Quickfur's renders

[quickfur]

The renders look like stereo pairs. (sigh). I really can't see stereo images. But they're beautiful all the same.

They are cross-eyed stereo pairs. The nice thing about it, is that if you can't see stereo pairs, you can still look at them individually as non-stereo images.

But cross-eyed viewing is really not that hard. I avoided it for a long time because it sounded a bit scary, but once I decided to learn it, it took me only a day or two to get accustomed to it. And once you see the images in full 3D, you just don't wanna go back again. Especially with 4D projections, you really want to see the projections with 3D depth, it helps a lot to perceive what's going on in the projected image.

[quickfur]

Hmm, do they really affect it that badly? How long are the renders taking?

Well OK, technically it doesn't take that long, but with the smaller polytopes it just takes a few seconds from writing the script to seeing the image on screen, whereas with the omnitruncated 120-cell, it's taking minutes for every small change to the scripts. It sorta dampens the momentum a lot (imagine when you're writing code and it takes two hours after a 1-line change to see whether or not it worked -- you can still do it, it's just not very pleasant).

OTOH, I was just being a bit whiny... If I intersperse it with other things, it's really not that bad. So maybe that's what I'll do, work on CRF crown jewels while waiting for the renders to run.

[quickfur]

Sighh... one week in February has already rolled by, and barely any progress on the omnitruncated-120cell page. There are some O(n^2) algorithms in my polytope viewer that's really killing rendering times with the insane number of vertices/edges that the omnitruncated 120-cell has. Maybe the Polytope of the Month for March should be the J92 rhombochoron instead. Or the castellated prism. That should give me enough buffer to get the omni-120cell done at last.

I vote for the J92-rhombochoron!! I would love to undertand it's structure in more depth

I've been making good progress on the omnitruncated 120-cell projections in the past day or so, after finally finding a good coloring scheme for the "mugshot" perspective projection. So it seems likely that March's Polytope of the Month will be the omnitruncated 120-cell after all.

But don't worry, the CRF crown jewels are definitely on the list for after March. I just thought it would be nice to finally have some closure for the uniform polychoron renders that I started years ago. Judging by the number of CRFs we're discovering every week recently, I'm not going to run out of things to render anytime soon! (Not to mention, even the good ole regular polychora have many more rendering possibilities that I just haven't had the time to visit yet. Like the vertex-first projections of the 120-cell, or, respectively, the cell-first projections of the 600-cell, which may indirectly lead to more CRF possibilities. And I'd like to render the 12 Hopf fibration rings of the 120-cell some time, too. So even within the uniform polychora alone, there's still a ton of stuff that can be done.)

[quickfur]

[...]
But don't worry, the CRF crown jewels are definitely on the list for after March.
[...]

And in preparation for that, I've already posted renders of the pentagonal rotunda (J6), the metabidiminished icosahedron, the bilunabirotunda (J91), and the triangular hebesphenorotunda (J92). These are probably old news for you forumites, but I'm posting here because I'm particularly pleased with this diagram that I made that shows the relationship between J92 and the icosidodecahedron:



As you can see, J92 is intimately connected with the icosidodecahedron; its hexagonal face is actually the 6 vertices of the icosidodecahedron lying on the bisected plane, scaled down by the Golden Ratio.

This diagram is also the precursor to the J92 rhombochoron: if we fold the two J92's into 4D around the grey plane until they make a 60° dichoral angle with each other, they would form half of the J92 rhombochoron (minus the 12 additional vertices needed to make this side of the polytope CRF).

[quickfur]

Welp, here it is! The final (convex) uniform polychoron on my website!



This one was quite a beast to render, due to its sheer number of surtopes. It exposed a lot of poorly-written algorithms in my polytope viewer program.

This is, of course, the omnitruncated 120-cell, aka the omnitruncated 600-cell. Some of my favorite images from this round of renders:







Really beautiful, if I say so myself.

And now that I'm finally done with the uniform polychora, next month we're gonna start on the cool stuff, the CRFs... And perhaps some of the Catalans as well? We shall see!

[Marek14]

I'm looking at your polychoron pages, and I think I found a mistake -- on the 24-cell page you say:

In 3D and beyond, the only regular space-tiling polytopes are the hypercubes, with one exception: the 24-cell.

This is, however, not true -- the 16-cell is also a regular space-tiling polytope; the 16-cell tiling is the dual of the 24-cell one (24-cell tiling is 3-4-3-3, the 16-cell is 3-3-4-3).

EDIT: Another minor bug - the grand antiprism page says:

In the same way, all the pentagonal prisms are identical in 4D, but the ones nearer the ends of the tube appear flattened because of the projection.

Should be "pentagonal antiprisms".

[quickfur]

Thanks for the corrections!! Sad to say, they were long-standing mistakes.

[quickfur]

Alright, here it is: the long-awaited renders of the J92 rhombochoron.



Well? What are you waiting for? Go to the J92 rhombochoron page and look at them. This month you guys get a bonus: I did renders of the J92 rhombochoron from two different 4D viewpoints! (Arguably the J92 rhombochoron deserves more than that, but it's already April and I don't have enough time to work on this all day. They can always be added later. )

Some of my favorite renders this time round:











This last one should ring bells among you Johnson solid fans. It's the tetrahedron + triangular prism (and 3 square pyramids) + tetrahedron sequence, serving an analogous role in 4D as the triangular + square + triangle sequence serves in 3D! I've a strong feeling that we haven't begun seeing all that this flexible cell combo can do just yet. They could be the key to making CRFs out of CVP 3 Johnson solids, by acting as a kind of macro-strut that is far more flexible in configurability than tetrahedra alone could ever hope to be.

[quickfur]

It's that time of the month again! (Sigh... that line is getting old.) This time we have some nice renders of the castellated rhombicosidodecahedral prism, the crown jewel that started CRFebruary:



The parallel projection centered on a rhombicosidodecahedron is an actual rhombic triacontahedron, with J91's serving as phi-rhombuses! Check it out:



Incidentally, it also shows an interesting way of inscribing a rhombicosidodecahedron inside a rhombic triacontahedron.

[Klitzing]

Incidentally, it also shows an interesting way of inscribing a rhombicosidodecahedron inside a rhombic triacontahedron.

In fact, this is how the rhombicosidodecahedron got its name:

• the squares are in the planes of a circumscribing rhombohedron,
• the triangles are in the planes of a circumscribing icosahedron, and
• the pentagons are in the planes of a circumscribing dodecahedron

--- rk

[quickfur]

Wow. I didn't know this before. Now the name makes a lot more sense. Thanks for sharing this tidbit!

[Keiji]

No PotM since May?

[ICN5D]

No, quickfur hasn't been around since May. That's probably his last post since then! Nice to see you back around here, Keiji.

[quickfur]

I apologize for the lack of updates... I was halfway through the renderings of spidrox, but got busy with other things and didn't have a chance to finish it. Unfortunately, I also made the mistake of working directly on the main branch of my website's version control repository for the spidrox renderings, so it would require additional effort to revert those changes and put them in a branch, and replace it with a simpler PotM, and only post spidrox when it's ready.

(Yeah my website is outdated -- it's still maintained by Subversion whereas most of my code projects are now on git, which is far more flexible in situations like this and would have been trivial to make this branch change. I just haven't had the time to work on 4D-related stuff at all, much less spend the time to migrate everything over to git -- it's not just the migration itself, but I also need to update a variety of scripts and stuff that I use for website maintenance; those would need some effort to update to use git instead of svn.)

I don't know when I'll get back to 4D stuff, but I'm hoping to, some day -- hopefully not too far into the future!

[quickfur]

Wow. It's been almost 4 years since my last major website update.

But! There's good news!! As of today, 8 Feb 2018, I'm officially resuming the PotM project! Check it out!

It's a modest resumption, to be sure, with something as simple as the tetrahedral ursachoron. But I didn't want to bite off too much to chew, so that things get stuck again. I've also started building up a queue of upcoming PotM's, the goal being to have a buffer of a few months (hopefully more) so that in the event I get too busy again, most of the work is done and it's just a matter of merging in one of the buffered PotM entries. In fact, I originally wanted to resume with spidrox, but it's proving to be a lot more work than I anticipated, so it's going to hafta wait. But rest assured, the spidrox PotM entry is on the way, and one day it will see the light of day!

Anyway, here's one of the renders of the tetrahedral teddy from this month's PotM:



Currently, I have another completed PotM entry in the buffer, so there should be another PotM next month. I'll try to build up a bigger buffer in the interim. Given the large number of highly-interesting CRFs we discovered especially from CRFebruary, there's no shortage of material to draw from. I'll probably just do a bunch of the simpler CRFs to be used as buffer material, while I work on more interesting things like spidrox.

[student91]

To build your buffer with interesting polytopes, I think the three pseudopyramids

line || bilbiro
triangle || thawro
pentagon || pocuro

could serve well, they are quite important in the EKF-research. Although the thread "Construction of BT-polytopes via partial Stott-expansion" does include huge amounds of interesting polytopes derived from ex and rox, these are quite big and thus a lot of work to render. I am happy to help you with something if you want, could you tell me what exactly makes the big polytopes so much work? (I could for example set up documents in the appropriate format for your rendering program, so you only have to take it as an imput, you only have to explain the format)

[quickfur]

The reason big polytopes are so much work is because of poor design decisions in my polyview program. There's a built-in polytope query mini-language that it uses for selecting subsets of surtopes (for highlighting, hiding/showing, assigning textures, etc.), which works quite well for simple to medium complexity polytopes, but not so well for larger polytopes where you need to split a subset into individual elements to color, or there's no easy way (in the mini-language) to express a certain symmetry subset.

In the case of spidrox, it's particularly tedious because there's currently no way to select individual cell rings, so the only way I can get them is to generate graphs and transcribe cell numbers by hand. (Which is already not bad; it's better than transcribing individual edge/face numbers, which would have brought the entire PotM project to a halt .)

That, and also because there's currently no way to save a particular subset / texture assignment for reuse, means that the rendition scripts have a lot of copy-n-pasting, which makes it error-prone, and also often has hidden dependencies on previous settings that aren't always reset later.

Also, when coloring a very large polytope, often the surtope subset that needs coloring contains elements that are touching each other, so individual elements have to be assigned different colors. The program currently has no automation for this, so I end up having to solve the graph coloring problem by hand over and over, once per subset of cells to color.

And it doesn't help that the code is quite old and I haven't really made any major changes to it for many years. And the original design wasn't really very well thought out, so there are some inefficiencies that makes rendering large polytopes somewhat slow, but fixing it would require a major rewrite. Anyway, I've been trying to get a revamped version of the polytope viewer program going, but so far haven't really succeeded. I do have a new program that can do vector arithmetic quite well, but it doesn't do anything with polytopes except load them into memory right now. I'll probably end up incrementally transcribing the old program over, so that I don't have to rewrite existing scripts, which would require tons of time that I don't have.

[student91]

Ah I see, I'm afraid I cannot help you that much, I am looking forward to seeing your new renders
It might happen that I find enough time to extend this program of mine to generate polyhedra files, but I understand this is not where most of your work goes into this, but rather into highlighting cells etc. (The quality of your website is way broader than the ability to render polytopes, keep up the good work!).

In the meantime, I manually made this cute animation of the ike-EKFs. It would of course be way better if this could be generated with a computer program one day. Anyhow, I might be a bit preoccupied with the EKFs, it could be one day I have a program that can help you with these, but this requires tons of time that I don't have . Don't forget to render smaller polytopes, it's good to have you back!

[quickfur]

What would help me is to update me on the current status of EKF research. I think in the aftermath of CRFebruary, I kinda lost track of the huge number of things we were finding, and didn't really manage to keep up with the latest EKF findings.

Did we actually end up enumerating all possibilities of EKF CRFs in 4D, or at least have some estimate on how many there might be? Also, does the current EKF scheme cover things like the tetrahedral D4.8.x CRFs that I found, or do those things belong in their own category?

And of course, any new ideas about how to find crown jewels containing strange Johnson solids (like sphenocorona, et al) would be nice. Actually, I've been wracking my brain on and off over the past years trying to think of a way to make a CRF containing a trigonal bipyramid, but so far haven't been able to come up with anything. It seems to be such a simple shape, nothing like the snub disphenoid with its degree 3 polynomials or sphenocorona with its degree 4 polynomials, yet it seems so hard to find a CRF that contains it! (Besides the boring prism, of course. But we're not interested in that. )

[Klitzing]

Have a look here: https://bendwavy.org/klitzing/explain/ekf.htm
--- rk

[quickfur]

Have a look here: https://bendwavy.org/klitzing/explain/ekf.htm
--- rk

Wow! That's a lot! Thanks for the link!

I'm guessing by the various "..." that the enumeration is not complete?

Also, did anyone figure out what exactly is going on with the D4.8.x CRFs? I still have no idea how to classify them, even though I constructed them myself.

[Klitzing]

Also, did anyone figure out what exactly is going on with the D4.8.x CRFs? I still have no idea how to classify them, even though I constructed them myself.

Those 4 CRF polychora where based on the following halfes, which then could be attached according to A_i+B_j:

Code: Select all

A1:
xf|oxF|...-3-ox|Fxo|...-3-oo|ofx|...
--------------------------------------
x.|...|...-3-o.|...|...-3-o.|...|... =  1 tet
..|...|...-3-ox|...|...-3-oo|...|... =  4 tet
xf|ox.|...-3-ox|Fx.|...-3-..|...|... =  4 thawro   3,4
..|...|...-3-..|...|...-3-.o|ofx|... = 12 peppy      5
..|...|...-3-.x|.xo|...-3-..|...|... = 12 squippy   2
..|...|...-3-.x|..o|...-3-.o|..x|... =  4 oct      1


Code: Select all

B1:
ox|Fxo|...-3-xo|oxF|...-3-of|xfo|...
--------------------------------------
o.|...|...-3-x.|...|...-3-o.|...|... =  1 oct
ox|...|...-3-xo|...|...-3-..|...|... =  4 oct
ox|Fxo|...-3-..|...|...-3-of|xfo|... =  6 bilbiro   3,5
..|...|...-3-xo|o..|...-3-of|x..|... =  4 teddi      1
.x|.x.|...-3-.o|.x.|...-3-..|...|... =  4 tricu      4
..|...|...-3-.o|ox.|...-3-..|...|... = 12 tet      2


Code: Select all

A2:
-o|oxF|...-3--x|Fxo|...-3--x|ofx|...
--------------------------------------
-o|...|...-3--x|...|...-3--x|...|... =  1 tut
-o|.x.|...-3--x|.x.|...-3--.|...|... =  4 tricu      4
-o|ox.|...-3--.|...|...-3--.|...|... = 12 tet      3
-.|...|...-3--x|Fxo|...-3--x|ofx|... =  4 thawro   1,2,5


Code: Select all

B2:
ox|Fxo|...-3-xf|oxF|...-3-oo|xfo|...
--------------------------------------
o.|...|...-3-x.|...|...-3-o.|...|... =  1 oct
ox|Fx.|...-3-xf|ox.|...-3-..|...|... =  4 thawro   2,4
..|...|...-3-xf|o..|...-3-oo|x..|... =  4 teddi      1
.x|.xo|...-3-..|...|...-3-..|...|... = 12 squippy   3
..|...|...-3-..|...|...-3-.o|xfo|... = 12 peppy      5


Now I just recognized that A₁ is the polar part (on either side) of some ex based EKF. And then it occured to me that the same connection surface bit between the pipes does occur in a further of those. Therefrom we would have to add here a further possible half CRF to that list:

Code: Select all

A3:
of|oxF|...-3-xx|Fxo|...-3-xo|ofx|...
--------------------------------------
o.|...|...-3-x.|...|...-3-x.|...|... =  1 tut
..|...|...-3-xx|...|...-3-xo|...|... =  4 tricu
..|...|...-3-..|...|...-3-xo|o..|... = 12 tet
..|...|...-3-xx|.x.|...-3-..|...|... = 12 trip
o.|.x.|...-3-x.|.x.|...-3-..|...|... =  4 tricu      4
o.|ox.|...-3-..|...|...-3-..|...|... = 12 tet      3
oo|oo.|...-3-oo|oo.|...-3-oo|oo.|... = 24 tet
..|...|...-3-.x|..o|...-3-.o|..x|... =  4 oct      1
..|...|...-3-.x|.xo|...-3-..|...|... = 12 squippy   2
..|...|...-3-..|...|...-3-.o|ofx|... = 12 peppy      5


Thereby pushing up the thus derived full CRFs from 2*2=4 up to 3*2=6!

Note that the Polygons of the wobbly connecting Surface have been labeled correspondingly in all 5 cases.

--- rk

[quickfur]

Nice find!

Still trying to wrap my head around the new A₃. Maybe I'll take a shot at constructing it in my polytope viewer tomorrow.

Now this is making me wonder, what's so special about that skew polygon that makes it possible to cut-n-paste different CRF pieces like this? What if there's nothing special about it, and there are other skew polygons that admit similar cut-n-paste combintations?

It seems that even within the realm of EKF polychora, there remains much unexplored territory.

[student91]

Very nice indeed!
The B-part is also derived of an ex-based EKF: when you take x3o3o || o3o3f || o3f3o || f3o3x || o3x3f and do (-x)3x3o || o3o3f || o3f3o || f3o3x || o3x3f and then o3x3o || x3o3f || x3f3o || F3o3x || x3x3f, you have a patch of an ex-EKF. Now when x3f3o is removed, you obtain B1, and when x3o3f is removed, you obtain B2.
Thus in summary, B1 and B2 are patchworks of ex-based EKFs

[quickfur]

Just a quick note that I finally got my act together today, and made my website work more nicely on a mobile browser. Basically, added a bunch of stuff to make it render more nicely by default, and a simple hack to hide the fat navbar inside a menu toggle, so that it won't crowd out everything else on a small screen. Also, upgraded to HTML5 (well, minimally ).

If you have a minute or two, check it out and let me know what you think. You may have to reload the page if it looks odd (if your browser has a stale copy of the stylesheet cached).

[quickfur]

Very nice indeed!
The B-part is also derived of an ex-based EKF: when you take x3o3o || o3o3f || o3f3o || f3o3x || o3x3f and do (-x)3x3o || o3o3f || o3f3o || f3o3x || o3x3f and then o3x3o || x3o3f || x3f3o || F3o3x || x3x3f, you have a patch of an ex-EKF. Now when x3f3o is removed, you obtain B1, and when x3o3f is removed, you obtain B2.
Thus in summary, B1 and B2 are patchworks of ex-based EKFs

Hmm. So this means all the crown jewels we've found so far are ultimately derived from EKFs. Well, excepting perhaps cube||icosahedron, which is truly novel, and perhaps the ursatopes, though the pentagonal ursatopes do still seem to relate to the EKFs quite closely (e.g., icosahedral teddy == diminished hemi-600-cell).

I just can't help wondering if there are any other directions of CRFs that we haven't searched yet, that might turn up more crown jewels. I've been thinking up various wild ideas again, but none of them show any promising signs.

[Klitzing]

Re-thought that recent post of mine. Esp. the pentagon labeled 5 there. At A3 there is a peppy attached. At B2 that is to be opposed by an other peppy. - But is that attachment still convex?

I'm reminding that in the sequel of ex-wedges, when attaching 2 of those, we encountered a similar case, where convexity thereby was lost. But then it could be re-established, when inserting 5-tet-rosettes into this cavity. - Would the similar concavity occur above as well? Would it need for the described fix there too?

And then, how about all the other attachments across that pentagon labeled 5? (Note that A1 does use the same peppies as A3!)

-> Could anyone have a quick look into that?


BTW, it just occured to me that A3 is nothing but A2 with the thawroes of the latter being augmented by thawro-pseudopyramids, i.e. a {3}||thawro each. In fact, those top triangles are given by that additional vertex layer f3x3o.

And even A1 and A3 are in a loosely sense quite similar: it is just that the final vertex layer is getting exchanged only. In A1 the axial stack ends with a tet, in A3 that stack ends with a (relatively inverted) tut. - Sure, this then requires for a different filling of being used cells as well.

--- rk

[quickfur]

It's March, and time for another Polytope of the Month! Spidrox, unfortunately, is still not ready. Well, I've really only been working on it slowly; but I did build up a buffer of simpler CRFs, so here's the next one:



That's the octahedral ursachoron, of course. A well-known CRF. Well, this time round my favorite projection is:



Anyway, y'all already know this polychoron really well already. So nothing new and exciting here. But I did like how some of the images turned out.

[quickfur]

Finally, 4 years later, my spidrox renders are finally ready.



Spidrox is also well-known, but I really like how some of its renders turned out. Here are some of my favorite images:







The spidrox page also has the most number of images I've squeezed into a single page on my website yet. Hope y'all enjoy it!

[Klitzing]

Amazing work!
--- rk