk
Klitzing
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### Re: Uniform Polyteron Sections and Verfs

Polyhedron Dude wrote:... Raccoth - retrocelli32-16teron - its facets are 32 firps (purple and green), 16 garpops (yellow), and 40 ohopes (red).

Verf (same vertex pattern as irl):
Code: Select all
`12 |  2  2  2  2 | 1  3  3  6 2 2 | 1 1 3  5---+-------------+----------------+--------- 2 | 12  *  *  * | 1  1  1  0 1 0 | 1 1 0  2  ab x 2 |  * 12  *  * | 0  2  0  2 0 0 | 1 0 1  2  ae q 2 |  *  * 12  * | 0  0  2  2 0 1 | 0 1 2  2  ah q 2 |  *  *  * 12 | 0  0  0  2 1 1 | 0 0 2  2  ak h---+-------------+----------------+--------- 3 |  3  0  0  0 | 4  *  *  * * * | 1 1 0  0  abc xxx = verf(tet) 3 |  1  2  0  0 | * 12  *  * * * | 1 0 0  1  abf xqq = verf(trip) 3 |  1  0  2  0 | *  * 12  * * * | 0 1 0  1  abi xqq = verf(trip) 3 |  0  1  1  1 | *  *  * 24 * * | 0 0 1  1  aeh qqh = verf(hip) 4 |  2  0  0  2 | *  *  *  * 6 * | 0 0 0  2  abjkxh(-x)h = verf(oho) 4 |  0  0  2  2 | *  *  *  * * 6 | 0 0 2  0  adhk qh(-q)h = verf(cho)---+-------------+----------------+--------- 6 |  6  6  0  0 | 2  6  0  0 0 0 | 2 * *  *  abcdef verf(firp) 6 |  6  0  6  0 | 2  0  6  0 0 0 | * 2 *  *  abcghi verf(firp) 6 |  0  2  4  4 | 0  0  0  4 0 2 | * * 6  *  abdeil verf(garpop) 5 |  2  2  2  2 | 2  1  1  2 1 0 | * * * 12  abfjk verf(ohope)`

And thus of raccoth itself:
Code: Select all
`80 |  12 |  12  12  12  12 |  4  12  12  24  6  6 |  2  2  6 12---+-----+-----------------+----------------------+------------ 2 | 480 |   2   2   2   2 |  1   3   3   6  2  2 |  1  1  3  5---+-----+-----------------+----------------------+------------ 3 |   3 | 320   *   *   * |  1   1   1   0  1  0 |  1  1  0  2 4 |   4 |   * 240   *   * |  0   2   0   2  0  0 |  1  0  1  2 4 |   4 |   *   * 240   * |  0   0   2   2  0  1 |  0  1  2  2 6 |   6 |   *   *   * 160 |  0   0   0   2  1  1 |  0  0  2  2---+-----+-----------------+----------------------+------------ 4 |   6 |   4   0   0   0 | 80   *   *   *  *  * |  1  1  0  0  tet 6 |   9 |   2   3   0   0 |  * 160   *   *  *  * |  1  0  0  1  trip 6 |   9 |   2   0   3   0 |  *   * 160   *  *  * |  0  1  0  1  trip12 |  18 |   0   3   3   2 |  *   *   * 160  *  * |  0  0  1  1  hip12 |  24 |   8   0   0   4 |  *   *   *   * 40  * |  0  0  0  2  oho12 |  24 |   0   0   6   4 |  *   *   *   *  * 40 |  0  0  2  0  cho---+-----+-----------------+----------------------+------------10 |  30 |  20  15   0   0 |  5  10   0   0  0  0 | 16  *  *  *  firp10 |  30 |  20   0  15   0 |  5   0  10   0  0  0 |  * 16  *  *  firp30 |  90 |   0  15  30  20 |  0   0   0  10  0  5 |  *  * 16  *  garpop24 |  60 |  16  12  12   8 |  0   4   4   4  2  0 |  *  *  * 40  ohope`

--- rk
Klitzing
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### Re: Uniform Polyteron Sections and Verfs

Klitzing wrote:
Polyhedron Dude wrote:... Today's polyteron is Gancpan and boy does it look "gancsta". This is the great spinocelliprismated penteract. It is one of seven members of the getitdin regiment - o(o'x"x)x = oGx. It is also non-orientable. Its facets are 10 girpdohs (yellow), 10 gnappoths (cyan), 32 tips (red), and 80 tuttips (blue). I just added the girpdo macros to my sectioning program a couple days ago. ...
...

--- rk

Hmmm, missed that you mentioned there getitdin, the regiment colonel, too.
So here comes getitdin's incidence matrix as well. This one being again Wythoffian.
Code: Select all
`x3x3o3o *b4/3x4*c. . . .      .    | 640 |   1   3   3 |   3   3   3   3   3 |   3  3   3   1  3  1 |  1  3  1  1------------------+-----+-------------+---------------------+----------------------+------------x . . .      .    |   2 | 320   *   * |   3   3   0   0   0 |   3  3   3   0  0  0 |  1  3  1  0. x . .      .    |   2 |   * 960   * |   1   0   2   1   0 |   2  1   0   1  2  0 |  1  2  0  1. . . .      x    |   2 |   *   * 960 |   0   1   0   1   2 |   0  1   2   0  2  1 |  0  2  1  1------------------+-----+-------------+---------------------+----------------------+------------x3x . .      .    |   6 |   3   3   0 | 320   *   *   *   * |   2  1   0   0  0  0 |  1  2  0  0x . . .      x    |   4 |   2   0   2 |   * 480   *   *   * |   0  1   2   0  0  0 |  0  2  1  0. x3o .      .    |   3 |   0   3   0 |   *   * 640   *   * |   1  0   0   1  1  0 |  1  1  0  1. x . . *b4/3x    |   8 |   0   4   4 |   *   *   * 240   * |   0  1   0   0  2  0 |  0  2  0  1. . o .      x4*c |   4 |   0   0   4 |   *   *   *   * 480 |   0  0   1   0  1  1 |  0  1  1  1------------------+-----+-------------+---------------------+----------------------+------------x3x3o .      .    |  12 |   6  12   0 |   4   0   4   0   0 | 160  *   *   *  *  * |  1  1  0  0  tutx3x . . *b4/3x    |  48 |  24  24  24 |   8  12   0   6   0 |   * 40   *   *  *  * |  0  2  0  0  quitcox . o .      x4*c |   8 |   4   0   8 |   0   4   0   0   2 |   *  * 240   *  *  * |  0  1  1  0  cube. x3o3o      .    |   4 |   0   6   0 |   0   0   4   0   0 |   *  *   * 160  *  * |  1  0  0  1  tet. x3o . *b4/3x4*c |  24 |   0  24  24 |   0   0   8   6   6 |   *  *   *   * 80  * |  0  1  0  1  gocco. . o3o      x4*c |   8 |   0   0  12 |   0   0   0   0   6 |   *  *   *   *  * 80 |  0  0  1  1  cube------------------+-----+-------------+---------------------+----------------------+------------x3x3o3o      .    |  20 |  10  30   0 |  10   0  20   0   0 |   5  0   0   5  0  0 | 32  *  *  *  tipx3x3o . *b4/3x4*c | 192 |  96 192 192 |  64  96  64  48  48 |  16  8  24   0  8  0 |  * 10  *  *  gichadox . o3o      x4*c |  16 |   8   0  24 |   0  12   0   0  12 |   0  0   6   0  0  2 |  *  * 40  *  tes. x3o3o *b4/3x4*c |  64 |   0  96  96 |   0   0  64  24  48 |   0  0   0  16  8  8 |  *  *  * 10  gittith`

--- rk
Last edited by Klitzing on Wed Oct 16, 2013 9:29 am, edited 1 time in total.
Klitzing
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### Re: Uniform Polyteron Sections and Verfs

Klitzing wrote:
Polyhedron Dude wrote:This one is Kafandoh - spikifacetospinodishexadecateron. It is one of the 37 members of the sirhin regiment and it's non-orientable. Sirhin's symbol is oxo6. It's facets are 16 garpops (cyan) and 16 ripdips (peach). I've rendered only a quadrant of the sections (like I did with gibtadin), the bottom right section is the center section. This one has an interesting structure to it. Below are the sections and verf. ...
...
--- rk

Haha, and here you did mention sirhin as its regiment colonel.
So here comes sirhin's incidence matrix too:
Code: Select all
`x3o3o *b3x3o. . .    . . | 160 |   3   6 |   3   6   6   3 |  1  6  3  2  3 |  2  3  1-------------+-----+---------+-----------------+----------------+---------x . .    . . |   2 | 240   * |   2   2   0   0 |  1  4  1  0  0 |  2  2  0. . .    x . |   2 |   * 480 |   0   1   2   1 |  0  2  1  1  2 |  1  2  1-------------+-----+---------+-----------------+----------------+---------x3o .    . . |   3 |   3   0 | 160   *   *   * |  1  2  0  0  0 |  2  1  0x . .    x . |   4 |   2   2 |   * 240   *   * |  0  2  1  0  0 |  1  2  0. o . *b3x . |   3 |   0   3 |   *   * 320   * |  0  1  0  1  1 |  1  1  1. . .    x3o |   3 |   0   3 |   *   *   * 160 |  0  0  1  0  2 |  0  2  1-------------+-----+---------+-----------------+----------------+---------x3o3o    . . |   4 |   6   0 |   4   0   0   0 | 40  *  *  *  * |  2  0  0  tetx3o . *b3x . |  12 |  12  12 |   4   6   4   0 |  * 80  *  *  * |  1  1  0  cox . .    x3o |   6 |   3   6 |   0   3   0   2 |  *  * 80  *  * |  0  2  0  trip. o3o *b3x . |   4 |   0   6 |   0   0   4   0 |  *  *  * 80  * |  1  0  1  tet. o . *b3x3o |   6 |   0  12 |   0   0   4   4 |  *  *  *  * 80 |  0  1  1  oct-------------+-----+---------+-----------------+----------------+---------x3o3o *b3x . |  32 |  48  48 |  32  24  32   0 |  8  8  0  8  0 | 10  *  *  ritx3o . *b3x3o |  30 |  30  60 |  10  30  20  20 |  0  5 10  0  5 |  * 16  *  srip. o3o *b3x3o |  10 |   0  30 |   0   0  20  10 |  0  0  0  5  5 |  *  * 16  rap`

--- rk
Klitzing
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### Re: Uniform Polyteron Sections and Verfs

Klitzing wrote:
Polyhedron Dude wrote:Now for an utter nightmare! Gipbin - great prismated biprismatopenteract. It is a member of the fawdint regiment, who's symbol is oGo or o(o'x"x)o. I suspect this one is orientable, so I rendered it as such. Its facets are 10 gittiths (cyan), 32 raps (blue), 40 goccopes (red), 80 opes (yellow), 80 tistodips (orange), and 80 tisdips (green). It also has 10 dippanoth shaped pseudofacets. Only the top-left quadrant of the section field has been rendered - therefore the center of the polytope is the bottom-right section. Now who's up for building a model? ...
...
--- rk

And here you did mention fawdint as its regiment colonel:
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  octo3x . . *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  *  *  rapo3x3o . *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
Klitzing
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### Re: Uniform Polyteron Sections and Verfs

Klitzing wrote:...
Polyhedron Dude wrote:This one is up to no good, for it is Bad - biprismatododecateron. It is in the dot regiment, dot being ooxoo. Its facets are 12 firps (cyan and yellow) and 20 triddips (red). Its verf has triddip (triangle duoprism) symmetry. ...
...
--- rk

This mail further did mention dot as its colonel:
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  rapor. . . . .    | 20 |  9 |  18 |  6  9 |  6  verf=triddip-------------+----+----+-----+-------+---. . x . .    |  2 | 90 |   4 |  2  4 |  4-------------+----+----+-----+-------+---. o3x . .  & |  3 |  3 | 120 |  1  2 |  3-------------+----+----+-----+-------+---o3o3x . .  & |  4 |  6 |   4 | 30  * |  2  tet. o3x3o .    |  6 | 12 |   8 |  * 30 |  2  oct-------------+----+----+-----+-------+---o3o3x3o .  & | 10 | 30 |  30 |  5  5 | 12  rap`

--- rk
Klitzing
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### Re: Uniform Polyteron Sections and Verfs

Klitzing wrote:
Polyhedron Dude wrote:Now for a weird looking one - Gloptin = great lapidoprismatotruncated penteract. It belongs to the 7 member quiptin regiment, quiptin's symbol is oxox"x. Gloptin's facets are 10 girpiths (purple), 32 sirdops (cyan), 10 gaqrits (red), and 80 tistodips (yellow). Its verf is a butterfly wedge pyramid. ...
...
--- rk

Here is the matrix of gloptin's mentioned colonel quiptin:
Code: Select all
`o3x3o3x4/3x. . . .   . | 960 |    4   2   1 |   2   2   4   4   1   2 |   1   2   2   2   2   4  1 |  1  1  2  2------------+-----+--------------+-------------------------+----------------------------+------------. x . .   . |   2 | 1920   *   * |   1   1   1   1   0   0 |   1   1   1   1   1   1  0 |  1  1  1  1. . . x   . |   2 |    * 960   * |   0   0   2   0   1   1 |   0   1   0   2   0   2  1 |  1  0  1  2. . . .   x |   2 |    *   * 480 |   0   0   0   4   0   2 |   0   0   2   0   2   4  1 |  0  1  2  2------------+-----+--------------+-------------------------+----------------------------+------------o3x . .   . |   3 |    3   0   0 | 640   *   *   *   *   * |   1   1   1   0   0   0  0 |  1  1  1  0. x3o .   . |   3 |    3   0   0 |   * 640   *   *   *   * |   1   0   0   1   1   0  0 |  1  1  0  1. x . x   . |   4 |    2   2   0 |   *   * 960   *   *   * |   0   1   0   1   0   1  0 |  1  0  1  1. x . .   x |   4 |    2   0   2 |   *   *   * 960   *   * |   0   0   1   0   1   1  0 |  0  1  1  1. . o3x   . |   3 |    0   3   0 |   *   *   *   * 320   * |   0   0   0   2   0   0  1 |  1  0  0  2. . . x4/3x |   8 |    0   4   4 |   *   *   *   *   * 240 |   0   0   0   0   0   2  1 |  0  0  1  2------------+-----+--------------+-------------------------+----------------------------+------------o3x3o .   . |   6 |   12   0   0 |   4   4   0   0   0   0 | 160   *   *   *   *   *  * |  1  1  0  0  octo3x . x   . |   6 |    6   3   0 |   2   0   3   0   0   0 |   * 320   *   *   *   *  * |  1  0  1  0  tripo3x . .   x |   6 |    6   0   3 |   2   0   0   3   0   0 |   *   * 320   *   *   *  * |  0  1  1  0  trip. x3o3x   . |  12 |   12  12   0 |   0   4   6   0   4   0 |   *   *   * 160   *   *  * |  1  0  0  1  co. x3o .   x |   6 |    6   0   3 |   0   2   0   3   0   0 |   *   *   *   * 320   *  * |  0  1  0  1  trip. x . x4/3x |  16 |    8   8   8 |   0   0   4   4   2   2 |   *   *   *   *   * 240  * |  0  0  1  1  stop. . o3x4/3x |  24 |    0  24  12 |   0   0   0   0   8   6 |   *   *   *   *   *   * 40 |  0  0  0  2  quith------------+-----+--------------+-------------------------+----------------------------+------------o3x3o3x   . |  30 |   60  30   0 |  20  20  30   0  10   0 |   5  10   0   5   0   0  0 | 32  *  *  *  sripo3x3o .   x |  12 |   24   0   6 |   8   8   0  12   0   0 |   2   0   4   0   4   0  0 |  * 80  *  *  opeo3x . x4/3x |  24 |   24  12  12 |   8   0  12  12   0   3 |   0   4   4   0   0   3  0 |  *  * 80  *  tistodip. x3o3x4/3x | 192 |  192 192  96 |   0  64  96  96  64  48 |   0   0   0  16  32  24  8 |  *  *  * 10  quiproh`

--- rk
Klitzing
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### Re: Uniform Polyteron Sections and Verfs

Today we have three polytera, all belonging to the scad regiment.

Scad - small cellated dodecateron, also called stericated hexateron - symbol is xooox. The scad regiment has 11 members. Its facets are 12 pens (yellow and magenta), 30 tepes(blue and red), and 20 triddips (green).

Dibhid - dodecabiprismatohemidodecateron. Dibhid is pronounced like DIV vid. Its facets are 12 pens (red and green), 20 triddips (yellow), and 6 spids (blue).

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

Whale Kumtu Dedge Ungol.
Polyhedron Dude
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### Re: Uniform Polyteron Sections and Verfs

Polyhedron Dude wrote:Today we have three polytera, all belonging to the scad regiment.

Scad - small cellated dodecateron, also called stericated hexateron - symbol is xooox. The scad regiment has 11 members. Its facets are 12 pens (yellow and magenta), 30 tepes(blue and red), and 20 triddips (green).

Scad is a convex Wythoffian, so rather immediate:
Code: Select all
`x3o3o3o3x. . . . . | 30 |  4  4 |  6 12  6 |  4 12 12  4 | 1  4  6  4 1----------+----+-------+----------+-------------+-------------x . . . . |  2 | 60  * |  3  3  0 |  3  6  3  0 | 1  3  3  1 0. . . . x |  2 |  * 60 |  0  3  3 |  0  3  6  3 | 0  1  3  3 1----------+----+-------+----------+-------------+-------------x3o . . . |  3 |  3  0 | 60  *  * |  2  2  0  0 | 1  2  1  0 0x . . . x |  4 |  2  2 |  * 90  * |  0  2  2  0 | 0  1  2  1 0. . . o3x |  3 |  0  3 |  *  * 60 |  0  0  2  2 | 0  0  1  2 1----------+----+-------+----------+-------------+-------------x3o3o . . |  4 |  6  0 |  4  0  0 | 30  *  *  * | 1  1  0  0 0  tetx3o . . x |  6 |  6  3 |  2  3  0 |  * 60  *  * | 0  1  1  0 0  tripx . . o3x |  6 |  3  6 |  0  3  2 |  *  * 60  * | 0  0  1  1 0  trip. . o3o3x |  4 |  0  6 |  0  0  4 |  *  *  * 30 | 0  0  0  1 1  tet----------+----+-------+----------+-------------+-------------x3o3o3o . |  5 | 10  0 | 10  0  0 |  5  0  0  0 | 6  *  *  * *  penx3o3o . x |  8 | 12  4 |  8  6  0 |  2  4  0  0 | * 15  *  * *  tepex3o . o3x |  9 |  9  9 |  3  9  3 |  0  3  3  0 | *  * 20  * *  triddipx . o3o3x |  8 |  4 12 |  0  6  8 |  0  0  4  2 | *  *  * 15 *  tepe. o3o3o3x |  5 |  0 10 |  0  0 10 |  0  0  0  5 | *  *  *  * 6  penor, using additional symmetry:. . . . .    | 30 |   8 |  12 12 |  8  24 |  2  8  6-------------+----+-----+--------+--------+---------x . . . .  & |  2 | 120 |   3  3 |  3   9 |  1  4  3-------------+----+-----+--------+--------+---------x3o . . .  & |  3 |   3 | 120  * |  2   2 |  1  2  1x . . . x    |  4 |   4 |   * 90 |  0   4 |  0  2  2-------------+----+-----+--------+--------+---------x3o3o . .  & |  4 |   6 |   4  0 | 60   * |  1  1  0  tetx3o . . x  & |  6 |   9 |   2  3 |  * 120 |  0  1  1  trip-------------+----+-----+--------+--------+---------x3o3o3o .  & |  5 |  10 |  10  0 |  5   0 | 12  *  *  penx3o3o . x  & |  8 |  16 |   8  6 |  2   4 |  * 30  *  tepex3o . o3x    |  9 |  18 |   6  9 |  0   6 |  *  * 20  triddip`

--- rk
Klitzing
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### Re: Uniform Polyteron Sections and Verfs

Polyhedron Dude wrote:... Dibhid - dodecabiprismatohemidodecateron. Dibhid is pronounced like DIV vid. Its facets are 12 pens (red and green), 20 triddips (yellow), and 6 spids (blue).

Matrix of vertex figure:
Code: Select all
`Vertex Pattern:    b              e       f                        h      a                        d       c              g    8 |  3  3 | 3  9 | 1 3 3--+-------+------+------2 | 12  * | 2  2 | 1 2 1  ab x2 |  * 12 | 0  4 | 0 2 2  ae q--+-------+------+------3 |  3  0 | 8  * | 1 1 0  abc xxx verf(tet)3 |  1  2 | * 24 | 0 1 1  abf xqq verf(trip)--+-------+------+------4 |  6  0 | 4  0 | 2 * *  abcd tet = verf(pen)6 |  6  6 | 2  6 | * 4 *  abcegh xo3ox&#q = verf(spid)4 |  2  4 | 0  4 | * * 6  abgh xo ox&#q = verf(triddip)`

And thus dibhid itself:
Code: Select all
`30 |   8 |  12 12 |  8  24 |  2 4  6---+-----+--------+--------+-------- 2 | 120 |   3  3 |  3   9 |  1 3  3---+-----+--------+--------+-------- 3 |   3 | 120  * |  2   2 |  1 2  1 4 |   4 |   * 90 |  0   4 |  0 2  2---+-----+--------+--------+-------- 4 |   6 |   4  0 | 60   * |  1 1  0  (tet) 6 |   9 |   2  3 |  * 120 |  0 1  1  (trip)---+-----+--------+--------+-------- 5 |  10 |  10  0 |  5   0 | 12 *  *  (pen)20 |  60 |  40 30 | 10  20 |  * 6  *  (spid) 9 |  18 |   6  9 |  0   6 |  * * 20  (triddip)`

--- rk
Klitzing
Pentonian

Posts: 1621
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

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