quickfur wrote:I've been preparing the renderings of the J92 rhombochoron for April's Polytope of the Month on my website, and it occurred to me that it should be possible to augment its J92's with J92 pseudopyramids in a CRF way (the J92 pseudopyramid has very low height: only 1/phi for edge length 2, so the result should be convex). This would complete the J62's and produce a CRF with 6 icosahedra in triangular antiprismic symmetry.
This augmentation also causes some octahedra to appear in positions matching some of the square pyramids, indicating a possible derivation of the J92 rhombochoron as a diminishing of two relatively flat patches of the o5o3x3o's surface glued together like some analogue of a bicupola (the square pyramids would be bisected octahedra, if this derivation is correct). Of course, this still doesn't fully explain some of the other parts of the J92 rhombochoron, such as the triangular prism surrounded by 3 square pyramids and 2 tetrahedra, that seems to be analogous to Johnson's "luna" (triangle+square+triangle).
this image is interesting!! The fact that it has augmentations (or, needs a diminishing in order to show the thawros) makes me think that it is comparable with the thawrod/bilbirod o5o3x3o. These had patches of o5o3x3o and x5o3o3x. If the triangular prism-tetrahedron dichoral angle is equal to the corresponding angle of x5o3o3x, we know where that came from.quickfur wrote:Alright. Just as Marek has checked, the tetraaugmented J92 rhombochoron exists, and is CRF. My polytope viewer confirms that all edges are unit length. Here's a projection of it:
Here, we see 3 icosahedra surrounding the triangle closest to the 4D viewpoint. This indeed looks like a fragment of the o5o3x3o's surface, if you consider the outline of a pentagonal prism between the front and left icosahedra: it's actually a pentuplet of 2 octahedra and 3 square pyramids, the latter of which may be considered as bisected octahedra from the o5o3x3o's surface.
that's all I can tell about it's structure as well[...]
So, I still can't figure out a derivation of the J92 rhombochoron from the uniform polychora. It seems to have quite a unique structure!
It does make sense, but do we really need a D-number for every augmentation? The bilbiro'd/thawro'd o5o3x3o also doesn't have listed all of it's possible diminishings. (Which is, in fact, just the inverse of what we're doing here) therefore, I suggest we only number the tetraaugmented one, call it D4.4.0 or D4.4.2Question: what D number should we assign the augmentations of the J92 rhombochoron? I'd like to use D4.4.1 - D4.4.6 (according to Marek's count of augmentations), would that make sense?
student91 wrote:this image is interesting!! The fact that it has augmentations (or, needs a diminishing in order to show the thawros) makes me think that it is comparable with the thawrod/bilbirod o5o3x3o. These had patches of o5o3x3o and x5o3o3x. If the triangular prism-tetrahedron dichoral angle is equal to the corresponding angle of x5o3o3x, we know where that came from.quickfur wrote:Alright. Just as Marek has checked, the tetraaugmented J92 rhombochoron exists, and is CRF. My polytope viewer confirms that all edges are unit length. Here's a projection of it:
Here, we see 3 icosahedra surrounding the triangle closest to the 4D viewpoint. This indeed looks like a fragment of the o5o3x3o's surface, if you consider the outline of a pentagonal prism between the front and left icosahedra: it's actually a pentuplet of 2 octahedra and 3 square pyramids, the latter of which may be considered as bisected octahedra from the o5o3x3o's surface.
But what's more interesting, when you look at the ike's vertex most left and up, this is connected to the ike next to it via an edge. This edge seems to be the apex of a bilbiro pseudopyramid!! This would mean the rhombochoron has both room for bilbiro's and thawro'sthat's all I can tell about it's structure as well[...]
So, I still can't figure out a derivation of the J92 rhombochoron from the uniform polychora. It seems to have quite a unique structure!
quickfur wrote:Klitzing wrote:Btw., here is the incmats of that figure (in its full hexadecachoral symmetry):
@Keiji: add this to the D4.11 page?
quickfur wrote:Question: what D number should we assign the augmentations of the J92 rhombochoron? I'd like to use D4.4.1 - D4.4.6 (according to Marek's count of augmentations), would that make sense?
quickfur wrote:It does make sense, but do we really need a D-number for every augmentation? The bilbiro'd/thawro'd o5o3x3o also doesn't have listed all of it's possible diminishings. (Which is, in fact, just the inverse of what we're doing here) therefore, I suggest we only number the tetraaugmented one, call it D4.4.0 or D4.4.2
1 0 0 0 10.472136
0 1 0 0 37.885438 *
0 0 0 1 54.832815 a 0 1 0 -1
0 0 1 0 82.249223 b 0 0 1 -1
1 1 0 0 86.249223 c 1 1 -1 0
1 0 0 1 109.665631 a
1 0 1 0 147.554175 b
0 1 0 1 181.442719 d 1 -1 1 -1
0 1 1 0 229.803398 b
0 0 1 1 270.164078 a
1 1 0 1 274.164078 c
1 1 1 0 332.996894 b
1 0 1 1 379.829710 a
0 1 1 1 506.439634
1 1 1 1 653.993788
x3o
o3x x3f F3o x3x
x3o x3f F3x o3F F3o x3o
F3o x3F X3o x3f
x3x o3F X3o F3f x3F F3x o3x
F3o x3F X3o x3f
x3o x3f F3x o3F F3o x3o
o3x x3f F3o x3x
x3o
x3o
o3x x3f F3o x3x
x3o x3f F3x
F3o x3F
x3x
x3o
o3x x3f F3o x3x
x3o x3f F3x o3F
F3o x3F f3x o3x
x3x o3F f3x o3x
o3x
o3x3o o3x3o x3(-x)3x x3x3o x3o3x
x3o3x -> (-x)3x3x x3 o 3x -> o3x3x x3x3x
o3x3o o3x3o x3(-x)3x x3x3o x3o3x
o3o5o o3o5o x3o5o
x3o5o (-x)3x3o o3x5o
o3o5x o3o5x x3o5x
f3o5o f3o5o F3o5o
o3x5o -> o3x5o -> x3x5o
f3o5o f3o5o F3o5o
o3o5x o3o5x x3o5x
x3o5o (-x)3x5o o3x5o
o3o5o o3o5o x3o5o
student5 wrote:student91 wrote:[...]quickfur wrote:Alright. Just as Marek has checked, the tetraaugmented J92 rhombochoron exists, and is CRF.
[...]
[...]
But what's more interesting, when you look at the ike's vertex most left and up, this is connected to the ike next to it via an edge. This edge seems to be the apex of a bilbiro pseudopyramid!! This would mean the rhombochoron has both room for bilbiro's and thawro's
[...]
this made me think the J92 rombochoron is similar to the D4.8.x series, they also have loads of options, together with a pretty similar orientation of bilbiro vs. thawro.
in fact, if you diminish the top, so that you get three bilbiroes centered around the top hexagon, and make a thawro from the bottom, you get tridiminished ikes, in the exact same orientation as D4.8.3
[...]edit: after further tought, they both seem to have something to do with the hexagonal rotunda, where the projection envelope of J92 is a hexagonal orthobirontunda
quickfur wrote:[...][...]edit: after further tought, they both seem to have something to do with the hexagonal rotunda, where the projection envelope of J92 is a hexagonal orthobirontunda
Yeah I've noticed that. So there's definitely some kind of relationship between these CRFs. The D4.8.x series have overlapping patches of the hexagonal rotunda that form tetrahedral symmetry, whereas here we have a pair of hexagonal rotunda end-to-end. So they must derive from some common source somewhere, but just put together in different ways.
In any case, looks like there's an entire family of augmented/diminished CRFs here, directly related to the J92 rhombochoron, and also more distantly related to the D4.8.x series of CRFs. Clearly, we need to be looking at the various surface patches that give rise to these combinations, because enumerating each individual CRF would produce a very long list!
x3x3o o3x3o //edit
o3x3f x3o3f x3x3o
o3F3o x3f3o o3x3f //augmentation of A2, maybe this part can also be bilbiro'd?
f3x3x F3o3x o3F3o
x3o3F x3x3f -> f3x3x //the point around which the second lace tower is rotated and stuck on the first
F3o3x F3x3o x3o3F
x3x3f o3x3f o3f3x //augmentation of B2, existent in B1
o3F3o x3f3o f3o3x //augmentation of B1, existent in B2
f3x3o F3x3o o3x3o
o3x3x x3o3x
Marek14 wrote:So, maybe we should consider the tetraaugmented version the basic polychoron with the others being various diminishings of it?
Marek14 wrote:So, maybe we should consider the tetraaugmented version the basic polychoron with the others being various diminishings of it?
x5o x5o
x5x
o5f o5f
x5o x5o
F5o
x5f x5f
x5x x5x
F5o F5o
x5o o5F x5o
x5f x5f
o5f o5f
F5o F5o
x5f
x5o x5o
x5x x5x
o5f
x5o x5o
quickfur wrote:Marek14 wrote:So, maybe we should consider the tetraaugmented version the basic polychoron with the others being various diminishings of it?
That's a good idea.
student5 wrote:edit: please tell me if I'm bloating the forum with these exapnsions and whether I should stop posting them until the first have been verified, or if I should start verification myself (I'm at loss as how to do this)
Marek14 wrote:In my opinion we shouldn't be afraid to change the numbers because of older posts. Those posts are only a month or two old, it's not as if it's a decades-old convention
Keiji wrote:Numbering as something associated to a previously discovered polytope is a little easier to remember. We've already established the numbers mean nothing, only that there is some vague similarity between ones under the same "main" number. Can we not make D4.12 into D4.4.1 and D4.13 into D4.4.2?
quickfur wrote:P.S. I skimmed over some older posts in this thread this morning, and realized that I'd forgotten to upload the software models for the D4.9.x polychora. Since I only actually constructed D4.9.1, I went and created a page for it, linked the render I made for it, and added the .def and .off files. Just in case people are interested to look at it. (Right, Marek? )
quickfur wrote:Klitzing wrote:Hehe, PolyhedronDude always thinks about non-convex conjugates as well... - But I cannot wrap my mind around that one so far. It then should include the conjugates of bilbiro and teddi for cells. Several cells potentially might be used retrograde then too. - But instead of a try to build that one from scratch, just by "translation" of the final polychoron, one well could start from the conjugate of sadi, and then apply the same rebuild and expansion onto that. Perhaps this might be easier to follow...
[...]
Interesting. What's the general procedure for constructing a conjugate? Does it apply to any polytope? So far I only know of the conversion of icosahedron -> great icosahedron, but I'm not sure I understand the general procedure yet.
//original
x3o3o (-x)3x3o o3x3o//this part is verified to be CRF, as in D.4.8
o3o3f o3o3f x3o3f
o3f3o o3f3o x3f3o
f3o3x f3o3x F3o3x
o3x3f o3x3f x3x3f
x3f3o (-x)3F3o o3F3o //until here it follows D.4.8.x, this is also the point after which things get weird.
F3o3o F3o3o X3o3o //this can't stack properly with o3F3o
f3o3f -> f3o3f -> F3o3f // this won't stack with o3F3o or X3o3o either
o3o3F o3o3F x3o3F//this won't stack with o3F3o either
o3f3x o3f3x x3f3x //this will stack, with a height of 0,7071067812
x3o3f (-x)3x3f o3x3f //stacks
o3f3o o3f3o x3f3o
f3o3o f3o3o F3o3o
o3o3x o3o3x x3o3x //stacks 0,2185080122 with F3o3o and 0,3535533906 with x3f3o
x3o3o (-x)3x3o o3x3o o3x3o
o3o3f o3o3f x3o3f x3o3f
o3f3o o3f3o x3f3o x3f3o
f3o3x f3o3x F3o3x F3o3x
o3x3f o3x3f x3x3f x3x3f
x3f3o (-x)3F3o o3F3o o3F3o //until here it is verified
F3o3o (-F)3F3o (-f)3F3o f3x3o //this stacks with o3F3o at a height of 0,1350453784 :-)
f3o3f = f3o3f -> F3o3f = F3o3f // this stacks with f3x3o at a height of 0,218...
o3o3F o3o3F x3o3F x3o3f //from now on it should be CRF, beacause it is just a normal partial stott, without any inversions
o3f3x o3f3x x3f3x x3f3x
x3o3f (-x)3x3f o3x3f o3x3f
o3f3o o3f3o x3f3o x3f3o
f3o3o f3o3o F3o3o F3o3o
o3o3x o3o3x x3o3x x3o3x
quickfur wrote:
quickfur wrote:
Marek14 wrote:The structure of D4.4.2 is as follows:
6 tetrahedra (reflection symmetry) - joined to 3 bilbiros and 1 triangular cupola
[...] So: Gaps between J91's -- they form 2 triplets which surround a triangular cupola and octahedron, with gaps filled by tetrahedra and big gaps between pentagonal faces filled by tridiminished icosahedra.
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