## Classifying the segmentochora

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

### Classifying the segmentochora

I noticed Keiji has started a project to classify Klitzing's segmentochora. I'd like to start this thread for discussing this.

First of all, a large number of segmentochora can be generated by choosing two uniform polyhedra with the same symmetry group and placing them in parallel hyperplanes. If the difference in circumradii is less than the edge length, then this will generate a segmentochoron. If the two polyhedra are equal, then the result is a prism; if one is a point, the result is a pyramid. Otherwise, the result is a kind of general cupola-like shape, which is a superset of what we traditionally considered cupolae in the past.

The segmentochoron cube||icosahedron is a notable exception, and is quite unique as far as segmentochoron constructions go.

Besides these, there are other sporadics such as line||orthogonal_3-prism (4.8.2), which occurs as a maximal diminishing of the rectified 5-cell, and a few other oddities. But the generalized cupola pretty fills up the bulk of Klitzing's segmentochora.

Note that due to the tetrahedron being the alternated cube, there is some overlap between the tetrahedron and cube family of uniform polyhedra, so there is somewhat a distinction between a cuboctahedron as the rectified cube (o4x3o), and a cuboctahedron as a "rhombitetrahedron" (x3o3x), for example. For the purposes of generating cupolae-like segmentochora with tetrahedral symmetry, we may regard the octahedron as a rectified tetrahedron (o3x3o), the cuboctahedron as a "rhombitetrahedron" (x3o3x) or, in Keiji's terminology, pyroperihedron, and the truncated octahedron as the omnitruncated tetrahedron (x3x3x).

One notable subclass of segmentochora worth classifying separately is the segmentochora in which the top cell's projection onto the bottom cell falls strictly within the bottom cell. These segmentochora can be used as augments of other polychora which have cells that match the shape of the bottom cell, and will produce a CRF if the dichoral angles add up to less than 180°. Segmentochora where the projection of the top cell protrudes outside the bottom cell cannot be used as augments, so it's a good distinction to draw when classifying them -- so that when we start considering stacks of monostratic CRFs or augmentations of uniform polychora, we will easily know which are possible augments and which aren't.
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### Re: Classifying the segmentochora

Also, when two uniform polyhedra (of the same symmetry) have circumradii that differ in more than the edge length, we may be able to construct bistratics in which the difference in circumradii is bridged by attaching pyramids or cupola to the outer cell. This may be a good direction to look in for finding more bistratic (or tristratic) CRFs.
quickfur
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### Re: Classifying the segmentochora

quickfur wrote:in Keiji's terminology, pyroperihedron

Speaking of which, I really would like a better name for this naming convention. So far I've called them Tamfang names because he provided the original inspiration for them, but IIRC he expressed disconsent at using his name to refer to any of my modifications. I just don't have anything else to call them, though

I will take a look at the classes you've mentioned shortly...

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### Re: Classifying the segmentochora

quickfur wrote:I noticed Keiji has started a project to classify Klitzing's segmentochora. I'd like to start this thread for discussing this.

First of all, a large number of segmentochora can be generated by choosing two uniform polyhedra with the same symmetry group and placing them in parallel hyperplanes. If the difference in circumradii is less than the edge length, then this will generate a segmentochoron. If the two polyhedra are equal, then the result is a prism; if one is a point, the result is a pyramid. Otherwise, the result is a kind of general cupola-like shape, which is a superset of what we traditionally considered cupolae in the past.

The segmentochoron cube||icosahedron is a notable exception, and is quite unique as far as segmentochoron constructions go.

Besides these, there are other sporadics such as line||orthogonal_3-prism (4.8.2), which occurs as a maximal diminishing of the rectified 5-cell, and a few other oddities. But the generalized cupola pretty fills up the bulk of Klitzing's segmentochora.

Note that due to the tetrahedron being the alternated cube, there is some overlap between the tetrahedron and cube family of uniform polyhedra, so there is somewhat a distinction between a cuboctahedron as the rectified cube (o4x3o), and a cuboctahedron as a "rhombitetrahedron" (x3o3x), for example. For the purposes of generating cupolae-like segmentochora with tetrahedral symmetry, we may regard the octahedron as a rectified tetrahedron (o3x3o), the cuboctahedron as a "rhombitetrahedron" (x3o3x) or, in Keiji's terminology, pyroperihedron, and the truncated octahedron as the omnitruncated tetrahedron (x3x3x).

One notable subclass of segmentochora worth classifying separately is the segmentochora in which the top cell's projection onto the bottom cell falls strictly within the bottom cell. These segmentochora can be used as augments of other polychora which have cells that match the shape of the bottom cell, and will produce a CRF if the dichoral angles add up to less than 180°. Segmentochora where the projection of the top cell protrudes outside the bottom cell cannot be used as augments, so it's a good distinction to draw when classifying them -- so that when we start considering stacks of monostratic CRFs or augmentations of uniform polychora, we will easily know which are possible augments and which aren't.

Well, as I assembled them by increasing 4d radius, all diminishings of some shape have already been assembled close to it.

That crown-jewel, cube || ike, as Wendy lately pointed out, in fact is not so especial after all. It is a diminished element from ex: cube is a faceted dodecahedron, and the next vertex layer of ex is just a tau-scaled icosahedron!

Those mentioned stackings of polyhedra belongign to a common symmetry, are just what Wendy calls lace prisms.

Sure, most segmentochora can be grouped that way: Take any lace prism (with common tetrahedral, octahedral, icosahedral or prismatic symmetry around its stacking axis), and consider all diminishings thereof. In few cases even gyrations might apply.

--- rk
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### Re: Classifying the segmentochora

I've already classified these:D I did it over a couple of months I think I posted it in the CRF thread I can't remember. I'll try and find the post when I get chance:D
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### Re: Classifying the segmentochora

wintersolstice wrote:I've already classified these:D I did it over a couple of months I think I posted it in the CRF thread I can't remember. I'll try and find the post when I get chance:D

Hmm.

We all really should be posting results to the wiki (CRF polychora discovery project, segmentochoron, etc.), so that everyone knows what's the up-to-date information. The CRF thread is too long and unwieldy now, and it's almost impossible to keep track of who has posted what, what's the latest results, what has been invalidated/disproven, etc., and we seem to be duplicating a lot of work. We should be spending time on new discoveries instead of redoing what has already been done (except for verification of previous results, of course, but that shouldn't be happening by accident!).

Keiji, is it possible to make the forum and the wiki share the same users/logins? That way we won't have the unnecessary additional barrier of making people sign up for another account on the wiki. The less barriers the more likely people will actually contribute.
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### Re: Classifying the segmentochora

We all really should be posting results to the wiki [...] so that everyone knows what's the up-to-date information.

Yes, this is what I am doing now. Who was it that brought this topic from the wiki into the forum?

We should be spending time on new discoveries instead of redoing what has already been done (except for verification of previous results, of course, but that shouldn't be happening by accident!).

On the contrary, trying to verify a previous incorrect result means you're more likely to repeat the same mistake because you think it's right. Independently rediscovering the same thing is far more likely to imply correctness.

One good tool to validate the structure of a candidate polytope is to calculate its incidence matrix and put it into the Polytope Explorer. At the moment I'm the only one able to enter things into that though, but feel free to send me imats to check. (Plus, it needs more polytopes! It doesn't even have the uniform polychora yet!)

Keiji, is it possible to make the forum and the wiki share the same users/logins? That way we won't have the unnecessary additional barrier of making people sign up for another account on the wiki. The less barriers the more likely people will actually contribute.

Not for the foreseeable future unfortunately. Anyone who would like a wiki account feel free to PM me, and I'll reply with an invite code.

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### Re: Classifying the segmentochora

what I posted were "the 29" which I'll repost here:

wintersolstice wrote:
here's what I call "the 29"

segmentotopes whose bases are either Platonic/Archimedean (excluding vertex transitive cases)

Key:

here are the catagories, underneath which are the Platonic solid/Platonic solid pairs that are Johnson cases, underneath that is why the others are invalid. Next to the name is a bracket showing the pair of bases for the segmentotope. the "X" isn't part of the name it just means you change it for the name of the Platonic Solids for it to name the shapes in that catagory.

meanings:

snubdis means truncate then alternate (based on snubdis 24-cell)

Birectate = Dual in 3D

Bitruncate = Truncate of the dual in 3D

Platonic = original Platonic solid

other notes:

I know the cupolas and antiprisms are already known (they're here just for completeness )

Cupoliprism was not my invention (based on the non-uniform scaliform polychoron name :
"Truncated tetrahedral cupoliprism"

the forward slashes mean that either platonic solid (or hypertruncate) can be used in the pair and it would be the same

the other names I did invent

the list:

X Cupola (Platonic||Cantellate)

1) Tetrahedron
2) Cube
3) Octahedron
4) Dodecahedron

Note: the icosahedron cupola is not CRF

X Antiprisms (Platonic||Birectate)

5) Octahedron/cube
6) Dodecahedron/icosahedron

Note: the tetrahedron antiprism is vertex transitive( 16-cell)

X Anticupola (Platonic||Rectate)

7) Cube
8 ) Octahedron
9) Dodecahedron
10) Icosahedron

Note: the tetrahedron anticupola is vertex transitive (rectified 5-cell)

Partially-base truncated X cupola (Truncate||Cantellate)

11) Tetrahedron
12) Cube
13) Octahedron
14) Dodecahedron
15) Icosahedron

Partially-base rectified X cupola ((Rectate||Cantellate)

16) Octahedron/cube
17) Dodecahedron/icosahedron

Note: the “Partially-base rectified tetrahedron cupola” = “octahedron anticupola”

Partially-base truncated X anticupola (Truncate|| Rectate)

18) Tetrahedron
19) Cube
20) Octahedron
21) Icosahedron

Note: the “Partially-base truncated dodecahedron anti cupola” is not CRF

Completely-base truncated X anticupola (Truncate||Cantitruncate)

22) Tetrahedron
23) Cube
24) Octahedron
25) Dodecahedron

Note: the “Completely-base truncated icosahedron anticupola” is not CRF

Truncated X cupoliprisms (Truncate||Bitruncate)

26) Octahedron/cube
27) Dodecahedron/icosahedrons

Note: “Truncated tetrahedron cupoliprism” is vertex transitive

Partially-base snubdis X antiprism (Snubdis||Birectate)

28) Octahedron

Note: only the octahedron can be “snubdis”ed

Semi cupola (Platonic|| Truncate)

29) Tetrahedron

Note: this shape is unique, the others in this group are not CRF
wintersolsticeTrionian Posts: 74Joined: Sun Aug 16, 2009 11:59 am

here's the rest (or most of them classified)

of the 251 on (orchidpalms.org)

200 are distinct

infinite sets (41)
prisms

antiprisms

3,n duoprisms

4,n duoprisms

antiduoprims (antiprism prisms)

antiprismatic rings

the 3,4,5,6,8,10,n members are listed

EDIT: 3-gon prism = 3,4 duprism

The 29 (29)
these are listed above

less than 4D (9)

3-gon, 4-gon, 5-gon pyramid and cupola (3-gon pyramid = tetrahedron)

line segement, 3-gon, 4-gon

uniform prisms (16)

prisms of the Archemedean and platonic solids (the cube and octahedron and there prisms are within the infinite sets)

Misc Scaliform (4)

(miscellaneos vertex transitive cases)

Recified 5-cell

Truncated Tetrahedral cupoliprism

16-cell

5-cell

(8 cell is member of infinte set)

Johnson solid prisms (25)

only 25 of the JS can be inscribed in a sphere

fragments (12)

3-gon, 4-gon, 5-gon ortho gyro magna bi cupolic rings (9 permutations)

gyro 5-gon cupola rotunda ring (the others aren't on the list)

aumented of rotunda ring

4-gon bipyramidal bicupolic ring

dimisishes and gyrations of the 29 (EDIT: 49)

the fragments are the 12 above plus the prism and antiprism pyramids

pyramids (12)

3-gon 4-gon 5-gon prism and antiprism pyramid

diminished, metabidiminished, tridiminished icosahedra, icosahedron, 4-gon and 5-gon pryramid (are the 6 other polyhedra)

Oddballs (3)

gyrated octahedral prism

diminished 3-gon antiprismatic ring/bidimished rectified 5 cell (dimished rectified 5 cell = 3-gon antiprismatic ring)

3-gon||tridiminished icosahedron (I haven't named this one)

when I can I'll post the 48:D aswell
Last edited by wintersolstice on Mon Aug 27, 2012 9:33 pm, edited 3 times in total.
wintersolstice
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### Re: Classifying the segmentochora

Thanks wintersolstice.

Is there any chance you can post the K4.x numbers for each polychoron you've classified?

(Only the ones listed under the two "as-yet-unclassified" categories at Segmentochoron#Classification are necessary)

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### Re: Classifying the segmentochora

Klitzing wrote:That crown-jewel, cube || ike, as Wendy lately pointed out, in fact is not so especial after all. It is a diminished element from ex: cube is a faceted dodecahedron, and the next vertex layer of ex is just a tau-scaled icosahedron!

Okay, I've been staring at projections of the hydrochoron for a while, and the icosahedron is obvious, but I can't see how you can get a cube out of it. I assume you're saying it's a parallel cross-section somehow?

I can see a dodecahedron, but that is bigger than the icosahedron, so surely wouldn't give equal edge lengths?

Keiji
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### Re: Classifying the segmentochora

Keiji wrote:
We all really should be posting results to the wiki [...] so that everyone knows what's the up-to-date information.

Yes, this is what I am doing now. Who was it that brought this topic from the wiki into the forum?

I brought it here 'cos I wanted to open up discussion about classification schemes for Klitzing's segmentochora. The way I see it, we'll use the forum for discussing/debating results, methods, and what-not, and the wiki for posting the results themselves. The wiki format doesn't lend itself very well to discussions (ever been to a wikipedia talk page that's 50 pages long?), just as the forum format doesn't lend itself very well for cross-referencing and looking up a specific topic/result.

We should be spending time on new discoveries instead of redoing what has already been done (except for verification of previous results, of course, but that shouldn't be happening by accident!).

On the contrary, trying to verify a previous incorrect result means you're more likely to repeat the same mistake because you think it's right. Independently rediscovering the same thing is far more likely to imply correctness.

Sure, but should we really have each one of us rediscovering the entire list of CRF polychora independently, just so we're sure the list is correct? I think it's beneficial to at least be aware of what others have found. And it's also good to know the overall status of the CRF project in a single place rather than having to wade through 100's of posts just to find out the progress for one particular sub-area.

One good tool to validate the structure of a candidate polytope is to calculate its incidence matrix and put it into the Polytope Explorer. At the moment I'm the only one able to enter things into that though, but feel free to send me imats to check. (Plus, it needs more polytopes! It doesn't even have the uniform polychora yet!)
[...]

I use my viewer to verify polytopes, because the convex hull algo will instantly give you strange results if you give it non-convex coordinates. Like a whole bunch of irregular tetrahedral cells where you expected a particular combination of CRFs. Plus, the viewer needs concrete coordinates, so that also means I have to actually calculate real coordinates for the polytope, so I'd notice if a structure makes it impossible to assign actual coordinates to the vertices.

As for the uniform polychora, I have coordinates for all of them (the convex ones, anyway). I think I'll start posting them to the wiki. As well as the CRFs I've rendered so far, since I have coordinates for those. I'm going to start putting up the Stella4D .off files that I made (I think I neglected to convert my polytopes for Marek for my recent CRF renders, I'll just put them up on the wiki from now on).
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### Re: Classifying the segmentochora

Keiji wrote:Thanks wintersolstice.

Is there any chance you can post the K4.x numbers for each polychoron you've classified?

(Only the ones listed under the two "as-yet-unclassified" categories at Segmentochoron#Classification are necessary)

yes but will take me sometime I also need to indentify the 48 diminishes and Gyrations aswell

I'll edit my post when I get the all:D
wintersolstice
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### Re: Classifying the segmentochora

quickfur wrote:[wiki versus forums]
[independently rediscovering everything]

I was just making a point (or two), you weren't supposed to take it completely seriously

As for the uniform polychora, I have coordinates for all of them (the convex ones, anyway). I think I'll start posting them to the wiki. As well as the CRFs I've rendered so far, since I have coordinates for those. I'm going to start putting up the Stella4D .off files that I made (I think I neglected to convert my polytopes for Marek for my recent CRF renders, I'll just put them up on the wiki from now on).

Well, I'm sure there are algorithms to create max-symmetry imats from coordinates, and a tool to add your coordinates along with automatically-calculated imats to the explorer would be extremely useful, but I don't think it would be the simplest thing for me to implement...

I've also thought about the possibility of an imat->FLD conversion tool, then I realized that FLD essentially portrays the exact same information as an imat but with potential ambiguity and messiness. So now I'm considering obsoleting FLD ( ) in favor of the "pure" imats. FLD does make it easier to identify the faces, though, but perhaps some simple formatting additions to imats, such as highlighting blocks of cells, would grant this...

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### Re: Classifying the segmentochora

Keiji wrote:
Klitzing wrote:That crown-jewel, cube || ike, as Wendy lately pointed out, in fact is not so especial after all. It is a diminished element from ex: cube is a faceted dodecahedron, and the next vertex layer of ex is just a tau-scaled icosahedron!

Okay, I've been staring at projections of the hydrochoron for a while, and the icosahedron is obvious, but I can't see how you can get a cube out of it. I assume you're saying it's a parallel cross-section somehow?

I can see a dodecahedron, but that is bigger than the icosahedron, so surely wouldn't give equal edge lengths?

Well, ex as a lace tower is just: pt || pseudo ike || pseudo doe || pseudo f-ike || pseudo id || pseudo f-ike || pseudo doe || pseudo ike || pt.
What I was refering to is the part: doe || ... || ... || f-ike (leaving out those intermediate layers), and use the faceting of the doe, i.e. the inscribed cube. (Several possibilities to do this, just one being selected.) Now reconnect those 2 layers with lacing edges of length f(=tau) as well (again being such secants of ex which connect the apices of 2 base-connected tets). Thus you just get a tau scaled cube || ike therein.

--- rk
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### Re: Classifying the segmentochora

Keiji wrote:
Klitzing wrote:That crown-jewel, cube || ike, as Wendy lately pointed out, in fact is not so especial after all. It is a diminished element from ex: cube is a faceted dodecahedron, and the next vertex layer of ex is just a tau-scaled icosahedron!

Okay, I've been staring at projections of the hydrochoron for a while, and the icosahedron is obvious, but I can't see how you can get a cube out of it. I assume you're saying it's a parallel cross-section somehow?

I can see a dodecahedron, but that is bigger than the icosahedron, so surely wouldn't give equal edge lengths?

I think he's referring to the cube inscribed in a dodecahedron (in 3D).
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### Re: Classifying the segmentochora

OK I'll make start with the numbering:D

Infinite sets

Already Classified

The 29

I'll get these as soon as I can

less than 4D

Not relevant

uniform prisms

already classified

Misc Scaliform

already classified apart from "Truncated tetrahedral cupoliprism" 55

Johnson solid prisms

already classified

fragments

9 of these (the bicupolic rings have been classified) the others:

gyro 5-gon cupolarotunda ring 146

It's augmentation 139

4-gon bipyramidal bicupolic ring 109

dimisishes and gyrations of the 29

I'll get these as soon as I can

pyramids

already classified

Oddballs (3)

gyrated octahedral prism 13

diminished 3-gon antiprismatic ring/bidimished rectified 5 cell 8

3-gon||tridiminished icosahedron (unnamed) 33

there also seems to be an issue with the cupolae.

according to wikipedia a 4D cupola is a platonic solid to it's cantellate. and if that is the case the icosahedron case isn't CRF

now Gyrobicupolae are made by taking the dual of the base so the "cube gyrobicupola" is actually a "cube cupola" and a "octahedron cupola" joined together. (it also means that "cube gyrobicupola" = "octahedron gyrobicupola")

since the icosahedron cupola isn't CRF it means that dodecahedron gyrobicupola (or the icosahedron gyrobicupola) isn't CRF

However on Klizting paper a cupola is a "platonic solid to rectate" and "rectate to cantellate" (this is meaning currently being used on here http://teamikaria.com/hddb/wiki/Segment ... sification)

it also says under the CRF article about the cupolae that there is only one for each Platonic solid (which appears to mean Platonic to cantellate but it isn't clear) and it counts ortho and gyro forms for all 5 even though some are the same and some not CRF

But it does say that the ability to construct them CRF needs to be checked.

There are 29 shapes (including the cube||icosahedron) that are cupola-like (their bases are Platonic and/or Archimedean). they include "the platonic||cantellate" (wikipedia definition and suposed definition on the CRF site) the 5 that were mentioned on "Segmentochoron classification" and several others

What I can suggest is for me to post the 29, so that all possible stacking of them base to base (to make the variations) can be worked out (along with elongations), then the count can be verified at last. with all 29 (or 28 if you exclude the cube||icosahedron) can be classed as cupola and antiprisms:D
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### Re: Classifying the segmentochora

wintersolstice wrote:[...]
according to wikipedia a 4D cupola is a platonic solid to it's cantellate. and if that is the case the icosahedron case isn't CRF
[...]

Well, I was proposing that we generalize the term "cupola" to cover all cases of A || B where A and B are of the same symmetry group, A ≠ B, and the difference in circumradius between A and B < edge length. There are some cases for which the difference in circumradius is exactly equal to the edge length, in which case you get a 3D tiling, not a full-dimensioned segmentochoron. If the edge length < difference in circumradius, then the shape cannot be made CRF.

I'm surprised the icosahedron case isn't CRF, though. Is it because edge length < difference in circumradius?
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### Re: Classifying the segmentochora

quickfur wrote:[...]
I'm surprised the icosahedron case isn't CRF, though. Is it because edge length < difference in circumradius?

Huh, this is really surprising for me. I calculated the difference in circumradius of an icosahedron and a rhombicosidodecahedron (both of edge length 2) to be approx 2.802, which is longer than the edge length. So icosahedron||rhombicosidodecahedron is not CRF! So the icosahedral cupola should be removed from the CRF page.

I found this very counterintuitive, so I did a little mental experiment where I attached triangular prisms to the faces of the icosahedron, as if I was going to construct the cupola, and then I realized that the reason this doesn't give a CRF cupola is because the dihedral angle of the icosahedron is too large: once you attach the triangular prisms to its faces, the angle between adjacent prisms is too narrow to fit in another triangular prism (which is necessary to construct the cupola). It's analogous to trying to make a CRF heptagonal cupola in 3D: once you put in the square faces there just isn't enough room between them to fit equilateral triangles. The best you can do is to use isosceles triangles. In the case of the icosahedral cupola, the prisms would have to be non-CRF in order to fit between two prisms attached to adjacent icosahedron faces.

This is very counterintuitive indeed. Just like the non-existence of the CRF dodecahedral pyramid.
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### Re: Classifying the segmentochora

quickfur wrote:
wintersolstice wrote:[...]
according to wikipedia a 4D cupola is a platonic solid to it's cantellate. and if that is the case the icosahedron case isn't CRF
[...]

Well, I was proposing that we generalize the term "cupola" to cover all cases of A || B where A and B are of the same symmetry group, A ≠ B, and the difference in circumradius between A and B < edge length. There are some cases for which the difference in circumradius is exactly equal to the edge length, in which case you get a 3D tiling, not a full-dimensioned segmentochoron. If the edge length < difference in circumradius, then the shape cannot be made CRF.

I'm surprised the icosahedron case isn't CRF, though. Is it because edge length < difference in circumradius?

when you say "symmetry groups" I'm guessing you mean a regular polytope and it's hypertruncates?:D) for these though I use the term "cupola relatives" and have come up with various names (containing the word "cupola" have you seen them above?
and a "polytope||expanded polytope" the word "Cupola" is used by itself:D

This is just how I would do it though, what do you think? (there still classed as cupola there's just a "true cupola" that's all:D)

quickfur wrote:I found this very counterintuitive, so I did a little mental experiment where I attached triangular prisms to the faces of the icosahedron, as if I was going to construct the cupola, and then I realized that the reason this doesn't give a CRF cupola is because the dihedral angle of the icosahedron is too large: once you attach the triangular prisms to its faces, the angle between adjacent prisms is too narrow to fit in another triangular prism (which is necessary to construct the cupola).

according to an observation I made, in order for a cupola (of the base being an expanded top) to be CRF the "top" needs to have ditope angles of less than 120 degrees, regardless of dimension!!!
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### Re: Classifying the segmentochora

wintersolstice wrote:
quickfur wrote:
wintersolstice wrote:[...]
according to wikipedia a 4D cupola is a platonic solid to it's cantellate. and if that is the case the icosahedron case isn't CRF
[...]

Well, I was proposing that we generalize the term "cupola" to cover all cases of A || B where A and B are of the same symmetry group, A ≠ B, and the difference in circumradius between A and B < edge length. There are some cases for which the difference in circumradius is exactly equal to the edge length, in which case you get a 3D tiling, not a full-dimensioned segmentochoron. If the edge length < difference in circumradius, then the shape cannot be made CRF.

I'm surprised the icosahedron case isn't CRF, though. Is it because edge length < difference in circumradius?

when you say "symmetry groups" I'm guessing you mean a regular polytope and it's hypertruncates?:D) for these though I use the term "cupola relatives" and have come up with various names (containing the word "cupola" have you seen them above?
and a "polytope||expanded polytope" the word "Cupola" is used by itself:D

This is just how I would do it though, what do you think? (there still classed as cupola there's just a "true cupola" that's all:D)

I kinda prefer to just class all of them the same way, especially after Klitzing pointed out that his definition of cupola (or was it antiprism?) is different from the one I always assumed would be most obvious. That caused me to rethink my definition of cupola, and why I should prefer that definition and not Klitzing's, or some other altogether. In the end, I think I decided to lean towards a more general category that includes all cupola-like shapes, which IMO is a cleaner definition that doesn't make an arbitrary choice to treat a certain subclass of objects in a special way, unless they stand out geometrically.

So in my new definition, a segmentotope A||B is:
- a prism if A=B;
- a pyramid if A=point (and B≠point);
- a wedge if A is subdimensional ((n-2)-dimensions or less) and B is full-dimensional ((n-1)-D);
- a cupola otherwise.

So a pyramid is just a subclass of a wedge where the tip is a point (as opposed to a line or a polygon, etc.). This gives a clean division of segmentotopes into prisms, wedges, and cupolae. Prisms take care of the special case where A and B are the same shape, wedges take care of the case where one of them is subdimensional, and everything else is lumped into the general category of cupolae.

In higher dimensions, having a general definition of cupola is much more useful, because the number of ways to put two things together just increases exponentially as the dimension increases. So rather than having to invent brand new categories for every dimension, might as well group them together.

quickfur wrote:I found this very counterintuitive, so I did a little mental experiment where I attached triangular prisms to the faces of the icosahedron, as if I was going to construct the cupola, and then I realized that the reason this doesn't give a CRF cupola is because the dihedral angle of the icosahedron is too large: once you attach the triangular prisms to its faces, the angle between adjacent prisms is too narrow to fit in another triangular prism (which is necessary to construct the cupola).

according to an observation I made, in order for a cupola (of the base being an expanded top) to be CRF the "top" needs to have ditope angles of less than 120 degrees, regardless of dimension!!!

Yep, you're right.

P.S. You might want to just post your categories on the wiki's CRF page, so that we don't forget about it and then you have to keep reposting. Plus, once it's on the wiki it's easier to refine/adjust it if need be. (Otherwise we have to keep posting modified lists and after a while nobody remembers which one is the "official" one.)
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### Re: Classifying the segmentochora

quickfur wrote:In the end, I think I decided to lean towards a more general category that includes all cupola-like shapes, which IMO is a cleaner definition that doesn't make an arbitrary choice to treat a certain subclass of objects in a special way, unless they stand out geometrically.

So in my new definition, a segmentotope A||B is:
- a prism if A=B;
- a pyramid if A=point (and B≠point);
- a wedge if A is subdimensional ((n-2)-dimensions or less) and B is full-dimensional ((n-1)-D);
- a cupola otherwise.

So a pyramid is just a subclass of a wedge where the tip is a point (as opposed to a line or a polygon, etc.). This gives a clean division of segmentotopes into prisms, wedges, and cupolae. Prisms take care of the special case where A and B are the same shape, wedges take care of the case where one of them is subdimensional, and everything else is lumped into the general category of cupolae.

To be blunt, I really can't see how this could be a good idea.

There's no point defining something as being "any X that is not Y nor Z nor W", because you can just call it an otherwise unclassified X.

I forget exactly how I defined cupolae when I made my list, but I do recall coming up with a wider definition than the "pure" one. I'll have a look at that this evening though.

I also don't agree with picking out wedges, writing something as a wedge often seems to hide its real symmetry (I struggled to construct the K4.8 from its being written as a wedge, when really it's a much more interesting shape than that, and your later observation that it's a diminishing of a certain uniform polychoron is a much easier way to understand it).

IIRC almost all the wedges in Klitzing's list are bicupolic or biantiprismatic rings and there are only three or four left over. Whereas there are infinitely many prisms and a decent number of pyramids.

Perhaps enumerating the 5D segmentotopes would give more reason to pick out "wedges", but so far I don't think it's appropriate.

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### Re: Classifying the segmentochora

Keiji wrote:To be blunt, I really can't see how this could be a good idea.

There's no point defining something as being "any X that is not Y nor Z nor W", because you can just call it an otherwise unclassified X.

I forget exactly how I defined cupolae when I made my list, but I do recall coming up with a wider definition than the "pure" one. I'll have a look at that this evening though.

I also don't agree with picking out wedges, writing something as a wedge often seems to hide its real symmetry (I struggled to construct the K4.8 from its being written as a wedge, when really it's a much more interesting shape than that, and your later observation that it's a diminishing of a certain uniform polychoron is a much easier way to understand it).

I think i agree here actually:D my definition of cupola is "polytope||expanded poytope" although I also regard anything with "Platonic and/or Archimedean solids" as being "Cupola-like" I don't see a problem with calling these cupola though:D so maybe the "polytope||expanded poytope" is just a type of cupola. and then there are diminishes and gyrations which form another catogory seperate from the cupola:D

my catorgarastion above shows how the "cupola" (form the definition in the above paragraph) can be subdivided

however maybe another catorgorisation system (completly independent) could be used to define cases by definition of dimension of the bases but come up with some new terms for that:D

Keiji wrote: Perhaps enumerating the 5D segmentotopes would give more reason to pick out "wedges", but so far I don't think it's appropriate

I have been investigating this actually:D I actually made a thread about extending "P||Expanded P" form if you want to have a look at the "multicupolic screw guages thread":D

as well a 5D Johnson cases:D
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### Re: Classifying the segmentochora

wintersolstice wrote:[...]there also seems to be an issue with the cupolae.

according to wikipedia a 4D cupola is a platonic solid to it's cantellate. and if that is the case the icosahedron case isn't CRF

As I outlined in my papaer, there are 2 different possibilities to extrapolate 3d cupola into 4d: either to use prisms instead of the latteral squares, or to use antiprisms. Both would be valid possibilities to choose a 4d definition from! (Or to allow both.) In my papaer I decded to use the antiprismatic version, as those work for all platonic solids (as xPoQo || oPxQo) and additionally for the quasiregular ones (as oPxQo || xPoQx). The prism-case (as xPoQo || xPoQx) would run into difficulties even for the ikosahedron! - I then mentioned that the latter ones OTOH (within the possible range) extrapolates the property of 3d cupolae to be a cap of a larger solid. In fact xPo || xPx (for P=2,3,4,5) is a cap of xPo3x. Similar is this prismatic choice (i.e. xPoQo || xPoQx) a cap of xPoQo3x. As this class of possible cupoloids therefore has an additional property (i.e. being a cap), I chose to call those ones just caps, while reserving the cupola name for the other ones.

now Gyrobicupolae are made by taking the dual of the base so the "cube gyrobicupola" is actually a "cube cupola" and a "octahedron cupola" joined together. (it also means that "cube gyrobicupola" = "octahedron gyrobicupola")

since the icosahedron cupola isn't CRF it means that dodecahedron gyrobicupola (or the icosahedron gyrobicupola) isn't CRF

You should be careful when applying names to kind of property-extrapolations into 4d, which do not conform with the very meaning of the word itself. This is a great deal esp. of Wendys polygloss, to try to cut all that historically wrong applied even. - The very word gyro just means rotated. Sure a rotated polygon looks like its dual, thus for segmentohedra this would be the same. But in 4d a rotated cube does not become an octahedron!

You might be even cf. my careful choose of wordings, esp. with such cases: I used tet || dual tet, but tut || inv(-erted) tut.

However on Klizting paper a cupola is a "platonic solid to rectate" and "rectate to cantellate" (this is meaning currently being used on here http://teamikaria.com/hddb/wiki/Segment ... sification)

it also says under the CRF article about the cupolae that there is only one for each Platonic solid (which appears to mean Platonic to cantellate but it isn't clear) and it counts ortho and gyro forms for all 5 even though some are the same and some not CRF

But it does say that the ability to construct them CRF needs to be checked.

This might relate to my statement "... and, if their circumradii do not differ too much, the result will be
a valid segmentochoron again." with respect to the table 2 of the article. I.e. table 2 provides the general building process, but you would have to check whether the outcome will be possible within spherical space (concentrically positioned base polyhedra would have a lacing vertex distance smaller than 1), to flat euclidean space (... equal to 1), or to hyperbolic space (... greater than 1).

Or it might relate to the slightly thereafter mentioned point "Even more generall one will have to take any 2 figures from Table 1..." (i.e. any 2 orbiform solids) "...in any possible relative orientation..." (this is the main point here!) "... and has to decide whether there would be a convex segmentochoron lying in
between". In fact this statement outlines what has been done in the research which terminated in that paper. So it is not an open issue!

(The only thing what still is open: I did all my research by carefully looking into any possibility. Thus I doubt that there would be any other cases. But I might have overlooked something. There is no proof that this set is complete. No computer research was done either.)
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### Re: Classifying the segmentochora

quickfur wrote:
wintersolstice wrote:[...]
according to wikipedia a 4D cupola is a platonic solid to it's cantellate. and if that is the case the icosahedron case isn't CRF
[...]

Well, I was proposing that we generalize the term "cupola" to cover all cases of A || B where A and B are of the same symmetry group, A ≠ B, and the difference in circumradius between A and B < edge length. There are some cases for which the difference in circumradius is exactly equal to the edge length, in which case you get a 3D tiling, not a full-dimensioned segmentochoron. If the edge length < difference in circumradius, then the shape cannot be made CRF.

I'm surprised the icosahedron case isn't CRF, though. Is it because edge length < difference in circumradius?

No need for getting even deeper into troubles of what should be or should not be considered a cupola. There is already a well established term for that subset of segmentotopes (those with the same symmetry group for both bases): the lace prisms! (Or, if you would like: the unit-edged ones.)

--- rk
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### Re: Classifying the segmentochora

Klitzing wrote:
quickfur wrote:[...]
Well, I was proposing that we generalize the term "cupola" to cover all cases of A || B where A and B are of the same symmetry group, A ≠ B, and the difference in circumradius between A and B < edge length. There are some cases for which the difference in circumradius is exactly equal to the edge length, in which case you get a 3D tiling, not a full-dimensioned segmentochoron. If the edge length < difference in circumradius, then the shape cannot be made CRF.
[...]

No need for getting even deeper into troubles of what should be or should not be considered a cupola. There is already a well established term for that subset of segmentotopes (those with the same symmetry group for both bases): the lace prisms! (Or, if you would like: the unit-edged ones.)
[...]

You're right, lace-prisms is a better name for the catch-all category. In this case, it would be the monostratic lace-prisms.

As for what is/isn't a cupola, I think that may be splitting hairs. As Klitzing says, there's no unique generalization of the 3D notion of cupola. IMO the real solution is to recognize that there is no unique 4D analogue of the 3D cupolae instead of trying to decide which of the equally-valid possibilities should be designated as "true" cupolae.
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### Re: Classifying the segmentochora

quickfur wrote:
wintersolstice wrote:
quickfur wrote:
wintersolstice wrote:[...]
according to wikipedia a 4D cupola is a platonic solid to it's cantellate. and if that is the case the icosahedron case isn't CRF
[...]

Well, I was proposing that we generalize the term "cupola" to cover all cases of A || B where A and B are of the same symmetry group, A ≠ B, and the difference in circumradius between A and B < edge length. There are some cases for which the difference in circumradius is exactly equal to the edge length, in which case you get a 3D tiling, not a full-dimensioned segmentochoron. If the edge length < difference in circumradius, then the shape cannot be made CRF.

I'm surprised the icosahedron case isn't CRF, though. Is it because edge length < difference in circumradius?

when you say "symmetry groups" I'm guessing you mean a regular polytope and it's hypertruncates?:D) for these though I use the term "cupola relatives" and have come up with various names (containing the word "cupola" have you seen them above?
and a "polytope||expanded polytope" the word "Cupola" is used by itself:D

This is just how I would do it though, what do you think? (there still classed as cupola there's just a "true cupola" that's all:D)

I kinda prefer to just class all of them the same way, especially after Klitzing pointed out that his definition of cupola (or was it antiprism?) is different from the one I always assumed would be most obvious. That caused me to rethink my definition of cupola, and why I should prefer that definition and not Klitzing's, or some other altogether. In the end, I think I decided to lean towards a more general category that includes all cupola-like shapes, which IMO is a cleaner definition that doesn't make an arbitrary choice to treat a certain subclass of objects in a special way, unless they stand out geometrically.

So in my new definition, a segmentotope A||B is:
- a prism if A=B;
- a pyramid if A=point (and B≠point);
- a wedge if A is subdimensional ((n-2)-dimensions or less) and B is full-dimensional ((n-1)-D);
- a cupola otherwise.

So a pyramid is just a subclass of a wedge where the tip is a point (as opposed to a line or a polygon, etc.). This gives a clean division of segmentotopes into prisms, wedges, and cupolae. Prisms take care of the special case where A and B are the same shape, wedges take care of the case where one of them is subdimensional, and everything else is lumped into the general category of cupolae.

In higher dimensions, having a general definition of cupola is much more useful, because the number of ways to put two things together just increases exponentially as the dimension increases. So rather than having to invent brand new categories for every dimension, might as well group them together.
[...]

But, using that definitionof yours for arbitrary dimension, and applying it back within 3d, you just would say that antiprism are cupolae... - ?
--- rk
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### Re: Classifying the segmentochora

quickfur wrote:
Klitzing wrote:
quickfur wrote:[...]
Well, I was proposing that we generalize the term "cupola" to cover all cases of A || B where A and B are of the same symmetry group, A ≠ B, and the difference in circumradius between A and B < edge length. There are some cases for which the difference in circumradius is exactly equal to the edge length, in which case you get a 3D tiling, not a full-dimensioned segmentochoron. If the edge length < difference in circumradius, then the shape cannot be made CRF.
[...]

No need for getting even deeper into troubles of what should be or should not be considered a cupola. There is already a well established term for that subset of segmentotopes (those with the same symmetry group for both bases): the lace prisms! (Or, if you would like: the unit-edged ones.)
[...]

You're right, lace-prisms is a better name for the catch-all category. In this case, it would be the monostratic lace-prisms.

No you are wrong here! All segmentotopes with a common non-trivial symmetrygroup in both bases are lace prisms. In fact those are exactly the unit-edged ones. And, a lace prism allways is monostratic (else it would be a lace tower)!
As for what is/isn't a cupola, I think that may be splitting hairs. As Klitzing says, there's no unique generalization of the 3D notion of cupola. IMO the real solution is to recognize that there is no unique 4D analogue of the 3D cupolae instead of trying to decide which of the equally-valid possibilities should be designated as "true" cupolae.

Well, a cupola in 3d has full symmetrical bases. It has triangles and squares connecting those.
So to extrapolate these figures into 4d, you would have to say what each of those components would become. Top-base polygons (the smaller ones) might become platonic solids. (But I extended that even to quasiregular cases after all.) The triangles most obviously generalize to pyramids. The squares should become somthing with axial symmetry again. So they either could become antiprisms (my choice, as there are more segmentochora which can be classified by that case) or prisms. In the latter case you would have additional things occuring within that new dimension: triangular prisms (kind as line atop square). Those cases do not even apply to ike (it would be hyperbolic!), nor to the quasiregulars. But those OTOH are caps again...

--- rk
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### Re: Classifying the segmentochora

Keiji wrote:[...] I also don't agree with picking out wedges, writing something as a wedge often seems to hide its real symmetry (I struggled to construct the K4.8 from its being written as a wedge, when really it's a much more interesting shape than that, and your later observation that it's a diminishing of a certain uniform polychoron is a much easier way to understand it).

IIRC almost all the wedges in Klitzing's list are bicupolic or biantiprismatic rings and there are only three or four left over. Whereas there are infinitely many prisms and a decent number of pyramids.

Perhaps enumerating the 5D segmentotopes would give more reason to pick out "wedges", but so far I don't think it's appropriate.

These shapes are wedges in the sense that they have two large cells that meet at a ridge, with other cells in between to fill up the gaps. In terms of their global shape, they can indeed serve the analogous role to 3D wedges: as a doorstop, as a coarse tool to pry things apart, etc., due to the shallow angle between the two large cells.

The catch is that in 4D, the extra dimension allows these wedges to exhibit a much richer geometry than 3D wedges -- instead of a left and right side, there's a whole circle of sides (to borrow gonegahgah's way of putting it). So again, as with (almost) any term generalized from 3D, "wedge" suffers from being an inadequate description of the 4D geometry.

Now to go a bit deeper into the actual shapes of these things: in 3D, a wedge can be thought of as two large flat faces with one face at the far end opposite the tip of the wedge that complete the trigonal cycle of faces, and then two sides that close up the shape. The dihedral angles in the trigonal cycle of faces are one smallish angle (the tip of the wedge) and two roughly equal, larger angles. In 4D, we have the same 3-membered cycle of two large cells with a cell opposite the tip of the wedge (i.e. the face where the two large cells meet). But instead of just two sides to close up the shape, we have an orthogonal ring of lateral cells, be they in the form of alternating prisms, alternating pyramids, or what-have-you. So basically they have a sort of irregular 3,n-duoprism symmetry to them, where n is the size of the lateral ring. The dichoral angles in the 3-membered ring are again, one small angle (between the two large cells) and two slightly larger angles, thus paralleling the 3D case. The n-membered ring can thus vary freely to some extent, so we get shapes like the 3-, 4-, 5-gonal bicupolic rings, etc..

In the case of K4.8, you can think of it as tetrahedron||square, in which the square is the tip of the wedge, with two anti-aligned trigonal prisms (the two slanted surfaces of the wedge) that meet at it, with the tetrahedron completing the 3-cycle. The orthogonal ring then comprises a 4-cycle of alternating square pyramids to close up the shape.

(So again you see the Hopf fibration at work here, in the form of two orthogonal rings that close up the surface of a 3-sphere. This is a very prominent feature of convex 4D shapes.)

Just as a 3D wedge can be cut from, say, a block of wood (of some initial shape) by slicing along two planes that meet at an edge, so a 4D wedge can be made from some initial polychoron with two cuts that meet at a face. K4.8 can be made by slicing the rectified 5-cell (oxoo) with two hyperplanes that meet at a face, i.e., a bidiminishing. The 600-cell wedges I posted earlier are made in the same way. You'll remember that "wedge #4" exhibits the same rich structure as the other 4D wedges: two large cells that meet at a face (pentagonal rotunda), a third cell to complete the 3 cycle (in this case it's a pentagonal antiprism pyramid, but truncating the pyramid gives you just the pentagonal antiprism), with a circle of cells that fill up the lateral gaps: a ring of tetrahedra near the antiprism (the "blunt end" of the wedge) and a ring of pentagonal pyramids near the decagonal face (the "sharp end" of the wedge).
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### Re: Classifying the segmentochora

Klitzing wrote:
quickfur wrote:[...]
You're right, lace-prisms is a better name for the catch-all category. In this case, it would be the monostratic lace-prisms.

No you are wrong here! All segmentotopes with a common non-trivial symmetrygroup in both bases are lace prisms. In fact those are exactly the unit-edged ones. And, a lace prism allways is monostratic (else it would be a lace tower)!

You're right, lace prisms implies monostratic.

As for what is/isn't a cupola, I think that may be splitting hairs. As Klitzing says, there's no unique generalization of the 3D notion of cupola. IMO the real solution is to recognize that there is no unique 4D analogue of the 3D cupolae instead of trying to decide which of the equally-valid possibilities should be designated as "true" cupolae.

Well, a cupola in 3d has full symmetrical bases. It has triangles and squares connecting those.
So to extrapolate these figures into 4d, you would have to say what each of those components would become. Top-base polygons (the smaller ones) might become platonic solids. (But I extended that even to quasiregular cases after all.) The triangles most obviously generalize to pyramids. The squares should become somthing with axial symmetry again. So they either could become antiprisms (my choice, as there are more segmentochora which can be classified by that case) or prisms. In the latter case you would have additional things occuring within that new dimension: triangular prisms (kind as line atop square). Those cases do not even apply to ike (it would be hyperbolic!), nor to the quasiregulars. But those OTOH are caps again...
[...]

But again, I think this is just splitting hairs. I could, for example, analyse a 3D cupola as a segmentohedron having a bottom face that is the result of marking an unmarked node in the CD diagram of the top face. For example, a square cupola would have a top face x4o, then the bottom face would be x4x. Then in the 4D case, I can allow any such combination: x4o3o || x4x3o can be considered a cupola, and so can x4x3o || x4x3x, or o4x3o || x4x3o, or any other such combination with the same relationship between the top/bottom cells. This would become a third definition of 4D cupola. (And I'm sure you can come up with more, if you wanted to.)

I could argue that this definition is more encompassing than yours, and therefore "better". But that is just a matter of opinion. My point is that there is more than one way to generalize the 3D concept, which aren't all compatible with each other, so maybe we should just call them all lace prisms and be done with it, instead of trying to argue over which definition of cupola is the "correct" one.
Last edited by quickfur on Wed Aug 29, 2012 6:53 pm, edited 1 time in total.
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### Re: Classifying the segmentochora

Klitzing wrote:You should be careful when applying names to kind of property-extrapolations into 4d, which do not conform with the very meaning of the word itself. This is a great deal esp. of Wendys polygloss, to try to cut all that historically wrong applied even. - The very word gyro just means rotated. Sure a rotated polygon looks like its dual, thus for segmentohedra this would be the same. But in 4d a rotated cube does not become an octahedron!

because a rotated square becomes it's dual this could mean that gyration could be redifined to mean both in 4D (like there being more than one meaning of "antiprism" in 4D) I'm not saying it should only that it could

I created a thread for a proposal to modify the definition of "Gyrated" for both analogies

it is only a suggestion and maybe there is a better one
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