axial CRFs

[quickfur]

Wow, this CRF is quite a pretty little thing! Here's a parallel projection along the axis of the join:



The green tetrahedron is the top cell of this CRF, and the antipodal truncated tetrahedron is outlined in magenta edges. Most importantly, you can see 2 of the 3 pairs of gyrobifastigium (J26) cells sharing edges with the central tetrahedron. The other pair is not shown because it would clutter the image too much.

Here they are, shown separately:



You can easily spot the octahedral cells at the 4 corners of the tetrahedral-like projection envelope, as well as the 4 tetrahedra touching the central one. The triangular face opposite the shared vertices are where these tetrahedra join with the trigonal cupolae that link them to the antipodal truncated tetrahedron. You can also see the 4 trigonal prisms sharing a face with the central tetrahedron, and nestled between the J26 cells.

Here are the coordinates I used (~ denotes vector concatenation; apacs/apecs are Wendy's usual abbreviations):

Code: Select all

# x3o3o:
apecs<1/√2, 1/√2, 1/√2> ~ <-√(5/2)>

# o3x3o:
apacs<0, √2, √2> ~ <0>

# o3x3x:
apecs<1/√2, 1/√2, -3/√2> ~ <√(5/2)>


All in all, a very neatly-assembled CRF, with 1+4=5 tetrahedra, 4 trigonal prisms, 4 octahedra, 6 J26's, 4 trigonal cupolae, and 1 truncated tetrahedron, for a total of 24 cells.

[Klitzing]

In other news, I've been searching a long time for a CRF that contains triangular bipyramid cells (other than the obvious prism), but so far haven't found anything. Maybe you have some ideas about some monostratics with tetrahedra that might be able to join up at the right angles to form a bipyramid?

When considering connvex monostratic lace prisms with across symmetries o3oPo, which use lacing tetrahedra, then we just have the following list:

• pen = pt||tet, relevant dihedral tet-3-tet = arccos(1/4) = 75.522488°
• octpy = pt||oct, relevant dihedral tet-3-oct = 60°
• rap = tet||oct, relevant dihedral tet-3-oct = arccos(-1/4) = 104.477512°
• hex = tet||-tet, relevant dihedral tet-3-tet = 120°
• tetatut = tet||tut, relevant dihedral tet-3-tut = 60°
• tetaco = tet||co, relevant dihedral tet-3-co = arccos(1/4) = 75.522488°
• octacube = oct||cube, relevant dihedral tet-3-oct = arccos[-(3 sqrt(2)-2)/4] = 124.101465°
• cubaco = cube||co, relevant dihedral tet-3-co = arccos[-(3-sqrt(8))/sqrt(8)] = 93.477707°
• cubasirco = cube||sirco, relevant dihedral tet-3-sirco = 60°
• ikepy = pt||ike, relevant dihedral tet-3-ike = arccos(sqrt[7+3 sqrt(5)]/4) = 22.238756°
• ikadoe = ike||doe, relevant dihedral tet-3-ike = arccos[-sqrt(5/8)] = 142.238756°
• doaid = doe||id, relevant dihedral tet-3-id = arccos[(3-sqrt(5))/sqrt(32)] = 82.238756°
• doasrid = doe||srid, relevant dihedral tet-3-srid = arccos(sqrt[7+3 sqrt(5)]/4) = 22.238756°
• trippy = pt||trip, relevant dihedral tet-3-trip = arccos(sqrt[3/8]) = 52.238756°

thus, at least this list does not leave any such possibilities for according combinations
--- rk

[Klitzing]

Ha, found further nice ones:

when stacking the segmentochora octasirco and cubasirco, i.e. producing the exterior blend of both (thereby blending out the sirco), then the obtained stack has the description oct||sirco||cube = xxo3ooo4oxx&#xt.

Sure, this one clearly remain a "mere" stacking of segmentochora, i.e. no lacing cells here become corealmic and would combine in turn. But, when actually calculating all those dihedrals across the 3 types of (blended out) sirco faces, then one gets surprised: all 3 independently happen to be 90 degrees exactly!

Furthermore, the around-symmetrical Stott expanded version thereof, i.e. toe||girco||tic = xxo3xxx4oxx&#xt, therefore would share this behaviour.

--- rk

[quickfur]

Interesting. Do all of the lacing cells meet the other lacing cells at 90°? If so, could this CRF be used to produce a tiling of 4-space somehow? Note, interestingly, that one can tile 3-space with x4o3x's, octahedra, and cubes, so if we lay out copies of this CRF such that the equatorial x4o3x's touch each other at the axial cube facets, then we could fill in at least the octahedral gaps with the oct cells of slightly displaced copies of this CRF.

Not 100% sure what would happen to the remaining cubical gaps, though, or what the 4D shape above them would be. But it seems like they ought to be CRF gaps! If I'm not mistaken, they ought to have 2 squippies joined at the square face, and something else to close up the gap.

Could it be possible that this tiling, if it exists, is some kind of CRF modification of a uniform 4-space tiling? An EKF tiling perhaps?

[Klitzing]

Wow, great idea of yours, quickfur!
And yes, all three dihedrals are 90 degrees exactly in octasircoacube: that across the triangle, that across the axial squares and that across the squares in rhombical positions of that equatorial (blended out) sirco.

So let's have a look. We need some tetracomb which is an infinite stacking of euclidean layers A = a4x3o4o, B = o4x3o4x, and C = b4o3o4x. Then the part in the column . o3o4o would be exactly our octasircoacube. Here "a" and "b" are to be chosen appropriately such that the lateral extend (distances between according cells) will match in all 3 layers.

So far I managed to calculate "b" to be b=q here. You know, one could provide sidpith = x3o3o4x also as oxxo3oooo4xxxx&#xt. Further we have cope = x o3x4o either oqo4xox xxx&#xt or oo4xx3oo&#x. And then tes = o3o34x being either oo4xx xx&#x or oqo ooo4xxx&#xt. And finally rit = o3o3x4o = oqo4xox3ooo&#xt. Using these tower descriptions of those polychora one then can describe the tetracomb sidpitit = x4o3o3x4o also as infinite stacking :BBC: of the layers B = o4x3o4x and C = q4o3o4x.

Therefore the lower half already is solved. Remains the upper one to be done ...

--- rk

[Klitzing]

Wow, great idea of yours, quickfur! [...]

So let's have a look. We need some tetracomb which is an infinite stacking of euclidean layers A = a4x3o4o, B = o4x3o4x, and C = b4o3o4x. Then the part in the column . o3o4o would be exactly our octasircoacube. Here "a" and "b" are to be chosen appropriately such that the lateral extend (distances between according cells) will match in all 3 layers.

So far I managed to calculate "b" to be b=q here. [...]
Therefore the lower half already is solved. Remains the upper one to be done ...

--- rk

In order to get the right distances in the parallel layers, one has to choose furthermore "a" to be a=x. Thus our layers to be considered are A = x4x3o4o = tich, B = o4x3o4x = srich, and C = q4o3o4x (some q,x-variant of chon with x-cubes, prolate x-square prisms of height q, oblate q-square prisms of heigth x, and q-cubes).

I did not see so far, from which uniform tetracomb the segment AB would derive out. In fact, it uses some oct||sirco (the one for our octasircoacube bistratic tower, which is the main point of interest of all this investigation), which happens to be the monostratic cap of spic = x3o4o3x and it uses also elsewhere co||tic, the monostratic cap of srico = x3o4x3o.

But then I desided not to bother about that. Rather we already have all necessary ingrediants for the searched for tetracomb! We just will have to stack those mentioned layers periodically as :ABCB:, i.e. consider the infinte tower of euclidean honeycombs  :xoqo:4:xxox:3:oooo:4:oxxx:&##x.

The to be used polychora in that tetracomb then are:

• oxo4xxx3ooo ...&#xt = coaticbicu (a mere convex stack of 2 segmentochora) with object frequency 1/22
• oqo4xox3ooo ...&#xt = rit = o3o3x4o with object frequency 1/22 too
• xo4xx .. ox&#x = squicuf (the well-known segmentochoron {8}||cube) with object frequency 6/22
• oqo4xox ... xxx&#xt = cope = o3x4o x with object frequency 3/22
• xo .. oo4ox&#x = squasc (the well-known segmentochoron line||ortho {4}) with object frequency 6/22
• oqo ... ooo4xxx&#xt = tes = o3o3o4x with object frequency 3/22
• ... xxo3ooo4oxx&#xt = octasircoacube (a mere convex stack of 2 segmentochora) with object frequency 2/22

Thus yes, that recently described octasircoacube indeed can be used to tile 4D space (together with some further constituents).

--- rk

[Klitzing]

By mere application of around-symmetrical Stott expansion wrt. that axial stacking of vertex layers one can obtain that "other" 4D filling (tetracomb) therefrom, which then incorporates the other recently mentioned special bistratic lace tower toagircoatic = xxo3xxx4oxx&#xt, which likewise has equatorial dihedrals of 90 degrees only.

In fact the respective polychora there would be accordingly:

• oxo4xxx3xxx ...&#xt = toagircobcu (a mere covex stack of 2 segmentochora), frequency 1/22
• oqo4xox3xxx ...&#xt = pabdirico (the parabidiminished o3x4o3o), frequency 1/22
• xo4xx .. ox&#x = squicuf (unchanged), frequency 6/22
• oqo4xox ... xxx&#xt = cope (unchanged), frequency 3/22
• xo .. xx4ox&#x = squicuf (further ones here), frequency 6/22 too
• oqo ... xxx4xxx&#xt = sodip = x4o x4x, frequency 3/22
• ... xxo3xxx4oxx&#xt = toagircoatic, frequency 2/22

And the vertex layers or crosssections in this periodic stacking :ABCB: here would be A = x4x3x4o (grico), B = o4x3x4x (grico with different orientation / alignment), C = q4o3x4x for sure.

--- rk

[username5243]

I came up with another segmentochoral stack with a blended lateral cell. This one is also a diminishing of a uniform polychoron, but I figured it belonged here.

The polychoron in question is xoo3oxx4xxx&#xt, sirco ||tic || tic. Its cells seem to be 1 sirco + 6 escues (J19) + 8 octs + 12+8 = 20 trips + 1 tic.

this iss not only a segmentochoral stack, but also a sirco-first diminishing of srit (x4o3x3o), which exposes a tic at the bottom of that cap. 6 of the sircoes each get a cupolaic cap removed, leaving J19s, while 8 octs and 12 trips get reduced to mere triangular faces and edges respectively.

In fact, this one reminds me of J19 itself, how it is both a segmentochoral stack and a diminishing of a uniform polychoron.

Just another one I thought of that I figured was worth a mention.

[quickfur]

@username5243: yes, this is a simple diminishing of x4o3x3o. It is essentially a Stott-expanded version of the elongated bisected 24-cell (aka the bisected rectified 16-cell), which contains elongated square pyramids, and has been known for a while now.

You can also make an elongated 24-cell, which contains elongated square bipyramids, by partial Stott expansion of the 24-cell (along 1 axis instead of the full symmetry). Conversely, you can get a 24-cell out of x4ox3o by Stott contraction, and various intermediate forms via partial Stott contraction and/or diminishings.

IOW, your elongated cupolae are just truncations of Stott-expanded octahedra.

[Klitzing]

Yes, there are several bistratic lace towers, which could be obtained primarily as a mere stack of 2 segmentochora, but having in addition some of their lacing facets corealmic thereby. E.g. in your case of xoo3oxx4xxx&#xt you attach sirco||tic to tic||tic. And it happens that the squacues of the former are corealmic to the ops of the latter. Thus those combine to escues there.

An alternate example would be obtained therefrom by means of Stott contraction. I.e. when you stack oct||co on co||co, yielding xoo3oxx4ooo&#xt. then the squippies of the first are corealmic to the cubes of the latter. Thus combining to esquippies then.

In both these cases those thus combined lacing cells are vertical. But there are also other examples where such combined lacings are still slanted! E.g. oox3ooo4oxx&#xt also shows up slanted lacing esquippies, and oox3xxx4oxx&#xt correspondingly shows up slanted lacing escues.

--- rk

[quickfur]

Wow! Very nice find!!! I would never have thought to look for this combination, because the trigonal prisms are sloping (not perpendicular to the axis of the join), so it's not immediately obvious that the combination would be CRF! Maybe I'll build a model for this later when I get some time.

Yes indeed they are,
as best can be seen in this lace city display:

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      x o   o x            -- tet
                    
                    
   x x  uo ou  x x         -- co
                    
                    
o x   x u   u x   x o      -- inv tut

                     \
                      +-- gybef


--- rk

@Klitzing, btw, did you ever name this nice little CRF? I'm thinking of putting it up on my website, but I wasn't sure what to call it. Naming it merely as a bistratic stack gives an overly-long title. Any good ideas?

[Klitzing]

That one is being provided here.

There I provided the Bowers style acronym "tetaco altut" extending the segmentotopal naming convention to bistratic stacks as well:
tet = tetrahedron  +  a = atop  +  co = cuboctahedron  +  a = atop  +  l = alternate (here: inverted)  +  tut = truncated tetrahedron

--- rk

[quickfur]

Hmm. That's still quite a mouthful of a name. Makes me wish we had a numerical naming system like Jxxx for the Johnson solids or K4.xxx for your segmentochora... But I suppose that's going to be a hard task, since we don't know the full extent of the 4D CRFs so far. And the numerical labels are liable to be very long anyway, given the sheer amount of 4D CRFs known to exist, even without counting all possibilities.

[quickfur]

WHOA. I just discovered something amazing. Today, I wondered what would happen if I tried to Stott-expand tetaco-altut by moving the gybefs apart by 1 edge length. The lace tower then becomes:

Code: Select all

x3x3o
x3x3x
o3u3x


which, of course, is non-CRF. But a careful examination of the projections seem to indicate that I can complete the incomplete hexagons, which, after working it out, reveals the next vertex layer as o3x3u. There are some new incomplete hexagons that show up, but it seems to be outlining 4 x3x3x's on the far side of the polytope...

and then it struck me: this is nothing other than an augmentation of x3x3x3o !!!

In other words, x3x3x3o can be augmented by x3x3o||x3x3x to produce an augmentation that contains 4 gybef cells! This seems to be some kind of analogue of the augmented truncated tetrahedron in 3D!

[username5243]

WHOA. I just discovered something amazing. Today, I wondered what would happen if I tried to Stott-expand tetaco-altut by moving the gybefs apart by 1 edge length. The lace tower then becomes:

Code: Select all

x3x3o
x3x3x
o3u3x


which, of course, is non-CRF. But a careful examination of the projections seem to indicate that I can complete the incomplete hexagons, which, after working it out, reveals the next vertex layer as o3x3u. There are some new incomplete hexagons that show up, but it seems to be outlining 4 x3x3x's on the far side of the polytope...

and then it struck me: this is nothing other than an augmentation of x3x3x3o !!!

In other words, x3x3x3o can be augmented by x3x3o||x3x3x to produce an augmented that contains 4 gybef cells! This seems to be some kind of analogue of the augmented truncated tetrahedron in 3D!

As soon as I read this, I wondered if a similar thing could be done to the original lace tower.

I figured out that the likely base in that case would be srip (x3o3x3o), augmented with tet ||co. Checking the dihedral angles it appears to still work out with the gybefs intact!

So, you could call that "Diminished augmented small rhombated pentachoron"... Not sure that's much of an improvement though

[quickfur]

P.S., here's the lace tower of the augmented x3x3x3o containing 4 gybefs:

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x3x3o
x3x3x
o3u3x
o3x3u
o3x3x


(I think I got the symbols right. Not 100% sure, was working directly with the vertex coordinates.)

This interesting CRF makes me wonder if other 5-cell family uniforms can be similarly augmented with augments containing lacing cells that blend into some kind of Johnson solid.

[Klitzing]

Yes, those are nice finds indeed!
Here is the lace city of Quickfur's find in that orientation, which shows up the gybefs nicely:

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      o x   x u   u x   x o            -- o3x3x (inv tut)
                                
                                
   x x   u u  xw wx  u u   x x         -- x3x3x (toe)
                                
                                
x o         w u   u w         o x      -- x3u3o ((x,u)-tut)
                                
                                
   u o   w x         x w   o u         -- u3x3o ((u,x)-tut)
                                
                                
      x o   u x   x u   o x             -- x3x3o (tut)


And this then is the according lace city of Username5243's find in according orientation, again featuring those gybefs quite nicely:

Code: Select all

      x o   o x            -- x3o3o (tet)
                    
                    
   x x  ou uo  x x         -- x3o3x (co)
                    
                    
o x   x u   u x   x o      -- o3x3x (inv tut)
                    
                    
   o o   x x   o o         -- o3x3o (oct)


The uppermost segment each then is the (correctly oriented) augmentation, the remainder then represents grip (above) resp. srip (below).

--- rk

[username5243]

<snip>

This interesting CRF makes me wonder if other 5-cell family uniforms can be similarly augmented with augments containing lacing cells that blend into some kind of Johnson solid.

A quick search of the first page of the thread (where Klitzing first created several such bistratic towers) gave me probably the simplest idea of how to make tihs work.

Point || tet ||co has etripies (J7) as lateral cells. Tet || co is just a half of spid (x3o3o3x), so spid should be augmentable by a tet pyramid (ie, pentachoron) to produce something with 4 J7s.

The Stott-expanded version of the bistratic tower above is oct || tut || toe, wich has etcues (J18) as cells. Tut || toe is a cap of prip (x3x3o3x), so augmenting prip with oct || tut should likewise produce something with 4 J18s.

[quickfur]

[...]
Point || tet ||co has etripies (J7) as lateral cells. Tet || co is just a half of spid (x3o3o3x), so spid should be augmentable by a tet pyramid (ie, pentachoron) to produce something with 4 J7s.

Unfortunately, point || tet || co is not CRF: the dichoral angle of point || tet is too steep compared to tet || co, causing the polytope to be concave at the tet joint.

So no J7-containing CRF with this construction.