Bilbirothawroids (D4.3 to D4.9)

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

Re: Johnsonian Polytopes

Postby Marek14 » Thu Feb 13, 2014 11:43 am

Klitzing wrote:
Marek14 wrote:OK, I explicitly constructed all augments of square biantiprismatic ring. And guess what? If you augment both antiprismatic cell, the result will be just a cube||octahedron antiprism :) Didn't expect that, though Klitzing probably knows that :D

So we have 6 polychora in this group:

Square biantiprismatic ring (also bidiminished cube||octahedron)
Cube-augmented square biantiprismatic ring (also augmented bidiminished cube||octahedron)
Antiprism-augmented square biantiprismatic ring (also diminished cube||octahedron)
Cube-antiprism-biaugmented square biantiprismatic ring (also augmented diminished cube||octahedron)
Antiprism-biaugmented square biantiprismatic ring (also cube || octahedron)
Triaugmented square biantiprismatic ring (also augmented cube||octahedron)

EDIT: Of course, in hindsight it's quite obvious that cube||gyrated square would have connection to cube||octahedron...

If you just augment one of the antiprisms, then I wouldn't be surprised if that comes out to be "gyro suippy || cube".
And yes, both antiprism augmented surely then is "oct || cube".
Whereas, whenever the cube will be augmented (besides potentially others), you'd get a bistratic figure (tower):
either "gyro {4} || cube || pt", or "gyro squippy || cube || pt", or "oct || cube || pt".
--- rk


Yes, augmenting one antiprism is gyro squippy || cube.
Marek14
Pentonian
 
Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

Re: Johnsonian Polytopes

Postby Klitzing » Thu Feb 13, 2014 12:11 pm

Marek14 wrote:I checked the tetrahedral ursachoron -- the dichoral angle at its "apex" tetrahedron (the one surrounded by 4 tridiminished icosahedra) is small enough that this tetrahedron can be augmented by pentachoron (resulting dihedral angle is 173.2837).

Nice observation Marek. reminds somehow to auteddi, the augmented version of teddi.

But then we would not only have ofxo3xooo3oooo&#xt, but also its expanded version too: ofxo3xooo3xxxx&#xt, i.e. by attaching onto the expanded tetrahedral ursachoron ofx3xoo3xxx&#xt a xo3oo3xx&#x segmentochoron (tet||co), i.e. half a spid = x3o3o3x.

Btw. the extended dihedral ursachoron (ofx3xoo2xxx&#xt) too can be augmented. But that one comes out to be not at all surprising. The un-augmented is just the teddi prism. The augmented one then will be the auteddi prism.

--- rk
Klitzing
Pentonian
 
Posts: 1637
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

Re: Johnsonian Polytopes

Postby Marek14 » Thu Feb 13, 2014 12:49 pm

Klitzing wrote:
Marek14 wrote:I checked the tetrahedral ursachoron -- the dichoral angle at its "apex" tetrahedron (the one surrounded by 4 tridiminished icosahedra) is small enough that this tetrahedron can be augmented by pentachoron (resulting dihedral angle is 173.2837).

Nice observation Marek. reminds somehow to auteddi, the augmented version of teddi.

But then we would not only have ofxo3xooo3oooo&#xt, but also its expanded version too: ofxo3xooo3xxxx&#xt, i.e. by attaching onto the expanded tetrahedral ursachoron ofx3xoo3xxx&#xt a xo3oo3xx&#x segmentochoron (tet||co), i.e. half a spid = x3o3o3x.

Btw. the extended dihedral ursachoron (ofx3xoo2xxx&#xt) too can be augmented. But that one comes out to be not at all surprising. The un-augmented is just the teddi prism. The augmented one then will be the auteddi prism.

--- rk


Yes, thought it's an 4D analogue of augmented tridiminished icosahedron, though I was a bit disappointed that the dichoral angle didn't come out as full 180 -- as that would make augmented tridiminished tetrahedra actual cells :)

However, it looks that our current state of CRF searching is really a mess. I think that the problem, ironically, is that we tend to be people with very good ability to visualize the higher dimensions. Almost any image is enough for us to see what we're talking about. But it's not systematic.

I think that first thing we need is to have a database of models. The problem here is that each of us is using different tools, but the *.off files people sometimes post for me seem generic enough -- I was able to write several from scratch in text editor, though that method is really not feasible for complicated polychora.

So, I think we need models -- for every CRF polychoron we find (with exception of huge families like augmented duoprisms, though they probably COULD be handled by an automated program) we should have a file with complete data necessary so anyone could reconstruct it. If our discoveries (and this thread has seen some amazing discoveries) are to spread, we MUST do this. Not everyone is capable to understand the structure from a still image (even stereoscopic). I am very much helped by Stella in this regard since I can do things like setting model in slow 4D rotation and watching it change, examine the sections etc., but that program is not much help in actually constructing the polychora in first place :)

I think I could compile most of the nonuniform segmentochora from the list, though it would be probably nice to have a list of coordinates :)
Marek14
Pentonian
 
Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

Re: Johnsonian Polytopes

Postby student91 » Thu Feb 13, 2014 3:54 pm

quickfur wrote:
Marek14 wrote:
quickfur wrote:
Marek14 wrote:
quickfur wrote:Hold on, when you have nested square roots, the CVP is no longer 2, because those values cannot be expressed as the root of a single quadratic polynomial.

Though you do have a point that cubics have no straightedge-compass constructions. Hmm. So cubics have a higher complexity than even nested square roots?


Maybe CVP 2 should not be thought as being root of single quadratic polynomial but rather as root of ONLY quadratic polynomials, i.e. never during search for these values are we forced to solve polynomial of degree 3 or higher.

In other words, CVP is the highest degree of a polynomial you have to solve to find coordinates for the given polytope, no matter how many polynomials of that type you have to solve.

EDIT: I basically say the same thing as Keiji, just in a bit different words.

After some further thoughts about this, I have come to agree with Marek. Consider, for example, the pentagon. Its coordinates when laid on the 2D plane involves nested square roots; yet in 3D, when it occurs as a cross-section of the icosahedron, its coordinates involve only the Golden Ratio, which has no nested square roots. Obviously, it makes no sense to say that the pentagon has CVP >2 in 2D, and CVP=2 in 3D. So this is strong proof that it's not about how many square roots you take, but what degree of polynomial you need to solve in order to find the values.


And consider equilateral triangle: in 2D it will always involve square roots, while in 3D it can be constructed between points (1,0,0),(0,1,0) and (0,0,1).

Or, for that matter, the n-simplex, which has integer coordinates in (n+1)-dimensions, but in n-dimensions involves square roots of triangular numbers (so not just roots of the same number, like √5 in things involving the Golden Ratio like the pentagonal polytopes).


I already proposed a definition that bypasses the coordinate problem:
student91 wrote:[...]
Furthermore, I thought of a definition of the CVP of a polytope as follows: Take a vertex figure of every vertex. calculate the minimal polynominal of the distances between every vertex inside this vertex figure. Get the biggest prime factor of the exponents. Do this for every distance and take the biggest number.
So basically you compute the biggest prime factor of the exponent for every two vertices that connect to the same vertex.

when you calculate the CVP this way, it is independent of any coordinate system. This means, the CVP gives the minimal number you would get if you place it in an idealized coordinate system.
How easily one gives his confidence to persons who know how to give themselves the appearance of more knowledge, when this knowledge has been drawn from a foreign source.
-Stern/Multatuli/Eduard Douwes Dekker
student91
Tetronian
 
Posts: 328
Joined: Tue Dec 10, 2013 3:41 pm

Re: Johnsonian Polytopes

Postby student91 » Thu Feb 13, 2014 5:15 pm

quickfur wrote:I think I may have found another possible CRF construction involving J91. Take a look at the following projection of the rectified 120-cell o5x3o3o:

Image

[...]

A more remote possibility, which I have to think about more carefully, is if you look at the o5x3o's in this projection and note that their outward-facing face is a triangle, that is, this is the part of their surface that matches that of J92. If we bisect the polychoron with a hyperplane that bisects these cells parallel to the triangular faces, we will get a non-CRF polystratic cup of the o5x3o3o, but hexagons may be inserted to turn the non-CRF bisected o5x3o's into J92's, and maybe some other CRF fragments can then be added to make the result CRF. The gaps where the exposed cyan cells lie, are where another 12 o5x3o's lie, and these are in just the right orientation to be bisected into pentagonal rotundae. So perhaps it is possible to make a CRF cup of o5x3o3o that contains 12 pentagonal rotundae and 20 J92's?

After thinking of this, I got a conclusion:
First look at the o3x5o-first buildup of the o5x3o3o. it looks like o3x5o||o3o5f||o3x5x||x3f5o||F3o5x||f3f5o (||o3F5o||f3x5x||.....).(Klitzing, your site is very helpfull to look things like this up, thanks :) ) The f3f5o are the hexagons you wanted to shrink. This means you get .....||F3o5x||x3x5?. it seems, according to my calculations, that ?=x works, which means the polytope can be capped of with a x3x5x, and we have a new CRF :D . Now I hope my calculations are correct :\
The new CRF should then look like oooxFx3xoxfox5ofxoxx&#xt.
of course you can take a cap of, giving oxFx3xfox5xoxx&#xt
it doesn't have your rotunda's. Instead, it has pentagonla cupola's. :)
How easily one gives his confidence to persons who know how to give themselves the appearance of more knowledge, when this knowledge has been drawn from a foreign source.
-Stern/Multatuli/Eduard Douwes Dekker
student91
Tetronian
 
Posts: 328
Joined: Tue Dec 10, 2013 3:41 pm

Re: Johnsonian Polytopes

Postby quickfur » Thu Feb 13, 2014 7:27 pm

Marek14 wrote:I checked the tetrahedral ursachoron -- the dichoral angle at its "apex" tetrahedron (the one surrounded by 4 tridiminished icosahedra) is small enough that this tetrahedron can be augmented by pentachoron (resulting dihedral angle is 173.2837).

Nice!! So this would be a direct analogue of the augmented tridiminished icosahedron (J64).

Next question: is this true of the n-simplex ursatopes in general? Can they all be augmented by n-simplexes? I think the answer is yes, though I don't have a rigorous proof: the n-simplex ursatopes can be considered as a modification of the truncated n-simplex where the base facet has been shrunken into a rectified (n-1)-simplex instead of the larger truncated (n-1)-simplex facet. This shrinking reduces the difacet angle at the top of the truncated n-simplex, so that if you augment it with an n-simplex, the (n-1)-simplex facets are no longer coplanar with what was originally the (n-1)-simplex facets (the shrinking has distorted them into (n-1)-ursatopes). So the result should be CRF.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Johnsonian Polytopes

Postby quickfur » Thu Feb 13, 2014 7:56 pm

Marek14 wrote:[...]
However, it looks that our current state of CRF searching is really a mess. I think that the problem, ironically, is that we tend to be people with very good ability to visualize the higher dimensions. Almost any image is enough for us to see what we're talking about. But it's not systematic.

I agree.

I think that first thing we need is to have a database of models. The problem here is that each of us is using different tools, but the *.off files people sometimes post for me seem generic enough -- I was able to write several from scratch in text editor, though that method is really not feasible for complicated polychora.

When I first started writing my polytope viewer program, I was writing polytope definitions by hand -- it was extremely tedious and error-prone. I managed to write the 5-cell, 16-cell, and tesseract definitions by hand, but when I got to the 24-cell, the 96 edges + 96 polygons defeated me. There are just far too many surtopes to write by hand. That's when I started to realize that a convex hull algorithm is a must. It doesn't work for non-convex polytopes, but fortunately, for CRFs, it is exactly what we need.

So, I think we need models -- for every CRF polychoron we find (with exception of huge families like augmented duoprisms, though they probably COULD be handled by an automated program)

I already have a program for producing duoprisms of any size, both uniform and non-uniform (all polygons are regular, though). :)

we should have a file with complete data necessary so anyone could reconstruct it. If our discoveries (and this thread has seen some amazing discoveries) are to spread, we MUST do this. Not everyone is capable to understand the structure from a still image (even stereoscopic). I am very much helped by Stella in this regard since I can do things like setting model in slow 4D rotation and watching it change, examine the sections etc., but that program is not much help in actually constructing the polychora in first place :)

I think I could compile most of the nonuniform segmentochora from the list, though it would be probably nice to have a list of coordinates :)

IMO the Stella4D format is not perfect because it cannot store coordinates in algebraic form. I think storing coordinates in algebraic form is a must, because otherwise we cannot prove that anything is CRF except when it has only integer coordinates, because we don't know whether the edge lengths are exactly 1, or 1.00000001 but we didn't notice because roundoff error in computer floating-point arithmetic is usually ignored (it has to be, otherwise irrational numbers like the golden ratio will never measure up exactly). Having algebraic coordinates is the only way we can rigorously prove that something is CRF, and not just a near-enough CRF that the computer failed to notice the discrepancy.

This is one of the reasons I have been rewriting a new version of my polytope viewer, which is able to parse algebraic coordinates. It is still using only floating-point, but the important thing is that the source file of the polytope contains algebraic expressions that can be mathematically verified to have unit distances by a mathematician. Since listing coordinates can be very tedious, I've also incorporated Wendy's apacs, epacs, etc., operators, plus conveniences like ± that generates all sign permutations of its argument, etc.. Most of the CRFs that I've rendered have source files that give algebraic coordinates.

OTOH, just because we post the algebraic coordinates, doesn't mean people are going to be able to see what the polytope is, because that would require solving the n-dimensional convex hull problem in our head. :P So I like Marek's idea of keeping a database of CRF models in Stella4D format -- the software is easily available and user-friendly (my polytope viewer is text-based, and not very user-friendly to point-n-click people :P).

So here is my proposal: since this forum has an associated wiki, which already has a page listing CRF polychora that have been discovered (well, at least those that somebody has taken the time to add to that page), and since this wiki has the ability to upload files, why not use it? Every CRF we discover should be posted on the wiki, linked from the CRF polychora discovery project page, along with a file containing its algebraic coordinates (or post the coordinates on the page itself), and another file containing the Stella4D .off file.

We should also talk about the canonical classification of CRFs, because that page right now, although it does have somewhat a similar organization to Johnson's list of 3D CRFs, still has a lot of overlaps between categories, and is not very navigable. Perhaps it is OK to have overlap between categories, since some CRFs can be understood in multiple ways, but there should be an official list of all known CRFs without repetition. There's also the official CRF count, which is hand-maintained right now and therefore not up-to-date. Ideally, there should be some way of automatically updating that count, based on which pages in the wiki are tagged with the 4D CRF category. (This is more tricky than it sounds, though, because we obviously don't want to have 1 page per CRF for the larger subsets like the duoprism augmentations or 600-cell diminishings, which would be too many; so for those cases we need to be able to specify a precomputed count for a particular subcategory of CRFs.)

In any case, please, people, go to the wiki's CRF polychora discovery project page and update it. IIRC, if you have a forum account you should also have a wiki account, so you can just start editing pages. Hopefully, it will not just be Keiji and myself who update the wiki!
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Johnsonian Polytopes

Postby Keiji » Thu Feb 13, 2014 8:22 pm

quickfur wrote:IIRC, if you have a forum account you should also have a wiki account, so you can just start editing pages. Hopefully, it will not just be Keiji and myself who update the wiki!


As I have previously mentioned, you don't automatically get a wiki account just by being on the forum. If anyone needs a wiki account, send me a PM and I will generate an invite code for you.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Johnsonian Polytopes

Postby Marek14 » Thu Feb 13, 2014 10:20 pm

quickfur wrote:
Marek14 wrote:So, I think we need models -- for every CRF polychoron we find (with exception of huge families like augmented duoprisms, though they probably COULD be handled by an automated program)

I already have a program for producing duoprisms of any size, both uniform and non-uniform (all polygons are regular, though). :)


Stella has also automatic generation of duoprisms and antiduoprisms -- I'm thinking about asking Webb to add generation of biantiprismatic rings as well.

we should have a file with complete data necessary so anyone could reconstruct it. If our discoveries (and this thread has seen some amazing discoveries) are to spread, we MUST do this. Not everyone is capable to understand the structure from a still image (even stereoscopic). I am very much helped by Stella in this regard since I can do things like setting model in slow 4D rotation and watching it change, examine the sections etc., but that program is not much help in actually constructing the polychora in first place :)

I think I could compile most of the nonuniform segmentochora from the list, though it would be probably nice to have a list of coordinates :)

IMO the Stella4D format is not perfect because it cannot store coordinates in algebraic form. I think storing coordinates in algebraic form is a must, because otherwise we cannot prove that anything is CRF except when it has only integer coordinates, because we don't know whether the edge lengths are exactly 1, or 1.00000001 but we didn't notice because roundoff error in computer floating-point arithmetic is usually ignored (it has to be, otherwise irrational numbers like the golden ratio will never measure up exactly). Having algebraic coordinates is the only way we can rigorously prove that something is CRF, and not just a near-enough CRF that the computer failed to notice the discrepancy.


Well, the *.off files have very precise numerical data. The compressed format Stella uses is not ideal because I don't think anything else can use it and the program is not free; however *.off or *.stel files would not be used for proof (that needs algebraic coordinates, at least for crown jewels), but for better visualization and, in my case, for easy checking of dichoral angles which allows for quick determination of possible augments, and for sections which allow for easy determination of diminishings.

This is one of the reasons I have been rewriting a new version of my polytope viewer, which is able to parse algebraic coordinates. It is still using only floating-point, but the important thing is that the source file of the polytope contains algebraic expressions that can be mathematically verified to have unit distances by a mathematician. Since listing coordinates can be very tedious, I've also incorporated Wendy's apacs, epacs, etc., operators, plus conveniences like ± that generates all sign permutations of its argument, etc.. Most of the CRFs that I've rendered have source files that give algebraic coordinates.

OTOH, just because we post the algebraic coordinates, doesn't mean people are going to be able to see what the polytope is, because that would require solving the n-dimensional convex hull problem in our head. :P So I like Marek's idea of keeping a database of CRF models in Stella4D format -- the software is easily available and user-friendly (my polytope viewer is text-based, and not very user-friendly to point-n-click people :P).


Well, I suggest *.off instead of Stella4D format since it's more universal, but of course we can post both. Another advantage of Stella is that no matter how the original file has defined colors, Stella is able to do automatic recolor based on equivalence of cells which, among other things, automatically determines the symmetry group and makes it easily visible.


So here is my proposal: since this forum has an associated wiki, which already has a page listing CRF polychora that have been discovered (well, at least those that somebody has taken the time to add to that page), and since this wiki has the ability to upload files, why not use it? Every CRF we discover should be posted on the wiki, linked from the CRF polychora discovery project page, along with a file containing its algebraic coordinates (or post the coordinates on the page itself), and another file containing the Stella4D .off file.

We should also talk about the canonical classification of CRFs, because that page right now, although it does have somewhat a similar organization to Johnson's list of 3D CRFs, still has a lot of overlaps between categories, and is not very navigable. Perhaps it is OK to have overlap between categories, since some CRFs can be understood in multiple ways, but there should be an official list of all known CRFs without repetition. There's also the official CRF count, which is hand-maintained right now and therefore not up-to-date. Ideally, there should be some way of automatically updating that count, based on which pages in the wiki are tagged with the 4D CRF category. (This is more tricky than it sounds, though, because we obviously don't want to have 1 page per CRF for the larger subsets like the duoprism augmentations or 600-cell diminishings, which would be too many; so for those cases we need to be able to specify a precomputed count for a particular subcategory of CRFs.)

In any case, please, people, go to the wiki's CRF polychora discovery project page and update it. IIRC, if you have a forum account you should also have a wiki account, so you can just start editing pages. Hopefully, it will not just be Keiji and myself who update the wiki!


It's just that I'm currently getting close to finishing a job (I'm translating a book about history of mathematics), so my time is limited at the moment. It might take a few days, I still have to translate the index.

Do you have a program available that could generate a full *.off file of convex hull just from the vertices? If I can generate my own, I can do a lot :)
Marek14
Pentonian
 
Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

Re: Johnsonian Polytopes

Postby Klitzing » Thu Feb 13, 2014 11:05 pm

student91 wrote:
quickfur wrote:A more remote possibility, which I have to think about more carefully, is if you look at the o5x3o's in this projection and note that their outward-facing face is a triangle, that is, this is the part of their surface that matches that of J92. If we bisect the polychoron with a hyperplane that bisects these cells parallel to the triangular faces, we will get a non-CRF polystratic cup of the o5x3o3o, but hexagons may be inserted to turn the non-CRF bisected o5x3o's into J92's, and maybe some other CRF fragments can then be added to make the result CRF. The gaps where the exposed cyan cells lie, are where another 12 o5x3o's lie, and these are in just the right orientation to be bisected into pentagonal rotundae. So perhaps it is possible to make a CRF cup of o5x3o3o that contains 12 pentagonal rotundae and 20 J92's?

After thinking of this, I got a conclusion:
...
The new CRF should then look like oooxFx3xoxfox5ofxoxx&#xt.
of course you can take a cap of, giving oxFx3xfox5xoxx&#xt
it doesn't have your rotunda's. Instead, it has pentagonla cupola's. :)

Whether that (oooxFx3xoxfox5ofxoxx&#xt) is what quickfur had in mind or not, I don't know.
But I had a hard time to recognise that this should be CRF!

First I just considered what you proclaimed. Edges might all be correct. I did not check them, but I calculated the respective heights between the layers. Those all vary between the values 0.5 (layers 2-3, 3-4, 5-6) and 0.309 (1-2 and 4-5). But then I considered the cells of that lace tower:
Code: Select all
a 3 b 5 c
---------
o   x   o   (1)
o   o   f   (2)
o   x   x   (3)
x   f   o   (4)
F   o   x   (5)
x   x   x   (6)

First consider the a3b column cells. Those are 20 tets (1-2), 20 tets (2-3), 20 J92's (3-6).
Then consider the b5c column cells. Those are 12 ids (1-5) and 12 pecues (5-6).
Now to the a2c column cells. Those are 30 flat {5}'s (1-3), 30 tets (3-4). But what then shall follow at (4-5) and (5-6) eluded me:

What should be those 30 xFx2oxx&#xt ? :o

But then I recognised that this "xFx" derives from the lower part of the J92. Accordingly this is NOT just 2 attached monostratic sections of a decagon (as it might seem at first sight), but rather it is a (bistratic) lune, i.e. the facial sequence {3} - {4} - {3} ! Therefore that requested bit should consist of squippy + trip + squippy each, heureka! 8)

Thus, after all, yes, it turns out to be indeed a further CRF! :nod:
The total cell count thus would be: 70 tets + 20 J92's + 13 ids + 12 pecues + 60 squippies + 30 trips + 1 grid.

Great find!


PS: it just occurs me, that quickfur might have thought of the diminished version of it, chopping off the layers (1) and (2) from the above, i.e. just to consider ..oxFx3..xfox5..xoxx&#xt only.
This then changes the corresponding total cell count into: 1 tid + 30 tets + 20 J92's + 12 peroes + 12 pecues + 60 squippies + 30 trips + 1 grid.
(Here they are again, the peroes or half-ids!)

Even better (more elementary) find! :D

--- rk

Edit: As pointed out above in bold: it should be tid, not ti, for sure.
Last edited by Klitzing on Sat Feb 15, 2014 12:06 am, edited 1 time in total.
Klitzing
Pentonian
 
Posts: 1637
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

Re: Johnsonian Polytopes

Postby quickfur » Thu Feb 13, 2014 11:06 pm

Marek14 wrote:
quickfur wrote:
Marek14 wrote:So, I think we need models -- for every CRF polychoron we find (with exception of huge families like augmented duoprisms, though they probably COULD be handled by an automated program)

I already have a program for producing duoprisms of any size, both uniform and non-uniform (all polygons are regular, though). :)


Stella has also automatic generation of duoprisms and antiduoprisms -- I'm thinking about asking Webb to add generation of biantiprismatic rings as well.

Ideally, you want a programmable interface instead, since otherwise we have to be bugging him every time we discover a new infinite family (or a finite family with enough members that automation is indispensible).

<soapbox> This is why I prefer CLI programs, because of the scriptability. Endow it with a sufficiently expressive scripting language, and your users can go wild with creating things you've never conceived of before, rather than having them bug you every single time they want something a little different. </soapbox> ;)

we should have a file with complete data necessary so anyone could reconstruct it. If our discoveries (and this thread has seen some amazing discoveries) are to spread, we MUST do this. Not everyone is capable to understand the structure from a still image (even stereoscopic). I am very much helped by Stella in this regard since I can do things like setting model in slow 4D rotation and watching it change, examine the sections etc., but that program is not much help in actually constructing the polychora in first place :)

I think I could compile most of the nonuniform segmentochora from the list, though it would be probably nice to have a list of coordinates :)

IMO the Stella4D format is not perfect because it cannot store coordinates in algebraic form. I think storing coordinates in algebraic form is a must, because otherwise we cannot prove that anything is CRF except when it has only integer coordinates, because we don't know whether the edge lengths are exactly 1, or 1.00000001 but we didn't notice because roundoff error in computer floating-point arithmetic is usually ignored (it has to be, otherwise irrational numbers like the golden ratio will never measure up exactly). Having algebraic coordinates is the only way we can rigorously prove that something is CRF, and not just a near-enough CRF that the computer failed to notice the discrepancy.


Well, the *.off files have very precise numerical data. The compressed format Stella uses is not ideal because I don't think anything else can use it and the program is not free; however *.off or *.stel files would not be used for proof (that needs algebraic coordinates, at least for crown jewels), but for better visualization and, in my case, for easy checking of dichoral angles which allows for quick determination of possible augments, and for sections which allow for easy determination of diminishings.

I'm sorry, I didn't realize Stella4D has its own format; I thought .off was the "official" format. Every where I referred to the Stella4D format you should read ".off" instead. ;)

[...] Do you have a program available that could generate a full *.off file of convex hull just from the vertices? If I can generate my own, I can do a lot :)

It's not a single program, but yes. There's a program that takes a list of floating-point coordinates and computes the convex hull for it, and outputs a .def file that my polytope viewer uses. Of course, that's not quite what you want, so there's also a program that converts a .def file into an .off file. That gives the basic functionality, but it can be a pain sometimes to manually specify every vertex, so for that, my (incomplete, and very much alpha-quality) new polytope viewer supports algebraic representations for the coordinates, including Wendy's handy apacs/epacs, etc., operators for generating permutations and sign changes.

The gotcha, though, is that these are all Linux command-line programs, and I've never tested them on Windows before (don't even have a Windows dev environment, in fact). If you have a Linux box, then I can send you the source code... but if not, I'll have to think of another way.

Alternatively, you can ask Keiji to hook it up to a web interface, since he has just recently managed to build working executables from the source code. :)

There are other convex hull programs out there that you can use, but probably they don't output .off files: most convex hull programs only compute the bounding hyperplanes, not the entire face lattice, which is what .off files need, though technically speaking, the convex hull algorithm itself does give you the incidence matrix, which is already enough to compute the entire face lattice, but I've never seen a program besides my own that actually bothers to do so. There's also the issue that not all convex hull algorithms are suitable for what we need: most applications of convex hulls are for linear optimization, and generally don't produce very nice output for highly-degenerate polytopes (i.e., polytopes with non-simplex facets). Some algorithms will simply abort if the input has degenerate (aka interesting :P ) facets, and some will break them up into simplexes or perturb the vertices to make them non-coplanar -- not very nice for what we need! The only algorithm I know of that nicely handles the kind of polytopes we're interested in is Motzkin's "double-description" algorithm, and this is what I use in my program. Another program that uses this algorithm is Nikolai Zolotykh's "Skeleton". I've not used this program before, but it's supposed to have some improvements that makes the algorithm run faster than Fukuda's cddlib, which is what my program uses.

Or you could just post the coordinates here and I'll run it through the convex hull for you. :mrgreen:
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Johnsonian Polytopes

Postby quickfur » Thu Feb 13, 2014 11:11 pm

Klitzing wrote:
student91 wrote:
quickfur wrote:A more remote possibility, which I have to think about more carefully, is if you look at the o5x3o's in this projection and note that their outward-facing face is a triangle, that is, this is the part of their surface that matches that of J92. If we bisect the polychoron with a hyperplane that bisects these cells parallel to the triangular faces, we will get a non-CRF polystratic cup of the o5x3o3o, but hexagons may be inserted to turn the non-CRF bisected o5x3o's into J92's, and maybe some other CRF fragments can then be added to make the result CRF. The gaps where the exposed cyan cells lie, are where another 12 o5x3o's lie, and these are in just the right orientation to be bisected into pentagonal rotundae. So perhaps it is possible to make a CRF cup of o5x3o3o that contains 12 pentagonal rotundae and 20 J92's?

After thinking of this, I got a conclusion:
...
The new CRF should then look like oooxFx3xoxfox5ofxoxx&#xt.
of course you can take a cap of, giving oxFx3xfox5xoxx&#xt
it doesn't have your rotunda's. Instead, it has pentagonla cupola's. :)

[...]
Thus, after all, yes, it turns out to be indeed a further CRF! :nod:
The total cell count thus would be: 70 tets + 20 J92's + 13 ids + 12 pecues + 60 squippies + 30 trips + 1 grid.

Great find!


PS: it just occurs me, that quickfur might have thought of the diminished version of it, chopping off the layers (1) and (2) from the above, i.e. just to consider ..oxFx3..xfox5..xoxx&#xt only.
This then changes the corresponding total cell count into: 1 ti + 30 tets + 20 J92's + 12 peroes + 12 pecues + 60 squippies + 30 trips + 1 grid.
(Here they are again, the peroes or half-ids!)

Even better (more elementary) find! :D

--- rk

You guys are going way too fast for me. ;) I'm still experimenting with a construction that involves J91's and J63's, and haven't gotten very far yet. Yesterday I was trying to compute the coordinates for axis-aligned pentagonal cupolae, and boy were the coordinates ugly! I think I spent most of the time trying to chase down algebraic errors I made rather than making any real progress on building the polychoron. :(

But anyway, if somebody can compute the coordinates for student91's new CRF, I can run it through my polyview viewer and render some nice images. :D
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Johnsonian Polytopes

Postby Marek14 » Thu Feb 13, 2014 11:14 pm

Well, I don't have a Linux box so web interface seems like best option. Bothering you with every set of vertices would probably not be very efficient :)

Got another interesting idea a moment ago: if you think about it, duoprisms are based on rings of the type (prism-prism-prism...) with arbitrary number of members. Antiduoprisms are based on rings (prism-antiprism-prism-antiprism). Biantiprismatic rings are based on rings (prism-antiprism-antiprism).

The question is, are there other general structures that could work? Obviously they have to contain even number of antiprisms, but how about rings formed by three prisms and two antiprisms? They wouldn't be monostratic so they wouldn't be found during examination of segmentochora. Can these be made into CRF polychora?
Marek14
Pentonian
 
Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

Re: Johnsonian Polytopes

Postby quickfur » Thu Feb 13, 2014 11:28 pm

Marek14 wrote:Well, I don't have a Linux box so web interface seems like best option. Bothering you with every set of vertices would probably not be very efficient :)

*glances at Keiji* :sweatdrop: ;) :P

OK OK, I'll take a stab at it tonight to see if I can throw something together.

Got another interesting idea a moment ago: if you think about it, duoprisms are based on rings of the type (prism-prism-prism...) with arbitrary number of members. Antiduoprisms are based on rings (prism-antiprism-prism-antiprism). Biantiprismatic rings are based on rings (prism-antiprism-antiprism).

The question is, are there other general structures that could work? Obviously they have to contain even number of antiprisms, but how about rings formed by three prisms and two antiprisms? They wouldn't be monostratic so they wouldn't be found during examination of segmentochora. Can these be made into CRF polychora?

That's what I've been wondering too! :) I've been thinking about whether any CRF other than the grand antiprism has rings of antiprisms. For some reason, the idea of rings of antiprisms just appeals to me a lot.

Also, I wonder if there are other CRFs that are similar to spidrox in having >2 rings with higher-order swirlprism symmetry. And what about the "Hopf function" of a non-uniform polyhedron? That would produce a non-uniform polytwister, and maybe some of them can be made discrete and CRF. AFAIK, this is an area that hasn't been studied in depth before (except perhaps by Jonathan?). I'd like to someday study what are the conditions under which a particular partitioning of the Hopf fibration can be made discrete and CRF. There's got to be a lot of interesting CRFs along the way!
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 14, 2014 4:43 am

It just occurred to me, that the truncated 120-cell x5x3o3o should be augmentable with x5o3x||x5x3o (i.e., a monostratic slice of x5o3x3o). If I'm not mistaken, the pentagonal cupola of the augment will be coplanar with the adjacent x5x3o's, so they will turn into augmented truncated dodecahedra (J68).

If this is correct, then the 120-cell itself should be augmentable with o5o3x||x5o3o. Again, the pentagonal pyramids of the augment should be coplanar with the adjacent dodecahedra, so they will turn into augmented dodecahedra (J58).

For both of these cases, the augments can only be added to non-adjacent cells, corresponding with the possible 3D augmentations of x5x3o, resp. x5o3o (no adjacent augments). Which corresponds with the non-adjacent diminishings of the 600-cell. Which, in turn, implies that there must be a huge number of CRF 120-cell augmentations (resp. x5x3o3o augmentations)!
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Johnsonian Polytopes

Postby Keiji » Fri Feb 14, 2014 6:48 am

quickfur wrote:
Marek14 wrote:Well, I don't have a Linux box so web interface seems like best option. Bothering you with every set of vertices would probably not be very efficient :)

*glances at Keiji* :sweatdrop: ;) :P

OK OK, I'll take a stab at it tonight to see if I can throw something together.


Oy, you know I've been planning to do that, don't go stealing my thunder! ;)

I'll probably have it up on Monday.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Johnsonian Polytopes

Postby wendy » Fri Feb 14, 2014 7:05 am

I'm just wondering if there is an extended list of edges that you're using here. 'F' apparently means 2.618etc, but if we get a list, i could add it to the page in 'notions and notations'.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 14, 2014 4:12 pm

Keiji wrote:
quickfur wrote:
Marek14 wrote:Well, I don't have a Linux box so web interface seems like best option. Bothering you with every set of vertices would probably not be very efficient :)

*glances at Keiji* :sweatdrop: ;) :P

OK OK, I'll take a stab at it tonight to see if I can throw something together.


Oy, you know I've been planning to do that, don't go stealing my thunder! ;)

I'll probably have it up on Monday.

Yay!

Basically, all you need to do is to have the web form generate data of the form:
Code: Select all
<1, 2, 3, 4>
<2, 3, 4, 5>
...

along with a name for the polytope (alphanumeric only, no spaces), then feed them to `makepoly -fperm <name> - <outputfile>`. The `-` means "standard input", you can replace that with a temporary input filename, this is where you feed the input vertex data. This should produce the .def file. Then you just run `def2off <outputfile>` and pipe standard output back to the CGI script. Be forewarned that, depending on how many surtopes your polytope has, makepoly may take a long time to run, so you might want to consider running it in the background and offering the user a way to download the resulting file afterwards (most default webserver installations come with a timeout for CGI scripts, that will kill any process running longer than a predefined threshold, so it wouldn't work with large polytopes). Some very large polytopes (notably the 120-cell family polychora) may take hours to complete, and produce very large files (the omnitruncated 120-cell .def file, as you may have noticed, is about 3.4MB in size).

One thing I should mention, is that the code I gave you is only the old version of the polytope viewer, and it doesn't support algebraic coordinates nor Wendy's permutation operators. So the input will have to be floating-point only, and each vertex will have to be explicitly given. (There is the apacs/epacs program in the code I gave you, but it requires input in a different format, supports only a single operation per run, and generally doesn't integrate very well with the other tools. It's probably more trouble than it's worth.)

Also, note that although makepoly can generate polytopes of any dimension, def2off only works with 4D polytopes (not even 3D polyhedra).

Also, makepoly does not try to eliminate redundant points in the input (even though cddlib itself offers that functionality, makepoly doesn't use it), and may produce faulty .def files if your input contains redundant points (i.e. points that lie inside the convex hull). One symptom of this is that polyview will produce .pov files that povray rejects. I don't know what def2off will do in this case, but it's likely to also produce garbled output. The new version of the polyview viewer is supposed to address all of these issues, but it will be a while before it's anywhere near ready for public consumption. :P
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Johnsonian Polytopes

Postby Marek14 » Fri Feb 14, 2014 4:28 pm

Hm, I was thinking about ursachora. Currently the ones we have are based on regular polyhedra with triangular faces. My question is: can a valid CRF polychoron be derived from a similar construction on Johnson solids with only square faces like pentagonal bipyramid or snub disphenoid?

Another question: if you have a octahedron||rhombicuboctahedron cupola, would it be CRF to have triangular bipyramid||elongated triangular orthobicupola or pentagonal bipyramid||elongated pentagonal orthobicupola?
Marek14
Pentonian
 
Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

Re: Johnsonian Polytopes

Postby Marek14 » Fri Feb 14, 2014 5:37 pm

Look at cube augments.

I decided to investigate various augments of uniform or prismatic polychora based on cube. Why cube? Because cube often appears in subsymmetric ways.

Let's look at 4 augments we'll investigate:

Cube pyramid: dichoral angle 45
Square pyramid prism: dichoral angle 90 (for square pyramid) and 54.7356 (for triangular prism)
4,3 duoprism: dichoral angle 90 (for triangular prism) and 60 (for cube)
Square biantiprismatic ring: dichoral angle 107.031 (for square pyramid) and 53,5156 (for square antiprism)

Gluing them together:
Cube pyramid + Cube pyramid: works.
Cube pyramid + Square pyramid prism: works.
Cube pyramid + 4,3 duoprism: works. Multiple cubes in the duoprism can be augmented.
Cube pyramid + Square biantiprismatic ring: works.
Square pyramid prism + Square pyramid prism: If it's glued in ortho position (pyramid onto pyramid), you'll get octahedral prism. If it's glued in gyro position (any other), it's a new shape.
Square pyramid prism + 4,3 duoprism: Two orientations are possible: either square pyramids are glued to cubes and triangular prisms to triangular prisms or there are two pairs square pyramid/triangular prism, two pairs triangular prism/triangular prism and two pairs triangular prism/cube. In second case two pairs square pyramid/triangular prism merge into augmented triangular prism (in fact, it's augmented triangular prismatic prism). Second augment doesn't preclude further augmentation by cube pyramids or square pyramid prisms.
Square pyramid prism + Square biantiprismatic ring: The only orientation that works is square pyramids against square antiprisms and triangular prisms against square pyramids.
4,3 duoprism + 4,3 duoprism: The only orientation that works is two triangular prisms against triangular prisms and four triangular prisms against cubes. The pairs of triangular prisms fuse into gyrobifastigiums -- in fact, this is gyrobifastigium prism.
4,3 duoprism + Square biantiprismatic ring: Impossible.
Square biantiprismatic ring + Square biantiprismatic ring: Impossible.

Augmenting other polychora

Tesseract: dichoral angle 90. Any combination of cells can be augmented with cube pyramid. When two adjacent cells are augmented, square pyramids merge into octahedra (known).
Tesseract can be also augmented with square pyramid prisms. Elongated and bielongated square pyramid prisms can result, but also other shapes (for example augmenting two opposite faces, but with different orientations of square pyramid prisms).
Truncated 24-cell: dichoral angle 135. No augment possible, as square pyramids would merge with truncated icosahedra.
Rectified 24-cell: dichoral angle 135. No augment possible, as square pyramids would merge with cuboctahedra.
Prismatorhombated tesseract: dichoral angles 144.736 and 135. No augment possible.
Small prismated tesseract: has two types of cubes, one with dichoral angle 135 and other with dichoral angles 135 and 144.736. The first type IS augmentable by cube pyramids: this will result in adjacent cubes becoming elongated square pyramids. Any combination of these 8 main cubes can be augmented, which might result in some "side" cubes becoming elongated square bipyramids.
Truncated octahedral prism: dichoral angles 90 and 125.264. Augmenting by cube pyramids is possible.
Cuboctahedral prism: dichoral angles are 125.264 and 90. Augmenting by cube pyramids is possible.
Rhombicuboctahedral prism: main cubes have dichoral angles 90 and 135, side cubes 90, 135 and 144.236. Main cube can be augmented by cube pyramids, which will change side cubes next to it in elongated square pyramids.
Rhombicosidodecahedral prism: dichoral angles 90, 159.095 and 148.283. No augmentation possible.
Great cuboctahedral prism: dichoral angles 90, 135 and 144.236. No augmentation possible.
Great icosidodecahedral prism: dichoral angles 90, 148.283 and 159.095. No augmentation possible.
Snub cube prism: dichoral angles 90 and 142.983. No augmentation possible.

Duoprisms:
3.4 duoprism and tesseract were already discussed. As for others, cube pyramids can augment 4,5 to 4,8 duoprisms (with 4,8 augments creating elongated square pyramids). Square pyramid prisms can augment 4,5 and 4,6 duoprisms, leading to prisms of augmented prisms.

Square antiduoprism can have cube prisms and square pyramid prisms augmented, which will lead to antiprisms becoming gyroelongated square pyramids/bipyramids.

As for Johnson solid prisms, cube pyramids work for augmenting prisms where no dihedral angle on square face of original solid is greater than 135 (excluding already mentioned cases), that is:
triangular cupola, square cupola (changing other cubes into elongated square pyramids), elongated and bielongated pentagonal pyramid, elongated square cupola and elongated square gyrobicupola (creating elongated square pyramids and/or augmented triangular prisms), triangular orthobicupola, square orthobicupola and gyrobicupola, gyroelongated square bicupola, sphenocorona.
Last edited by Marek14 on Fri Feb 14, 2014 7:15 pm, edited 1 time in total.
Marek14
Pentonian
 
Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 14, 2014 5:49 pm

Marek14 wrote:Hm, I was thinking about ursachora. Currently the ones we have are based on regular polyhedra with triangular faces. My question is: can a valid CRF polychoron be derived from a similar construction on Johnson solids with only square faces like pentagonal bipyramid or snub disphenoid?

You mean only triangular faces? Sure, you can make ursachora from them... but remember that at the bottom of the ursachoron you'll need the rectified version of the top, and pyramids of the vertex figures. Only if the latter two are CRF, will the entire polychoron be CRF.

Of course, you may figure out a way to extend the bottom a little more to make room for more cells that turns irregular vertex figure pyramids into something CRF, but I'm guessing that will be hard, because there's very limited room for wiggling here.

Another question: if you have a octahedron||rhombicuboctahedron cupola, would it be CRF to have triangular bipyramid||elongated triangular orthobicupola or pentagonal bipyramid||elongated pentagonal orthobicupola?

Wait, isn't octahedron||rhombicuboctahedron flat? If I'm not mistaken, it's just a fragment of the 3D tessellation of octahedra and triangular prisms?

For triangular bipyramid||elongated triangular orthobicupola (resp. pentagonal), I'm not sure if the lacing cells will be CRF; wouldn't they be sheared sideways? Or maybe it's possible? Try it out and see. :)
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 14, 2014 6:21 pm

Marek14 wrote:[...]
Square pyramid prism + Square pyramid prism: If it's glued in ortho position (pyramid onto pyramid), you'll get octahedral prism. If it's glued in gyro position (any other), it's a new shape.
[...]

Cute, this is a 4D analogue of the gyrobifastigium (probably not the only analogue, though!), in the sense of being a gyro-bi-(square pyramid prism), analogous to the gyro-bi-(digon pyramid prism) = gyrobifastigium. Here's a render of this cutie, as seen from <5,0,0,0>:

Image

Here's another render, seen from <0,5,0,0>:

Image

And it appears that it is orbiform as well, so it should be counted among the orbiform bistratic CRFs.

Here are the coordinates I used:
Code: Select all
<±1, ±1, 0, ±1>
<0,  0, √2, ±1>
<0, ±1, -√2, 0>

And here's the .off file.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Johnsonian Polytopes

Postby student91 » Fri Feb 14, 2014 6:46 pm

quickfur wrote:And it appears that it is orbiform as well, so it should be counted among the orbiform bistratic CRFs.

It's even monostratic!!, Cf. K4.13
I had a hard time understanding it's structure as well, when I had to calculate it's dichoral angles.
It's lace city looks a bit like this:
Code: Select all
x2o   x2x

x2x   o2x
How easily one gives his confidence to persons who know how to give themselves the appearance of more knowledge, when this knowledge has been drawn from a foreign source.
-Stern/Multatuli/Eduard Douwes Dekker
student91
Tetronian
 
Posts: 328
Joined: Tue Dec 10, 2013 3:41 pm

Re: Johnsonian Polytopes

Postby Marek14 » Fri Feb 14, 2014 6:50 pm

Oh, yes -- it's the triangular prism||gyrated triangular prism :) And you're right, octahedron || rhombicuboctahedron is flat...
Marek14
Pentonian
 
Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 14, 2014 7:32 pm

quickfur wrote:It just occurred to me, that the truncated 120-cell x5x3o3o should be augmentable with x5o3x||x5x3o (i.e., a monostratic slice of x5o3x3o). If I'm not mistaken, the pentagonal cupola of the augment will be coplanar with the adjacent x5x3o's, so they will turn into augmented truncated dodecahedra (J68).

If this is correct, then the 120-cell itself should be augmentable with o5o3x||x5o3o. Again, the pentagonal pyramids of the augment should be coplanar with the adjacent dodecahedra, so they will turn into augmented dodecahedra (J58).

For both of these cases, the augments can only be added to non-adjacent cells, corresponding with the possible 3D augmentations of x5x3o, resp. x5o3o (no adjacent augments). Which corresponds with the non-adjacent diminishings of the 600-cell. Which, in turn, implies that there must be a huge number of CRF 120-cell augmentations (resp. x5x3o3o augmentations)!

Hmm. It turns out that I was wrong. The 120-cell cannot be augmented by x5o3o||o5o3x; the height is too large and makes the result non-convex. :\ This was a wrong generalization from the first case, :oops: which is also invalid. :(
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 14, 2014 7:38 pm

Whoa, guys, I just ran into this today: based on the picture on this webpage, it should be possible to tile 3D space with icosidodecahedra, dodecahedra, and tridiminished icosahedra? I'm not sure if the pattern can be continued indefinitely; could this be the beginning of a 3D Penrose tiling with icosahedral symmetry???

Edit: Looks like you need triangular prisms and J91's to continue the pattern, otherwise the exposed middle triangular face of the tridiminished icosahedra will need a non-convex cell on top of it. But is it possible to continue this tiling in a CRF way? Could it be the first example of a CRF penrose tiling???
Last edited by quickfur on Fri Feb 14, 2014 7:47 pm, edited 1 time in total.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 14, 2014 7:40 pm

Marek14 wrote:Oh, yes -- it's the triangular prism||gyrated triangular prism :) And you're right, octahedron || rhombicuboctahedron is flat...

Whoa. That blew my mind for a minute... I had always wrongly thought that triangular prism||gyrated triangular prism = octahedral prism. :oops: Now I know it's actually something else, even a 4D gyrobifastigium!
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Johnsonian Polytopes

Postby Marek14 » Fri Feb 14, 2014 7:49 pm

quickfur wrote:
Marek14 wrote:Oh, yes -- it's the triangular prism||gyrated triangular prism :) And you're right, octahedron || rhombicuboctahedron is flat...

Whoa. That blew my mind for a minute... I had always wrongly thought that triangular prism||gyrated triangular prism = octahedral prism. :oops: Now I know it's actually something else, even a 4D gyrobifastigium!


That would be if the triangular prism was gyrated by rotating around the triangular faces. But this one is rotated against a horizontal axis.
Marek14
Pentonian
 
Posts: 1191
Joined: Sat Jul 16, 2005 6:40 pm

Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 14, 2014 7:54 pm

Marek14 wrote:
quickfur wrote:
Marek14 wrote:Oh, yes -- it's the triangular prism||gyrated triangular prism :) And you're right, octahedron || rhombicuboctahedron is flat...

Whoa. That blew my mind for a minute... I had always wrongly thought that triangular prism||gyrated triangular prism = octahedral prism. :oops: Now I know it's actually something else, even a 4D gyrobifastigium!


That would be if the triangular prism was gyrated by rotating around the triangular faces. But this one is rotated against a horizontal axis.

Right, now I know that. :) I guess "gyro" is a bit ambiguous when you're dealing with an object of reduced symmetry like the triangular prism, since there are non-equivalent ways to rotate it.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Johnsonian Polytopes

Postby Klitzing » Fri Feb 14, 2014 10:49 pm

Marek14 wrote:Let's look at 4 augments we'll investigate:

Cube pyramid: dichoral angle 45
Square pyramid prism: dichoral angle 90 (for square pyramid) and 54.7356 (for triangular prism)
4,3 duoprism: dichoral angle 90 (for triangular prism) and 60 (for cube)
Square biantiprismatic ring: dichoral angle 107.031 (for square pyramid) and 53,5156 (for square antiprism)


Hmmm, those neither are specified exact enough (which angle is truely meant - and - where does this numerical value derive from as closed mathematical term), nor provide the complete list of possible angles of those polychora.

Cube pyramid (cubpy):
at {3} between squippy and squippy: 120°
at {4} between cube and squippy: 45°

Square pyramid prism (squippyp):
at {4} between trip and trip: arccos(-1/3) = 109.471221°
at {4} between squippy and cube: 90°
at {3} between squippy and trip: 90°
at {4} between cube and trip: arccos(1/sqrt(3)) = 54.735610°

4,3 duoprism (tisdip):
at {4} between cube and trip: 90°
at {3} between trip and trip: 90°
at {4} between cube and cube: 60°

Square biantiprismatic ring ({4} || gyro cube):
at {3} between squippy and tet: arccos[-(3 sqrt(2)-2)/4] = 124.101465°
at {4} between cube and squippy: arccos[-(2-sqrt(2))/2] = 107.031248°
at {4} between cube and squap: ???
at {4} between squap and squap: ???
at {3} between squap and tet: ???
at {3} between squap and squippy: ???

(I suppose that squap calculations would include cubical roots. This is why I omitted calculations of corresponding values so far. - The 2 provided ones at {4} || gyro cube were derived when calculating oct || cube instead.)

--- rk
Klitzing
Pentonian
 
Posts: 1637
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

PreviousNext

Return to CRF Polytopes

Who is online

Users browsing this forum: No registered users and 13 guests

cron