Uniform Polyteron Sections and Verfs

[Polyhedron Dude]

These past few months, I've been rendering fields of sections for many of the uniform polytera - so far I have rendered all of the ones in the 19 hixic (hexateron) and the 8 hinnic (demipenteractic) regiments as well as hitting all of the penteractic regiments including the starry ones! In other words 91 of the known 94 regiments have been hit with cross sections (the remaining three regiments are the two antiprisms found by Norman Johnson and a recent regiment that I found by blending 6 demipenteracts). I render them by rendering a 2-D field of 3-D sections and I use a "natural filling" which renders zero density areas as hollow and gives non-orientables a binary density map. Some have a nice pleasant look, while others are crazy looking intricate nightmares and everywhere in between. I decided to make this thread to display some of these by releasing a few polytera each week. I'll also provide verfs of them rendered in sections as well. Due to the size of the "section field" renders, I decided to show a smaller render with fewer sections on the pics, but the larger one can be accessed by clicking the link near the smaller pic.

The first polyteron to be revealed is a convex one that many of you may recognize - Spix - Aka runcinated hexateron or small prismated hexateron. Its symbol is oxoox. Spix heads up a regiment with 37 members plus one fissary. Its facets are 6 raps (blue), 6 spids (red), 15 opes (green), and 20 triddips (orange). Its verf is a trigonal antifastegium which is pictured below, followed by the field of cross sections - don't forget to click on the link near the field of sections for a much larger render with more sections in it.


http://pages.suddenlink.net/hedrondude/spix.png

[Klitzing]

I decided to make this thread to display some of these by releasing a few polytera each week.

Very good idea!
I'm tuned to see them all.
Or, taken in fb-jargon: like!
--- rk

[Polyhedron Dude]

I decided to make this thread to display some of these by releasing a few polytera each week.

Very good idea!
I'm tuned to see them all.
Or, taken in fb-jargon: like!
--- rk

I'll be doing a starry one next.

[Klitzing]

Good to have a sneek preview section here on what to be released on your website http://www.polytope.net/hedrondude/polytera.htm only in days to come ...
(Spix clearly belongs to the there still not provided category 8.)
--- rk

[Polyhedron Dude]

Our next polyteron is Gibtadin - great biprismato32dispenteract. Its facets are 10 gichadoes (cyan), 10 gaqrits (green), 32 decas (red), and 80 tisdips (blue). Its symbol is (o'x"x)xo which can be shortened to Gxo where G stands for gocco. The gibtadin regiment has three members, the others are quiprin and glaprin. The conjugate regiment is the prin regiment oxxo'x. The most prominent facet is gichado, which turns out to have several zero density cavities inside and therefore generating interesting snowflake like openings into the interior of gibtadin, these can be seen in the sections. I've also added a second link to "zoom" in to the center region of the field of sections to see the structure of these holes better. The verf of gibtadin is a trapezoid-dyad disphenoid which is pictured below.



http://pages.suddenlink.net/hedrondude/gibtadin.png
http://pages.suddenlink.net/hedrondude/gibtadinz.png

[Polyhedron Dude]

This one is Kafandoh - spikifacetospinodishexadecateron. It is one of the 37 members of the sirhin regiment and it's non-orientable. Sirhin's symbol is oxo6. It's facets are 16 garpops (cyan) and 16 ripdips (peach). I've rendered only a quadrant of the sections (like I did with gibtadin), the bottom right section is the center section. This one has an interesting structure to it. Below are the sections and verf.



http://pages.suddenlink.net/hedrondude/kafandoh.png

[Klitzing]

Wow, that one Looks dramatic!

Btw. both garpop and ripdip lack linked pictures at your 4D website...

--- rk

[Polyhedron Dude]

Wow, that one Looks dramatic!

Btw. both garpop and ripdip lack linked pictures at your 4D website...

--- rk

As a matter of fact, there are many more dramatic ones like this - several of the larger regiments have that one non-orientable member with a toroidal spiky verf usually with two types of facets - many of them with garpops. There's even a noble polyteron in the nit regiment with 32 garpops for its facets, it is called nat - I'll put this one in soon.

I just added the links to my site for the remaining pennic and tessic polychora (see cat.6, 10, 11, 12, and 14), I already had the pics on the site - just forgot to add the links (also forgot to add sto and gotto sections, I'll add those later).

[Polyhedron Dude]

Now for a rather simple convex one - with a very complex regiment - Nit. Nit is the birectified penteract, aka penteractitriacontaditeron. Its symbol is ooxoo, also oo9o. The facets of nit is 10 icoes (green) and 32 raps (purple). The nit regiment is the largest known regiment among the uniform polytera (except for the idcossid and dircospid prisms). It has 149 members (assuming I didn't miss any). It also has 139 fissaries and many dyadic coincidics - if we included all of these, there would be over 500 members in this regiment. Its verf is a tisdip, seen below. I've rendered the whole field this time instead of a quadrant. Enjoy.



http://pages.suddenlink.net/hedrondude/nit.png

[Polyhedron Dude]

A very interesting polyteron in the nit regiment is Nat, aka spinotriacontaditeron. This one is noble with 32 garpops as its facets. Nat generates a very mesmerizing field of sections.



http://pages.suddenlink.net/hedrondude/nat.png

[Klitzing]

Indeed it does!
And the bone-like color, you've chosen here, even adds to this effect ...

Btw., would be somewhere a listing of noble uniforms (or even better: scaliforms) accessible?

(In 3D those would be just the regulars, both, the convex ones and the non-convex ones.
For 4D I'd started collecting those at http://bendwavy.org/klitzing/explain/noble.htm. But there might be some more.
For 5D this is the first one I know of, beyond the regulars for sure.)

--- rk

[Polyhedron Dude]

There are three known non-regular uniform nobles, they are:

Dot - dodecateron - ooxoo - has 12 raps
Nat - spinotriacontaditeron - in nit regiment - has 32 garpops
Hit - hexadecateron - in hin regiment - has 16 pinnips.

No current known noble non-uniform scaliforms yet.

On the 4-D list, there's one missed - Affic in afdec regiment, it is a fissary with 48 cotcoes.

[Polyhedron Dude]

Now for an utter nightmare! Gipbin - great prismated biprismatopenteract. It is a member of the fawdint regiment, who's symbol is oGo or o(o'x"x)o. I suspect this one is orientable, so I rendered it as such. Its facets are 10 gittiths (cyan), 32 raps (blue), 40 goccopes (red), 80 opes (yellow), 80 tistodips (orange), and 80 tisdips (green). It also has 10 dippanoth shaped pseudofacets. Only the top-left quadrant of the section field has been rendered - therefore the center of the polytope is the bottom-right section. Now who's up for building a model?



http://pages.suddenlink.net/hedrondude/gipbin.png

[Polyhedron Dude]

This one is up to no good, for it is Bad - biprismatododecateron. It is in the dot regiment, dot being ooxoo. Its facets are 12 firps (cyan and yellow) and 20 triddips (red). Its verf has triddip (triangle duoprism) symmetry.



http://pages.suddenlink.net/hedrondude/bad.png

Time to cue Michael Jackson's hit song "It's Bad, it's Bad,..."

[Polyhedron Dude]

Now for a weird looking one - Gloptin = great lapidoprismatotruncated penteract. It belongs to the 7 member quiptin regiment, quiptin's symbol is oxox"x. Gloptin's facets are 10 girpiths (purple), 32 sirdops (cyan), 10 gaqrits (red), and 80 tistodips (yellow). Its verf is a butterfly wedge pyramid.



http://pages.suddenlink.net/hedrondude/gloptin.png

[Polyhedron Dude]

Now comes Danbitot - dispenteractibitruncated triacontaditeron. It is a "lone operative", it forms a one-member regiment. Its symbol is o(x'x"x)o which can be shortened to oCo where C stands for cotco. Speaking of cotco, danbitot is the medial 5-D version of cotco, which is in between the blocky version (nottant - ooC) and the spiky version (naquitant - Coo). Its facets are 32 decas (cyan), 10 thaquitoths (green), and 10 thatoths (magenta). Thatoth - oC is the 4-D blocky cotco variant and thaquitoth is the spiky version - Co. Thaquitoth sections can be seen as the top row or leftmost column in the section field, while thatoth is the second to the last row or second-right column.



http://pages.suddenlink.net/hedrondude/danbitot.png

[Polyhedron Dude]

This is the tenth polyteron in this thread, so which one can we use for polyteron number ten, hmmm , lets see, number ten, ... AH Got it!

Tin - triacontadipenteract. It is a non-orientable member of the rin regiment, rin being ooox'o. Its facets are 10 firts (pink) and 32 tips (blue). I've rendered the whole field instead of a quadrant.



http://pages.suddenlink.net/hedrondude/tin.png

So, out of the first ten polytera, which one is your favorite?

[quickfur]

I like spix. It's nice and simple, yet intricate in its own way.

Of course, I'm partial to convex polytopes... but the others are really nice too. I've been busy and haven't been able to work on my projection-based renders recently, but I've still been thinking at the back of my mind about how to deal with rendering projections of non-convex polytopes. Well, in principle I know what needs to be done, but it's just a matter of how to implement it. I do find the projection method more intuitive (for me, anyway) than sectioning.

[Klitzing]

Haha, right you are, quickfur! To me sectioning kind of is like dealing with reals by mere providing sequences of rational approximants: the final step (of taking the limit resp. of interpolating those sections - esp. of "seeing" the edges, which merely connect vertices of individual sections) is evidently missing in display and has always to be done in one owns mind.

The last one, i.e. tin, looks best to me. Esp. because of its chessboard-like framework pattern ...

--- rk

[wendy]

I think these are gui lace cities? I don't completely follow...

[Klitzing]

I think these are gui lace cities? I don't completely follow...

More or less. Those are based on your lace city idea. But those do not show only the special vertex positions, where you usually provide the orthogonal symbol display, rather provide additional intermediate steps.

Eg. here is your lace city of rin (rectified penteract, o3o3x4o):

Code: Select all

o3x4o  o3o4q  o3x4o
                   
                   
                   
o3o4q         o3o4q
                   
                   
                   
o3x4o  o3o4q  o3x4o

and here is hedrondudes corresponding field of section: http://www.polytope.net/hedrondude/polytera/RIN.JPG.

--- rk

[Polyhedron Dude]

I like spix. It's nice and simple, yet intricate in its own way.

Of course, I'm partial to convex polytopes... but the others are really nice too. I've been busy and haven't been able to work on my projection-based renders recently, but I've still been thinking at the back of my mind about how to deal with rendering projections of non-convex polytopes. Well, in principle I know what needs to be done, but it's just a matter of how to implement it. I do find the projection method more intuitive (for me, anyway) than sectioning.

I find your renders quite fascinating, they really help bring out the 4-D-ness of the polychora. Sectioning works well with star polytopes to bring out the intersection details, but you do lose that 4-D effect.

The last one, i.e. tin, looks best to me. Esp. because of its chessboard-like framework pattern ...

Many non-orientable polytera have these cool chessboard patterns, I've seen some that are so intricate that individual sections have more squares than the chessboard itself!

It was hard for me to pick a favorite, they all look interesting for different ways, but Nat seems to stand out for being noble and intricate, the field of sections of nat looks like an intricate design that might be found in a king's palace - made of gold.

I think these are gui lace cities? I don't completely follow...

The lace cities can act like a map of where major transitions take place on the field of sections, like the boundaries and ridges of the field, or locations where an instant change can be seen instead of a gradual change, they are also the locations where actual vertices are - this can be seen on the danbitot sections where there is a quith in the top left corner, it seems to grow as you move right, then it hits a place where it starts to truncate and then it immediately turns into a cotco and it shrinks at the top right (which is actually the top center of danbitot). Further down towards the bottom right (near center of polyteron) there is an instant change to a tic, another important location in the lace city.

Now for our next polyteron, this one will noqyapants off, its Noquapant. Noquapant is the penteractiquasiprismated penteracti32teron. Its symbol is (x'x"x)xx = Cxx for short. It is a lone operative and is the most amazing looking of the simplex-verfed polytera. Its facets are 10 thaquitpaths (yellow), 10 gaquidpoths (cyan), 32 gippids (green), 80 hodips (red-orange), and 40 cotcopes (lavender).



http://pages.suddenlink.net/hedrondude/noquapant.png

[quickfur]

I like spix. It's nice and simple, yet intricate in its own way.

Of course, I'm partial to convex polytopes... but the others are really nice too. I've been busy and haven't been able to work on my projection-based renders recently, but I've still been thinking at the back of my mind about how to deal with rendering projections of non-convex polytopes. Well, in principle I know what needs to be done, but it's just a matter of how to implement it. I do find the projection method more intuitive (for me, anyway) than sectioning.

I find your renders quite fascinating, they really help bring out the 4-D-ness of the polychora.

Thanks!

Sectioning works well with star polytopes to bring out the intersection details, but you do lose that 4-D effect.

Sectioning does have one advantage over projections: they are not prone to illusions and ambiguity caused by obscuring of elements that overlap in the field of vision. They also let you get "inside" the polytope, esp. a non-convex one, without making it hard or impossible to see where the elements are.

I am trying to figure out a way of dealing with non-convex polytopes in my renderer, though that's probably not going to happen for a while yet, since I'm currently in the process of rewriting the software for it. The current program I wrote, while it works, has a lot of hacks and limitations, and worst of all, performance problems with large polytopes. My last polytope-of-the-month renders, for example, took excruciatingly long to render, and I'm not even dealing with the most complex 4D polychoron yet. Its smooth-surface handling is also hacky, which means thus far it can only properly deal with a limited number of curvy shapes (duocylinder, and various circle-polygon cartesian products, but anything much beyond that and it starts to produce strange/wrong output). I'd like to fix up this part as well, to make it possible to render, say, your wonderful regular swirlprisms in projection.

All in all, it adds up to a pretty tall order, so I'm not holding my breath for it to be ready anytime soon.

[...]
Now for our next polyteron, this one will noqyapants off, its Noquapant. Noquapant is the penteractiquasiprismated penteracti32teron. Its symbol is (x'x"x)xx = Cxx for short. It is a lone operative and is the most amazing looking of the simplex-verfed polytera. Its facets are 10 thaquitpaths (yellow), 10 gaquidpoths (cyan), 32 gippids (green), 80 hodips (red-orange), and 40 cotcopes (lavender).



http://pages.suddenlink.net/hedrondude/noquapant.png

Whoa. It is pretty amazing-looking. Some of those sections show incredibly intricate starry patterns where I wouldn't expect it from a 5-cube symmetry! Incredible.

Just out of curiosity, how do you handle coordinates for these polytopes? Do you use full Cartesian coordinates, or just a point and a generating symmetry (via a Wythoff construction perhaps)? Currently, my renderer uses explicit coordinates for vertices, and vertex sets for j-faces. To avoid having to manually specify vertex sets (which is impractical beyond the simplest polychora -- the 24-cell, for example, took me 2 days to write out in full), I use a convex hull algorithm to generate the j-faces. Unfortunately, this means it can't handle non-convex polytopes.

I've been thinking about some kind of automated faceting algorithm for generating non-convex polytopes, though. Is it true that the hyperplanes of all cells in a non-convex uniform polytope always corresponds with (i.e. parallel to) some element of a convex uniform polytope? Or are there cases where the hyperplanes are non-parallel to all j-faces of any convex uniform polytope? If the former, I may have an easy way of generating non-convex polytopes; if the latter, then how in general do you construct these things?

(In any case, even if I can generate coordinates for non-convex polytopes, it will still take a while before the renderer can actually handle them correctly, since to produce the correct projection images I'd need full 3D volumetric clipping, and right now I don't have a good way of doing that yet.)

[Klitzing]

All those polytopes, which can be obtained by Wythoff's kaleidoskopical construction (i.e. those which have Dynkin diagrams with o- and x- nodes only) do have that desired property: the most what can happen are the omnitruncates, i.e. the Dynkin symbols with all nodes using x-marks (being ringed).

Beyond those Wythoffians there are only few sporadic uniforms. Most of those are snubs. Or Miller's Monster (3D), or the great antiprism (4D), ...


Convex hull algorithms can be used for non-convex ones too. You just would have to know the respective army general, that is the topological form and the respective edge lengths of the convex hull. And then doing facetings.

--- rk

[Polyhedron Dude]

Just out of curiosity, how do you handle coordinates for these polytopes? Do you use full Cartesian coordinates, or just a point and a generating symmetry (via a Wythoff construction perhaps)? Currently, my renderer uses explicit coordinates for vertices, and vertex sets for j-faces. To avoid having to manually specify vertex sets (which is impractical beyond the simplest polychora -- the 24-cell, for example, took me 2 days to write out in full), I use a convex hull algorithm to generate the j-faces. Unfortunately, this means it can't handle non-convex polytopes.

I've been thinking about some kind of automated faceting algorithm for generating non-convex polytopes, though. Is it true that the hyperplanes of all cells in a non-convex uniform polytope always corresponds with (i.e. parallel to) some element of a convex uniform polytope? Or are there cases where the hyperplanes are non-parallel to all j-faces of any convex uniform polytope? If the former, I may have an easy way of generating non-convex polytopes; if the latter, then how in general do you construct these things?

(In any case, even if I can generate coordinates for non-convex polytopes, it will still take a while before the renderer can actually handle them correctly, since to produce the correct projection images I'd need full 3D volumetric clipping, and right now I don't have a good way of doing that yet.)

I generate macros to draw the polygonal slice sequences and then do a symmetry operation on them. I might right in something like "cubize (obj)" where obj = object {thaquitpath4(m,n) translate z*(sq2+1)}, where the thaquitpath macro generates the polygonal shaped slices of thaquitpath on the grid at coordinates (m,n) then moves them up from the center. Cubize macro then makes 6 copies of it into a cube like pattern.

The only cells that might not parallel to the cells of a convex polytope would be snub cells, all normal cells are normal to some main axis.

[wendy]

Miller's monster is actually a CT snub, the 'wythoff symbol' is in fact correct for it.

Most of the regular snubs are generated by Wythoff-Stott constructions by alternating an omnitruncate.

The GAP is classified as a toridal laminate.

[Polyhedron Dude]

Now for a polyteron that puts a whole new meaning to the phrase "weird looking". This one is Gadencorn - great dispinocellirhombated penteract. It belongs to the wacbinant regiment, wacbinant's symbol is (o'x"x)ox = Gox, it contains 27 members. Gadencorn is non-orientable. Its facets are 10 gaqripts (cyan), 10 girpiths (red), 32 garpops (pinkish), 40 grohps (orange), and 40 quitcopes (green). I just recently named the members of this regiment and first rendered gadencorn yesterday.



http://pages.suddenlink.net/hedrondude/gadencorn.png
http://pages.suddenlink.net/hedrondude/gadencornf.png <-- check here for a 24 x 24 render!

[Klitzing]

Wow, even the vertex figure of gadencorn looks quite intricate.
--- rk

[quickfur]

Wow. This last polyteron just blew my mind. The complexity of some of the slices is just ... incredible.

[Polyhedron Dude]

Gadencorn liked to freak me out when I first seen it being rendered, it appears it is not alone - other wacbinant members (I've rendered four so far) are taking on a crazy look to them, Wacbinant itself looks more normal, but it is still quite intricate. Glarcat is quite dramatic, and Garcornit, whose symbol is (o'x"(x)o,x) has a more calmer appearance, but still interesting. Gadencorn is also a tame polyteron (as opposed to feral or wild) - hard to believe due to its appearance.