## Thawro'd 600-cell

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### Thawro'd 600-cell

So, it's been rather boring here lately. So I decided to construct another CRF.

I've been meaning to get around to exploring what happens when you take two J92's and join them at a dichoral angle of 120° at their hexagonal face (instead of the 60° in the J92 rhombochoron). Well, I finally got around to it today. It turns out that the o3F vertices of the J92's are exactly a unit edge length apart, so 3 tetrahedra neatly fits into that space, and the rest are filled in with pentagonal pyramids and square pyramids:

It's quite a nice arrangement of cells, nice and simple (though the pairs of square pyramids do raise some eyebrows). Only 3 additional vertices were needed to complete this near side of the polytope.

On the far side, it turns out that the polytope closes up in a CRF way with only 3 more layers of vertices, which produce a configuration of 8 metabidiminished icosahedra in trigonal prismic symmetry, with florets of 5 tetrahedra that suggest that this is nothing but a diminishing of a 600-cell:

There are a bunch more pentagonal pyramids that fill up the gaps with the near side, but otherwise, this looks very much like a 600-cell diminishing. A very interesting CRF, nevertheless!

Here's the lace city of this CRF:

Code: Select all
`                   x3o               f3x    f3o           o3F       x3x       F3o x3f   o3x           o3F               f3x    f3o                   x3o`

I'm not sure, but this CRF may have been previously considered? I seem to remember student91 or student5 some time ago posting some lace cities for some modified 600-cells, but I never got around to actually looking into them carefully.
Last edited by quickfur on Fri Apr 18, 2014 3:06 am, edited 1 time in total.
quickfur
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### Re: Thawro'd 600-cell

That must've been student5, hè has been posting expansions of ex according to trigonal Subsymmetry. Those are rather interesting, as I think Those expansions give patches that can easilly be combined. The combination youe have posted hasn't been considered before. It seems to be a unexpanded ex and an expanded ex fused together. I'm quite busy lately, so I haven't been able to post much.
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student91
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### Re: Thawro'd 600-cell

quickfur wrote:So, it's been rather boring here lately. So I decided to construct another CRF.

[...] Here's the lace city of this CRF:

Code: Select all
`                   x3o               f3x    f3o           o3F       x3x       F3o x3f   o3x           o3F               f3x    f3o                   x3o`

Hmm, either you mixed up here 60 and 120 degrees, or the online representation of "code" tags is somehow squashed, compared to that what you typed in.

Here is the (non-CRF) 2/6-lune of ex:
Code: Select all
`        o3o                                           x3o   o3f f3o   o3x                                                                 f3x       x3f   f3o o3o                                 o3F   F3o       o3f                                                                 f3f   o3F f3x   x3o`

And here the reduced one according to your new find.
Code: Select all
`x3o   o3f       o3x                                                                 f3x       x3f                                           o3F   F3o       o3f                                                                 x3x   o3F f3x   x3o`

This then displays not only the thawro-ing (f3f -> x3x in the lower left corner), but also which ex vertices are to be rejected: the 2 o3o ones (i.e. the tips of teddi-pyramids) and 2x3 vertices of f3o (i.e. the tips of mibdi-pyramids).

In fact, it thus rather looks like that these 8 diminishings come completely independent of that thawro-ing here. Then we would have a thawro-ed 2/6-lune of ex (only through the thawro-ing this lune itself becomes CRF!), and several diminishings thereof. - Yours then just is an octadiminishing. - Right?

--- rk
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### Re: Thawro'd 600-cell

Klitzing wrote:[...]
Hmm, either you mixed up here 60 and 120 degrees, or the online representation of "code" tags is somehow squashed, compared to that what you typed in.

Well, I wanted to save some space, so I didn't draw the diagram to actually measure 120° around the left ridge.

[...]
In fact, it thus rather looks like that these 8 diminishings come completely independent of that thawro-ing here. Then we would have a thawro-ed 2/6-lune of ex (only through the thawro-ing this lune itself becomes CRF!), and several diminishings thereof. - Yours then just is an octadiminishing. - Right?
[...]

Sounds about right. I constructed it as a folding of thawro's into 4D, so I went for the minimum number of additional vertices to make it close up in a CRF way. So probably adding more vertices will produce the full thawro'd 2/6-lune. I didn't look at your lace cities in detail, because I've just constructed something new that's a candidate for a J92 hexachoron. I'm guessing I should post that in a different thread once I get some nice renders up (and work out the lace city -- I have the coordinates but need to sit down and work out where each node in the lace city should go).

But anyway, does this mean that it should be possible to also thawro the 3/6 and 1/6 lunes of the 600-cell to get CRFs?? The next thing you know, we're gonna be making tetrahedron-centered 600-cell lunes, perhaps made CRF by pocuro'ing?
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### Re: Thawro'd 600-cell

quickfur wrote:But anyway, does this mean that it should be possible to also thawro the 3/6 and 1/6 lunes of the 600-cell to get CRFs?? The next thing you know, we're gonna be making tetrahedron-centered 600-cell lunes, perhaps made CRF by pocuro'ing?

The 3/6-lune being thawro-ed clearly won't be CRF, as the 2 bounding thawroes become co-realmicly attached at their hexagons.

The 1/6-lune of ex looks like being thawro-able however:
Code: Select all
`        o3x                                                   x3f   f3o o3o                       F3o       o3f                                         x3x   o3F f3x   x3o`

In fact it then should be closely related to thawrorh, at least in the surroundings of the hexagon.

--- rk
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### Re: Thawro'd 600-cell

quickfur wrote:[...] I didn't look at your lace cities in detail, because I've just constructed something new that's a candidate for a J92 hexachoron. I'm guessing I should post that in a different thread once I get some nice renders up (and work out the lace city -- I have the coordinates but need to sit down and work out where each node in the lace city should go).

But anyway, does this mean that it should be possible to also thawro the 3/6 and 1/6 lunes of the 600-cell to get CRFs?? The next thing you know, we're gonna be making tetrahedron-centered 600-cell lunes, perhaps made CRF by pocuro'ing?

The hexachoron might just be what student5 has been posting. his lace-city looks like this:
Code: Select all
`                 x3o                                                           o3x   x3f F3o   x3x                                                                              x3o x3f   F3x      o3F    F3o x3o                                       F3o       x3F  A3B        x3f                                                                         x3x  o3F A3B   F3f   x3F f3x   o3x                                                                         F3o       x3F  A3B        x3f                                      x3o x3f   F3x      o3F    F3o x3o                                                                               o3x   x3f F3o   x3x                                                           x3o     `
. you can clearly see a hexagon of thawros there, augmented with pseudopyramid apices. If your hexachoron is different, it surely is awesome. if it's not, it would be nice to have some renders. I don't think you need a new topic for it, as it is still about thawro'd hexacosachora.
EDIT: A3B most probably is A3o with A=F+x=f+2x
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### Re: Thawro'd 600-cell

Alright, added the software models for the thawro'd 600-cell here: D4.13.1. I numbered it with a .1 suffix 'cos I think the hexachoron seems to be related to student5's construction, which is ultimately a thawro'ing of the 600-cell, so that one will get a .2 suffix.
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### Re: Thawro'd 600-cell

Don't have time to check it now, but can these two shapes be augmented with J92 pseudopyramids?
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### Re: Thawro'd 600-cell

I'm pretty sure it's possible, due to the low height of the J92 pseudopyramid. Also, judging from the J92-J92-J62 configuration around the edges, which are identical to what occurs in the J92 rhombochoron, I'm pretty sure the augmenting pseudopyramid would have its pentagonal pyramids merge with the J62's to form icosahedra (or monodiminished icosahedra).
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### Re: Thawro'd 600-cell

I've got some questions.

quickfur wrote:

The upper pentagons of the first picture would obviously join the thawro cells to peppies.
: But for the frontal pentagonal struts this is much lesser "obvious". But I guess that there too one tet face, 2 thawro faces and 2 squippy faces would connect to a further peppy, and that this very peppy then would connect to a second peppy, which in the second picture then would connect to 2 mibdi faces and to 3 tet faces. Right? I even suppose that the angle between these 2 peppies at the pentagon would neither be 180 degrees, nor such that it could be replaced by a 5 tet rosette? Right or wrong?

Then upon the augmentation, I did mention. That is, adding vertices atop / within the teddies resp. mibdis:
Klitzing wrote:In fact, it thus rather looks like that these 8 diminishings come completely independent of that thawro-ing here. Then we would have a thawro-ed 2/6-lune of ex (only through the thawro-ing this lune itself becomes CRF!), and several diminishings thereof. - Yours then just is an octadiminishing. - Right?
: Would that then replace the so far pentagons at the teddi-mibdi join resp. at the mibdi-mibdi join by 5 tet rosettes or rather by 2 peppy pairs (or even by a single pedpy?) at each occurency?

My guess here would be that the former question about the peppy-peppy join (between those pics) the existance of the changed base vertices of the thawroes might imply some strange dihedral angles and so the 5 tet rosette would come out of scope here. But for the second cases, as being then parts of an ex sub-structure, these 5 tet rosettes might be implied. Note moreover that we would consider CRFs.

Both are just guesses. I did not calculate any dihedral angle here. But you might simply derive that from your vertex calculations... And thus could answer these questions much more quickly?

--- rk
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### Re: Thawro'd 600-cell

Klitzing wrote:[...]The upper pentagons of the first picture would obviously join the thawro cells to peppies.

Correct.

:?: : But for the frontal pentagonal struts this is much lesser "obvious". But I guess that there too one tet face, 2 thawro faces and 2 squippy faces would connect to a further peppy, and that this very peppy then would connect to a second peppy, which in the second picture then would connect to 2 mibdi faces and to 3 tet faces. Right? I even suppose that the angle between these 2 peppies at the pentagon would neither be 180 degrees, nor such that it could be replaced by a 5 tet rosette? Right or wrong?

Yes, these pentagonal pyramids are in pairs, with one of the pair on the near side (first image) and the other on the far side (second image). They cannot be 180° to each other, because otherwise they'd be pentagonal bipyramids.

Furthermore, since these models are produced by a convex hull algorithm, in no case can a pair of pentagonal pyramids be replaced with a rosette of 5 tetrahedra: that is only possible if the dichoral angle between the two pentagonal pyramids are > 180°. Or to be precise, the dichoral angle will have to be exactly (360° - A), where A = dichoral angle of pentagonal pyramids in the segmentochoron pentagonal_pyramid || point.

Basically, the analogy here is with two isosceles triangles joined at a phi-edged base (resp. pentagonal pyramids joined at the pentagon). If their dichoral angle is < 180°, that is, the phi-edged base (resp. pentagonal face) protrudes inwards towards the centroid of the polytope, that is, they are concave, then you can close them up with two triangles (resp. rosette of 5 tetrahedra) to make the result convex. If they are 180° to each other, they form a phi-rhombus (resp. pentagonal bipyramid): the phi-edge is coplanar with the triangles (resp. pentagonal face with the two pyramids) and becomes internal to a single merged face. If their dichoral angle is > 180°, that is, the phi-edged base (resp. pentagonal face) protrudes outwards, then they form a wedge on the surface of the polytope, and any closure with two triangles (resp. rosette of 5 tetrahedra) will be internal to the polytope -- they will lie behind the phi-scaled edge (resp. pentagonal face).

Then upon the augmentation, I did mention. That is, adding vertices atop / within the teddies resp. mibdis:
Klitzing wrote:In fact, it thus rather looks like that these 8 diminishings come completely independent of that thawro-ing here. Then we would have a thawro-ed 2/6-lune of ex (only through the thawro-ing this lune itself becomes CRF!), and several diminishings thereof. - Yours then just is an octadiminishing. - Right?
: Would that then replace the so far pentagons at the teddi-mibdi join resp. at the mibdi-mibdi join by 5 tet rosettes or rather by 2 peppy pairs (or even by a single pedpy?) at each occurency?

I believe what happens is that the augments on adjacent J62 / J63's will become non-convex, that is, the pentagonal pyramids introduced by the augments will have dichoral angle > 180°. The convex hull algorithm would then, in effect, cover over these concave gaps with rosettes of 5 tetrahedra. Or rather, it inserts pentagonal_pyramid||point segmentochora between the augments so that the result becomes convex.

My guess here would be that the former question about the peppy-peppy join (between those pics) the existance of the changed base vertices of the thawroes might imply some strange dihedral angles and so the 5 tet rosette would come out of scope here. But for the second cases, as being then parts of an ex sub-structure, these 5 tet rosettes might be implied. Note moreover that we would consider CRFs.

Both are just guesses. I did not calculate any dihedral angle here. But you might simply derive that from your vertex calculations... And thus could answer these questions much more quickly?

Well, the peppy-peppy join here is basically an indication that the two halves are thin caps of the 600-cell, where after stripping off the rosettes of 5 tetrahedra, the pentagonal pyramids thus exposed still have < 90° dichoral angle with the bisecting hyperplane. So they can be glued together with two pentagonal pyramids exposed on the surface of the result. If they were exactly 90°, then putting the two halves together will produce pentagonal bipyramids; if they were > 90°, then putting the halves together will be non-convex, and the convex hull algorithm will introduce lacing edges between the tips of the pentagonal pyramids, which, in effect, inserts the rosettes of 5 tetrahedra (and the pentagonal pyramids become internal to the polytope).
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### Re: Thawro'd 600-cell

so you basically second all my guesses...
--- rk
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### Re: Thawro'd 600-cell

Yes.
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### Re: Thawro'd 600-cell

Klitzing wrote:The 1/6-lune of ex looks like being thawro-able however:
Code: Select all
`        o3x                                                   x3f   f3o o3o                       F3o       o3f                                         x3x   o3F f3x   x3o`

In fact it then should be closely related to thawrorh, at least in the surroundings of the hexagon.

--- rk

Meanwhile got it through. It indeed exists right in this way.
Its cell consist would be: 2 thawroes, 6 squippies, 6 peppies, and 74 tets.

Vertex coordinates would be:
• acs[1], 0, 0, 0
• acs[1/2, sqrt(3)/2], 0, 0
• 0, ecs[tau2/sqrt(3), -1/2sqrt(3)], 1/2
• acs[tau2/2], ecs[tau2/2sqrt(3), 1/2sqrt(3)], 1/2
• acs[1/2], ecs[tau3/2sqrt(3), tau/2sqrt(3)], tau/2
• acs[tau2/2],ecs[ tau-1/2sqrt(3), -tau/2sqrt(3)], tau/2
• acs[tau/2], ecs[(tau+2)/2sqrt(3), -tau/2sqrt(3)], tau/2
• 0, ecs[1/sqrt(3), tau2/2sqrt(3)], tau2/2
• acs[1/2], ecs[1/2sqrt(3), -tau2/2sqrt(3)], tau2/2
• 0, ecs[tau/sqrt(3), -tau-1/2sqrt(3)], tau2/2
• acs[tau/2], ecs[tau/2sqrt(3), tau-1/2sqrt(3)], tau2/2
• 0, 0, 0, tau
(acs[...] = all changes of sign in all covered coordinates / ecs[.., ..] = even changes of sign of a pair of coordinates)

Obviously the last vertex could be diminished. This then would result in a multi-wedge (the lace city becomes triangular, featuring one icosahedron and 2 thawroes on its outline). Then the resulting cell count would become 2 thawroes, 6 squippies, 6 peppies, 54 tets, and 1 ike.

Finally, you could consider an external blend of 2 of these latter diminished figures, thus blending out the icosahedron. The lace city then becomes quite similar to that of thawrorh. Most probably this is just how that one could be achieved: as a mere convexification of that blend.

--- rk
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### Re: Thawro'd 600-cell

Hmmm,
just am wondering about the peppy pairs of thawrorh, eg. clearly visualised in a sepparate pic in that post:

Seem that those are needed for being convex, and that the corresponding replacing 5-tet-rosettes then would make it non-convex? - Is that right? or are those accordingly still augmentable in a convex way?

Well, if not, then the cell consist of the thawro-wedged 1/6-lune of ex should become 2 thawroes, 6 squippies, 18 peppies, and 44 tets instead...

--- rk
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### Re: Thawro'd 600-cell

To understand the arrangement, it helps to note that the pentagonal face shared by the peppies protrudes outwards in the 4th direction, and the apices are receded in the 4th direction. In order to replace the peppies with rosettes of tetrahedra in a convex manner, it would require that the apices protrude outwards in the 4th direction, and the pentagon of vertices recede inwards, i.e., the opposite of the current arrangement. Or put another way, this replacement will have the tetrahedra slanting in the opposite direction of a convex arrangement, so the result will be a concave depression in the surface of the polytope.

Another way to look at this, is to consider the segmentochoron pentagonal_pyramid||point, which consists of 2 peppies and 5 tetrahedra. The current vertices constituting the peppy pairs in the J92 rhombochoron correspond with the vertices of pentagonal_pyramid||point in the orientation where the 2 peppies face the current 4D viewpoint. Therefore, the arrangement of 5 tetrahedra with the same vertices corresponds exactly with the 5 tetrahedral cells on the far side of pentagonal_pyramid||point, which face away from the 4D viewpoint. So obviously, replacing the peppies with the rosette of 5 tetrahedra in the J92 rhombochoron is equivalent to deleting the outward-facing cells of embedded pentagonal_pyramid||point segmentochora, thus exposing the internal cavity of the segmentochoron as a concave hole in the polytope. Equivalently, this is the same as cutting out a chunk in the shape of a pentagonal_pyramid||point from the 4D bulk of the J92 rhombochoron.
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### Re: Thawro'd 600-cell

That is, when you'd render the coordinates provided in my prior post, then this same arrangement of pyramid pairs should show up.

Or, taken in a different way, my prior assumption with the 74 tets would have some cavities, which can be filled up by 6 of these 5-tet-rosettes (= line || ortho {5}), in order to become a true CRF.

--- rk
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### Re: Thawro'd 600-cell

Wouldn't that render be mostly the same as this:

This is thawrorh before we knew how to close it up. It's lace-city looks like this:
Code: Select all
`        o3x                                                   x3f       o3o                       F3o                                                   x3x   o3F f3x   x3o`
That's almost the same as your lace-city:
Code: Select all
`        o3x                                                   x3f   f3o o3o                       F3o       o3f                                         x3x   o3F f3x   x3o`
The two f3o-triangles will fill up the teddi's at the other side: You can see the teddi's here. Unfortunately there's an f-edge here. The f-edge can be seen in the lace-city as the connection between F3o and o3F. In other lace-cities this f-edge is internal to some pentagon, but at the 1/6-lune all pentagons to which this f-edge is internal are destroyed. This means it gets revealed, and the polytope isn't CRF. Of course it could be possible that by extruding pyramids on the teddi's, something awesome makes it CRF again, but I doubt that.
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student91
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### Re: Thawro'd 600-cell

ehh no - wind back! (To my initial post.)

The head vertex of those peppy-pair pentagons, as well as the arm vertices of those, do belong to the thawro'ed 1/6-lune. But not so the 2 feet vertices. Where the thawrorh lace city uses x3F, the thawro'ed 1/6-lune would use o3f instead. That is, there is not the full pentagon at all. Accordingly from this 5-tet-rosettes only 2 tets each are contained. Thus it might be possible after all that the former tip to tip edge might still be exterior.

At least the former subtraction of 6*5 tets, and the addition of 6*2 peppies instead, would be not allowed.

Thus it might be great, if you could render those vertices. At least in order to see what their convex hull would look like. (So far I considered just all unit edges between. But I have not evaluated global convexity. Nor any hull calculations, which might introduce "false" edges, which might then have othen than unit size.)

--- rk
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### Re: Thawro'd 600-cell

Thus it looks like that the thawro'ed 1/6-lune of ex would not be convex, after all.

But then I think, this same problem does not occur in the corresponding thawro'ed 2/6-lune of ex. I.e. in
Code: Select all
`        o3o                                           x3o   o3f f3o   o3x                                                                 f3x       x3f   f3o o3o                                 o3F   F3o       o3f                                                                 x3x   o3F f3x   x3o`

which clearly is related to the octa-diminished version
Code: Select all
`x3o   o3f       o3x                                                                 f3x       x3f                                           o3F   F3o       o3f                                                                 x3x   o3F f3x   x3o`

The reason here ought be that the offending f-edges, occuring as lacings of to o3F||F3o lace prism, - so even not being covered in the way of thawrorh by 6 pentagons, defined by the positions of x3x, o3F, F3o, x3F, F3x, - would rather be covered by different 6 pentagons, defined by the positions of o3F, o3F, f3x, f3x, F3o of the above lace cities. (The corresponding tips of the incident peppies then are at positions x3x resp. x3f.)

So, we conclude, that a thawro'ed 3/6-lune of ex surely is not possible just because of the thawroes being corealmic.
That a thawro'ed 2/6-lune of ex then is possible to be done convex.
But a corresponding thawro'ed 1/6-lune of ex would not be possible to become convex itself. (It then would require for the extension into thawrorh, which is convex again.)

Sadly we have to say farewell to the 1/6-lune.

--- rk
Klitzing
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### Re: Thawro'd 600-cell

Note, however, that the J92 rhombochoron merely represents the most obvious convexification of the 1/6-lune at the time (because the non-CRF part lies completely within a hyperplane, so the most obvious way to make it CRF is to paste a mirror image copy of the shape on the other side). There may be other ways of convexifying it. Of course, the additional vertices probably won't come from the 600-cell proper, but from some other surface patch.

One thought is to start with the non-CRF 1/6-lune (configuration of two J92's), then add the 12 vertices to get rid of the f-edges and make the near side of the polytope convex, then on the far side find the minimal number of additional vertices that will make it CRF.

P.S. Another thought. I wonder how many other non-convex derivations from the uniforms (e.g., lunes, or some other cuttings thereof) are amenable to convexification in a similar way? Perhaps we could study some non-convex cuttings of uniform polytopes, and see if there are ways of convexifying them, via additional vertices, or pasting copies of themselves together, etc..
quickfur
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