Pseudopyramids

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

Pseudopyramids

Postby quickfur » Mon Mar 31, 2014 3:01 am

So, we have recently discovered the J91 and J92 pseudopyramids. The existence of these appeared to be entirely coincidental at first, but afterwards, we discovered that J91 and J92 can be derived by Stott expansion of certain facetings of the icosahedron. The shape of the pseudopyramid apex is exactly correlated with the symmetry underlying this faceting+expansion.

Then student91 (iirc) discovered a similar faceting+expansion that creates a pentagonal orthocupolarotunda. This construction suggests that perhaps there's also a pseudopyramid of the pentagonal orthocupolarotunda? :) (It would not have been in Klitzing's list of segmentochora, because it's non-orbiform.)

So, does this pseudopyramid exist? :)
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Re: Pseudopyramids

Postby Keiji » Mon Mar 31, 2014 5:52 am

My immediate thought is - pecu || pentagon is pentagonal orthobicupolic ring - so pecuro || pentagon would exist as something stuck onto it by either of the pecu cells. That something would then be another CRF in its own right :)
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Re: Pseudopyramids

Postby Klitzing » Mon Mar 31, 2014 10:45 am

Well, first of all, we have the segmentochoron K-4.146: {5} || pero with cell count: 5 peppies + 5+5 squippies + 1 pip + 1 pecu + 1 pero.

Next we have the segmentochoron K-4.133: {5} || gyro pecu (alternatively described as K-4.133.1: {10} || pap) with cell count: 10 squippy + 2 pecu + 1 pap.

Now build the external blend of those, blending out 1 pecu of either these, you should come out with exactly what you are looking for, at least with a cell count: 5 peppies + 5+5+5+5 squippies + 1 pip + 1 pap + 1 pecu + 1 pero.

You then would just have to establish that that final pecu and pero would be co-realmic. If so, you could unite these 2 into a single replacing cell.

Anyone having GreatStella could check that dihedral angle (or else would like to calculate it)?

But then, we should look at the relative orientations a bit closer: K-4.146 has the pecu-squares attached to the pero-triangles at the decagon-sides. K-4.133 attaches the pecu-squares at the pecu-triangles at the decagon-sides. Now, blending out the joining pecu, would leave pecu-squares attached to pero-pentagons at the decagon-sides! Therefore this blend, if the encasing pero and pecu are co-realmic, would result in the unified cell being a pegycuro (J33) rather than a pocuro (J32).

Quickfur then surely would like to provide pics, I suppose. :D

--- rk


PS-1: now, when I think about it, we could alternatively check K-4.154: {5} || pecu (alternatively described as K-4.154.1: {10} || pip). That one has cell count: 1 pip + 5 tet + 5 trip + 2 pecu.

Using that one in that external blend instead of K-4.133, would result in K-4.146 + K-4.154, i.e in a cell count: 5 peppies + 5+5 squippies + 2 pips + 5 tet + 5 trip + 1 pecu + 1 pero.

And, if here the pero and pecu would become co-realmic, it indeed would join into a pocuro!


PS-2: Well, do not need for any dihedral angle calculations! Managed it by my own: From my segmentochoron-PDF I just checked the respective heights (because co-realmic cells should result in the same heights to the (common) opposite base):

height( {5} || pero ) = 0.309
height( {5} || gyro pecu ) = 0.5
height( {5} || pecu ) = 0.309

Thus, it seems that the second choice (that of "PS-1") would be indeed the one to be chosen. And so, pocuro indded is correct as joined cell.

Haha :P , thus finally not only having found the thawro-wedge (aka thawro pseudopyramid), but now also the pocuro-wedge!

--- rk
Last edited by Klitzing on Mon Mar 31, 2014 5:18 pm, edited 1 time in total.
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Re: Pseudopyramids

Postby student91 » Mon Mar 31, 2014 2:24 pm

And this pseudopyramid can too be found in a modified o5o3x3o. (the thawro-pseudopyramid could be found in the thawro'd o5o3x3o, and the bilbiro pseudopyramid could be found in the bilbiro'd o5o3x3o). This one can be found in the pecuro'd o5o3x3o, aka bilbiro'd o5o3x3o aka D4.9.0.
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Re: Pseudopyramids

Postby quickfur » Mon Mar 31, 2014 6:26 pm

And Klitzing was right, the apex is {5}, not gyro {5}. Here's a render of our pretty new CRF:

Image

It's interesting that the J91 and J92 pseudopyramids, as well as this one, all have height E/(2*phi) (where E = edge length). I suppose it's the result of all three being related to the Stott expansion of some faceting of the icosahedral pyramid.

Here are the coordinates (I wanted a "nice" orientation of the J32, so I calculated the coordinates based on the .5. orientation rather than the usual .4s3. coordinates which are algebraically nicer but harder to orient w.r.t. the coordinate planes):
Code: Select all
#
# J32:
#

# x5o:
<-√((10+2*√5)/5), 0,    √((20+8*√5)/5), 0>
<-√((5-√5)/10),   ±phi, √((20+8*√5)/5), 0>
< √((5+2*√5)/5),  ±1,   √((20+8*√5)/5), 0>

# o5f:
< √((20+8*√5)/5),   0,      √((10+2*√5)/5), 0>
<-√((25+11*√5)/10), ±phi,   √((10+2*√5)/5), 0>
< √((5+√5)/10),     ±phi^2, √((10+2*√5)/5), 0>

# x5x:
<±√(3+4*phi), ±1,     0, 0>
<±√(2+phi),   ±phi^2, 0, 0>
<0,           ±2*phi, 0, 0>

# x5o:
<-√((10+2*√5)/5), 0,    -2*√((3-phi)/5), 0>
<-√((5-√5)/10),   ±phi, -2*√((3-phi)/5), 0>
< √((5+2*√5)/5),  ±1,   -2*√((3-phi)/5), 0>

#
# Apex
#

# x5o:
<-√((10+2*√5)/5), 0,    √((2+phi)/5), 1/phi>
<-√((5-√5)/10),   ±phi, √((2+phi)/5), 1/phi>
<√((5+2*√5)/5),   ±1,   √((2+phi)/5), 1/phi>

The computation of the √((2+phi)/5) in the 3rd coordinate of the apex was quite a long nasty derivation involving denesting and re-nesting square roots. :glare: Had I known this earlier, I would've settled with the .4s3. coordinates instead. :angry: :XP: I almost had to do the same for the 1/phi, except that after having worked with phi-based coordinates for so long, I almost instantly recognized the value of 1/phi^2 in the squared edge lengths, so I could skip most of the derivation and just check the end result with the original vertex coordinates. :lol:

P.S. These coordinates assume edge length = 2.

EDIT 2: Added .def and .off files to the Pentagonal orthocupolarotunda pseudopyramid wiki page.
Last edited by quickfur on Tue Apr 01, 2014 3:34 am, edited 1 time in total.
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Re: Pseudopyramids

Postby quickfur » Mon Mar 31, 2014 7:13 pm

Looking at this projection, and the fact that the pentagon lies exactly midway between the top and bottom pentagons, it suggests that perhaps J38 (elongated pentagonal bicupola)||pentagon should also be CRF, and would be a 4th pseudopyramid?

Along another line, J40 (elongated pentagonal orthocupolarotunda) || pentagon prism should be CRF too, as a Stott expansion of the J32 pseudopyramid, right? (Or do the other lacing cells turn non-CRF under this expansion?)
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Re: Pseudopyramids

Postby Marek14 » Mon Mar 31, 2014 7:38 pm

Would it work with pentagonal gyrocupolarotunda?
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Re: Pseudopyramids

Postby quickfur » Mon Mar 31, 2014 9:06 pm

Marek14 wrote:Would it work with pentagonal gyrocupolarotunda?

I thought about that (in fact, in my initial construction I accidentally started with the gyrocupolarotunda instead of the ortho one). But it doesn't work, because a pentagonal antiprism would be required to lace the pentagons of different orientations, but the pentagonal antiprism is significantly shorter than the pentagonal prism, so either you have to break the bottom cell into a bent rotunda + cupola pair, or you have to stretch some edges into non-CRF lengths.

But maybe if we stretched out the apex into a full cell, we might be able to get a CRF out of it. It wouldn't be a pseudopyramid anymore, though, since the apex would be full-dimensioned!
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Re: Pseudopyramids

Postby Klitzing » Mon Mar 31, 2014 9:24 pm

quickfur wrote:And Klitzing was right, the apex is {5}, not gyro {5}. Here's a render of our pretty new CRF:

Image

It's interesting that the J91 and J92 pseudopyramids, as well as this one, all have height E/(2*phi) (where E = edge length). I suppose it's the result of all three being related to the Stott expansion of some faceting of the icosahedral pyramid.

Not at all. Just consider 2 pentagons and the underlying edge: that arrangement occures in the same way in the bilbiro-wedge, in the thawro-wedge, and in the pocuro-wedge!

--- rk
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Re: Pseudopyramids

Postby quickfur » Tue Apr 01, 2014 1:01 am

Hmm. Also based on the projection, it would appear that J34 (pentagonal orthobirotunda) || pentagonal_prism should be a segmentotope? It should be in Klitzing's list? Or is it non-CRF?
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Re: Pseudopyramids

Postby Klitzing » Tue Apr 01, 2014 5:45 am

Poscuro is not orbiform, thus poscuro||{5} cannot be a segmentochoron.
Pero||pip then might be better suited. But I fear they won't be aligned directly axial, but rather with a slight offset.
(I think, you have already the coordinates, so you could check.)

---rk
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Re: Pseudopyramids

Postby student91 » Thu Apr 03, 2014 3:11 pm

It just occured to me that a 600-cell lune can be augmented with a pocuro-pseudopyramid, in a way that makes a pocuro occur!
As you might know, the pocuro-pseudopyramid is part of D4.9.0 To show you what I mean, look at the top part of the lace-city of D4.9.0:
Code: Select all
                        o5x             o5x                       
                o5o             x5x            o5o               
                                                                                           
                    f5o                     f5o                   
            o5x                                    o5x           
                                o5F                               
                        f5x             f5x                       
                                                                                           
            x5x                                    x5x           
    x5o             o5F                     o5F             x5o   
                                                                                           
                F5o             o5V             F5o               
o5o                     f5f             f5f                     o5o
    o5f                                                     o5f   
            x5f                                    x5f           
                                                                                         
x5o                     V5o             V5o                     x5o
                                x5F                               
            F5o                                    F5o           
    x5x             f5f                     f5f             x5x   
            o5F                                    o5F           
                                F5x                               
o5x                     o5V             o5V                     o5x
                                                                                       
            f5x                                    f5x           
    f5o                                                     f5o   
o5o                     f5f             f5f                     o5o
                o5F             V5o             o5F               
                                                                                           
    o5x             F5o                     F5o             o5x   
            x5x                                     x5x           
                                                                                           
                        x5f             x5f                       
                                F5o                               
            x5o                                     x5o           
                    o5f                     o5f                   
                                                                                         
                o5o             x5x             o5o               
                        x5o             x5o
The top part looks like this:
Code: Select all
                        o5x             o5x                       
                o5o             x5x            o5o               
                                                                                           
                    f5o                     f5o                   
            o5x                                    o5x
That is, two rotunda's at an angle of 144, with some more augments. Now exactly these augments can be placed on the 144-degrees lune of the 600-cell:
Code: Select all
                 x5o           x5o
        o5o             x5x            o5o
                                                   
            o5f                     o5f           
     x5o                                   x5o     
                                                   
                 f5o           f5o                 
                        o5f                       
     o5o                                   o5o     
            o5x                     o5x           
                                                   
                        x5o                       
                 o5o           o5o

Now if you delete one of the pentagonal pyramids, you get a pocuro! :D
(in fact, you can also further diminish the o5o, so the rotunda is augmented with only a pentagon, and you get a rotunda||pentagon placed on top of the 144-lune)
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Re: Pseudopyramids

Postby quickfur » Sun Apr 06, 2014 12:22 am

student91 wrote:It just occured to me that a 600-cell lune can be augmented with a pocuro-pseudopyramid, in a way that makes a pocuro occur![...]

I've been meaning to get to this, but have been busy with other things... but this is a very interesting development. I'm going to try to construct this CRF sometime, hopefully tonight but might be busy with something else. Tomorrow I'll likely be busy as well, but maybe tomorrow night...

In any case, again this is showing that "underlying grid" that I kept alluding to... looks like it's not just the 4D tiling with the 600-cell + teddi's, but it's something more going on here. I've been trying to construct CRFs that contain J12 (triangular bipyramid), for example, and it seems that just about every combination I try doesn't work out. But with 600-cell related CRFs, you just randomly shoot in the dark and half the time you hit a CRF!

Now I'm wondering if there's a way to "back-construct" a larger chunk of the 600-cell from this augmented 600-cell 144°-lune. Like add in a few more vertices overlying the pentagonal cupolae and fill it out with CRF overlays. How many ways can you add vertices to a 144°-lune in a CRF way? or, for that matter, the hemi-600-cell (180°-lune)? I also note that this pocuro augmentation involves adding a rather thin augment to the 144°-lune (the height of a pocuro pseudopyramid is only 1/phi for edge length 2, whereas the nearest deleted layer of 600-cell vertices is phi units away (for edge length 2). Are thicker augments possible in a CRF way?

On another thought, this implies a non-convex polychoron with 600-cell symmetry and pocuro cells. :P (If you imagine the full 600-cell with all possible pocuro's embedded inside.)
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Re: Pseudopyramids

Postby quickfur » Sun Apr 06, 2014 2:46 pm

I have to run right now, but just a quick note: I've successfully constructed the pocuro'd 600-cell 144°-luna! Here's a very quick and rough render:

Image

I got stuck last night trying to find the coordinates, 'cos it wasn't obvious how to compute the vertices of the augmenting pentagon, until I realized this morning that pentagonal_rotunda||{5} is just a diminishing of o5x3o||o5o3x. So once I identified the coordinates of the pentagonal rotunda (which was nicely lying in a hyperplane with w=0, so it was a subset of the canonical o5x3o coordinates!), it was simple to identify the pentagonal section in o5o3x that corresponded to it. That was all I needed, since the height of the pentagon above the hyperplane of the rotunda is already known to be 1/phi (edge length=2). I confirmed with polyview that all edges are equal and there is a cell with 25 vertices, shown in yellow above. :D

I'll upload the files and coordinates later, gotta go! :mrgreen:
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Re: Pseudopyramids

Postby quickfur » Mon Apr 07, 2014 1:22 am

OK, uploaded software models to D4.12 page.

In the meantime, did you notice something interesting about the above projection? Especially the part where two square pyramids link the square face of the pocuro to the pentagonal prism in the pocuro pseudopyramid augment, surrounded by pairs of pentagonal pyramids? The J92 rhombochoron has a similar formation of two square pyramids linking the square faces of the J92 to the triangular prisms, also surrounded by pairs of pentagonal pyramids. :D There seems to be some underlying connection here, though I'm not sure what, because in the case of the J92 rhombochoron, there are 6 pairs of pentagonal pyramids and 6 formations of the double square pyramids, whereas here, there are only 5 pairs of pentagonal pyramids and 5 corresponding formations of the double square pyramids between them.
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Re: Pseudopyramids

Postby Marek14 » Mon Apr 07, 2014 6:53 am

quickfur wrote:OK, uploaded software models to D4.12 page.


Seems you've made a mistake -- the OFF file is just a copy of the DEF file.
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Re: Pseudopyramids

Postby quickfur » Mon Apr 07, 2014 2:36 pm

Marek14 wrote:
quickfur wrote:OK, uploaded software models to D4.12 page.


Seems you've made a mistake -- the OFF file is just a copy of the DEF file.

Oops, sorry!! :oops: Fixed.
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Re: Pseudopyramids

Postby Marek14 » Mon Apr 07, 2014 3:04 pm

Now I can see it :) Stella found 41 distinct kinds of cells there (32 kinds of tetrahedra, 2 kinds of square pyramids, 5 kinds of pentagonal pyramids and 1 each for pentagonal prisms and pocuros).
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