Uniform Polychoron #1850

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

Uniform Polychoron #1850

Postby _Geometer » Sun Sep 27, 2020 7:58 pm

Recently, I discovered two new uniform polychora, which, to my knowledge, are the first since the Ondips in 2006.

It started back in August last year, after I found two scaliform polychora in the Gudap regiment: each has 50 tetrahedra and 50 square pyramids, with the first having 10 pentagrammic prisms and the other 10 pentagonal prisms - I respectively called these the Toroidal Cycloantiprism and Great Cycloantiprism, with the Cyclo- part referring to their each having a “ring” of prisms among their cells. I then remembered that 12 Gudap could form a compound in Sishi’s regiment, and realized that these two scaliform antiprisms - let’s respectively shorten their names to Tocap and Gocap so I don’t have to continually refer to them as “the scaliform antiprisms” - could also form compounds; these would be the first Sishi-regiment polychora to contain facetings of octahedra - in this case, square pyramids - and not be Dox or either of the compounds of 25 Ihi. This discovery about the square pyramids in Sishi led me to find at least ten swirlprisms with those cells, plus two that didn’t have square pyramids and were formed by replacing the antiprisms in Ditsop and Tipsop with pentagrammic and pentagonal pyramids. However, I would soon forget about my discoveries. . .

Slow-forward to a few weeks ago, when I had finally gotten the message that I could save the vertex-figures of polychora in Stella4D: I decided to go through the various folders of polychora, looking for things I had saved there, and came across the scaliform swirlprisms. This time, I would create swirlprisms using more sophisticated techniques than just throwing together a bunch of facets and hoping they would create a closed figure. One of my methods was blending Tocap and Gocap with the uniform Sishi-swirlprisms wherever I found pentagrammic or pentagonal prisms - I found several scaliform through this, including a strange fissary.
This fissary swirlprism was strange both in that it contained tetrahemihexahedra and tetrahedra, the latter of which came in two groups, one of 240 and another of 360. That led me back into faceting mode, this time to try creating a scaliform with one group of tetrahedra and thahs - it worked; it had those cells, as well as two groups of triangular prisms, one of 120 and another of 360. I saved the polychoron to the scaliform swirlprism folder before realizing that it, in fact, was not scaliform; all of its cells were uniform polyhedra. Soon afterwards, I created another polychoron by reorienting the thahs in the first, as well as a weird compound by replacing the thahs with cuboctahedra this was a weird compound in that it was uniform, but none of its components were.

Altogether, with my revised search methods, I found over 40 swirlprisms, two of which were noble and two of which were ditetrahedral - I actually have the list if you want to see it.
Swirlprisms (5).pdf
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Last edited by _Geometer on Mon Oct 05, 2020 5:00 pm, edited 7 times in total.
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Re: Uniform Polychoron #1850

Postby _Geometer » Tue Sep 29, 2020 9:40 pm

On a subject seemingly unrelated to swirlprisms, I was experimenting with blending the compound of 50 Prarsi with various Dircospids - I created a few weird scaliforms; has anyone else done this? Anyway, I was going to start trying to create chiral Dircospids by blending various Gidipthi troops with the 25-Prarsi compound, and started out with 5 Quipdohis. What resulted was a compound of what looked like Prarsis and Quipdohis, only to notice that the "Quipdohis" looked sort of off. So, I isolated one of them, only to discover that it, in fact, was some sort of swirlprism. Unfortunately, I couldn't get it to generate from its verf; the only way it can be created is by blending 5 Quipdohis with the 25-Prarsi. Of course, this thing has its conjugate in the Padohi regiment, created by blending Padohi with the 5-Prarsi.

I ended up finding 24 swirlprisms, eight of which were fissary; I'll update my list tomorrow to show those discoveries.
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Re: Uniform Polychoron #1850

Postby Mecejide » Wed Sep 30, 2020 3:38 am

We’ve actually been recently discussing the blends of prissi and prarsi with members of categories 21, 22, 28, and 29 on the Discord server. You can actually get the sidpapixhi-5 prissi blend to work in Stella, among others.
Edit: the blends aren’t swirlprisms. They have what Bowers calls ixitic symmetry.
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Re: Uniform Polychoron #1850

Postby Mecejide » Wed Sep 30, 2020 7:18 am

What symmetry do the verfs of the two uniforms have?
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Re: Uniform Polychoron #1850

Postby _Geometer » Wed Sep 30, 2020 12:35 pm

Mecejide wrote:What symmetry do the verfs of the two uniforms have?


Chiral 3-fold dihedral - if I can figure out how embed files from Stella4d, I will upload all of the verfs I have found.
Last edited by _Geometer on Wed Sep 30, 2020 9:03 pm, edited 2 times in total.
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Re: Uniform Polychoron #1850

Postby _Geometer » Wed Sep 30, 2020 12:40 pm

Mecejide wrote:We’ve actually been recently discussing the blends of prissi and prarsi with members of categories 21, 22, 28, and 29 on the Discord server. You can actually get the sidpapixhi-5 prissi blend to work in Stella, among others.
Edit: the blends aren’t swirlprisms. They have what Bowers calls ixitic symmetry.


I got at least 22 blends - 11 Padohis and 11 Gidipthis - to work; four from each regiment were fissary. Interestingly, the Padohi-5-Prissi and its conjugate can be created, but only as compound Idcossids and Dircospids, repsectively.

I guess my mistaking these for swirlprisms came from the blending-of-5-Prissis part, seeing that the 5-Prissi had the same symmetry as the 5-Ico - which I also thought to be a swirlprism. Also, what is “ixitic symmetry” and what are its naming conventions?
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Re: Uniform Polychoron #1850

Postby username5243 » Wed Sep 30, 2020 4:47 pm

Ixitic symmetry is like hyic where each of the 120 dodecahedral elements instead has chiral tet symmetry instead.

we actually have a Discrod server now that is quite active, with lots of new discoveries this year. Here's the link if you want to join:

https://discord.gg/V9FUnH
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Re: Uniform Polychoron #1850

Postby _Geometer » Wed Sep 30, 2020 6:03 pm

username5243 wrote:Ixitic symmetry is like hyic where each of the 120 dodecahedral elements instead has chiral tet symmetry instead.

we actually have a Discrod server now that is quite active, with lots of new discoveries this year. Here's the link if you want to join:

https://discord.gg/V9FUnH

1 - Thanks for that, I had never seen the term "ixitic" before today.
2 - I joined this morning, but haven't had an opportunity to really do anything yet.
3 - When did people start talking about blending Prissis with Padohi? Also, has anyone considered yet the possibilty of a uniform ixitic Padohi or Gidipthi troop, or possibly even new uniform Idcossids or Dircospids?
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Re: Uniform Polychoron #1850

Postby username5243 » Wed Sep 30, 2020 6:55 pm

Mecejide is on the Discord and had suggested the idea before for a while, not sure anyone tried a systematic approach though.

There's another related symmetry called ixoic, which is similar, but instead of 120 chiral tet elements, there's 120 pyritohedral elements. We have one member over there trying to find as many convex isogonal polychora as they can, and there's some with those symmetry groups.
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Re: Uniform Polychoron #1850

Postby Robert Webb » Sun Oct 04, 2020 5:17 am

Great discovery _Geometer! I will add these to a future version of Stella4D.

Models in Stella can have a comment attached, so I like to include a short note to say who discovered new findings. Are you willing to share your real name so I can attribute their discoveries to you?

Also, is there a decision about the numbering? Which one is 1850 and which is 1851?

Looks like Jonathan Bowers has used these as a springboard to find two additional ones too. I guess these will be 1852 & 1853, but again, in which order?

I'm expecting the small ones to come before the great ones, but Bowers is best to answer this. Not sure if he's here, but I'll try to reach out to him too.

A thread was started about the new discoveries on the Stella forum too, and I posted some pictures of cross-sections:
https://www.software3d.com/Forums/viewt ... ?f=3&t=503
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Re: Uniform Polychoron #1850

Postby _Geometer » Sun Oct 04, 2020 9:40 pm

_Geometer was the name I announced the discoveries under, so call me _Geometer.
Also, Gidditsphit was discovered first, but it would make sense to list Sidditsphit first, keeping with the rest of the listing. Could you leave a note on Gidditsphit that it was the first discovered?
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Re: Uniform Polychoron #1850

Postby ndl » Mon Oct 05, 2020 3:46 am

_Geometer wrote:
Mecejide wrote:What symmetry do the verfs of the two uniforms have?


Chiral 3-fold dihedral - if I can figure out how embed files from Stella4d, I will upload all of the verfs I have found.


Just put them all in a .zip file. I would love to see all of your discoveries.
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Re: Uniform Polychoron #1850

Postby username5243 » Tue Oct 20, 2020 1:28 am

Okay, back to the thread, where we have breaking news at this hour.

For those who are not on the discord yet, Bowers has just blended sidditsphit with odho (a demitessics ico member) and it came out uniform. That means there are even more sishi members tha we thought before, probably enough that Bowers has designated these to be category 30 of uniform polychora. The counting is still in progress, so we don't have exact numbers for now.

That is all, folks. This iss a developing story, so stay tuned for updates.
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Re: Uniform Polychoron #1850

Postby Robert Webb » Mon Nov 16, 2020 7:18 am

username5243 wrote:For those who are not on the discord yet, Bowers has just blended sidditsphit with odho (a demitessics ico member) and it came out uniform. That means there are even more sishi members tha we thought before, probably enough that Bowers has designated these to be category 30 of uniform polychora. The counting is still in progress, so we don't have exact numbers for now..


I tried the discord invitation above, but it didn't work. How do I join the discord?

I've been writing some code to automate the search, and I think I've found two new ones, but I need to see what Bowers has found first.

So 1850 & 1851 were discovered, then the rest were blends?

What are the other new ones blends of?

I tried the one you mentioned above, blending sidditsphit with odho, and it does indeed work, if blending one copy of each vertex figure, not multiples to maintain symmetry. The vertex figure ends up with no symmetry at all. So is this one 1856?

Presumably he also tried blending gidditsphit with odho, which also works, to get 1857?
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Re: Uniform Polychoron #1850

Postby Polyhedron Dude » Mon Nov 16, 2020 2:48 pm

Try this link, it seems to be more recent.

https://discord.gg/MMnVMTdn

The odho blends look to be false positives. I checked them more closely, although they have identical vertex figures, they are not truly vertex transitive, sort of like the pseudorhombicuboctahedron. Currently there are six uniform finds for this year, the first two by _Geometer (in sishi regiment), the next two found by me by blending (also in sishi), and two more by _Geometer which has the grand antiprism as a convex hull and are derivitives of the sishi regiment. Also of note is the 'scaliform explosion' that has recently happened due to _Geometer, Mecejide, and myself. There are now 22 scaliform categories plus one infinite one.
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Re: Uniform Polychoron #1850

Postby Robert Webb » Wed Nov 18, 2020 3:24 am

Polyhedron Dude wrote:Try this link, it seems to be more recent.

https://discord.gg/MMnVMTdn

The odho blends look to be false positives. I checked them more closely, although they have identical vertex figures, they are not truly vertex transitive, sort of like the pseudorhombicuboctahedron. Currently there are six uniform finds for this year, the first two by _Geometer (in sishi regiment), the next two found by me by blending (also in sishi), and two more by _Geometer which has the grand antiprism as a convex hull and are derivitives of the sishi regiment. Also of note is the 'scaliform explosion' that has recently happened due to _Geometer, Mecejide, and myself. There are now 22 scaliform categories plus one infinite one.

Discord link says it's invalid.

Can the .stel files be posted somewhere for the vertex figures of the new uniforms? Either here and/or on the related thread on my forum: https://www.software3d.com/Forums/viewt ... ?f=3&t=503

What did you blend with to find the other new ones?

I blended with odho and it seemed to work. Stella4D reported only one type of vertex, but in 4D this is established by heuristics rather than full symmetry as it is in 3D. Is there a way I can tell it's not uniform?

I believe I have found two additional new ones. I doubt they will be repeats of any of the new ones you've mentioned and I think they probably have higher symmetry than any of the recent new ones. But you'll be better to judge all that! Before announcing them though, I want to see if I can automate a search to see if they can be blended to yield additional ones.
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Re: Uniform Polychoron #1850

Postby Polyhedron Dude » Wed Nov 18, 2020 7:57 am

Here is a perminate link to the discord, the other one seems to change daily.

https://discord.gg/zMRu7T4

I'll go to the Stella Forums to upload the stella files, it won't let me do it here.
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Re: Uniform Polychoron #1850

Postby Robert Webb » Wed Nov 18, 2020 11:46 am

Polyhedron Dude wrote:Here is a perminate link to the discord, the other one seems to change daily.

https://discord.gg/zMRu7T4

I'll go to the Stella Forums to upload the stella files, it won't let me do it here.

Thanks, the Discord link worked this time.

Alas it seems the "new" uniforms I found are the first two new blends that were found. I didn't expect the blended vertex figures would have tetrahedral symmetry (greater symmetry than what was blended).
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