Isogonal polytopes

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

Isogonal polytopes

So recently I was thinking about isogonal (vertex-transitive) polychora and came up with one set that I thought may be of interest. I'm not sure of these have been thought of before, but it doesn't look like they have.

To construct these polytopess, start with any pennic or icoic convex uniform (except the double-symmetric cases). Now, take two of these uniform polychora in "dual" positions (for example, if you were doing the srip case, it would be x3o3x3o and o3x3o3x). This would give a compound that, as far as I know, should be uniform in all cases. Now, take the convex hull of this compound. The resul should be vertex transitive. These could be represented with Klitzing's "tegum sum" notation; for instance, the srip case would be xo3ox3xo3ox&#zy, for some unknown edge length y.

This process can be applied to the following uniform polychora: pen, rap, tip, srip, grip, prip, ico, rico, tico, srico, grico, and prico. THe double symmetric cases (deca, spid, gippid, cont, spic, gippic) won't work with the uniform variatns, though it may be possible to construct vertex transitive cases from non-uniform variatns (for example, o3x3y3o + o3y3x3o for some non-unit edge y).

So, my question is: What would these polychora look like geometrically, I haven't really tried to look at their elements much if at all. The pen and ico cases are easy - they're the dual of deca (10 vertices, 30 disphenoid cells) and the dual of cont (48 vertices, 288 disphenoid cells) - but what of the rest? And how would one go aobut naming these objects?
Trionian

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Re: Isogonal polytopes

I have studied these kinds of polychora for months now, and I can give you the observations:
1. They are not uniform; there is no means of resizing such double symmetric polychora to have unit edges.
2. Yes, you can deform those with symmetrically ringed nodes, then make a double symmetry polychoron, if and only if the components do not have the extended symmetry.
3. Personally, I name them with the bi- prefix, followed by the operation, and then the type of the polychoron.

I will try to list all of them:
Bitruncatodecachoron (blend of 2 tips): 40 tets (10 regular, 30 D2d), 20 octs (D3d). Verf is a hexakis triangular cupola.
Bimesotruncatodecachoron (blend of 2 non-uniform decas): 10 tuts, 20 hips (D3d). Verf is a triangular bipyramid with C2v symmetry.
Biambodecachoron (blend of 2 raps): 10 tets, 20 octs (D3d). Verf is a triangular bifrustum.
Birhombatodecachoron (blend of 2 srips): 70 octs (10 regular, 20 D3d, 40 C3v), 30 tets (D2d). Verf is a cube with a triangular prism attached to one face.
Bicantitruncatodecachoron (blend of 2 grips): 10 tuts, 20 hips (D3d), 80 trips (20 D3h, 60 C2v), 30 tets (D2d). Verf is an octahedron with Cs symmetry.
Biruncinatodecachoron (blend of 2 non-uniform spids): 10 tets, 20 trips, 30 cubes (D2d). Verf is a triangular pyramid with its side faces augmented by tetrahedra.
Biruncitruncatodecachoron (blend of 2 prips): not explored yet. Verf is an augmented triangular prism.
Biomnitruncatodecachron (blend of 2 gippids whose toes do not have regular hexagons): 10 toes, 40 hips (20 D3h and 20 D3d), 90 cubes (30 D2d, 60 C2v). Verf is an irregular triangular bipyramid.

Bitruncatotetracontaoctachoron (blend of 2 ticoes): 48 cubes, 144 squaps, 288 tets (D2d). Verf is a hexakis triangular cupola.
Bimesotruncatotetracontaoctachoron (blend of 2 non-uniform decas): 48 tics, 144 ops (D4d). Verf is a triangular bipyramid with C2v symmetry.
Biambotetracontaoctachoron (blend of 2 ricoes): 48 cubes, 144 squaps. Verf is a triangular bifrustum.
Birhombatotetracontaoctachoron (blend of 2 sricoes): 48 coes, 144 squaps, 384 octs (C3v). Verf is a cube with a triangular prism attached to one face.
Bicantitruncatotetracontaoctachoron (blend of 2 gricoes): 48 tics, 144 ops (D4d), 768 trips (192 D3h, 576 C2v), 288 tets (D2d). Verf is an octahedron with Cs symmetry.
Biruncinatotetracontaoctachoron (blend of 2 non-uniform spics): 48 octs, 192 trips, 288 cubes (D2d). Verf is a triangular pyramid with its side faces augmented by tetrahedra.
Biruncitruncatotetracontaoctachoron (blend of 2 pricoes): not explored yet. Verf is an augmented triangular prism.
Biomnitruncatotetracontaoctachoron (blend of 2 gippics whose gircoes do not have regular octagons): 48 gircoes, 144 ops (D4d), 192 hips, 864 cubes (288 D2d, 576 C2v). Verf is an irregular triangular bipyramid.
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Trionian

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Re: Isogonal polytopes

Neat.

I actually discovered alternate ways to derive the rap and rico cases, based on certain Catalan polychora.

To generate the rap case, start with the dual of spid. This thing has 20 cells which are the dual of a trigonal antiprism (it's a kind of rhombohedron). Now, spid can be given as the tegum sum of two pens and two raps (which, per Klitzing's site, all have the same edge length of sqrt(10)/2 for unit lacing edges). Now to generate the hull of the 2 rap compound, take the dual of spid and diminish the 10 vertices which correspond to the pens in the tegum sum. You'll be left with the vertices of the two raps. The resulting object will have as cells 10 tets (from the diminished vertices) + 20 trigonal antiprisms (essentially a stretched oct, formed by removing two opposite points from each original cell).

Now you can derive the rico case in a similar way. Start with the dual of spic, which has 144 square antitegums (dual of a square antiprism, has 8 kite shaped faces) cells. Klitzing doesn't give exact measurements here but the dual of spic can be given as the tegum sum of two icoes (of size a) and two ricoes (of possibly different size b). Now, remove the vertices of the icoes. You'll be left with something that has 48 cubes (from the diminished vertices - remember, spic has octs, so its dual has cube verfs in those places) and 144 (presumably non-uniform) square antiprisms (by removing two points from each cell).

Unfortunately, you can't construct the rest of the polychora in a similar way...

It also occurred to me that you might be able to do something interesting with certain tessics under demitessic symmetry - take a compound of 2 (or even 3) identical demitessics and take the hull. Unfortunately, if I had to guess, the 2-compounds will just be semi-uniform variants of tessic polychora, while the 3-compounds will just be similar variants of icoics. This is similar to how in 3D the analogous 2-tut hull is just a semi-uniform variation of sirco.
Trionian

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Re: Isogonal polytopes

I calculated the edges of the antiprism in the figure above.

The side of the cube or square base is 1, and the edges making the lacing of the square antiprism is 2q-2 = 0.82842.

These were found from the matrix norms of the vectors (0,1,-1,0) and (-1,2,-q,0) under the unreduced Stott matrix for {3,4,3}.
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Re: Isogonal polytopes

rectified decachoron

it is possible to describe it in a tegum sum representation too. And this then makes it possible to provide the full incidence matrix together with the element descriptions too. Obviously it is an isogonal polychoron only, as it uses edges of size x=1 and h=sqrt(3). Here it comes:

Code: Select all
`xo3od3do3ox&#zh   → height = 0                    d = 3 (pseudo)(tegum sum of 2 inverted (x,d)-srips)o.3o.3o.3o.     & | 60 |  2   4 |  1   6  2 |  3  2------------------+----+--------+-----------+------x. .. .. ..     & |  2 | 60   * |  1   2  0 |  2  1  xoo3oo3oo3oo&#h    |  2 |  * 120 |  0   2  1 |  2  1  h------------------+----+--------+-----------+------x.3o. .. ..     & |  3 |  3   0 | 20   *  * |  2  0  x-{3}xo .. .. ..&#h  & |  3 |  1   2 |  * 120  * |  1  1  isot.. od3do ..&#zh   |  6 |  0   6 |  *   * 20 |  2  0  h-{6}------------------+----+--------+-----------+------xo3od3do ..&#zh & | 18 | 12  24 |  4  12  4 | 10  *  ambo-tutxo .. .. ox&#h    |  4 |  2   4 |  0   4  0 |  * 30  disphenoid`

--- rk
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Re: Isogonal polytopes

rectified tetracontoctachoron

it is possible to describe it in a tegum sum representation too. And this then makes it possible to provide the full incidence matrix together with the element descriptions too. Obviously it is an isogonal polychoron only, as it uses edges of size x=1 and k=sqrt[2+sqrt(2)]. Here it comes:

Code: Select all
`xo3oK4Ko3ox&#zk   → height = 0                    K = kk = 2+sqrt(2) = 3.414214 (pseudo)                    k = x(8,2) = sqrt[2+sqrt(2)] = 1.847759(tegum sum of 2 inverted (x,K)-sricos)o.3o.4o.3o.     & | 576 |   2    4 |   1    6   2 |  3   2------------------+-----+----------+--------------+-------x. .. .. ..     & |   2 | 576    * |   1    2   0 |  2   1  xoo3oo4oo3oo&#k    |   2 |   * 1152 |   0    2   1 |  2   1  h------------------+-----+----------+--------------+-------x.3o. .. ..     & |   3 |   3    0 | 192    *   * |  2   0  x-{3}xo .. .. ..&#k  & |   3 |   1    2 |   * 1152   * |  1   1  isot.. oK4Ko ..&#zk   |   8 |   0    8 |   *    * 144 |  2   0  k-{8}------------------+-----+----------+--------------+-------xo3oK4Ko ..&#zk & |  36 |  24   48 |   8   24   6 | 48   *  ambo-ticxo .. .. ox&#k    |   4 |   2    4 |   0    4   0 |  * 288  disphenoid`

--- rk
Last edited by Klitzing on Mon Jul 13, 2020 4:21 pm, edited 1 time in total.
Klitzing
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Re: Isogonal polytopes

rectified small prismated decachoron

it is possible to describe it in a tegum sum representation too. And this then makes it possible to provide the full incidence matrix together with the element descriptions too. Obviously it is an isogonal polychoron only, as it uses edges of size x=1 and q=sqrt(2). Here it comes:

Code: Select all
`uo3ox3xo3ou&#zq   → height = 0                    u = 2 (pseudo)(q-laced tegum sum of 2 inverted (u,x)-srips)o.3o.3o.3o.     & | 60 |   4   4 |  2  2  2   6 |  1  3  2------------------+----+---------+--------------+---------.. .. x. ..     & |  2 | 120   * |  1  1  0   1 |  1  1  1oo3oo3oo3oo&#q    |  2 |   * 120 |  0  0  1   2 |  0  2  1------------------+----+---------+--------------+---------.. o.3x. ..     & |  3 |   3   0 | 40  *  *   * |  1  0  1  x-{3}.. .. x.3o.     & |  3 |   3   0 |  * 40  *   * |  1  1  0  x-{3}uo .. .. ou&#zq   |  4 |   0   4 |  *  * 30   * |  0  2  0  q-{4}.. ox .. ..&#q  & |  3 |   1   2 |  *  *  * 120 |  0  1  1  xqq------------------+----+---------+--------------+---------.. o.3x.3o.     & |  6 |  12   0 |  4  4  0   0 | 10  *  *  octuo3ox .. ou&#zq & |  9 |   6  12 |  0  2  3   6 |  * 20  *  rect-trip.. ox3xo ..&#q    |  6 |   6   6 |  2  0  0   6 |  *  * 20  tall (x,q)-3ap`

--- rk
Klitzing
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Re: Isogonal polytopes

rectified small prismated tetracontoctachoron

it is possible to describe it in a tegum sum representation too. And this then makes it possible to provide the full incidence matrix together with the element descriptions too. Obviously it is an isogonal polychoron only, as it uses edges of size x=1 and q=sqrt(2). Here it comes:

Code: Select all
`uo3ox4xo3ou&#zq   → height = 0                    u = 2 (pseudo)(q-laced tegum sum of 2 inverted (u,x)-sricos)o.3o.4o.3o.     & | 576 |    4    4 |   2   2   2    6 |  1   3   2------------------+-----+-----------+------------------+-----------.. .. x. ..     & |   2 | 1152    * |   1   1   0    1 |  1   1   1oo3oo3oo3oo&#q    |   2 |    * 1152 |   0   0   1    2 |  0   2   1------------------+-----+-----------+------------------+-----------.. o.4x. ..     & |   4 |    4    0 | 288   *   *    * |  1   0   1.. .. x.3o.     & |   3 |    3    0 |   * 384   *    * |  1   1   0uo .. .. ou&#zq   |   4 |    0    4 |   *   * 288    * |  0   2   0.. ox .. ..&#q  & |   3 |    1    2 |   *   *   * 1152 |  0   1   1------------------+-----+-----------+------------------+-----------.. o.4x.3o.     & |  12 |   24    0 |   6   8   0    0 | 48   *   *  couo3ox .. ou&#zq & |   9 |    6   12 |   0   2   3    6 |  * 192   *  rect-trip.. ox4xo ..&#q    |   8 |    8    8 |   2   0   0    8 |  *   * 144  tall (x,q)-4ap`

--- rk
Klitzing
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Re: Isogonal polytopes

triangle based biambodecachoron

If you'd mark the centers of the 20 triangles of a decachoron and would consider the convex hull, you'd get this further isogonal polychoron.

It is possible to describe it in a tegum sum representation too. And this then makes it possible to provide the full incidence matrix together with the element descriptions too. It is an isogonal polychoron only, as it uses edges of size x=1 and y = sqrt(2/5). Here it comes:

Code: Select all
`oo3xo3ox3oo&#zy   → height = 0                    y = sqrt(2/5) = 0.632456(tegum sum of 2 inverted raps)o.3o.3o.3o.    & | 20 |  6  3 |  6  9 |  2  6-----------------+----+-------+-------+------.. x. .. ..    & |  2 | 60  * |  2  1 |  1  2oo3oo3oo3oo&#y   |  2 |  * 30 |  0  4 |  0  4-----------------+----+-------+-------+------.. x.3o. ..    & |  3 |  3  0 | 40  * |  1  1.. xo .. ..&#y & |  3 |  1  2 |  * 60 |  0  2-----------------+----+-------+-------+------.. x.3o.3o.    & |  4 |  6  0 |  4  0 | 10  *  tet.. xo3ox ..&#y   |  6 |  6  6 |  2  6 |  * 20  (x,y)-3ap`

--- rk
Last edited by Klitzing on Mon Jul 13, 2020 4:35 pm, edited 1 time in total.
Klitzing
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Re: Isogonal polytopes

triangle based biambotetracontoctachoron

If you'd mark the centers of the 192 triangles of a tetracontoctachoron and would consider the convex hull, you'd get this further isogonal polychoron.

It is possible to describe it in a tegum sum representation too. And this then makes it possible to provide the full incidence matrix together with the element descriptions too. It is an isogonal polychoron only, as it uses edges of size x=1 and y = 2-sqrt(2). Here it comes:

Code: Select all
`oo3xo4ox3oo&#zy   → height = 0                    y = 2-sqrt(2) = 0.585786(tegum sum of 2 inverted ricos)o.3o.4o.3o.    & | 192 |   6   3 |   6   9 |  2   6-----------------+-----+---------+---------+-------.. x. .. ..    & |   2 | 576   * |   2   1 |  1   2oo3oo4oo3oo&#y   |   2 |   * 288 |   0   4 |  0   4-----------------+-----+---------+---------+-------.. x.4o. ..    & |   4 |   4   0 | 288   * |  1   1.. xo .. ..&#y & |   3 |   1   2 |   * 576 |  0   2-----------------+-----+---------+---------+-------.. x.4o.3o.    & |   8 |  12   0 |   6   0 | 48   *  cube.. xo4ox ..&#y   |   8 |   8   8 |   2   8 |  * 144  (x,y)-4ap`

--- rk
Klitzing
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Re: Isogonal polytopes

duoantiprism

If you'd alternate the vertices of the general xnx xmx (2n,2m)-duoantiprism, you'd get this further isogonal polychoron. That one in fact can be made metrically correct for any semi-uniform variant therefrom.

It even is possible to describe it in a tegum sum representation too. And this then makes it possible to provide the full incidence matrix together with the element descriptions too. It is an isogonal polychoron only, as it uses 3 edge sizes in general. Here it comes:

Code: Select all
`ao-n-oa bo-m-ob&#zc   → height = 0,                        a = A*x(2n),                        b = B*x(2m),                        c = sqrt(A^2+B^2)(c-laced tegum sum of 2 bidual (a,b)-sized (n,m)-duoprisms)o.-n-o. o.-m-o.     | NM  * |  2  2   4  0  0 | 1 1   4   2   4   2 0 0 |  2  2  2  2.o-n-.o .o-m-.o     |  * NM |  0  0   4  2  2 | 0 0   2   4   2   4 1 1 |  2  2  2  2--------------------+-------+-----------------+-------------------------+------------a.   .. ..   ..     |  2  0 | NM  *   *  *  * | 1 0   2   0   0   0 0 0 |  2  1  0  0..   .. b.   ..     |  2  0 |  * NM   *  *  * | 0 1   0   0   2   0 0 0 |  0  0  1  2oo-n-oo oo-m-oo&#c  |  1  1 |  *  * 4NM  *  * | 0 0   1   1   1   1 0 0 |  1  1  1  1..   .a ..   ..     |  0  2 |  *  *   * NM  * | 0 0   0   2   0   0 1 0 |  2  0  1  0..   .. ..   .b     |  0  2 |  *  *   *  * NM | 0 0   0   0   0   2 0 1 |  0  1  0  2--------------------+-------+-----------------+-------------------------+------------a.-n-o. ..   ..     |  N  0 |  N  0   0  0  0 | M *   *   *   *   * * * |  2  0  0  0..   .. b.-m-o.     |  M  0 |  0  M   0  0  0 | * N   *   *   *   * * * |  0  0  0  2ao   .. ..   ..&#c  |  2  1 |  1  0   2  0  0 | * * 2NM   *   *   * * * |  1  1  0  0..   oa ..   ..&#c  |  1  2 |  0  0   2  1  0 | * *   * 2NM   *   * * * |  1  0  1  0..   .. bo.  ..&#c  |  2  1 |  0  1   2  0  0 | * *   *   * 2NM   * * * |  0  0  1  1..   .. ..   ob&#c  |  1  2 |  0  0   2  0  1 | * *   *   *   * 2NM * * |  0  1  0  1.o-n-.a ..   ..     |  0  N |  0  0   0  N  0 | * *   *   *   *   * M * |  2  0  0  0..   .. .o-m-.b     |  0  M |  0  0   0  0  M | * *   *   *   *   * * N |  0  0  0  2--------------------+-------+-----------------+-------------------------+------------ao-n-oa ..   ..&#c  |  N  N |  N  0  2N  N  0 | 1 0   N   N   0   0 1 0 | 2M  *  *  *  axially scaled n-apao   .. ..   ob&#c  |  2  2 |  1  0   4  0  1 | 0 0   2   0   0   2 0 0 |  * NM  *  *  disphenoid..   oa bo   ..&#c  |  2  2 |  0  1   4  1  0 | 0 0   0   2   2   0 0 0 |  *  * NM  *  disphenoid..   .. bo-m-ob&#c  |  M  M |  0  M  2M  0  M | 0 1   0   0   M   M 0 1 |  *  *  * 2N  axially scaled m-ap`

Note that the above matrix is correct only if both N and M are different from 2, else a-2-o and/or b-2-o would become degenerate and contributes to a different dimension. But for those cases similar matrices are available as well, for sure. It occurs that 2 uniform figures are given also for special cases: gudap is the case for n=5, m=5/3 (ie. N=M=5) and the other one would be obtained as the mentioned special case n=m=2 (i.e. N=M=2), which is nothing but hex.

--- rk
Klitzing
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Re: Isogonal polytopes

sniccap

snub-cubic antiprism, alternation of gircope:

Code: Select all
`s2s3s4sdemi( . . . . ) | 48 |  1  1  1  1  2  2 |  1  1  3  3  3  3 | 1  1 1 1  4----------------+----+-------------------+-------------------+------------      s2s . .   |  2 | 24  *  *  *  *  * |  0  0  2  2  0  0 | 1  1 0 0  2      s 2 s .   |  2 |  * 24  *  *  *  * |  0  0  2  0  2  0 | 1  0 1 0  2      s  2  s   |  2 |  *  * 24  *  *  * |  0  0  0  2  2  0 | 0  1 1 0  2      . s 2 s   |  2 |  *  *  * 24  *  * |  0  0  0  2  0  2 | 0  1 0 1  2sefa( . s3s . ) |  2 |  *  *  *  * 48  * |  1  0  1  0  0  1 | 1  0 0 1  1sefa( . . s4s ) |  2 |  *  *  *  *  * 48 |  0  1  0  0  1  1 | 0  0 1 1  1----------------+----+-------------------+-------------------+------------      . s3s .   |  3 |  0  0  0  0  3  0 | 16  *  *  *  *  * | 1  0 0 1  0      . . s4s   |  4 |  0  0  0  0  0  4 |  * 12  *  *  *  * | 0  0 1 1  0sefa( s2s3s . ) |  3 |  1  1  0  0  1  0 |  *  * 48  *  *  * | 1  0 0 0  1sefa( s2s 2 s ) |  3 |  1  0  1  1  0  0 |  *  *  * 48  *  * | 0  1 0 0  1sefa( s 2 s4s ) |  3 |  0  1  1  0  0  1 |  *  *  *  * 48  * | 0  0 1 0  1sefa( . s3s4s ) |  3 |  0  0  0  1  1  1 |  *  *  *  *  * 48 | 0  0 0 1  1----------------+----+-------------------+-------------------+------------      s2s3s .   |  6 |  3  3  0  0  6  0 |  2  0  6  0  0  0 | 8  * * *  *      s2s 2 s   |  4 |  2  0  2  2  0  0 |  0  0  0  4  0  0 | * 12 * *  *      s 2 s4s   |  8 |  0  4  4  0  0  8 |  0  2  0  0  8  0 | *  * 6 *  *      . s3s4s   | 24 |  0  0  0 12 24 24 |  8  6  0  0  0 24 | *  * * 2  *sefa( s2s3s4s ) |  4 |  1  1  1  1  1  1 |  0  0  1  1  1  1 | *  * * * 48`

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Re: Isogonal polytopes

sniddap

snub-dodecahedral antiprism, alternation of griddip:

Code: Select all
`s2s3s5sdemi( . . . . ) | 120 |  1  1  1  1   2   2 |  1  1   3   3   3   3 |  1  1  1 1   4----------------+-----+---------------------+-----------------------+---------------      s2s . .   |   2 | 60  *  *  *   *   * |  0  0   2   2   0   0 |  1  1  0 0   2      s 2 s .   |   2 |  * 60  *  *   *   * |  0  0   2   0   2   0 |  1  0  1 0   2      s  2  s   |   2 |  *  * 60  *   *   * |  0  0   0   2   2   0 |  0  1  1 0   2      . s 2 s   |   2 |  *  *  * 60   *   * |  0  0   0   2   0   2 |  0  1  0 1   2sefa( . s3s . ) |   2 |  *  *  *  * 120   * |  1  0   1   0   0   1 |  1  0  0 1   1sefa( . . s5s ) |   2 |  *  *  *  *   * 120 |  0  1   0   0   1   1 |  0  0  1 1   1----------------+-----+---------------------+-----------------------+---------------      . s3s .   |   3 |  0  0  0  0   3   0 | 40  *   *   *   *   * |  1  0  0 1   0      . . s5s   |   5 |  0  0  0  0   0   5 |  * 24   *   *   *   * |  0  0  1 1   0sefa( s2s3s . ) |   3 |  1  1  0  0   1   0 |  *  * 120   *   *   * |  1  0  0 0   1sefa( s2s 2 s ) |   3 |  1  0  1  1   0   0 |  *  *   * 120   *   * |  0  1  0 0   1sefa( s 2 s5s ) |   3 |  0  1  1  0   0   1 |  *  *   *   * 120   * |  0  0  1 0   1sefa( . s3s5s ) |   3 |  0  0  0  1   1   1 |  *  *   *   *   * 120 |  0  0  0 1   1----------------+-----+---------------------+-----------------------+---------------      s2s3s .   |   6 |  3  3  0  0   6   0 |  2  0   6   0   0   0 | 20  *  * *   *      s2s 2 s   |   4 |  2  0  2  2   0   0 |  0  0   0   4   0   0 |  * 30  * *   *      s 2 s5s   |  10 |  0  5  5  0   0  10 |  0  2   0   0  10   0 |  *  * 12 *   *      . s3s5s   |  60 |  0  0  0 30  60  60 | 20 12   0   0   0  60 |  *  *  * 2   *sefa( s2s3s5s ) |   4 |  1  1  1  1   1   1 |  0  0   1   1   1   1 |  *  *  * * 120`

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Re: Isogonal polytopes

bhidtex

bi-hecatonicosidiminished truncated hexacosichoron - Similar to bidex, the diminishing of ex at the vertices of 2 inscribed f-icoes, this isogonal polychoron is obtained as a diminishing of tex at the vertices of 2 inscribed sqrt[6+sqrt(5)]-exes. In contrast to the former will the latter neither be noble, in fact it uses 3 types of cells, nor scaliform, in fact it uses 2 edge sizes.

A general vairiant might be used so, having 3 edge sizes a, b, and c = a+b. Still, that size class b further divides into 3 classes by incidence, cf. below. The specific variant here, relating to a diminishing of the uniform tex variant, only uses 2 edge sizes so, in fact a = b = x and hence c = u. – A different variant however would be obtained when using a = 0. That one then becomes spidrox.

Code: Select all
`1200 |   1    2   1   1   1 |    3   2   2   1   3 |   5   1   1-----+----------------------+----------------------+------------   2 | 600    *   *   *   * |    0   2   0   0   3 |   5   0   0  a,  eg. x   2 |   * 1200   *   *   * |    1   0   1   1   1 |   2   1   1  b1, eg. x   2 |   *    * 600   *   * |    2   0   0   0   1 |   2   1   0  b2, eg. x   2 |   *    *   * 600   * |    2   1   0   0   0 |   2   1   0  b3, eg. x   2 |   *    *   *   * 600 |    0   1   2   0   0 |   2   0   1  c,  eg. u-----+----------------------+----------------------+------------   3 |   0    1   1   1   0 | 1200   *   *   *   * |   1   1   0  b3o = b1,b2,b3   4 |   2    0   0   1   1 |    * 600   *   *   * |   2   0   0  bc&#a = a,b3,a,c   4 |   0    2   0   0   2 |    *   * 600   *   * |   1   0   1  b2c = b1,c,b1,c   5 |   0    5   0   0   0 |    *   *   * 240   * |   0   1   1  b5o = b1,b1,b1,b1,b1   6 |   3    2   1   0   0 |    *   *   *   * 600 |   2   0   0  a3b = a,b1,a,b1,a,b2-----+----------------------+----------------------+------------  10 |   5    4   2   2   2 |    2   2   1   0   2 | 600   *   *  shaved tut  10 |   0   10   5   5   0 |   10   0   0   2   0 |   * 120   *  pap  10 |   0   10   0   0   5 |    0   0   5   2   0 |   *   * 120  pip`

The shaved tut here is a shape where a single hexagon-hexagon-edge is diminished completely - down to the neighbouring vertex layer.

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Re: Isogonal polytopes

spidrico

Rico can be obtained as hull of kitapna, which is the compound of 4 icoes. Spidrico is obtained therefrom by chopping off 24 of its vertex pyramids in a swirl-symmetric distribution, that is, it is the hull of just 3 of those icoes. All cubes thereby get reduced to qo3oq&#x, any co gets chirally tridiminished at every alternate vertex of its medial hexagon. Additionally the former vertex figures (x q3o) come in. (Thus it is an isogonal polychoron only.)

(In fact, it is the icoic version of spidrox, which then even happens to become scaliform.)

Code: Select all
`72 |  2  2   4 |  1   6  4  2 |  3  2  2---+-----------+--------------+--------- 2 | 72  *   * |  1   2  0  0 |  2  1  0  x 2 |  * 72   * |  0   2  2  0 |  2  1  1  x 2 |  *  * 144 |  0   1  1  1 |  1  1  1  q---+-----------+--------------+--------- 3 |  3  0   0 | 24   *  *  * |  2  0  0  xxx 3 |  1  1   1 |  * 144  *  * |  1  1  0  xxq 4 |  0  2   2 |  *   * 72  * |  1  0  1  xqxq 3 |  0  0   3 |  *   *  * 48 |  0  1  1  qqq---+-----------+--------------+--------- 9 |  6  6   6 |  2   6  3  0 | 24  *  *  chiral tridim. co 6 |  3  3   6 |  0   6  0  2 |  * 24  *  qo3oq&#x 6 |  0  3   6 |  0   0  3  2 |  *  * 24  x q3o`

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Re: Isogonal polytopes

truncated decachoron

Truncation of the decachoron would result in 3 different edge sizes in the outcome isohedral polychoron. That one here is scaled such so that the shorter specified one becomes unity. Then the larger specified edge will have size h=sqrt(3). The third one would be the arbitrary expansion size y (wrt. the rectified extremum, i.e. corresponding to the arbitrary truncation depth). In fact, for y=0 this results again in redeca, while y → ∞ results again in the pre-image deca (rescaled back down accordingly).

Code: Select all
`xo3yb3by3ox&#zh   → height = 0                    y > 0 (depending on truncation depth)                    b = y+3(h-laced tegum sum of 2 inverted (x,y,b)-grips)o.3o.3o.3o.     & | 120 |  1  1   2 |  1   3  2 |  3  1 ------------------+-----+-----------+-----------+------x. .. .. ..     & |   2 | 60  *   * |  1   2  0 |  2  1  x.. y. .. ..     & |   2 |  * 60   * |  1   0  2 |  3  0  yoo3oo3oo3oo&#h    |   2 |  *  * 120 |  0   2  1 |  2  1  h------------------+-----+-----------+-----------+------x.3y. .. ..     & |   6 |  3  3   0 | 20   *  * |  2  0  (x,y)-{6}xo .. .. ..&#h  & |   3 |  1  0   2 |  * 120  * |  1  1  xhh.. yb3by ..&#zh   |  12 |  0  6   6 |  *   * 20 |  2  0  (y,h)-{12}------------------+-----+-----------+-----------+------xo3yb3by ..&#zh & |  36 | 12 18  24 |  4  12  4 | 10  *  trunc-tutxo .. .. ox&#h    |   4 |  2  0   4 |  0   4  0 |  * 30  disphenoid`

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Re: Isogonal polytopes

truncated tetracontoctachoron

Truncation would result in 3 different edge sizes in the outcome isohedral polychoron. That one here is scaled such so that the shorter specified one becomes unity. Then the larger specified edge will have size k=sqrt[2+sqrt(2)]. The third one would be the arbitrary expansion size y (wrt. the rectified extremum, i.e. corresponding to the arbitrary truncation depth). In fact, for y=0 this results again in recont, while y → ∞ results again in the pre-image cont (rescaled back down accordingly).

Code: Select all
`xo3yb4by3ox&#zk   → height = 0                    k = x(8,2) = sqrt[2+sqrt(2)] = 1.847759                    y > 0 (depending on truncation depth)                    b = y+2+sqrt(2) (pseudo)(k-laced tegum sum of 2 inverted (x,y,b)-gricoes)o.3o.4o.3o.     & | 1152 |   1   1    2 |   1    3   2 |  3   1------------------+------+--------------+--------------+-------x. .. .. ..     & |    2 | 576   *    * |   1    2   0 |  2   1  x.. y. .. ..     & |    2 |   * 576    * |   1    0   2 |  3   0  yoo3oo4oo3oo&#k    |    2 |   *   * 1152 |   0    2   1 |  2   1  k------------------+------+--------------+--------------+-------x.3y. .. ..     & |    6 |   3   3    0 | 192    *   * |  2   0  (x,y)-{6}xo .. .. ..&#k  & |    3 |   1   0    2 |   * 1152   * |  1   1  xkk.. yb4by ..&#zk   |   16 |   0   8    8 |   *    * 144 |  2   0  (y,k)-{16}------------------+------+--------------+--------------+-------xo3yb4by ..&#zk & |   72 |  24  36   48 |   8   24   6 | 48   *  trunc-ticxo .. .. ox&#k    |    4 |   2   0    4 |   0    4   0 |  * 288  disphenoid`

It shall be pointed out here that the former isogonal polychoron features semi-regular dodecagons, while this one even shows up semiregular hexadecagons! Both are quite uncommon within spherical geometry outside of (multi)prismatic cases.

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Re: Isogonal polytopes

truncated small prismated decachoron

Truncation would result in 3 different edge sizes in the outcome isohedral polychoron. That one here is scaled such so that the shorter specified one becomes unity. Then the larger specified edge will have size q=sqrt(2). The third one would be the arbitrary expansion size y (wrt. the rectified extremum, i.e. corresponding to the arbitrary truncation depth). In fact, for y=0 this results again in respid, while y → ∞ results again in the pre-image spid (rescaled back down accordingly).

Code: Select all
`by3ox3xo3yb&#zq   → height = 0                    y > 0 (depending on truncation depth)                    b = y+2 (pseudo)(q-laced tegum sum of 2 inverted (b,x,y)-prips)o.3o.3o.3o.     & | 120 |   2  1   2 |  1  2  2   3 |  1  3  1------------------+-----+------------+--------------+---------.. .. x. ..     & |   2 | 120  *   * |  1  1  0   1 |  1  1  1  x.. .. .. y.     & |   2 |   * 60   * |  0  2  2   0 |  1  3  0  yoo3oo3oo3oo&#q    |   2 |   *  * 120 |  0  0  1   2 |  0  2  1  q------------------+-----+------------+--------------+---------.. o.3x. ..     & |   3 |   3  0   0 | 40  *  *   * |  1  0  1  x-{3}.. .. x.3y.     & |   6 |   3  3   0 |  * 40  *   * |  1  1  0  (x,y)-{6}by .. .. yb&#zq   |   8 |   0  4   4 |  *  * 30   * |  0  2  0  (y,q)-{8}.. ox .. ..&#q  & |   3 |   1  0   2 |  *  *  * 120 |  0  1  1  xqq------------------+-----+------------+--------------+---------.. o.3x.3y.     & |  12 |  12  6   0 |  4  4  0   0 | 10  *  *  (x,y)-tutby3ox .. yb&#zq & |  18 |   6  9  12 |  0  2  3   6 |  * 20  *  trunc-trip.. ox3xo ..&#q    |   6 |   6  0   6 |  2  0  0   6 |  *  * 20  tall (x,q)-3ap`

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Re: Isogonal polytopes

truncated small prismated tetracontoctachoron

Truncation would result in 3 different edge sizes in the outcome isohedral polychoron. That one here is scaled such so that the shorter specified one becomes unity. Then the larger specified edge will have size q=sqrt(2). The third one would be the arbitrary expansion size y (wrt. the rectified extremum, i.e. corresponding to the arbitrary truncation depth). In fact, for y=0 this results again in respic, while y → ∞ results again in the pre-image spic (rescaled back down accordingly).

Code: Select all
`by3ox4xo3yb&#zq   → height = 0                    y > 0 (depending on truncation depth)                    b = y+2 (pseudo)(q-laced tegum sum of 2 inverted (b,x,y)-pricos)o.3o.4o.3o.     & | 1152 |    2   1    2 |   1   2   2    3 |  1   3   1------------------+------+---------------+------------------+-----------.. .. x. ..     & |    2 | 1152   *    * |   1   1   0    1 |  1   1   1  x.. .. .. y.     & |    2 |    * 576    * |   0   2   2    0 |  1   3   0  yoo3oo4oo3oo&#q    |    2 |    *   * 1152 |   0   0   1    2 |  0   2   1  q------------------+------+---------------+------------------+-----------.. o.4x. ..     & |    4 |    4   0    0 | 288   *   *    * |  1   0   1  x-{3}.. .. x.3y.     & |    6 |    3   3    0 |   * 384   *    * |  1   1   0  (x,y)-{6}by .. .. yb&#zq   |    8 |    0   4    4 |   *   * 288    * |  0   2   0  (y,q)-{8}.. ox .. ..&#q  & |    3 |    1   0    2 |   *   *   * 1152 |  0   1   1  xqq------------------+------+---------------+------------------+-----------.. o.4x.3y.     & |   24 |   24  12    0 |   6   8   0    0 | 48   *   *  (x,y)-toeby3ox .. yb&#zq & |   18 |    6   9   12 |   0   2   3    6 |  * 192   *  trunc-trip.. ox4xo ..&#q    |    8 |    8   0    8 |   2   0   0    8 |  *   * 144  (x,q)-squap`

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Re: Isogonal polytopes

hull of truncated stellated decachoron

Code: Select all
`xo3xo3ox3ox&#zc - height = 0                  c = sqrt(7/5)o.3o.3o.3o.     & | 40 |  1  3   6 |  3   9   9 |  1  6  3  3------------------+----+-----------+------------+------------x. .. .. ..     & |  2 | 20  *   * |  0   6   0 |  3  3  0  0  x.. x. .. ..     & |  2 |  * 60   * |  2   0   2 |  1  1  0  2  yoo3oo3oo3oo&#c    |  2 |  *  * 120 |  0   2   2 |  0  2  1  1  c------------------+----+-----------+------------+------------.. x.3o. ..     & |  3 |  0  3   0 | 40   *   * |  1  0  0  1  x-{3}xo .. .. ..&#c  & |  3 |  1  0   2 |  * 120   * |  0  1  1  0  xcc.. xo .. ..&#c  & |  3 |  0  1   2 |  *   * 120 |  0  1  0  1  xcc------------------+----+-----------+------------+------------.. x.3o.3o.     & |  4 |  0  6   0 |  4   0   0 | 10  *  *  *  x-tetxo .. ox ..&#c  & |  4 |  1  1   4 |  0   2   2 |  * 60  *  *  tall (x,c)-2apxo .. .. ox&#c    |  4 |  2  0   4 |  0   4   0 |  *  * 30  *  tall (x,c)-2ap.. xo3ox ..&#c  & |  6 |  0  6   6 |  2   0   6 |  *  *  * 20  tall (x,c)-3ap`

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