Johnsonian Polytopes

[Klitzing]

Klitzing's paper says 4.73, the square || square cupola, has cells: 4 tetrahedra + 2 square pyramids + 4 trigonal prisms + 1 cube.

Shouldn't that be 2 square cupolae, not square pyramids? 4.73 should be the square orthobicupolic ring.

Haha, right you are!
--- rk

[quickfur]

I've found another bistratic: the bidiminished rectified 24-cell:



This polychoron is constructed by deleting the vertices of two antipodal cuboctahedra from the rectified 24-cell. It has 2 truncated octahedra, 6 cuboctahedra, 12 cubes, and 16 triangular cupolae.

I'll be posting the coordinates and stuff to the wiki; just posting it here for now in case it leads to discovery of more bistratics.

[Keiji]

Bidiminished 3-xylotomochoron, please

(Rectified is ambiguous, or at least misleading, so I want to move away from that word)

Good work though! If that's a bistratic, does that mean it can be divided into two (in this case identical) monostratic halves? What shape would each half be?

[quickfur]

Bidiminished 3-xylotomochoron, please

(Rectified is ambiguous, or at least misleading, so I want to move away from that word)

If we want to be truly unambiguous, we should use Wendy's notation for it: bidiminished x3x4o3o (or my equivalent notation: bidiminished F1100, where F denotes the symmetry group of the 24-cell).

But "bidiminished" isn't really that unambiguous either, so we might as well use Wendy's lacing notation: the layers are o4x3x || q4o3x || o4x3x, (q denotes sqrt(2) at the corresponding node in the Coxeter-Dynkin diagram, so the middle layer is a non-uniform rhombicuboctahedron with edge length sqrt(2) between two rectangles, and length 1 between rectangle/triangle). So the lacing notation would be: oqo4xox3xxx&#xt. (I'm not 100% confident with the notation yet, wendy/klitzing please correct me if I messed it up.)

Good work though! If that's a bistratic, does that mean it can be divided into two (in this case identical) monostratic halves? What shape would each half be?

It'd be rhombicuboctahedron||truncated octahedron, but the rhombicuboctahedron has uneven edge lengths (q4o3x, so the 6 axial squares have edge length sqrt(2), the 12 rectangles have edge lengths 1 and sqrt(2), and the triangles have edge length 1).

(Side-note: the 4-teddy is also bistratic, you can cut it into tetrahedron||tetrahedron2 and tetrahedron2||octahedron, where tetrahedron2 is tetrahedron scaled by the golden ratio.)

[quickfur]

Bidiminished 3-xylotomochoron, please [...]

Oh btw, where does the 3 come from? The rectified 24-cell I'm talking about is oxoo (CD diagram), not xxoo.

[Klitzing]

Bidiminished 3-xylotomochoron, please

(Rectified is ambiguous, or at least misleading, so I want to move away from that word)

Good work though! If that's a bistratic, does that mean it can be divided into two (in this case identical) monostratic halves? What shape would each half be?

Quite easy: the figure displayed is xxx3xox4oqo&#xt. Accordingly the monostratic half would be xx3xo4oq&#x.
The main crux here is that you have cuboctahedra reaching with their squares from one base to the other. If those are bisected you would get sqrt2-squares in the middle. Also the cubes will be dissected, while reaching with opposite edges from base to base, into 1 x sqrt2 rectangles.
Thus yes, there is a monostratic half. But that one is no longer unit-edged only.

--- rk

[Keiji]

If we want to be truly unambiguous, we should use Wendy's notation for it: bidiminished x3x4o3o

But that's hard to read, and easier to get one bit wrong when writing.

Speaking of which... 3-tome = oxoo, known to the rest of the world as "rectified"... but the notation you wrote there is xxoo, which is truncated, or 2-tome!

Which shape did you actually mean, oxoo or xxoo? This is why we should use unambiguous names

Oh btw, where does the 3 come from? The rectified 24-cell I'm talking about is oxoo (CD diagram), not xxoo.

Glad I saw this before hitting submit. So the xxoo was a typo? (or does wendy's notation just do weird things and change the first o to an x? )

The tome number is not the Dx number, and it does not represent all Dx numbers.

1-tome = xoooo...
2-tome = xxooo...
3-tome = oxooo...
4-tome = oxxoo...
5-tome = ooxoo...
6-tome = ooxxo...
etc.

Simply put, the tome number is how "far" you are between the parent (1-tome) and its dual ((2n-1)-tome). The tomotopes are the polytopes that can be written with tome numbers. Note that the pantome isn't a tomotope, it's a mesotruncate (doubled symmetry polytope).

we might as well use Wendy's lacing notation [...] oqo4xox3xxx&#xt

That still looks alien to me. I'd love to learn it, but... in before wendy tries her best to explain it and it still goes way over my head.

It'd be rhombicuboctahedron||truncated octahedron

So K4.75? Please, if it's a convex segmentochoron, give its K4.x number, just so we can be on the same page.

Also, rhombicuboctahedron is a mouthful, the word you want this time is stauroperihedron.

[quickfur]

If we want to be truly unambiguous, we should use Wendy's notation for it: bidiminished x3x4o3o

But that's hard to read, and easier to get one bit wrong when writing.

Speaking of which... 3-tome = oxoo, known to the rest of the world as "rectified"... but the notation you wrote there is xxoo, which is truncated, or 2-tome!

Ahhhahahaha... I made yet another slip-up. I meant o3x4o3o. Or, in my (slightly more readable, if I say so myself) notation, F0100.

Which shape did you actually mean, oxoo or xxoo? This is why we should use unambiguous names

Yeah, if we could only agree on which naming scheme to use.

Oh btw, where does the 3 come from? The rectified 24-cell I'm talking about is oxoo (CD diagram), not xxoo.

Glad I saw this before hitting submit. So the xxoo was a typo? (or does wendy's notation just do weird things and change the first o to an x? )

It's yet another typo from yours truly.

The tome number is not the Dx number, and it does not represent all Dx numbers.

1-tome = xoooo...
2-tome = xxooo...
3-tome = oxooo...
4-tome = oxxoo...
5-tome = ooxoo...
6-tome = ooxxo...
etc.

Simply put, the tome number is how "far" you are between the parent (1-tome) and its dual ((2n-1)-tome). The tomotopes are the polytopes that can be written with tome numbers. Note that the pantome isn't a tomotope, it's a mesotruncate (doubled symmetry polytope).

Yikes! I find this inordinately confusing. IMO we should stick to CD diagram derivations, as those unambiguously and neatly covers all possibilities.

we might as well use Wendy's lacing notation [...] oqo4xox3xxx&#xt

That still looks alien to me. I'd love to learn it, but... in before wendy tries her best to explain it and it still goes way over my head.

I'll try to explain (and wendy/klitzing please correct me if I'm wrong): the way you do it is by first writing out the layers in your lace tower:

Code: Select all

o4x3x
q4o3x
o4x3x


(the q in the second layer, as I said, means sqrt(2), this means the corresponding node on the CD diagram is effectively scaled by sqrt(2) to produce a non-equal edge rhombicuboctahedron.)

Since all layers have the same symmetry, we can abbreviate it by grouping columns together (with elements in each column read from top to bottom), so you have oqo (1st column) to the left of 4, then xox (2nd column) between 4 and 3, then xxx (3rd column) to the right of 3. This lets us compress it into a linear notation: oqo4xox3xxx. The suffix &#t means you make a lace tower from the preceding string (this page may help, see the Notation section at the bottom).

It'd be rhombicuboctahedron||truncated octahedron

So K4.75? Please, if it's a convex segmentochoron, give its K4.x number, just so we can be on the same page.

Also, rhombicuboctahedron is a mouthful, the word you want this time is stauroperihedron.

It's only an improvement in 1 syllable. I like my own notation better: C101 (C represents n-cube symmetry, and 1 means a ringed node, so C101 means cube symmetry, CD diagram xox, that is, x4o3x). Or my pet name for it, "rhombicube".

I think using Klitzing's double-bar notation is clear enough, since his paper has an index in the back where you can look up the page where the segmentochoron is described. It's kinda troublesome to keep looking up these numbers if the double bar notation itself already gives you an unambiguous construction of the shape (I'm very bad at remembering numbers).

[Keiji]

Yeah, if we could only agree on which naming scheme to use.

I guess I'll make a note to add a label-translator to the next version of this forum/wiki, so you can post your name for something inside label-tags, and it'll appear as my name for it when I read what you wrote.

tomotopes

Yikes! I find this inordinately confusing. IMO we should stick to CD diagram derivations, as those unambiguously and neatly covers all possibilities.

But they're not exactly pronuncible.

Wendy's lacing notation [...] oqo4xox3xxx&#xt

I'll try to explain...

Ah, that's nice and simple actually

What if you want a sqrt(2)-scaled ringed node, though? (or is that just unnecessary?)

Also, rhombicuboctahedron is a mouthful, the word you want this time is stauroperihedron.

It's only an improvement in 1 syllable. I like my own notation better: C101 (C represents n-cube symmetry, and 1 means a ringed node, so C101 means cube symmetry, CD diagram xox, that is, x4o3x). Or my pet name for it, "rhombicube".

C-one-oh-one might be fine, but you wait til 5D... C-one-oh-oh-oh-one? C-one-zero-zero-zero-one? No, I prefer stauroperiteron, thank you very much

I think using Klitzing's double-bar notation is clear enough, since his paper has an index in the back where you can look up the page where the segmentochoron is described. It's kinda troublesome to keep looking up these numbers if the double bar notation itself already gives you an unambiguous construction of the shape (I'm very bad at remembering numbers).

Yes, you can look them up, however the advantage of giving the K4.x number is that it's then immediately obvious that it's an actual convex segmentochoron (as opposed to something that is, somehow, nonconvex, or orbiform, or without all edge lengths equal). Since you were just talking about a shape that didn't have all edge lengths equal, I could not be sure that it could be deformed to have equal edge lengths without looking up the double-bar notation.

[quickfur]

Yeah, if we could only agree on which naming scheme to use.

I guess I'll make a note to add a label-translator to the next version of this forum/wiki, so you can post your name for something inside label-tags, and it'll appear as my name for it when I read what you wrote.

lol... that'll be neat if you can pull it off. And maybe have a wendy-mode for converting to wendy's notation too. Better yet, have a toggle that will translate between them upon clicking on the name.

tomotopes

Yikes! I find this inordinately confusing. IMO we should stick to CD diagram derivations, as those unambiguously and neatly covers all possibilities.

But they're not exactly pronuncible.

What I meant was a nomenclature that maps directly to the CD diagram.

And BTW, it's not the "tomotopes" that I find confusing, but the fact that you have both CD-diagram derived things like peritope, and a separate scheme for tomotopes based on the series of regular-to-dual truncations (3-tomotope, where the 3 has a non-trivial mapping to the CD diagram).

Wendy's lacing notation [...] oqo4xox3xxx&#xt

I'll try to explain...

Ah, that's nice and simple actually

What if you want a sqrt(2)-scaled ringed node, though? (or is that just unnecessary?)

The q node is a ringed node.

Or should I say, ringed nodes in the CD diagram traditionally implies an expansion of the appropriate distance such that edge length becomes 1. Wendy extended this concept to include expansions that generate non-equal edge lengths. Unmarked CD nodes don't have any associated scale. This is perhaps best understood by the mirrors interpretation of the CD diagrams: an unmarked node means the point lies on the mirror, i.e., o means distance = 0; x means distance = 1; q means distance = sqrt(2); and f means distance = golden ratio (1+sqrt(5))/2. Once a point is placed in this way according to the ringings of the CD diagram, then the (recursive) reflection images of the point generates the vertices of the polytope.

Also, rhombicuboctahedron is a mouthful, the word you want this time is stauroperihedron.

It's only an improvement in 1 syllable. I like my own notation better: C101 (C represents n-cube symmetry, and 1 means a ringed node, so C101 means cube symmetry, CD diagram xox, that is, x4o3x). Or my pet name for it, "rhombicube".

C-one-oh-one might be fine, but you wait til 5D... C-one-oh-oh-oh-one? C-one-zero-zero-zero-one? No, I prefer stauroperiteron, thank you very much

And what would you call C10110 then? Or C10111?

I think using Klitzing's double-bar notation is clear enough, since his paper has an index in the back where you can look up the page where the segmentochoron is described. It's kinda troublesome to keep looking up these numbers if the double bar notation itself already gives you an unambiguous construction of the shape (I'm very bad at remembering numbers).

Yes, you can look them up, however the advantage of giving the K4.x number is that it's then immediately obvious that it's an actual convex segmentochoron (as opposed to something that is, somehow, nonconvex, or orbiform, or without all edge lengths equal). Since you were just talking about a shape that didn't have all edge lengths equal, I could not be sure that it could be deformed to have equal edge lengths without looking up the double-bar notation.

Right, so a reference to a K4.x number would be inaccurate, since one might imagine that the edge lengths were actually equal.

[quickfur]

Hmm. Just realized that there are two bidiminishings of the rectified 24-cell (henceforth: F0100); the one I showed is a parabidiminishing. It's also possible to have a metabidiminishing.

(What's meta? If we cut the 24-cell into 5 layers: point, cube, octahedron, cube, point, then if the first vertex falls on the first point, and the 2nd vertex falls on the second cube (i.e. separated by the first cube and octahedron layers), then the two vertices are meta. Now if we map the 24 vertices of the 24-cell to the 24 cuboctahedra of the F0100, then the metabidiminishing is when two cuboctahedra which correspond to a pair of meta vertices are deleted from the polychoron.)

I haven't checked, but I think anything closer than meta will produce a non-CRF, so these are the only possible bidiminishings.

However, it is possible to have a metatridiminishing, produced by deleting the vertices of three cuboctahedra which are meta to each other. Each diminishing produces a gap that can be filled in by a truncated octahedron; the metatridiminishing (which I believe is a maximal diminishing) produces 3 truncated octahedra that are adjacent to each other, and lie on a great circle. The gaps in between are filled in by an interesting band of interconnected trigonal cupolae and cubes: 18 cupolae and 6 cubes.



I colored two of the truncated octahedra green and red, respectively, and outlined the third with red edges. The cubes are rendered in yellow. The triangular cupolae are a bit hard to see, but there are 3 around every adjacent pair of cubes.

I believe this is a maximal diminishing of the F0100 (but I could be wrong). It appears to have some kind of symmetry derived from the 3,6-duoprism, but with truncated octahedra in one ring and a complicated structure of cubes and trigonal cupola in the other. Most fascinating!

[Keiji]

(What's meta? If we cut the 24-cell into 5 layers: point, cube, octahedron, cube, point, then if the first vertex falls on the first point, and the 2nd vertex falls on the second cube (i.e. separated by the first cube and octahedron layers), then the two vertices are meta. Now if we map the 24 vertices of the 24-cell to the 24 cuboctahedra of the F0100, then the metabidiminishing is when two cuboctahedra which correspond to a pair of meta vertices are deleted from the polychoron.)

No, this is meta

On a more serious note though, I would have thought possible diminishings would increase by dimension? We have (meta)tridiminishings in 3D Johnson solids...

I believe this is a maximal diminishing of the F0100 (but I could be wrong). It appears to have some kind of symmetry derived from the 3,6-duoprism, but with truncated octahedra in one ring and a complicated structure of cubes and trigonal cupola in the other. Most fascinating!

Very nice

Try rendering it once for each ring, oriented so that the ring in question lies in the xy plane so we "look down the middle". That might look nice (or it might not, but it's worth trying )

[Klitzing]

we might as well use Wendy's lacing notation [...] oqo4xox3xxx&#xt

That still looks alien to me. I'd love to learn it, but... in before wendy tries her best to explain it and it still goes way over my head.

Whats hard to read here? Those are just stacked Dynkin symbols:
oqo4xox3xxx&#xt just means the following:
It is a lace tower (last "t") of 3 consecutive vertex layers (a tower of just 2 layers is called a lace prism, the "t" thus will be dropped).
Those vertex layers are described as o..4x..3x.. (= o4x3x) || .q.4.o.3.x. (= q4o3x) || ..o4..x3..x (= o4x3x).
Here "o" means an unringed node of that Dynkin graph, "x" a ringed one, that is, representing a unit-edge.
As some special non-unit edge length do occure often, Wendy has named those as well:
x = unit edge
q = sqrt2 edge
h = sqrt3 edge
f = tau edge
v = 1/tau edge
u = 2*unit edge
Accordingly q4o3x looks like a sirco, but the edges of the squares in cubical position are sqrt2 times larger than those of the triangles.
Finally there is a "&#x" part in that symbol, meaning that in addition (&) the lacings (#) have unit edges (x) only as well.

--- rk

[Klitzing]

Hmm. Just realized that there are two bidiminishings of the rectified 24-cell (henceforth: F0100); the one I showed is a parabidiminishing. It's also possible to have a metabidiminishing.

(What's meta? If we cut the 24-cell into 5 layers: point, cube, octahedron, cube, point, then if the first vertex falls on the first point, and the 2nd vertex falls on the second cube (i.e. separated by the first cube and octahedron layers), then the two vertices are meta. Now if we map the 24 vertices of the 24-cell to the 24 cuboctahedra of the F0100, then the metabidiminishing is when two cuboctahedra which correspond to a pair of meta vertices are deleted from the polychoron.)

I haven't checked, but I think anything closer than meta will produce a non-CRF, so these are the only possible bidiminishings.

However, it is possible to have a metatridiminishing, produced by deleting the vertices of three cuboctahedra which are meta to each other. Each diminishing produces a gap that can be filled in by a truncated octahedron; the metatridiminishing (which I believe is a maximal diminishing) produces 3 truncated octahedra that are adjacent to each other, and lie on a great circle. The gaps in between are filled in by an interesting band of interconnected trigonal cupolae and cubes: 18 cupolae and 6 cubes.



I colored two of the truncated octahedra green and red, respectively, and outlined the third with red edges. The cubes are rendered in yellow. The triangular cupolae are a bit hard to see, but there are 3 around every adjacent pair of cubes.

I believe this is a maximal diminishing of the F0100 (but I could be wrong). It appears to have some kind of symmetry derived from the 3,6-duoprism, but with truncated octahedra in one ring and a complicated structure of cubes and trigonal cupola in the other. Most fascinating!

Nice observation, indeed.
But I suppose you were refering to pt || cube || q-oct || cube || pt instead.
para-distances are then those 2 pt, or likewise any opposite vertex pair of that equatorial q-oct.
meta-distances are any 3 pairwise neighboured vertices of that q-oct, or likewise 2 such and either of the points.

As you observed metabidiminishings will be allowed, thus the maximal diminishing would cut off all those tips, i.e. any of the 6 vertices of the q-oct and those 2 points. I.e. 8-diminished in total...

--- rk

[quickfur]

[...]
(What's meta? If we cut the 24-cell into 5 layers: point, cube, octahedron, cube, point, then if the first vertex falls on the first point, and the 2nd vertex falls on the second cube (i.e. separated by the first cube and octahedron layers), then the two vertices are meta. Now if we map the 24 vertices of the 24-cell to the 24 cuboctahedra of the F0100, then the metabidiminishing is when two cuboctahedra which correspond to a pair of meta vertices are deleted from the polychoron.)
[...]

Nice observation, indeed.
But I suppose you were refering to pt || cube || q-oct || cube || pt instead.

Well, I was referring to "octahedron" in general, scaled appropriately.

para-distances are then those 2 pt, or likewise any opposite vertex pair of that equatorial q-oct.
meta-distances are any 3 pairwise neighboured vertices of that q-oct, or likewise 2 such and either of the points.

When I studied the diminishings of the 24-cell, I classified vertex distances into 4 categories. Since distances are defined by a pair of points, we can assume without loss of generality that the first point falls on the first pt in the above decomposition. Then the vertices on the cube next to it are adjacent; the vertices on the q-oct are ortho (I use this in my listing of 24-cell diminishings, though it doesn't apply here), the vertices on the second cube are meta, and the opposite pt is para.

[Marek14]

Tetrahedral vertices in CRF polychora:

I used Mathematica to check on various tetrahedra for the purpose of CRF polychora:
Even though the list only shows tetrahedral verfs, it already points to several promising areas of research, I think.

They are in form of abcdef, where faces are abd, ace, bcf and def. Allowed faces are:
333, 334, 335, 344, 346, 348, 340, 355, 350, 366, 388, 300, 444, 445, 446, 447, 448, 449, 440, 450, 466, 468, 460, 555 and 566. I'll refer to them with the simplest polyhedron that has them, if there are several.
Seems that euclidean tilings tetrahedra appear here as well (for example 466664). Not sure how to weed them out.

333333: four tetrahedra (verf of pentachoron)
333334: two tetrahedra, two square pyramids
333335: two tetrahedra, two pentagonal pyramids
333344: tetrahedron, two square pyramids, triangular prism
333355: tetrahedron, two pentagonal pyramids, metabidiminished icosahedron
333444: three square pyramids, cube
333445: two square pyramids, pentagonal pyramid, pentagonal prism
334344: tetrahedron, three triangular prisms (verf of tetrahedral prism)
334346: tetrahedron, triangular prism, two triangular cupolas
334348: tetrahedron, triangular prism, two square cupolas
334340: tetrahedron, triangular prism, two pentagonal cupolas
334366: tetrahedron, two triangular cupolas, truncated tetrahedron
334388: tetrahedron, two square cupolas, truncated cube
334434: two square pyramids, two triangular prisms
334436: two square pyramids, two triangular cupolas
334438: two square pyramids, two square cupolas
334430: two square pyramids, two pentagonal cupolas
334444: square pyramid, two triangular prisms, cube
334446: square pyramid, triangular prism, triangular cupola, hexagonal prism
334448: square pyramid, triangular prism, square cupola, octagonal prism
334440: square pyramid, triangular prism, pentagonal cupola, decagonal prism
334466: square pyramid, two triangular cupolas, truncated octahedron
334530: square pyramid, pentagonal pyramid, pentagonal cupola, pentagonal rotunda
334544: pentagonal pyramid, two triangular prisms, pentagonal prism
335355: tetrahedron, three metabidiminished icosahedra
335350: tetrahedron, metabidiminished icosahedron, two pentagonal rotundas
335300: tetrahedron, two pentagonal rotundas, truncated tetrahedron
335450: square pyramid, metabidiminished icosahedron, pentagonal rotunda, diminished rhombicosidodecahedron
335535: two pentagonal pyramids, two metabidiminished icosahedra
335530: two pentagonal pyramids, two pentagonal rotundas
335555: pentagonal pyramid, two metabidiminished icosahedra, dodecahedron
336366: tetrahedron, three truncated tetrahedra (verf of truncated pentachoron)
336444: square pyramid, two triangular cupolas, cube
336446: square pyramid, triangular cupola, truncated tetrahedron, hexagonal prism
336466: square pyramid, two truncated tetrahedra, truncated octahedron
336544: pentagonal pyramid, two triangular cupolas, pentagonal prism
336566: pentagonal pyramid, two truncated tetrahedra, truncated icosahedron
338388: tetrahedron, three truncated cubes (verf of truncated tesseract)
338444: square pyramid, two square cupolas, cube
338448: square pyramid, square cupola, truncated cube, octagonal prism
330300: tetrahedron, three truncated dodecahedra (verf of truncated 120-cell)
330445: square pyramid, pentagonal cupola, pentagonal rotunda, pentagonal prism
330440: square pyramid, pentagonal cupola, truncated dodecahedron, decagonal prism
330450: square pyramid, pentagonal rotunda, truncated dodecahedron, diminished rhombicosidodecahedron
330540: pentagonal pyramid, pentagonal cupola, truncated dodecahedron, diminished rhombicosidodecahedron
330555: pentagonal pyramid, two pentagonal rotundas, dodecahedron
344443: four triangular prisms (verf of 3,3-duoprism)
344444: two triangular prisms, two cubes (verf of 3,4-duoprism)
344445: two triangular prisms, two pentagonal prisms (verf of 3,5-duoprism)
344446: two triangular prisms, two hexagonal prisms (verf of 3,6-duoprism)
344447: two triangular prisms, two heptagonal prisms (verf of 3,7-duoprism)
344448: two triangular prisms, two octagonal prisms (verf of 3,8-duoprism)
344449: two triangular prisms, two enneagonal prisms (verf of 3,9-duoprism)
344440: two triangular prisms, two decagonal prisms (verf of 3,10-duoprism)
34444n: two triangular prisms, two n-gonal prisms (verf of 3,n-duoprism)
344463: two triangular prisms, two triangular cupolas
344464: triangular prism, triangular cupola, cube, hexagonal prism
344466: triangular prism, triangular cupola, hexagonal prism, truncated octahedron
344468: triangular prism, triangular cupola, octagonal prism, truncated cuboctahedron
344460: triangular prism, triangular cupola, decagonal prism, truncated icosidodecahedron
344483: two triangular prisms, two square cupolas
344484: triangular prism, square cupola, cube, octagonal prism
344486: triangular prism, square cupola, hexagonal prism, truncated cuboctahedron
344404: triangular prism, pentagonal cupola, cube, decagonal prism
344405: triangular prism, pentagonal cupola, pentagonal prism, diminished rhombicosidodecahedron
344663: triangular prism, two triangular cupolas, truncated tetrahedron
344664: two triangular cupolas, cube, truncated octahedron
344665: two triangular cupolas, pentagonal prism, truncated icosahedron
344684: triangular cupola, square cupola, cube, truncated cuboctahedron
344883: triangular prism, two square cupolas, truncated cube
345454: triangular prism, metabidiminished icosahedron, two pentagonal prisms
345450: triangular prism, metabidiminished icosahedron, two diminished rhombicosidodecahedra
345404: triangular prism, pentagonal rotunda, pentagonal prism, decagonal prism
345054: pentagonal cupola, metabidiminished icosahedron, pentagonal prism, diminished rhombicosidodecahedron
346444: triangular prism, triangular cupola, cube, hexagonal prism
346446: triangular prism, triangular cupola, hexagonal prism, truncated octahedron
346448: triangular prism, triangular cupola, octagonal prism, truncated cuboctahedron
346440: triangular prism, triangular cupola, decagonal prism, truncated icosidodecahedron
346464: triangular prism, truncated tetrahedron, two hexagonal prisms (verf of truncated tetrahedral prism)
346466: triangular prism, truncated tetrahedron, two truncated octahedra (verf of great rhombated pentachoron)
346468: triangular prism, truncated tetrahedron, two truncated cuboctahedra (verf of great rhombated tesseract)
346460: triangular prism, truncated tetrahedron, two truncated icosidodecahedra (verf of great rhombated 120-cell)
346644: two triangular cupolas, two hexagonal prisms
346646: two triangular cupolas, two truncated octahedra
346648: two triangular cupolas, two truncated cuboctahedra
346640: two triangular cupolas, two truncated icosidodecahedra
346663: two triangular cupolas, two truncated tetrahedra
346664: triangular cupola, truncated tetrahedron, hexagonal prism, truncated octahedron
346844: triangular cupola, square cupola, hexagonal prism, octagonal prism
346846: triangular cupola, square cupola, truncated octahedron, truncated cuboctahedron
346864: square cupola, truncated tetrahedron, hexagonal prism, truncated cuboctahedron
346044: triangular cupola, pentagonal cupola, hexagonal prism, decagonal prism
346046: triangular cupola, pentagonal cupola, truncated octahedron, truncated icosidodecahedron
346064: pentagonal cupola, truncated tetrahedron, hexagonal prism, truncated icosidodecahedron
348444: triangular prism, square cupola, cube, octagonal prism
348446: triangular prism, square cupola, hexagonal prism, truncated cuboctahedron
348484: triangular prism, truncated cube, two octagonal prisms (verf of truncated cubic prism)
348486: triangular prism, truncated cube, two truncated cuboctahedra (verf of great rhombated 24-cell)
348644: triangular cupola, square cupola, hexagonal prism, octagonal prism
348646: triangular cupola, square cupola, truncated octahedron, truncated cuboctahedron
348684: triangular cupola, truncated cube, octagonal prism, truncated cuboctahedron
348846: two square cupolas, two truncated cuboctahedra
348883: two square cupolas, two truncated cubes
348046: square cupola, pentagonal cupola, truncated cuboctahedron, truncated icosidodecahedron
340444: triangular prism, pentagonal cupola, cube, decagonal prism
340445: triangular prism, pentagonal cupola, pentagonal prism, diminished rhombicosidodecahedron
340454: triangular prism, pentagonal rotunda, pentagonal prism, decagonal prism
340404: triangular prism, truncated dodecahedron, two decagonal prisms (verf of truncated dodecahedral prism)
340405: triangular prism, truncated dodecahedron, two diminished rhombicosidodecahedra
340644: triangular cupola, pentagonal cupola, hexagonal prism, decagonal prism
340646: triangular cupola, pentagonal cupola, truncated octahedron, truncated icosidodecahedron
340656: triangular cupola, pentagonal rotunda, truncated icosidodecahedron, truncated icosahedron
340604: triangular cupola, truncated dodecahedron, decagonal prism, truncated icosidodecahedron
340846: square cupola, pentagonal cupola, truncated cuboctahedron, truncated icosidodecahedron
340046: two pentagonal cupolas, two truncated icosidodecahedra
340003: two pentagonal cupolas, two truncated dodecahedra
355553: four metabidiminished icosahedra
355555: two metabidiminished icosahedra, two dodecahedra
355503: two metabidiminished icosahedra, two pentagonal rotundas
355003: metabidiminished icosahedron, two pentagonal rotundas, truncated dodecahedron
356566: metabidiminished icosahedron, truncated tetrahedron, two truncated icosahedra
350054: two pentagonal rotundas, two diminished rhombicosidodecahedra
350003: two pentagonal rotundas, two truncated dodecahedra
366663: four truncated tetrahedra (verf of decachoron)
366664: two truncated tetrahedra, two truncated octahedra (verf of bitruncated tesseract)
366665: two truncated tetrahedra, two truncated icosahedra (verf of bitruncated 120-cell)
368684: truncated tetrahedron, truncated cube, two truncated cuboctahedra [mysterious!]
388883: four truncated cubes (verf of 48-cell)
444444: four cubes (verf of tesseract)
444445: two cubes, two pentagonal prisms (verf of 4,5-duoprism)
444446: two cubes, two hexagonal prisms (verf of 4,6-duoprism)
444447: two cubes, two heptagonal prisms (verf of 4,7-duoprism)
444448: two cubes, two octagonal prisms (verf of 4,8-duoprism)
444449: two cubes, two enneagonal prisms (verf of 4,9-duoprism)
444440: two cubes, two decagonal prisms (verf of 4,10-duoprism)
44444n: two cubes, two n-gonal prisms (verf of 4,n-duoprism)
444450: cube, pentagonal prism, decagonal prism, diminished rhombicosidodecahedron
444466: cube, two hexagonal prisms, truncated octahedron (verf of truncated octahedral prism)
444468: cube, hexagonal prism, octagonal prism, truncated cuboctahedron (verf of truncated cuboctahedral prism)
444460: cube, hexagonal prism, decagonal prism, truncated icosidodecahedron (verf of truncated icosidodecahedral prism)
444555: three pentagonal prisms, dodecahedron (verf of dodecahedral prism)
444566: pentagonal prism, two hexagonal prisms, truncated icosahedron (verf of truncated icosahedral prism)
445544: four pentagonal prisms (verf of 5,5-duoprism)
445540: two pentagonal prisms, two diminished rhombicosidodecahedra
445644: two pentagonal prisms, two hexagonal prisms (verf of 5,6-duoprism)
445640: pentagonal prism, hexagonal prism, diminished rhombicosidodecahedron, truncated icosidodecahedron
445744: two pentagonal prisms, two heptagonal prisms (verf of 5,7-duoprism)
445844: two pentagonal prisms, two octagonal prisms (verf of 5,8-duoprism)
445944: two pentagonal prisms, two enneagonal prisms (verf of 5,9-duoprism)
445044: two pentagonal prisms, two decagonal prisms (verf of 5,10-duoprism)
445n44: two pentagonal prisms, two n-gonal prisms (verf of 5,n-duoprism)
446466: cube, three truncated octahedra (verf of truncated 24-cell)
446468: cube, truncated octahedron, two truncated cuboctahedra
446566: pentagonal prism, two truncated octahedra, truncated icosahedron (verf of great rhombated 600-cell)
446644: four hexagonal prisms (verf of 6,6-duoprism)
446646: two hexagonal prisms, two truncated octahedra (verf of great prismated decachoron)
446648: two hexagonal prisms, two truncated cuboctahedra (verf of great prismated 48-cell)
446744: two hexagonal prisms, two heptagonal prisms (verf of 6,7-duoprism)
446844: two hexagonal prisms, two octagonal prisms (verf of 6,8-duoprism)
446846: hexagonal prism, octagonal prism, truncated octahedron, truncated cuboctahedron (verf of great prismated tesseract)
446944: two hexagonal prisms, two enneagonal prisms (verf of 6,9-duoprism)
446044: two hexagonal prisms, two decagonal prisms (verf of 6,10-duoprism)
446046: hexagonal prism, decagonal prism, truncated octahedron, truncated icosidodecahedron (verf of great prismated 120-cell)
446n44: two hexagonal prisms, two n-gonal prisms (verf of 6,n-duoprism)
447744: four heptagonal prisms (verf of 7,7-duoprism)
447844: two heptagonal prisms, two octagonal prisms (verf of 7,8-duoprism)
447944: two heptagonal prisms, two enneagonal prisms (verf of 7,9-duoprism)
447044: two heptagonal prisms, two decagonal prisms (verf of 7,10-duoprism)
447n44: two heptagonal prisms, two n-gonal prisms (verf of 7,n-duoprism)
448844: four octagonal prisms (verf of 8,8-duoprism)
448846: two octagonal prisms, two truncated cuboctahedra
448944: two octagonal prisms, two enneagonal prisms (verf of 8,9-duoprism)
448044: two octagonal prisms, two decagonal prisms (verf of 8,10-duoprism)
448n44: two octagonal prisms, two n-gonal prisms (verf of 8,n-duoprism)
449944: four enneagonal prisms (verf of 9,9-duoprism)
449044: two enneagonal prisms, two decagonal prisms (verf of 9,10-duoprism)
449n44: two enneagonal prisms, two n-gonal prisms (verf of 9,n-duoprism)
440044: four decagonal prisms (verf of 10,10-duoprism)
440n44: two decagonal prisms, two n-gonal prisms (verf of 10,n-duoprism)
44nn44: four n-gonal prisms (verf of n,n-gonal duoprism)
44mn44: two m-gonal prisms, two n-gonal prisms (verf of m,n-gonal duoprism)
466664: four truncated octahedra
555555: four dodecahedra (verf of 120-cell)

Tetrahedra that are too large or impossible:
333555, 334300, 334468, 334460, 334533, 334540, 334566, 335533, 338544, 330444, 330544, 344403, 344406, 344604, 344003, 346843, 346043, 348843, 348043, 348044, 340446, 340406, 340844, 340043, 340044, 340053, 350053, 360604, 300003, 446460, 446640, 448566, 448046, 440566, 440046, 450054, 450064, 466665, 466684, 466604, 468864, 468064, 460064, 556566, 566665

Tetrahedra that come out planar:
334433, 346643, 348844, 340045

[quickfur]

[...]

I believe this is a maximal diminishing of the F0100 (but I could be wrong). It appears to have some kind of symmetry derived from the 3,6-duoprism, but with truncated octahedra in one ring and a complicated structure of cubes and trigonal cupola in the other. Most fascinating!

Very nice

Try rendering it once for each ring, oriented so that the ring in question lies in the xy plane so we "look down the middle". That might look nice (or it might not, but it's worth trying )

Alright, this took me a while, but finally I figured out a way to make the structure (of the metatridiminished rectified 24-cell) clear. Here's the ring of cupolae:



This is not very clear, because there are 18 touching cupolae here. To make it easier to see, I separated them out into 3 hexagonal sub-rings:





And here are the 6 cubes:



This is all looking down through the ring of cupolae. I'll do the ring of truncated octahedra later.

[wendy]

This is a set of polar teddies.

It works with xfo3ooxPooo&t, for P=3,4,5.

These are aranged about a capping 3P (tetrahedron, octahedron, icosahedron), with 4, 8, 20 teddies. The base consists of an octahedron, CO, or ID. There are 4, 6, 12 P-pyramids to fill in the reamining face.

Putting two of the icosahedral teddies together, one gets xfofx3ooxoo5ooooo&#t. This is what you get by dropping opposite vertices of the 3,3,5, along with the vertices of the dodecahedral section. The pentagon-pyramid faces are replaced by a cluster of 5 tetrahedra (to make it convex), so you get 40 teddies crawling around on the surface of the 3,3,5

[quickfur]

This is a set of polar teddies.

It works with xfo3ooxPooo&t, for P=3,4,5.

These are aranged about a capping 3P (tetrahedron, octahedron, icosahedron), with 4, 8, 20 teddies. The base consists of an octahedron, CO, or ID. There are 4, 6, 12 P-pyramids to fill in the reamining face.

Yep, I rendered these in a few earlier posts.

Putting two of the icosahedral teddies together, one gets xfofx3ooxoo5ooooo&#t. This is what you get by dropping opposite vertices of the 3,3,5, along with the vertices of the dodecahedral section. The pentagon-pyramid faces are replaced by a cluster of 5 tetrahedra (to make it convex), so you get 40 teddies crawling around on the surface of the 3,3,5

That's just one of the diminishings of the 600-cell, isn't it? The 40 teddies sit at the vertices of a dodecahedral prism inscribed in the 600-cell.

On the other hand, the apiculations of the P=3,4 cases look to be interesting as pseudo-analogues of the icosahedron in 4D (apiculating the teddi makes an icosahedron; apiculating xfo3oox5ooo&t doesn't make a 600-cell, but something else with a bunch of tetrahedra and pentagonal pyramids).

[wendy]

A short notation was devised for the uniforms, based on the dynkin symbols. It covers all dimensions.

The first letter is the dimension of the element. A=1, B=2, C=3, D=4, etc.

The second letter is the view, as t=3,..,3 o=3,,4 c =4,,3 q =3,4,3 i =3,,5 and d=5,,3. These are the letters that start the 3d name, eg dodecahedron.

The number is the sum of nodes, in the form 1,2,4,8,16,.... One can reverse the symbol to minimise the number, eg Cd1 and Ci4 are both the dodecahedron, but in the forms x5o3o vs o3o5x.

The snubs are handled by the next node, eg 8 in 3d, 16 in 4d. The specific examples are Cc8 snub cube, Cd8 snub dodecahedron, Dq16 snub 24choron, Di16 grand antiprism.

Polygons are Bn, eg B5. The prisms in 3d are ABn the prism product of A and Bn. The antiprisms are Cn, like C5 pentagonal ap.

Prism product is by running the symbols together, so B10B12 is a decagon dodecagon prism.

The half cubes, the gossets are handled by the non-3 branch, being e, a and for the gossets, g, b. The 2_21 is then Fb1, ie x3o3o3o3oBo. The nodes count a to f for the F, has a b node at the end, and sums to 1.

The polytope 0 is as always the central point. You can use Klitzing's idea of eg xo3oo5ox as Ci1/4. meaning Ci one on four. It is a bichen more sayable than x and o.

[quickfur]

I just stumbled on this page today, that claims that the only convex 4D deltatopes (4-polytopes containing only tetrahedra) are the 5-cell, 16-cell, 600-cell, tetrahedral bipyramid, and icosahedral bipyramid. Is this true? Are we sure that there aren't any analogues of, say, the snub disphenoid?

[Klitzing]

I just stumbled on this page today, that claims that the only convex 4D deltatopes (4-polytopes containing only tetrahedra) are the 5-cell, 16-cell, 600-cell, tetrahedral bipyramid, and icosahedral bipyramid. Is this true? Are we sure that there aren't any analogues of, say, the snub disphenoid?

Yep, that result was found by the "blind couple" (hehe , a german couple with familyname Blind) in the end 70s and 80s. In fact those did the complete Johnson solid research within higher dimensions. - Just that they extrapolated that definition not in the current sense of CRF, but in the sense of convex regular facets, i.e. regular D-1 dimensional elements. Deltatopes are just a small sub-result thereof. The complete list of non-uniform findings is listed e.g. at the lower part of http://bendwavy.org/klitzing/explain/johnson.htm, note esp. the asterix footnote. (A special page on deltatopes (in fact: isotopes) could be found there as well: http://bendwavy.org/klitzing/explain/delta.htm.)

Some of their related publications:
- R. Blind, Konvexe Polytope mit kongruenten regulären (n-1)-Seiten im R^n (n>=4) (Convex Polytopes with Congruent Regular (n-1)-Facets in R^n (n>=4)), Comment. Math. Helvetici, 54, 304-308 (1979)
- R. Blind, Konvexe Polytope mit regulären Facetten im R^n (n>=4) (Convex Polytopes with Regular Facets in R^n (n>=4)), in: Contributions to Geometry, J. Tölke and J. M. Wills eds. Birkhäuser, Basel 1979
- G. Blind and R. Blind, Die konvexen Polytope im R^4, bei denen alle Facetten reguläre Tetraeder sind (All Convex Polytopes in R^4 the Facets of Which Are Regular Tetrahedra), Mh. Math. 89, 87-93 (1980)
- G. Blind and R. Blind, Über die Symmetriegruppen von regulärseitigen Polytopen (On the Symmetry Groups of Regular-Faced Polytopes), Mh. Math. 108, 103-114 (1989)
- G. Blind and R. Blind, The semiregular polytopes, Comment. Math. Helvetici, 66, 150-154 (1991)

[quickfur]

Weird! So there are only a very limited number of 4D deltachora. I wonder if perhaps the number of "unusual" CRF polychora (the analogues of the snub disphenoid, sphenocorona, etc.) are actually quite limited? Or is the cause of the limitation the fact that we restrict ourselves to regular polyhedra, and once we admit general Johnson polychora many more "unusual" CRFs will be possible?

I'm starting to question whether there will be many unusual CRF polychora with, say, snub disphenoid cells or sphenocorona cells -- the main problem is that the almost-unique dihedral angles in these shapes may make them very difficult to fit in a shape that closes up in 4D. There seems to be some kind of inherent property in any dimension that certain CRF shapes close up more easily than others -- that's why you get only a limited number of regular polytopes, and their derivatives seem to account for most of the CRFs. That's also why things like square pyramids show up everywhere, but more unusual members of the Johnson solids only rarely turn up. I wonder if the number of CRF outliers (those that can't be made from cut-n-paste operations from regular polytope or uniform prism derivatives) will actually approach zero as the dimension increases.

Here's an interesting possible line of investigations: how many 4D CRFs are there whose cells are all square pyramids? What about if you admit tetrahedra as well as square pyramids?

[Klitzing]

Weird! So there are only a very limited number of 4D deltachora. I wonder if perhaps the number of "unusual" CRF polychora (the analogues of the snub disphenoid, sphenocorona, etc.) are actually quite limited? Or is the cause of the limitation the fact that we restrict ourselves to regular polyhedra, and once we admit general Johnson polychora many more "unusual" CRFs will be possible?

There are just 5 regulars in 3d, but lots more convex polyhedra with regular faces: not only the Johnson solids and the Archimedeans, but also 2 infinite series! So yes, there is a lot more possible beyond. And even consider those regulars: there are 3 of them with triangular faces, but just one each with squares resp. pentagons. So those latter ones could connect just to themselves. In order to get "new" figures, we thus are left with only 3! (The same argument works for any other dimension above as well.)

I'm starting to question whether there will be many unusual CRF polychora with, say, snub disphenoid cells or sphenocorona cells -- the main problem is that the almost-unique dihedral angles in these shapes may make them very difficult to fit in a shape that closes up in 4D. There seems to be some kind of inherent property in any dimension that certain CRF shapes close up more easily than others -- that's why you get only a limited number of regular polytopes, and their derivatives seem to account for most of the CRFs. That's also why things like square pyramids show up everywhere, but more unusual members of the Johnson solids only rarely turn up. I wonder if the number of CRF outliers (those that can't be made from cut-n-paste operations from regular polytope or uniform prism derivatives) will actually approach zero as the dimension increases.

Here's an interesting possible line of investigations: how many 4D CRFs are there whose cells are all square pyramids? What about if you admit tetrahedra as well as square pyramids?

Somewhere in this thread I've already given a sketch on how to run thru. But even that preliminar step of constructing all possible vertex figure solids was much too huge to been handled without further ideas. So your restriction to just the very smallest possible facets, i.e. tetrahedron and square pyramids, would clearly be such!

tet has as verf: regular unit edged triangle;
squippy has as verfs: unit-edged square and right triangle with edges 1:1:sqrt(2).

So we next would have to construct all possible convex solids from these 3 faces. (E.g. we get for free, that the unique sqrt(2)-edge clearly has to join 2 right triangles. But OTOH this set clearly contains all Platonics (4), all Archimedeans (3), all (not already counted) Prismatics (2), and all Johnsonians (28 - if I counted correctly), which are subject to triangular and square faces only. - And in addition those which contain at least one (in fact at least 2!) right triangular faces...) And finally you'd have to construct out of the thus obtained set of possible vertex figure polyhedra the searched for tiny subset of CRF polychora.

--- rk

[Klitzing]

4D riddle:

srico = x3o4x3o can be shown to have caps of the form co || tic = xx3ox4xo&#x.

Now, could anyone visualize what its according full diminished version would be?

--- rk

[quickfur]

4D riddle:

srico = x3o4x3o can be shown to have caps of the form co || tic = xx3ox4xo&#x.

Wait, shouldn't that be oo3xx4ox&#x?

Now, could anyone visualize what its according full diminished version would be?
[...]

The x3o4x3o actually has two different kinds of diminishings. You can cut off o4x3o||x4x3o as you have above, and then do another deeper cut at the same place of a x4x3o||x4x3x.

For the former case (cutting off o4x3o||o4x3x), you can consider the 24-cell that corresponds with the 24 o4x3o's that are being cut off. Each cut precludes any further diminishing of the o4x3o's that correspond with adjacent cells on the 24-cell. So the maximal diminishings correspond with the maximal diminishings of the 24-cell, or dually, the largest set of non-adjacent vertices on the 24-cell. This corresponds with the octadiminished 24-cell (which is the tesseract). When the octadiminishing is performed on x3o4x3o, you get the tesseractic uniform polychoron x4x3o3x (runcitruncated tesseract).

This means that x3o4x3o is just an octa-augmented x4x3o3x; however, due to the higher degree of symmetry of the latter, it is possible to diminish the latter such that it does not correspond with any augmented x4x3o3x. There are 5 of these, corresponding with the metadiminished 24-cells (where meta means given two deleted vertices, one lies 1 edge away from the antipode of the other): the metabidiminished 24-cell, the metatridiminished 24-cell, two metatetradiminished 24-cells, and the metapentadiminished 24-cell.

P.S. I haven't fully explored the deeper cuts yet, so I can't say for sure what their structure is. I believe they correspond with the bisected 24-cells, but unlike the 24-cell case, there is more than just half left after the deeper cut, so it may show a more complicated structure than the diminishings of the bisected 24-cell. I'll have to get back to examining this sometime; it's a very interesting area to explore.

[Klitzing]

Hy quickfur,

you've won the challange. Not only that you've been the first to solve my riddle in this list. Also within the other list I've posted that riddle as well (cf. http://www.mathconsult.ch/lists/cgi/private/polyhedron/2012/msg00367.html) no-one has come before. Moreover, you are right for sure with respect to my typo:

4D riddle:

srico = x3o4x3o can be shown to have caps of the form co || tic = xx3ox4xo&#x.

Wait, shouldn't that be oo3xx4ox&#x?

Now, could anyone visualize what its according full diminished version would be?
[...]

[...]
When the octadiminishing is performed on x3o4x3o, you get the tesseractic uniform polychoron x4x3o3x (runcitruncated tesseract).
[...]

Icoic symmetry clearly is richer than the tessic one. But an application of that diminishing at all 24 positions would overlap. At most one could use just 8 diminishings within the tessic subsymmetry. And that leads to the searched for proh = x3o3x4x (Bowers name: (small) prismatorhombated hexadecachoron). Or, as you put it:

This means that x3o4x3o is just an octa-augmented x4x3o3x.

I don't know how you actually derived this result. There are several ways to do this. - I yesterday just stumbled upon this by the consideration of the corresponding lace cities (orthogonal to a common 4-fold axis):

Code: Select all

srico = x3o4x3o:
----------------
              o4x   q4o   o4x             
                                           
          x4x   w4o     w4o   x4x         
                                           
                                           
    x4x   x4u   w4x     w4x   x4u   x4x   
                                           
o4x           o4X   q4w   o4X           o4x
    w4o   w4x                 w4x   w4o   
                                           
q4o           q4w         q4w           q4o
                                           
    w4o   w4x                 w4x   w4o   
o4x           o4X   q4w   o4X           o4x
                                           
    x4x   x4u   w4x     w4x   x4u   x4x   
                                           
                                           
          x4x   w4o     w4o   x4x         
                                           
              o4x   q4o   o4x             


Code: Select all

co || tic = ox3xx4oo&#x:
------------------------
    o4x   q4o   o4x   
                       
x4x   w4o     w4o   x4x


Code: Select all

proh = x3o3x4x:
---------------
      x4x   w4o     w4o   x4x     
                                   
                                   
x4x   x4u   w4x     w4x   x4u   x4x
                                   
                                   
w4o   w4x                 w4x   w4o
                                   
                                   
                                   
w4o   w4x                 w4x   w4o
                                   
                                   
x4x   x4u   w4x     w4x   x4u   x4x
                                   
                                   
      x4x   w4o     w4o   x4x     


And yes, you are right, there is

another deeper cut at the same place of a x4x3o||x4x3x

too. I.e.

Code: Select all

tic || girco = ox3xx4xx&#x:
---------------------------
      x4x   w4o     w4o   x4x     
                                   
                                   
x4x   x4u   w4x     w4x   x4u   x4x


But, as could be seen already from these lace cities as well, those caps of proh resp. supra-caps of srico would overlap if applied at all 8 positions of proh. In fact, those could be applied there only at diametrically opposite positions. (If applied to srico directly, there are other multi-diminishings too. But there will be none with such a high symmetry as the above mentioned tessic one.)

--- rk

[quickfur]

Hy quickfur,

you've won the challange. Not only that you've been the first to solve my riddle in this list. Also within the other list I've posted that riddle as well (cf. http://www.mathconsult.ch/lists/cgi/private/polyhedron/2012/msg00367.html) no-one has come before. [...]

Thanks for the compliment! Though I've to admit I "cheated" a bit... I just happened to have studied the diminishings of x3o4x3o before, so I've already known of its connection with the x4x3o3x.

[...]

This means that x3o4x3o is just an octa-augmented x4x3o3x.

I don't know how you actually derived this result. [...]

Actually, I derived it visually, and used my polyview viewer to verify the intermediate results.

I usually work with the 3D perspective and parallel projections, which are surprisingly effective at doing 4D geometry "intuitively". I can't say I'm very good at it yet, but at least with the uniform polytopes I do reasonably well at visualizing where the cutting hyperplane intersects the polytope at various "nice" angles (nice as it, it lines up with vertices). From this, I make a quick list of candidate diminishings, which can then be analysed to determine which vertices to delete. My polytope viewer's ability to filter vertices based on various face lattice based criteria helps a lot, as I don't have to work directly with coordinates but can manipulate the shapes based on the connectivity of the various elements. Once a candidate diminishing is determined, I make a copy of the original polytope's vertices and remove the vertices manually (I haven't automated this part yet), and run it through a convex hull algorithm to make sure that the result is what I think it is. Sometimes I make a mistake, and the result is non-CRF, but this is immediately noticeable both from the edge length computations my viewer can do, and from the unexpected projection images once the diminished polytope is rendered. (Since I derive most of my results visually, I do have some idea of what to expect in the resulting projection; if I see something unexpected, that's an indication that either I made a mistake in assuming that a particular diminishing is CRF, etc., or the result is correct but I visualized it incorrectly.)

[Klitzing]

Did you consider spic = small prismatotetracontoctachoron = runcinated 24-cell = x3o4o3x?
It has a layer-structure with several unit-edged sections. Here its lace city, which shows those directly:

Code: Select all

                o4o         x4o         o4o                      = x3o4o = oct
                                                           
                                                           
                                                           
        x4o         x4x             x4x         x4o              = x3o4x = sirco
                                                           
                                                           
                                                           
o4o                         w4o                         o4o      = w3o4o = w-oct (w = 1+sqrt2)
                                                           
        x4x         w4o             w4o         x4x              = o3x4x = tic
                                                           
                                                           
                                                           
x4o             w4o         q4q         w4o             x4o      = q3x4o = toe-variant (q = sqrt2)
                                                           
                                                           
                                                           
        x4x         w4o             w4o         x4x       
                                                           
o4o                         w4o                         o4o
                                                           
                                                           
                                                           
        x4o         x4x             x4x         x4o       
                                                           
                                                           
                                                           
                o4o         x4o         o4o               


Accordingly you should suppose to get true bistratic CRFs in here:

• sirco || pseudo w-oct || tic -- total height = 1/sqrt2
  Facets being: 1 sirco + 6 squacu + 8 oct + 12 trip + 1 tic; accordingly, after all, this comes out to be monostratic only
  In fact this is due to reduced internal structure of the sirco itself, more precisely: of those used diminished ones, the squacues. Filling those in again would require a squippy at its 4-fold square, but also 4 (greater) facated trips, 4 (smaller) faceted trips, and an 8-fold pyramid. Here those faceted trips in turn are the vertex-pyramid of trip (lacings are unit-edges, base is its vertex figure, i.e. a x-q-q triangle!) for the smaller one resp. its complement for the greater one. And the 8-fold pyramid has the unit-edged regular octagon for base, but lacing edges being of size q! - Hence this section, when taken as a true 3-layer thingy, won't be a CRF.

• tic || pseudo q3x4o || tic -- total height = 1
  Facets being: 2 tic + 8 trip + 6 op; accordingly, after all, this is neither bistratic
  Again, this is due to the missing internal structure of those ops. But this time we are luckier
  Re-introducing that structure would ask for 4 squippies + 4 (complete) trips attached to the squares of the ops. And then again some dissected trips... But we could replace the rest by 2 squacues. Thus we get "4g || 8p"s instead of those ops. Implementing those, we would result in some augmented tic-prism: having for facets 2 tic + 8+6*4(=32) trip + 6*4(=24) squippy + 6*2(=12) squacu.

Did you ever consider such strange augmentations, leading from mere monostratics to true bistratics?

So we could move on, using this positive result from 4-fold onto 2-fold, 3-fold, and 5-fold.

• 2-fold: augment a trip-prism (= tisdip, i.e. the 3,4-duoprism) by "line || cube" instead of the used cubes.
• 3-fold: augment a tut-prism (= tuttip) by "3g || 6p" instead of the used hips.
• 4-fold (the above just derived case): augment a tic-prism (= ticcup) by "4g || 8p" instead of the used ops.
• 5-fold: augment a tid-prism (= tiddip) by "5g || 10p" instead of the used dips.

Now, why does that 4-fold case occure within the icoic symmetry (i.e. in spic), but the others not in (some) related ones? - It is that the added vertex layers generally would have different circumradii than those of the starting 4D prisms; only for the 4-fold case those would coincide:

• 2-fold: r(tisdip) = sqrt(5/6) = 0.912871 > r(line || cube) = sqrt(3/4) = 0.866025
• 3-fold: r(tuttip) = sqrt(13/8) = 1.274755 > r(3g || 6p) = sqrt(7/5) = 1.183216
• 4-fold: r(ticcup) = sqrt(2+sqrt(2)) = 1.847759 = r(4g || 8p) = sqrt(2+sqrt(2)) = 1.847759
• 5-fold: r(tiddip) = sqrt((39+15*sqrt(5))/8) = 3.011250 < r(5g || 10p) = sqrt(23+10*sqrt(5)) = 6.735034

But, in this CRF project, orbiformity never was asked for. (Rather it was to find cases outside.) Thus all these 4 such augmented cases (= exterior blends) should be wellcome.
Have those been encountered so far?

You might ask wether those are possible at all? Esp. with respect to those different radii? Well, to me this looks similar to the following: rectangles are valide 2D polygons even so their side-lengths differ. Correspondingly here we have structures with different curvatures attached side-by-side, kind of similar to the Johnson solid J92 = thawro = triangular hebesphenorotunda. - Am I wrong? (Perhaps someone might like to calculate coordinates for those, in order to prove their existance... )

--- rk

[quickfur]

Did you consider spic = small prismatotetracontoctachoron = runcinated 24-cell = x3o4o3x?
It has a layer-structure with several unit-edged sections. [...]

I haven't gotten to it yet. But yeah, from a quick glance at my projections of it shows that it can be cut into some layers. I'll have to look more carefully into it to figure out the details, though. But maybe some other time ... busy with other stuff at the moment.