## New Uniform Polychoron (I Think)

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

### New Uniform Polychoron (I Think)

It has 840 cells (120 small stellated dodecahedra and 720 pentagrammic antiprisms), 6000 faces (2400 pentagrams and 3600 triangles), 5400 edges, and 720 vertices. The pentagrammic antiprisms are attached to the small stellated dodecahedra by their pentagrammic faces and to each other by their traingular faces. Its vertex figure is a pentagonal bifrustum. In Richard Klitzing’s notation, it is a quarter (β5o5/2o5β).
Mecejide
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### Re: New Uniform Polychoron (I Think)

Welcome Mecejide!

At least I finally get what you are after, when speaking of a "quarter(ß5o5/2o5ß)":

x5o5/2o5x happens to be a double cover of sophi.
ß5o5/2o5ß then is the holosnub of that wrt. vertex alternation.

Thus when starting only with sophi itself and applying vertex alternation (in a holosnubbed manner) thereon, you would be kind of speaking of "half(ß5o5/2o5ß)".
But it happens that this alternation would result in doubled up elements in turn, that is, you could consider to reduce these once more - and this is, where your "quarter" derives from.

But you should be aware that the mere holosnubbing aka vertex alternation of a unit-edged sophi would produce pentagrams of side length f=1.618 both as faces for sissid (= holsnubbed gad) and stap (= holosnubbed pip), and of isosceles triangles with base f and lacing sides q=1.414 for those stap triangles.

When you now proclaim that the result ought be a new uniform polychoron, then you assume that both types of edges should be simultanuously resizable back to unit length. The existance of the alternating surely is the trivial part. That one always exists as desired. Just apply the construction device. But the possibility for such a simultanuous resizement surely is the non-trivial part. And in fact I don't see for now, whether that one will be possible or not. You surely should point out an according clue, if you'd feel that it will be possible.

--- rk
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### Re: New Uniform Polychoron (I Think)

Klitzing wrote:When you now proclaim that the result ought be a new uniform polychoron, then you assume that both types of edges should be simultanuously resizable back to unit length. The existance of the alternating surely is the trivial part. That one always exists as desired. Just apply the construction device. But the possibility for such a simultanuous resizement surely is the non-trivial part. And in fact I don't see for now, whether that one will be possible or not. You surely should point out an according clue, if you'd feel that it will be possible.

--- rk

So, in order to make this uniform, the staps need to be shortened. The pentagrams are already regular, so there’s no problem there. The triangles, though, would need to be squashed. However, the edges which need to be made shorter are only attached to other triangles, so they all get shortened by the same amount.
Mecejide
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### Re: New Uniform Polychoron (I Think)

The point here is, that when changing the relative sizes of edges in the staps, you also change the dihedral angles between the bases and the lacing faces. I still have no clue whether this affects the overall geometry of the polychoron or not.
--- rk
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### Re: New Uniform Polychoron (I Think)

Looks like this one has two edge lengths and is therefore not truly uniform, we could call this type 'uniform-like'. The staps would look like sophi's verf and have a tau-sq2-sq2 isosceles triangle instead of equalateral. This shape could also be blended with any righi regiment member with sissids to give a few more 'uniform-like' polychora, one has ids and staps and would have a hole going through the verf.
Whale Kumtu Dedge Ungol.
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### Re: New Uniform Polychoron (I Think)

But couldn’t it be made uniform by changing the edge lengths?
Mecejide
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### Re: New Uniform Polychoron (I Think)

Mecejide wrote:But couldn’t it be made uniform by changing the edge lengths?

As I've just stated above, when you try to change the relative lengths of the edges, you also would change the dihedral angles in those staps. And doing so you need to prove independantly that this new figure would close up as desired (instead of running around and around infinitely). - Once again: so far I have no clue here. Could be possible or could be wrong. I don't know.

But, from the posted pics of yours I know that you have GreatStella. So just check there. As I remember, there you ought have the possibility to give in the vertex figure and let the program try to produce the according polychoron...

--- rk
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### Re: New Uniform Polychoron (I Think)

Okay, I now realize that doesn't work, but I think I've discovered several new scaliform polychora. They are the sidtaps and gidtaps that use inter-regimental compounds as blend components. For example, there is the blend of 5 roxes, 5 sriphis, and 5 nipixadies.

EDIT:
I just now realized they’re all fissary.
EDIT 2:
There are four of them:
The blend of 5 roxes, 5 sriphis, and 5 nipixadies; of sidsadixox and 5 sriphis; or of sissixfix and 5 roxes
The blend of 5 sophis, 5 rixhis, and 5 lifpipixhis; of sidsosadex and 5 sophis; or of sasfusadex and 5 rixhis
The blend of 5 raggixes, 5 graphis, and 5 mif pixadies; of gidsadixox and 5 graphis; or of gissixfix and 5 raggixes
The blend of 5 quiphis, 5 frogfixes, and 5 ofpipixhis; of gidsosadex and 5 quiphis; or of gasfusadex and 5 frogfixes
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### Re: New Uniform Polychoron (I Think)

I should probably give them UBSAs (Unofficial Bowers-Style Acronyms), too.
1. Sidsdix fasix (Small disnub snub dishexacosifusihexacosichoron)
2. Sidsafsdixox (Small disnub fusisnub dishexacosihexacosichoron)
3. Gidsdix fasix (Great disnub snub dishexacosifusihexacosichoron)
4. Gidsafsdixox (Great disnub fusisnub dishexacosihexacosichoron)
Mecejide
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